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Lets imagine 5 plots of different size are sampled for a target species:

`plot# count area(m^2) plot_density 1 1 5 0.2 2 3 2 1.5 3 0 10 0.0 4 5 1 5.0 5 2 6 0.33`

What is this species' density? I see two ways to calculate the density that give completely different values.

The first way averages the density at each plot:

$$ frac{Sigma(frac{count_i}{area_i})}{5} = 1.41/m^2 $$

This seems *OK* but it doesn't do much to control for changing plot sizes. For example if plot 3 above was 10 times larger the density would still be the same. In a situation where plot sizes are determined by the environment (say, under natural cover objects) this seems less than ideal.

The second way totals the counts and the search area and divides them:

$$ frac{Sigma(count_i)}{Sigma(area_i)}=0.46/m^2 $$

I prefer the second method because it seems to describe the animals' density more accurately. And I would be happy to use the second method, but my issue is that I am unsure how to calculate a summary statistic for method two since a mean was never calculated. Method one gives a standard deviation of 2.1. What is the standard deviation for method two??

One possible solution I've come up with involves breaking the larger plots into 1m^2 plots and dividing the number of animals across those smaller plots. So now I have 24 1m^2 plots with the following "counts":

`plot# count area(m^2) plot_density 1-5 0.2 1 0.2 6-7 1.5 1 1.5 8-17 0.0 1 0.0 18 5.0 1 5.0 19-24 0.33 1 0.33`

Now, using the first equation above I get: $$ frac{Sigma(frac{count_i}{area_i})}{24} = 0.73/m^2 $$ With a standard deviation of 1.26.

Is this a reasonable approach? Is there an established solution to this problem?

To me, there are two issues that are mixed up here (if I understand you correctly). First, do you want to estimate the mean and variance for a statistical population (i.e. to characterize a larger population by independent samples), or do you want to calculate the actual density for a particular area, where you have counted all occurences in that entire area (but maybe divided the area into subareas out of convenience when counting)? This is not clear from your question.

In the second case, your second option of pooling counts and areas is suitable. However, then you have only calculated the actual density in that particular area (ignoring issues of detectability of the organism when counting), and you cannot draw any inferences about a larger statistical population.

If your aim is to draw inferences for a larger statistical population, a start is to calculate the mean and standard deviation (sd) of your sample. In that case, I assume that your samples are chosen randomly and independently from a larger statistical population. Your first option is then the right approach. However, since your samples have different sizes, you might want to attach more weight to larger samples, since they can be assumed to better describe the population average than smaller samples. This is called a weighted mean.

Generally, the weighted arithmetic mean is defined as:

$$ar{x}_w = dfrac{sum_{i=1}^nw_ix_i}{sum_{i=1}^nw_i}$$

where $w_i$ are the weights for each sample ($x_i$) and $n$ are the number of samples. Using this formula, you will arrive at exactly the same value for the weighted mean as you calculated in your second attempt (0.458), if you use the areas as weights.

The weighted standard deviation is a bit more problematic, in the sense that there doesn't exist one single standard way of calculating this. However, a commonly used formula is:

$$ sigma_{w_1} = sqrt{ frac{ sum_{i=1}^n w_i (x_i - ar{x}_w)^2 }{ frac{(M-1)}{M} sum_{i=1}^n w_i } },$$

where $M$ are the number of nonzero weights. Other definitions of the weighted standard deviation can be found at Wikipedia: weighted sample variance.

Another version is defined as (called "reliability weights" on the wiki page):

$$ sigma_{w_2} = sqrt{ frac{ sum_{i=1}^n w_i (x_i - ar{x}_w)^2 }{ sum_{i=1}^n w_i - frac{sum_{i=1}^n w_i^2}{sum_{i=1}^n w_i}}},$$

If I haven't made a mistake, these will give the standard deviations 1.15 and 1.22, using your data.

As for the **biological interpretation**, all calculations of average densities indicate a clumped distribution, since the coefficient of variation (CV) is larger than 1 (CV = sd/mean), but more so if you use a weighted mean. The CV using the arithmetric mean is 1.5, while it is 2.5-2.65 for the weighted mean, which is reasonable since you are giving more weight to the large area sample with a zero count. However, I should also note that you should be cautious about using a weighted mean when you have a strongly clumped distribution, since your run the risk that e.g. a large sample plot land on an area with no occurences, which might bias your density estimate. Generally, when you have a clumped distribution, you need to sample more intensively to get a good estimate of average density, and many small samples is often better than a few large ones.

## Evaluating Bayesian spatial methods for modelling species distributions with clumped and restricted occurrence data

Affiliations Centre for Biodiversity and Environment Research, Department of Genetics, Evolution and Environment, University College London, London, United Kingdom, Big Data Institute, University of Oxford, Oxford, United Kingdom

Roles Conceptualization, Funding acquisition, Writing – review & editing

Affiliations Centre for Biodiversity and Environment Research, Department of Genetics, Evolution and Environment, University College London, London, United Kingdom, Institute of Zoology, Zoological Society of London, London, United Kingdom

Roles Conceptualization, Funding acquisition, Resources, Writing – review & editing

Affiliations Centre for Biodiversity and Environment Research, Department of Genetics, Evolution and Environment, University College London, London, United Kingdom, Institute of Zoology, Zoological Society of London, London, United Kingdom

## Negative binomial distribution

The negative binomial distribution , like the normal distribution, arises from a mathematical It is commonly used to describe the distribution of count data, such as the numbers of parasites in blood specimens. Also like the normal distribution, it can be completely defined by just two parameters - its mean (m) and shape parameter (k). However, unlike the normal distribution, the negative binomial does not naturally result from the use of large samples - nor does it arise from a single causal model.

- Inverse binomial sampling:

Let a proportion P of individuals possess a certain character. If observations are selected at random, the number of observations in excess of k that are taken in order to obtain just k individuals with the character has a negative binomial distribution, with exponent k.

This model is the least relevant biologically, but the one given by most standard texts.

To what extent any of these models might plausibly apply to your data is another matter - particularly since it is possible to devise models that yield an aggregated distribution, whose observations are not distributed negative binomially. Various two parameter models give rise to count Depending upon how they are set up, a number of these alternate models can have more than one mode, and some can also yield a negative binomial.

### Properties

Sometimes the occurrence of an event is not independent of other events in the same sampling unit. For example many organisms are not distributed randomly, or are not sampled randomly, and thus the Poisson distribution does not provide a good description of their pattern of dispersion. The most common pattern of spatial dispersion is aggregated, rather than random or regular. The same may also happen for events occurring through time - one event may 'spark off' other events resulting in a contagious distribution. The negative binomial distribution is one of several probability distributions that can be used to describe such a pattern of dispersion.

So what does the distribution look like? The distribution is shown below for values of k ranging from 8 to 0.1.

With k having a value of 8 the distribution is more or less symmetrical. But for all values of k less than about 8, the distribution is right skewed indicating an aggregated distribution. In fact the parameter (k) is often used as a measure of the degree of clumping or aggregation. It can range from 0 to infinity, with the lower the value of k, the higher the degree of contagion. Note that the distribution is always unimodal - the only reason for the apparent bimodality at lower values of k is the pooling of classes above 140.

- For example, its parametric (population) variance is Hence, unlike with the Poisson distribution, the variance is always greater than the mean. But if k is sufficiently large, approaches m, and the negative binomial converges to Poisson.
- Where k=1, because of its mathematical form the negative binomial is said to be a 'geometric distribution'. If however, k is very small and the zero category is ignored, the negative binomial converges to a 'log-series' distribution. Unfortunately, although it is widely employed to provide an index of species richness, there is no plausible causal model for the log-series distribution.

If a set of organisms conforms to the negative binomial model, if it is maximally clumped k approaches zero, and if it is random k approaches infinity - unfortunately the converse is not true.

The negative binomial model may be described as being 'versatile, but without carrying too deep a causative commitment'. Very often it is used as a fairly arbitrary, but convenient, approximation to how counts are distributed - and, provided the data have a negative binomial distribution, k is used as a measure of that distribution's shape.

In generalized linear models (which we meet in ) the negative binomial is sometimes used as a convenient, if somewhat arbitrary, 'error structure'. More problematically, k and 1 */*_{k} are used as measures of 'aggregation' - and, in some cases, aggregation is defined as such.

Be aware however that, whilst a particular model produces a certain value of k, this does not mean the converse is true. Furthermore, there is as yet no unambiguous biological definition of aggregation, nor any agreement on how best to quantify it.

Thus the several existing ways of measuring aggregation are not different ways of measuring the same thing:

they measure different things." **Pielou** (1977)

Mathematical ecology - Wiley, New York

### Assumptions

Since there are so many different models that can give rise to the negative binomial, it is hard to give a simple set of assumptions for its validity. However, we can look at what factors may lead to misleading results:

For random samples, because the negative binomial distribution is unimodal, any additional modes may be ascribed to statistical noise - assuming the discrepancy is not too great. In which case, the observed mean is taken to be an estimate of the true mean - the sample variance, calculated using the ordinary sample variance formula, is an estimate of the parametric variance (albeit this is usually an underestimate) - then k can be treated as a sample statistic, and its standard error (or confidence limits) estimated.

If the samples were not taken at random this reasoning does not apply, and the population for which inferences are made should be confined to the actual data at hand - extending its results to other populations (or 'superpopulations') can be horribly misleading - and its standard errors (or confidence limits) equally so.

Using k as an index of aggregation assumes all your samples have the same sample size, and the same population density.

- Few indices of aggregation are truly independent of density, and k of the negative binomial is very much so. For entirely mathematical reasons, the negative binomial k is both scale and density dependent, but not linearly. Added to this, when the density of organisms relative to the sampling unit drops very low, the distribution will appear to tend towards randomness (that is a Poisson distribution) irrespective of the true situation - this is sometimes described as an 'integer effect'.

As its name implies, the negative binomial shape parameter , k, describes the shape of a negative binomial distribution. In other words, k is only a reasonable measure to the extent that your data represent a negative binomial distribution. However testing observations against this distribution must be viewed with caution because small samples are unlikely to be shown to be significantly different, and very large sets of real data are almost certain to be. One way to address this is to compare the fit of one or more of the alternate models.

Other long-tailed distributions, such as the Neyman type A and Polya-Aeppli, yield the relationship However, whilst single-cause models may be criticised as overly specific, the negative binomial model is anything but.

Because the negative binomial distribution can arise for many different reasons, the likelihood that a particular one is responsible is correspondingly small. Using k as a measure of aggregation assumes this is the most likely model and that 'aggregation' can be unambiguously defined. Defining 'aggregation' as what it is that k measures is a circular argument, albeit a common one.

Whilst they can be useful descriptively, neither the negative binomial nor its shape parameter can tell you which - if any - biological, statistical or sampling process was responsible for your results being distributed in the way that you observe. A number of ecological studies have shown k is neither constant within a species, nor consistently different between species when compared across a wide range of population densities. Nevertheless, there have been repeated attempts to use k as a parameter for population models, and to support particular ecological and behavioural models. Sadly perhaps, k is not a fundamental biological constant, and does not describe any sort of general ecological or behavioural property - it is merely a rough and ready measure, which assumes an arbitrary model.

### Negative binomial population parameters

- The

**mean**frequency of failures, m, can also be calculated as - where k is the mean number of successes.

- As P and Q, where and
- As p and q, where and
- k may be described as the 'shape', or 'size' parameter.

### Mass probability function

The expected probability of obtaining a given value of a count, r, can be expressed as shown below.

## What is species distribution?

Species distribution is the manner in which a biological taxon is spacially arranged.

#### Explanation:

There are three basic types of species distribution within an area:

1) Uniform Species Distribution: in this form the species are evenly spaced. Uniform distributions are found in populations in which the distance between neighbouring individuals is maximised. Example - penguins often exhibit uniform spacing by aggressively defending their territory among their neighbours.

2) Random Species Distribution: this is the least common form of distribution in nature and occurs when the members of a given species are found in environments in which the position of each individual is independent of the other individual.

Example - when dandelion seeds are dispersed by wind , a random distribution occurs as the seedlings land in random places determined by uncontrollable factors.

3)Clumped Species Distribution: this the most common type of dispersion wherein the distance between neighbouring individuals is minimised.

Example - the bald eagles nest of eaglets exhibits a clumped distribution because all the offsprings are in a small subset of a survey area before they learn to fly.

The pattern of distribution is not permanent for each species. The distribution of species depends upon various biotic and abiotic factors. The distribution pattern can change seasonally, in response to the availability of resources and also depending upon the scale at which they are viewed. Various factors like speciation,extinction, continental drift, glaciation, variations of sea levels, river capture, and available resources are useful in understanding species distribution.

## 19.1 Population Demographics and Dynamics

Populations are dynamic entities. Their size and composition fluctuate in response to numerous factors, including seasonal and yearly changes in the environment, natural disasters such as forest fires and volcanic eruptions, and competition for resources between and within species. The statistical study of populations is called demography : a set of mathematical tools designed to describe populations and investigate how they change. Many of these tools were actually designed to study human populations. For example, life tables , which detail the life expectancy of individuals within a population, were initially developed by life insurance companies to set insurance rates. In fact, while the term “demographics” is sometimes assumed to mean a study of human populations, all living populations can be studied using this approach.

### Population Size and Density

Populations are characterized by their population size (total number of individuals) and their population density (number of individuals per unit area). A population may have a large number of individuals that are distributed densely, or sparsely. There are also populations with small numbers of individuals that may be dense or very sparsely distributed in a local area. Population size can affect potential for adaptation because it affects the amount of genetic variation present in the population. Density can have effects on interactions within a population such as competition for food and the ability of individuals to find a mate. Smaller organisms tend to be more densely distributed than larger organisms (Figure 19.2).

### Visual Connection

As this graph shows, population density typically decreases with increasing body size. Why do you think this is the case?

#### Estimating Population Size

The most accurate way to determine population size is to count all of the individuals within the area. However, this method is usually not logistically or economically feasible, especially when studying large areas. Thus, scientists usually study populations by sampling a representative portion of each habitat and use this sample to make inferences about the population as a whole. The methods used to sample populations to determine their size and density are typically tailored to the characteristics of the organism being studied. For immobile organisms such as plants, or for very small and slow-moving organisms, a quadrat may be used. A quadrat is a wood, plastic, or metal square that is randomly located on the ground and used to count the number of individuals that lie within its boundaries. To obtain an accurate count using this method, the square must be placed at random locations within the habitat enough times to produce an accurate estimate. This counting method will provide an estimate of both population size and density. The number and size of quadrat samples depends on the type of organisms and the nature of their distribution.

For smaller mobile organisms, such as mammals, a technique called mark and recapture is often used. This method involves marking a sample of captured animals in some way and releasing them back into the environment to mix with the rest of the population then, a new sample is captured and scientists determine how many of the marked animals are in the new sample. This method assumes that the larger the population, the lower the percentage of marked organisms that will be recaptured since they will have mixed with more unmarked individuals. For example, if 80 field mice are captured, marked, and released into the forest, then a second trapping 100 field mice are captured and 20 of them are marked, the population size (*N*) can be determined using the following equation:

Using our example, the population size would be 400.

These results give us an estimate of 400 total individuals in the original population. The true number usually will be a bit different from this because of chance errors and possible bias caused by the sampling methods.

### Species Distribution

In addition to measuring density, further information about a population can be obtained by looking at the distribution of the individuals throughout their range. A species distribution pattern is the distribution of individuals within a habitat at a particular point in time—broad categories of patterns are used to describe them.

Individuals within a population can be distributed at random, in groups, or equally spaced apart (more or less). These are known as random, clumped, and uniform distribution patterns, respectively (Figure 19.3). Different distributions reflect important aspects of the biology of the species they also affect the mathematical methods required to estimate population sizes. An example of random distribution occurs with dandelion and other plants that have wind-dispersed seeds that germinate wherever they happen to fall in favorable environments. A clumped distribution, may be seen in plants that drop their seeds straight to the ground, such as oak trees it can also be seen in animals that live in social groups (schools of fish or herds of elephants). Uniform distribution is observed in plants that secrete substances inhibiting the growth of nearby individuals (such as the release of toxic chemicals by sage plants). It is also seen in territorial animal species, such as penguins that maintain a defined territory for nesting. The territorial defensive behaviors of each individual create a regular pattern of distribution of similar-sized territories and individuals within those territories. Thus, the distribution of the individuals within a population provides more information about how they interact with each other than does a simple density measurement. Just as lower density species might have more difficulty finding a mate, solitary species with a random distribution might have a similar difficulty when compared to social species clumped together in groups.

### Demography

While population size and density describe a population at one particular point in time, scientists must use demography to study the dynamics of a population. Demography is the statistical study of population changes over time: birth rates, death rates, and life expectancies. These population characteristics are often displayed in a life table.

#### Life Tables

Life tables provide important information about the life history of an organism and the life expectancy of individuals at each age. They are modeled after actuarial tables used by the insurance industry for estimating human life expectancy. Life tables may include the probability of each age group dying before their next birthday, the percentage of surviving individuals dying at a particular age interval (their mortality rate , and their life expectancy at each interval. An example of a life table is shown in Table 19.1 from a study of Dall mountain sheep, a species native to northwestern North America. Notice that the population is divided into age intervals (column A). The mortality rate (per 1000) shown in column D is based on the number of individuals dying during the age interval (column B), divided by the number of individuals surviving at the beginning of the interval (Column C) multiplied by 1000.

For example, between ages three and four, 12 individuals die out of the 776 that were remaining from the original 1000 sheep. This number is then multiplied by 1000 to give the mortality rate per thousand.

As can be seen from the mortality rate data (column D), a high death rate occurred when the sheep were between six months and a year old, and then increased even more from 8 to 12 years old, after which there were few survivors. The data indicate that if a sheep in this population were to survive to age one, it could be expected to live another 7.7 years on average, as shown by the life-expectancy numbers in column E.

A | B | C | D | E |
---|---|---|---|---|

Age interval (years) | Number dying in age interval out of 1000 born | Number surviving at beginning of age interval out of 1000 born | Mortality rate per 1000 alive at beginning of age interval | Life expectancy or mean lifetime remaining to those attaining age interval |

0–0.5 | 54 | 1000 | 54.0 | 7.06 |

0.5–1 | 145 | 946 | 153.3 | — |

1–2 | 12 | 801 | 15.0 | 7.7 |

2–3 | 13 | 789 | 16.5 | 6.8 |

3–4 | 12 | 776 | 15.5 | 5.9 |

4–5 | 30 | 764 | 39.3 | 5.0 |

5–6 | 46 | 734 | 62.7 | 4.2 |

6–7 | 48 | 688 | 69.8 | 3.4 |

7–8 | 69 | 640 | 107.8 | 2.6 |

8–9 | 132 | 571 | 231.2 | 1.9 |

9–10 | 187 | 439 | 426.0 | 1.3 |

10–11 | 156 | 252 | 619.0 | 0.9 |

11–12 | 90 | 96 | 937.5 | 0.6 |

12–13 | 3 | 6 | 500.0 | 1.2 |

13–14 | 3 | 3 | 1000 | 0.7 |

#### Survivorship Curves

Another tool used by population ecologists is a survivorship curve , which is a graph of the number of individuals surviving at each age interval versus time. These curves allow us to compare the life histories of different populations (Figure 19.4). There are three types of survivorship curves. In a type I curve, mortality is low in the early and middle years and occurs mostly in older individuals. Organisms exhibiting a type I survivorship typically produce few offspring and provide good care to the offspring increasing the likelihood of their survival. Humans and most mammals exhibit a type I survivorship curve. In type II curves, mortality is relatively constant throughout the entire life span, and mortality is equally likely to occur at any point in the life span. Many bird populations provide examples of an intermediate or type II survivorship curve. In type III survivorship curves, early ages experience the highest mortality with much lower mortality rates for organisms that make it to advanced years. Type III organisms typically produce large numbers of offspring, but provide very little or no care for them. Trees and marine invertebrates exhibit a type III survivorship curve because very few of these organisms survive their younger years, but those that do make it to an old age are more likely to survive for a relatively long period of time.

## How should species density be calculated for a clumped distribution? - Biology

Measuring density with plots & quadrats

### What is a Quadrat?

Quadrats do not have to be square but their area must be known. Other quadrat shapes commonly include circles and rectangles.

Square quadrats can be any size. Common sizes include: 25 by 25 cm, 50 by 50 cm, 1 by 1 m and similar sizes in feet. Quadrats are used in many different scientific disciplines from vegetation assessment to archeological investigations. Quadrats are required for estimating several vegetation attributes including:

**Density** - for counting the number of objects within the unit area of the quadrat.

**Biomass** - achieved by "clipping" all the material of a given type (e.g., grass, shrub or forb) or species within a quadrat.

**Cover** - often accomplished by estimating the area of a quadrat that is covered by a plant's canopy.

**Frequency** - the proportion of quadrats in which a species occurs is called frequency, thus quadrats are required to estimate plant frequency.

### Design of Quadrat-based Sampling

Four properties of a monitoring protocol are essential to consider before starting a sampling protocol using quadrats (Bonham 1989). They are:

*1. What is the distribution of the plants across the landscape under assessment?*

- Is the distribution of plants clumped and variable,
**heterogeneous**, across the landscape? - Or, are plants rather evenly,
**homogeneously**, distributed across the landscape? - Do the features of interest occur in linear strips, clumps, and what average area does one individual occupy?

**2. What size quadrat should be selected?**

- In selecting an appropriate quadrat size, we need to ensure that the quadrats are
**big enough**to contain at least one plant of interest and should include enough plants to get a good estimate of density. - Conversely, the quadrat needs to be
**small enough**that the count can be conducted in a reasonable amount of time. In other words, you don't want to measure hundreds of individuals per quadrat.

**Rules of Thumb for Quadrat Size:**

- A quadrat is too large if the 2 most abundant species are found in every plot.
- A quadrat is too small if the most abundant species is not found in a majority of the plots.
- If more than 5% of sampling units have none of the plants of interest, increase the plot size.

**Quadrat Size Depends on Plant Size** :

- The larger the average sized plant the larger the necessary frame.
- A plot should be larger than the average-sized plant and larger than the average space between plants.
- It is difficult to sample plants of different life forms (i.e., shrubs and grasses) with the same plot frame. Therefore, nested techniques are often used where shrubs are measured with one plot (e.g., a longbelt transect) and herbaceous plants are measured with a separate, often smaller, plot.
- Select the size of quadrat based on species of greatest interest.

**Perimeter to Area Ratio:**

- The perimeter:area ratio decreases as plot size increases.
- If borderline decisions (i.e., is a plant in or out of plot) are difficult to make, then select a plot size that reduces the perimeter:area ratio.

- Sparse vegetation requires larger quadrats than dense vegetation.
- Uniform vegetation requires fewer and smaller quadrats than diverse and heterogeneously distributed vegetation.

*3***. What shape of quadrat should be selected?**

Many quadrat shapes exist for vegetation assessment -- from squares to rectangles to circles.

- More likely to cut across plants or clumps of plants rather than be completely occupied by plants. So, generally best for "clumped" vegetation.
- Rarely completely occupied by bare spaces.
- Often have lower variance than squares or circles.
- Can reduce plot to plot variability in sparsely vegetated communities.
- Easier to estimate % cover than in circles or squares.

Elongated rectangles have often been shown to work effectively in ecological studies (Bonham 1989 Elzinga et al. 2001), because vegetation frequently occurs in clumps. Belt transects are rectangular in shape, where the "long" end is very much longer than the short end. Belt transects, sometimes called strip quadrates, differ from line transects as they have a larger and specifically defined width. A line transect is a narrow line (< 1 inch wide) stretched across a plot and it is not a "quadrat" because it has no area. Whereas a belt transect is several feet or meters wide and is a quadrat of sorts.

- Greater perimeter:area ratio than circles but less than rectangles.
- Most typically used to estimate frequency because presence/absence is easy to estimate.
- Squares are easier to estimate % cover than circles but not as easy as rectangles.

- Less perimeter (per area) than square or rectangle.
- Often used in clipping because perimeter decisions are difficult to make when clipping.
- Reducing perimeter:area ratio is also good in communities with large sod-forming plants.

In the figure to the right, the red square would clearly be too small a quadrat because it is too small to capture even a single individual. Likewise, the green circle in this example although is big enough to capture >1 individual its shape is not optimal in capturing the density of these clumped plants. In this example. the elongated rectangle is the most appropriate quadrat as both a representative number are sampled and the shape captures the typical arrangement of plants.

*4. How many observations are needed to accurately estimate the density of the species?*

- Small quadrats (or plots) tend to have higher variability or express greater differences from plot to plot.
- Small plots are usually faster to read.
- But, the number of plots you must examine depends on variability. The more variation among plots, the more plots you need to estimate.
- Therefore, there is a trade-off between number needed and quadrat size.

### Sources of Uncertainty

One of the main sources of error in quadrat based sampling occurs when deciding whether an individual is within or outside the quadrat frame. These types are errors are called "boundary decisions" and protocols should be discussed to ensure consistent decisions (Elzinga et al. 1998). For example, would you include a species that entered the quadrat or just those species with roots in the quadrat. What then happens with species that roots cross the quadrat but terminate outside it? Can you achieve this assessment without damaging the plant? A further common source of error is due to differences in observers opinion. Again this can be minimized by having clear consistent standards. For more details on boarder decision study figure above and text in Elzinga et al. 1998.

## Methods of Sampling Plant Communities

When the vegetation is to be studied along an environmental gradient or eco-tone (e.g. tropical to temperate, high or low rainfall areas or precipitation gradient, adjacent areas with different types of soil, etc.) a line is laid down across a stand or several stands at right angles. This method of linear sampling of the vegetation is called transect.

**Depending upon the object of study, two types of transect can be drawn:**

(1) Line Transect or Line Intercept and

The extent of area determines the number and size of transects. When transects are used to sample the vertical distribution of vegetation (i.e. stratification) they are called ‘bisects’.

In this type of transect the vegetation is sampled only over a line (without any width). A line is laid over the vegetation with a metric steel tape or steel chain or long rope and kept fixed with the help of pegs or hooks. This line will touch some plants on its way from one point to the other. The observer will start recording these plants from one end and will gradually move towards the other end.

**From this type of transect following information could be collected:**

(a) The number of times each species appears along the line,

(b) The trend of increase or decrease of distance between the individuals of a species,

(c) The percentage of occurrence of different species in relation to the total species,

(d) The gradual disappearance or appearance of different species along the line, etc.

From the observations in a number of such parallel line transects, comments can be made on the habitat and other environmental conditions on different portions of the transect. Every species has its own ecological amplitude and tentatively expresses the status of available water and other edaphic conditions, atmospheric humidity, availability of light, grazing and other biological pressures, etc.

**Demonstrate the Gradual Floristic Change in Two Different Types of Adjacent Plant Communities:**

Date: _____________________________ Day Temperature:

Locality: ___________________________ Climatic Zone:

Total Area:_________________________ Relative Humidity:

Altitude:___________________________ Annual Precipitation:

When two different types of vegetation develop side by side a gradual change of species content is generally seen in the intermediate region. For understanding the mode of such change the communities are generally studied by line transect method.

(i) A long thread or a rope,

(iii) Two surveyor’s hooks or long nails.

A thread or a rope or a long measuring tape is laid across the stand or stands in the communities under study and fixed with two hooks at two ends. Record individual plants touching the thread and the distance from a particular end.

**Every individual plant touching the rope from one end to the other is recorded in Table 3:**

From the gradual change of the concentration of different species in different portions of the transect and the new arrival or disappearances of species, comments can be made on the habitat conditions of two communities and the transitional region.

The belt is a long strip of vegetation of uniform width. The width of the belt is determined according to the type of vegetation or the stratum of vegetation under study. In close herbaceous vegetation it is usually 10 cm, but it varies from 1 to 10 m in woodland.

The length of the vegetation is determined according to the purpose of the study. If a transect is essential then the lines should be marked using deep-seated wooden pegs at regular intervals. A belt could be kept isolated by installing tall wire-net fence on all its sides keeping safety-space from lines.

A belt is generally studied by dividing it into some equal sized segments. The length of each segment is generally equal to the width of the transect. These segments are sometimes called quadrats. Belt transects are used in determining and understanding the gradual change in abundance, dominance, frequency and distribution of different species in the transitional region between two different types of vegetation.

**Demonstrate the Gradual Change of Abundance and Frequency of Different Species in a Transitional Zone following the Belt Transect Method:**

For understanding the gradual change in density and frequency of different species in the transitional region between two different types of vegetation the area is generally studied by Belt Transect method (Fig. 1.10).

(iii) Surveyor’s hooks or nails and

Place two hooks 50 cm apart at both ends of the transect (A & B and C & D). Connect these two sets of nails by long threads (A & C and B & D). Place more nails along these two lines at every 50 cm (F, G, J, etc. and E, H, I, etc.). Connect these nails crosswise with threads.

Now, a series of quadrate (e.g. ABEF, EFGH, GHD, etc.) have been demarcated along the transect. Distribute the quadrats into three distinct zones: I: the first vegetation type II: the transition region and III: the second vegetation type.

**The Density (D) and Frequency (F) of different species in different zones can be calculated using following formulae:**

D = No. of individuals of the species in all the sample plots/No. of sample plots studied

F = No. of points of occurrences of the species/No. of sample plots studied

(Also refer Exercise Nos. 8 and 9.)

Record all the species along with their numbers in all quadrats (if a sharp change is apprehended) or the alternate quadrats (if the change appears to be very slow) in the following Table 4 and calculate their Density and Frequency.

By transect method, one can estimate different qualitative and quantitative characters of the vegetation and can correlate the findings with the different environmental conditions.

Further, in Belt-Transect, it is possible to determine the basal area or cover (by introducing another column in the Table) of all the recorded species from which Density, Frequency and Importance Value Index also can be calculated.

#### 2. Bisect:

The structure of vegetation with regard to the relative height, depth and lateral spread of plants in both aerial and underground parts could be determined by the use of bisects. It is essentially a line transect along which a trench has been dug to a depth greater than that of the deepest root systems.

The extent of different aerial and underground parts are carefully measured and plotted to scale on coordinate graph paper. This method reveals the form and interrelationship of underground systems of different species growing in the community and also their relationship to different types and/or layers of soil.

**So, the bisect studies provide the following information:**

(a) A rough floristic picture of the community,

(b) Stratigraphic distribution of different species,

(c) Utilization of space by different species,

(d) Underground structures of plants,

(e) Arrangement and extent of root-system, etc.

#### 3. Trisect:

It is the photographic method of recording the dynamic characters of plant community. In this technique a particular plot of the vegetation is photographed periodically by keeping the camera in the same direction and at the same height. This is done by permanently fixing three wooden pegs at a place in the vegetation so that the bases of a tripod camera-stand can be set on these pegs.

The technique is effectively used in monitoring the degradation or the recovery of rangeland, secondary succession of a denuded place, spread of a disease or some newly introduced weed into the area, etc. As these changes take place gradually and very slowly it is essential to keep detailed and permanent record for comparison. A series of photographs very nicely provides that record.

#### 4. Ring Counts:

The age of different types of woody plants (e.g. trees, shrubs, liana etc.) may be determined by counting the annual growth rings of the aerial or subterranean stems.

Growth rings can also reveal the climatic history of a place chronologically like the years of high rainfall or drought, presence of some chemical in the soil or atmosphere, forest fire, heavy snowfall etc. The method is also important in determining the successive stages of development of a vegetation and specially the sequence of dominants and subdominants.

#### 5. Quadrat Method:

The quadrat is a square sample area of varying size marked-off in the plant community for the purpose of detailed study. Generally a number of quadrats are studied to acquire reasonably faithful data to realise different analytic and synthetic characters of the plant community.

It is also effectively used to determine the exact differences or similarities in the structure and composition between two or more plant communities of related or unrelated vegetation.

**Quadrats can be of four types:**

Enlisting the names of different species growing in the quadrat.

Records the number of individuals of each species represented in each quadrat.

Records the position and areas covered by bunches, mats or tufts of grasses, mosses, etc. on the coordinated or graph paper. These graphs help to compare any change in structure of community in future.

For the study of biomass or weight of each species, all individuals are uprooted (but when the weight of a particular organ, e.g., branch, leaf, fruit, etc., is to be determined only the concerned organ is clipped or harvested) and its fresh or dry weight is recorded.

Demarcation or laying out of different types of quadrats are basically same. Generally, an adjustable wooden frame is prepared with perforations at regular intervals on each arm. Four arms are fixed in the field with the help of long nails or surveyor’s hooks and it is ready to provide data necessary for list, list- count and clip quadrat.

But, in chart quadrat more nails or hooks are fixed to the perforations on quadrat arms at regular intervals. Nails of opposite arms are connected by threads to divide the plot into a number of smaller quadrats to facilitate the recording of the area covered by individual plants on a coordinate paper in scale. When such wooden frames are not easily available it can be replaced by long threads or ropes.

The best size of quadrat to use in a community should be determined with care. It should be large enough and enough quadrats should be studied to produce reliable results.

The size of quadrats to be used in a given community is determined by constructing a species area curve. This is done by sampling the vegetation with nested quadrat method.

**Nested quadrats are a series of quadrats, laid one over the other with gradually increasing size and can be practiced in the following way:**

(ii) Surveyor’s hooks or long nails,

Put two nails ‘O’ and ‘ Y’ 5 m apart. Place the nail ‘X’ 5 m away from ‘O’ nail at right angle with the OY arm. Connect YO and OX by a long thread. Place the nails A and B on OX and OY, respectively, 50 cm away from ‘O’. Using another nail make a 50 cm x 50 cm square (Quadrat No. 1). Record all species growing in this quadrat.

Put another set of three nails increasing the length of arms 50 cm each (Quadrat No. 2). Record only newly found species in the list. Similarly, demarcate Quadrat Nos. 3, 4, 5 etc. increasing 50 cm arm length at every step. Continue the process so long as a recognisable number of new species is added each time (Fig. 1.11).

If the total number of species in every Quadrat (e.g. 4, 7, 9 etc. as in the table) are plotted on a graph paper against the area and number, respectively, for OX and OY axes, it will yield a sigmoid curve which is known as ‘Species area curve’ (Fig. 1.12).

The size of the Quadrat which recorded the highest number of species should be selected as the size of Quadrat for sampling the community under study.

[For general practice a 1 m × 1 m Quadrat sample is used for herbaceous vegetation, 5 m × 5 m for shrubby vegetation and 20 m × 20 m for trees.]

**Selection of Quadrats:**

For studying any plant community a number of quadrats should be studied. As the collected data will be processed statistically, the quadrats should be layed at random, with no bias for any particular region within the community. There are a number of methods for such random selection of quadrats.

I. Collect or prepare a map of the area under study. Draw a number of vertical or horizontal lines and number them separately. The numbers of vertical and horizontal lines are to be written separately on small pieces of paper and keep these two sets of paper squares in two separate beakers.

Mix these numbers in each beaker. Draw one number from each beaker and mark the place where lines representing these two numbers have crossed. Draw such number pairs repeatedly to find out the positions of a desired number of quadrats and mark the places properly.

II. Enter the area with blindfolded eyes and a stick in hand. Throw the stick over your shoulder at different parts of the vegetation. Each such point where the stick falls should be selected as a sample area.

For experimental purposes sometimes quadrats are marked permanently with the help of deep-seated wooden-pegs at four corners and studied at different times according to the need of the working programme. To understand the biotic pressure on the vegetation like grazing, etc. or to record its developmental history, some sample plots are needed to be kept isolated by fencing them properly with wire-nets.

**For practising these methods of studying vegetation following exercises may be worked out:**

**Determine the Ground Cover Flora of an Area by Quadrat Sampling:**

Name of the Place: _______________ Annual Precipitation:

Altitude:________________________ Climatic Zone:

Day Temperature: ________________ Humidity:

To determine the flora of a piece of vegetation the area should be sampled with a number of quadrats so that all the species growing there can get a chance to be recorded to give a total floristic picture.

Randomly select five sample plots. Lay out one 1m × 1m (1sq.m) quadrat at a sample plot. Find out and record different species growing inside it. Repeat the process for all other four sample plots.

**Different plant species growing in all the quadrats are now recorded in Table 6:**

**Now, the determined flora of the plant community under study is represented in Table 7:**

Comment on the species richness of the vegetation and also on some common and/or rare plants recorded in the area. Also, comment on the environment in which the vegetation has been developed, as reflected by the flora.

**Determine the Flora of a Forested Area by Quadrat Sampling****:**

Date: __________________________Day Temperature:

Locality: _______________________ Climatic Zone:

Total Area: _____________________ Relative Humidity:

Altitude: _______________________ Annual Precipitation:

To determine the flora of a forested area sampling should be done separately using quadrats of different sizes for trees, shrubs and herbaceous plants. Standard size of the quadrats for sampling trees, shrubs and herbs are 20 × 20 m (400 sq m), 5 × 5 m (25 sq m), and 1 × 1 m (1 sq m), respectively.

Lay one 400 sq m quadrat in each of the 3-5 randomly selected sample plots. Within each such quadrat demarcate two 25 sq m and four 1 sq m quadrats. Mark each set of quadrats as in Fig. 1.13. Record canopy forming plants from 400 sq m quadrats, shrubs, shruby climbers and trees saplings from 25 sq m quadrats and herbaceous plants from 1 sq m quadrate.

Record different species of plants growing in different quadrats separately in Table 8:

Density and frequency of the recorded species of plants can be easily determined by introducing two more columns in Table 1 for number of individuals and points of occurrences.

Now, the determined flora of the plant community under study is represented in Table 2 (as in Table 9 of Exercise 4).

Comment on the species richness of the vegetation and their stratigraphic distribution. Discuss the abundance, association etc. of different elements of the flora. Also, comment on the environment in which the vegetation has been developed, as reflected by the flora.

**Determine Coverage and Dominance by Different Ground Covering Plants in a Sample Area****:**

Name of the Place: Climatic Zone:

Day Temperature: Annual Precipitation:

The aerial portion of each and every individual plant covers some area in the vegetation. The total area covered by a species within the sample area tells for its dominance or importance to the community. In a multi-storeyed plant community such a study is conducted for every stratum of vegetation separately. The cover is generally determined by Chart Quadrat method.

I. Lay one 1 m × 1 m (i.e. 1 sq m) quadrat. Place nails at every 10 cm on each arm. Connect nails of opposite arms with threads to divide the 1 sq m quadrat into 100 small sq cm quadrats. Draw a replica of the layout in the scale of 1 cm = 10 cm on the graph paper.

Carefully draw the area occupied by individual species on the graph paper and mark them using separate symbols for each species. Find out the area covered by each species on the graph paper and multiply the numbers with 10.

For better results a number of areas should be studied. The area covered in a unit sample area could be multiplied by the area of the vegetation to determine the total area covered by a species or by different species (Fig. 1.14).

II. Lay one 1 m × 1 m quadrat. Fix the graph paper on a drawing board, set the pantograph on it and record the area covered by each species.

**Calculate the Relative Dominance of different species using the formula:**

Relative Dominance (R. Dm.) = Total coverage of the species/Total coverage of all the species × 100

**Area occupied by each species in the sample area and percentage of total cover are recorded in the Table 10:**

Considering the natural adaptability and dominance of each species comment on the environment of the area and the vegetation structure.

**Determine the Basal Area of Trees in a Forest****:**

Day Temperature: Annual Precipitation:

Basal area refers to the ground actually penetrated by the stems, and is readily seen when the leaves and stems are clipped at the ground surface. It is one of the chief characteristics to determine the relative dominance of a species and the nature of the community. It also helps to determine the yield.

Demarcate a quadrat of 20 m x 20 m in size or larger, if essential, with the help of wooden pegs and ropes. Measure the circumferences of each stand (tree) at breast height with the help of a measuring tape.

**The radius of each plant is calculated with the formula:**

r = Circumference of the tree/2π

and then, the basal area of each plant is obtained by the formula

Basal area = πr 2 or, 4πc 2 (c = circumference of the tree).

The total basal area is obtained by summing up the basal areas of all the species.

**The mean basal area per tree can be calculated as:**

Mean Basal Area/Tree = Total Basal Area/Number of trees

The mean area of one stand multiplied by density (i.e. number of individuals/unit area) produce the basal cover/unit area.

The area coverage is used to express the dominance. The higher the coverage area the greater is the dominance. Relative Dominance (R. Dm.) of a species is the proportion of basal area covered by the species to the total basal coverage of all the species in the area:

Relative Dominance (R. Dm.) =Total Basal Area of the species/Total Basal Area of all the species × 100

**Data recorded for computation and the results are presented in the following Table 11:**

Comment on the dominance of the determined dominant (i.e. with high R. Dm. value) trees and the major association they represent within the community. Also, try to inculcate the causes of such dominance, specially if the cause appears to be biotic.

**Determine the Relative Density of Different Herbaceous Plants Growing in a Community****:**

Name of the Place: Climatic Zone:

Day Temperature: Annual Precipitation:

The density of a species expresses its numerical strength within the community in relation to a definite area. Herbaceous species are extremely sensitive to different micro-climatic conditions and that is why their density varies greatly even in different portions of a particular type of vegetation.

In order to determine the Relative Density of the floristic members of a vegetation, the area should be sampled with List-cunt Quadrat method.

Randomly select five sample plots. Lay out one 1 m x 1 m (1 sq m) quadrat in a sample plot. Find all the species growing inside the quadrat and record their population number (individuals). Repeat the process for four other sample plots.

**Density of a species can be determined according to the formula:**

Density (D) = No. of individuals of the species in all the sample plots/Total number of sample plots studied

Relative Density (R.D.) = No. of individuals of the species/No. of individuals of all the species × 100

Density of a species indicates its abundance — so determine the abundance class using this data. Result

**Different plant species growing in the five quadrats, their population number and the determined Relative Density are recorded in the following Table 12:**

Considering the natural adaptability of each species and their determined RD in the community, comment on the environment of the area and the vegetation. Also, comment on the abundance of different species. Prepare a list of species recorded with high RD.

**Determine the Relative Frequency of Different Herbaceous Species Growing in an Area****:**

Name of the Place: Climatic Zone:

Day Temperature: Annual Precipitation:

The individuals of all the species growing in an area are not evenly distributed. The distribution patterns of individuals of different species indicate their reproductive capacity as well as their adaptability to the environment. Frequency refers to the degree of dispersion in terms of percentage occurrence.

Randomly select five sample plots. Lay out one 1 sq m quadrat in a sample plot. Find all the species growing inside the quadrat as in list quadrat method. Repeat the process for four other sample plots.

**Frequency (F) and Relative Frequency (RF) can be calculated with the help of the following formulae:**

Frequency (F) = No. of points of occurrences of the species/Total number of quadrats studied

Relative Frequency (RF) = No. of points of occurrences of the species/Total number of quadrats studied × 100

[Points of occurrences mean the number of samples or quadrats in which the species is growing.]

**Now, a constancy class (or Presence) can be determined for each species using the classification:**

**Record different species of plants growing in five quadrats in the Table 13:**

Comment on the total number of species and their distribution within the vegetation.

**Determine the Importance Value Index for Different Species Growing in a Herbaceous Plant Community****:**

Altitude: Climatic Zone: Total Area:

Humidity: Annual Precipitation: Day Temperature:

Individual values of density, cover and/or frequency do not give a total picture of any species growing in a plant community. But the sum of these three values can give a better picture about their ecological importance.

Randomly select five sample plots and study them for the determination of Cover, Relative Density, Relative Frequency (as in Exercises 2, 3& 4). Find our the Importance Value Index (IVI) of all the recorded species with the formula IVI = R. Dm. + R.D. + R.F.

IVI of a species can also be presented in a Phyto-graph. Draw a circle and divide it into four equal quarters by two radial lines laying at right angles to each other. Divide the three radii from centre to circumference into 100 equal parts and the fourth radius into 300 parts.

## Top 7 Attributes of Population

**The following points highlight the top seven attributes of population. The attributes are: 1. Population Density 2. Natality 3. Mortality 4. Population Growth 5. Age Structure 6. Patterns of Distribution 7. Population Genetics.**

### Attribute # 1. Population Density:

Population density is defined as the size of a population in relation to a definite unit of space. It is expressed as the total number of individuals or the population biomass per unit area or volume. For example, we can express the population size as 500 rabbits per square mile or 500 rabbits in a square block or 500 rabbits per hectare.

Populations are always in a dynamic state. As they change over time and space, no population has a single structure. It is important to understand the factors that regulate population size. The population size has two components—Local density of individuals, and total range of the population.

In a particular habitat, the density of individuals of a particular species depends upon the intrinsic quality of their habitat, and on the net movement of individuals into that habitat from other habitats. It is obvious that individuals are numerous where resources are most abundant.

**Thus local density provides us with:**

(1) Information about the interaction of a population with its environment, and

(2) Changes in density reflect changing local conditions.

The factors that regulate population size can be classified as extrinsic and intrinsic. The populations own response to density is said to be intrinsic, while the interaction with the rest of the community is said to be the extrinsic factor. Intrinsic factors include intraspecific competition, immigration, emigration and physiological and behavioural changes affecting reproduction and survival. Extrinsic factors are interspecific competition, predation, parasitism and disease.

#### Important Indexes Used:

It is the number (or biomass) per unit of total space.

Ecological density is the number (or biomass) per unit of habitat space (area or volume available that can be colonised by the population)

It is used to denote changing (increasing or decreasing) population and is time relative. For example, the number of birds seen on a tree per hour.

**4. Frequency of occurrence:**

Frequency of occurrence is the percentage of sample plots occupied by a species.

The importance value of each species are formed by combining density, dominance and frequency during the descriptive studies of vegetation.

#### Methods for Estimating Population Densities:

The estimation of population density is important for the study of population dynamics. Populations, as we know, contain too many individuals distributed over too large an area. This poses problems to make a complete count of the population, particularly in case of mobile individuals.

A very popular way of estimation of population density is through a method called mark-recapture method. This method involves capturing of a fraction of the population and marking with tags, paint, radio collars etc. and releasing them back into the population enough time is allowed for the marked individuals to recover and mingle with the rest of the population. A second sample is taken, after a certain time period, from the mingled population.

**The ratio of the marked to unmarked is noted and the estimate of the population size can be calculated by the following equation:**

where x is the number of marked individuals recaptured,

n is the total number or size of the second sample,

M is the number of individuals marked initially (first sampling), and N is the total size of the population.

If the above equation is rearranged, then we get

**Thus, the estimate of the total population density is called Lincoln index. The validity of the above equation is based upon the factors listed:**

1. The marking technique has no negative effect on the mortality of marked individuals.

2. The marked individuals were released at the same site of capture and were allowed to mix with the population, based on their natural behaviour.

3. The marking technique does not affect the probability of being recaptured.

4. The markings should be clear and firmly fixed so that they are not lost or overlooked.

5. There should not be any significant natality or mortality during the time interval under study.

6. There should not be any significant immigration or emigration of marked or unmarked individuals during the time interval under study.

Supposing 10 tigers (M) were radio collared and was allowed to mix with the population. After a certain time period 23 tigers (n) came to a particular site for licking salts. Of these 23 tigers, the numbers of radio collared animals were 4(x). Thus, the density of tiger population is that area is estimated to be

However, the estimate of tiger population is generally done by the pug mark method.

Many techniques and methodology for population sampling has been tried. This sampling methodology is itself an important field of research. The other methods generally used are minimum known alive (MKA) total counts, quadrate or transect sampling, removal sampling, plotless methods etc.

#### Sources, Sinks and Meta-populations:

As reproductive success varies among habitats, the local population density is often influenced by interaction with other populations. When there is abundant resources in habitats, more offsprings are produced by individuals than required to replace themselves. In such cases, the surplus offspring may disperse to other areas.

Such populations are said to be source populations (Fig. 4.35). The reverse occurs in case of poor habitats. Here, few offsprings are produced locally, to replace loss due to mortality. Thus, to maintain the population, individuals immigrate from other habitats. These populations are said to be sink populations (Fig. 4.35).

There are populations that exist as a set of subpopulations referred to as meta-populations by Richard Levins (1970). These meta-populations are more or less isolated but there exist some exchange of individuals (and genes) by way of dispersal. These concepts of sources, sinks and meta-populations are important as they serve as a framework for studying many of our threatened and endangered species.

### Attribute # 2. Natality:

Populations increase by the addition of individuals in two ways, by birth and immigration. On the other hand, individuals leave populations by two ways—death and emigration. The birth of new individuals is referred to as natality.

Natality is a broad term and it covers the production of new individuals of any organisms, whether they are born, hatched, germinate or arise by division. The theoretical maximum production of new individuals under ideal conditions (no stress) is said to be maximum natality.

It is constant for a given population. Ecological or realised natality is the increase of population under an actual or specific environmental field condition and is not a constant. It is variable depending on the size and age composition of the population and with the physical environmental conditions. Natality is expressed as rate, that is, as numbers in a given time. For example, if there is 320 births in a population during a year, then the natality rate is 320 per year.

#### Natality Rate is of Two Types:

1. Crude or absolute natality rate is obtained by dividing the number of new individuals produced by a specific unit of time.

2. Specific natality rate is obtained by dividing the number of new individuals per unit time by a unit of population. It is also referred to as average rate of change per unit population.

To illustrate the difference between crude and specific natality, let us consider a population of 1,000 fishes in a pond that has increased by reproduction to 3,500 in a year. The crude natality is 2,500 per year and the specific natality is 3.5 per year per individual.

#### Sex Ratios and Dispersal:

Competitions often result in biased sex ratios. Female- biased sex ratios are formed due to competition among males for mates. On the other hand, sex ratios toward more males (Male- biased) takes place when there is competition for resources, both among siblings and between offspring and parent.

Births often result in dispersal of offsprings to avoid over-crowding. Young males of mammals and a few birds disperse far away from their birth places than do females. This phenomena is called natal dispersal. Daughters have a tendency to remain near their mother and thus compete with her and among themselves for resources necessary for reproduction. This tendency to remain near their birthplace is called philopatry (home-loving).

### Attribute # 3. Mortality:

Mortality or death value of individuals is more or less the opposite of natality. The death rate is the number of individuals dying during a given time interval (deaths per unit time), or it can be expressed as a specific rate in terms of units of the total population or any part of it.

The loss of individuals under a given environmental condition is referred to as ecological or realised mortality. It, like ecological natality, is also not a constant, but may vary with the size and age composition of the population and environmental conditions. However, the minimum loss under ideal or non-limiting conditions is a constant for a population and is referred to as minimum mortality.

Biologists are interested not only why organisms die but also the reasons of their death at a given age. The converse of mortality is survivals or longevity, which focuses on the age of death of individuals in a population. Often the survival rate is of great importance than the death rate.

**Longevity can be said to be of two types:**

(1) Potential longevity, and

**1. Potential Longevity:**

The maximum lifespan attained by an individual of a particular species is said to be potential longevity. It depends upon the physiological condition of plants and animals (also referred to as physiological longevity) and the organisms die simply due to old age. Potential longevity can also be described as the average longevity of individuals living under optimum conditions. In nature, few organisms live in optimum conditions.

It is the actual lifespan of an organism. It is the average longevity of the individuals of any population living under real environmental conditions. In nature, most animals and plants die from disease or are eaten by predators, or succumb to natural hazards. Thus, this longevity is measured in the field and is also referred to as ecological longevity.

The above two types of longevity can be best exemplified by the work of Lack (1954). He observed that the European robin has an average life expectancy of 1 year in the wild, whereas in captivity it can live at least 11 years.

Measurements of mortality can be done directly or indirectly. Direct measurements can be achieved by marking a series of organisms and observing how many of these marked organism survive from time ‘t’ to time ‘t +1’.

As normality varies with age, specific mortality studies are of much interest, as they enable ecologists to determine the forces underlying the overall population mortalities. In such cases the indirect measurements of survivality is of importance, as one can estimate the mortality between successive age groups in a given population. Such measurements are widely used in fisheries work in the analysis of catch curves.

**The survival rate can be indirectly estimated from the decline in relative abundance from age group to age group in the following way:**

Survival rate between IInd and IIIrd year fishes (any particular species) =

relative abundance of IIIrd year fishes/relative abundance of IInd year fishes

#### Life Tables and Survivorship Curves:

Population has a spatial structure. It also has a genetic structure. A third aspect is to do with the rates of births and deaths and the pattern of distribution of individuals among different age classes. Ecologists are not the only ones to be interested in the third aspect.

The insurance and health care professionals are keenly interested in such dynamics of human populations. Ecologists and business professionals thus constructed a life table which could be used to describe the demographic characteristics of the population they study.

Life table is a tabular accounting of the birth rates and probabilities of death for each age class in the population. It, thus, gives a statistical account of death and survival of a population by age. Raymond Pearl and Parker (1921) were the first to introduce the life table into general biology, for a laboratory population of the fruitfly, Drosophila melanogaster.

The individuals from birth (born at approximately the same time) to the end of the life cycle, form a group known as a cohort and their investigation is turned as cohort analysis.

To understand the construction of a life table, one must have knowledge of the age structure of the population. It comprises of different age classes and the number of individuals in each age class residing at the same time.

In a life table, age is designated by the symbol × (Table 4.10). The first or youngest age class is x = 0. Ages are depicted in years (for some organism it may even be months, days or hours). The age specific variables are indicated by the subscript x.

L_{x} is survivorship (the number of individuals alive at the start) d_{x} is mortality, q_{x} is mortality rate (number dying divided by the number alive at the beginning of the time interval) and e_{x} is the life expectation (the average time left to an individual at the beginning of the interval).

Table 4.10 shows the life table of Mckinley Murre population. At the start (age 0) 100 individuals were taken to be born at the beginning of the interval. Over one half (55) died during the first interval. The mortality was 55 /_{100} × 100 = 55%. As in the first year 55 animals died, so 100 – 55 = 45 survived to begin the second year.

During the second period (from age one to two), 30 died. Mortality rate in the second year was found to be slightly higher i.e. 30 /_{45} x 100 = 67%. The life expectation at birth was, on an average, just over one year (1.15). The number aged one year has a life expectation of slightly less than one year (0.94).

At age 0 life expectation is the same as mean natural longevity. Physiological longevity is another aspect of longevity which is the age reached by individuals dying of old age. Individuals living under conditions where death results due to predation, accident, poor nutrition and infection are not factors.

Using different types of information, life tables can be constructed in several ways. In general, there are two basic types of life tables: age-specific and time-specific. Age specific or dynamic life tables are simple and the data of a cohort can be obtained by keeping track of their ages at death. Once the columns of age (x) and deaths (d_{x}) are known the rest can be calculated.

This method can be readily applied to populations of plants, sessile animals or populations of mobile animals located on a small area, where the marked individuals can be continually resampled throughout their lifespan.

**Limitations of this method are that:**

(1) It can take a long time to collect the data, and

(2) It is difficult to apply to highly mobile animals.

Time-specific or static life tables are a bit complicated. In this case, information from a single time period is used to estimate one of the columns of the table. For example, from the census of a population an estimate of the survivorship (l_{x}) column can be obtained. From this, one can calculate the other columns. In another case, if the death rates (q_{x}) of a year can be estimated from the census report, then this information can be used to calculate the other columns.

The second approach is more useful and easy for human populations as for other animals it is difficult to get relevant data. Limitations of this method is that (1) one must know the ages of individuals, through growth rings, tooth wear or some other reliable index, and (2) the sizes of the cohorts and the survivorship must remain the same from year to year.

Cohort life table data may be very instructive when plotted to form a survivorship curve for a particular population. When the data from column l_{x} (survivorship) are plotted against the x (age) column the resulting curve thus formed is called a survivorship curve.

This curve is convenient for use as a visual aid to detect changes in survivorship (and mortality) by period of life. Using actual numbers would make comparison of life tables difficult. So, a second way of presenting survivorship curves is to use a log scale for the number of individuals (Fig. 4.36).

The main advantage of using a semi-log plot is that in case of any population where the proportion of individuals dying in each unit of time (each day, week or year) is a constant, then the plot will always show a straight line.

Survivorship curves are hypothetically of three types (Fig. 4.36). It was first introduced by Pearl (1927). The three survivorship curves are called Types I, II and III, or better known as convex, diagonal and concave. The convex or Type I curve indicates high survivorship or very low mortality among younger individuals up to a particular age, after which most of the population dies.

This situation is characteristic of some human population, many species of large animals and Dall mountain sheep. Such a situation would happen if environmental factors were unimportant and most of the organisms lived out their full physiological longevity. The abrupt drop in survivorship would depend on how variable the population was in genetic factors affecting length of life.

A highly concave or Type III curve results when mortality is high during the young stages. This pattern is typical in the case of oak trees, marine invertebrates (oysters), many fishes and some human population. Mortality is extremely high during the free-swimming larval stage or the acorn seedling stage.

This results from such factors like inexperience in foraging and avoiding predators and lack of immunity to disease. Once an individual is well-established on a favourable substrate, life expectancy improves considerably.

A diagonal or Type II curve indicates a constant probability of dying. It can be otherwise stated that a constant percentage of the population is lost in each time period. Probably no population in the natural world has a constant age-specific survival rate throughout its whole life span.

However, a slightly concave curve, approaching a diagonal straight line on a semilog plot (Type II), is characteristic of many birds, mice, rabbit and deer (Fig. 4.37). In these cases, the mortality rate is high in young but low and more nearly constant in the adults (1 year or older).

Most probably no population for which adequate birth-to-death information is available actually displays any one of the above idealized curves. The real pattern most frequently observed seems to be one in which there is a juvenile segment of high mortality, followed by an adult segment of low or nearly constant mortality.

The final senile segment is one where mortality again rises (Fig. 4.38). Deviation from this pattern is often the result of omitting part of the organism’s lifespan. For example, survivorship curves for wild birds rarely show the final senile segment. Thus, shortfall for most individual animals takes place between the possible and the actual survivorship curve.

**Shape of Survivorship curve:**

**The shape of survivorship curve is related to the following factors:**

1. Shape related to the degree of parental care or other protection given to the young. For example, survivorship curve for honey bees, robins etc. who protect their young ones, are less concave than those occurring for grasshopper, sardines etc. who do not protect their young ones.

2. Shape related with the density of the population. For example, survivorship curves for two mule deer (Odocoileus hemi-onus) populations living in the chaparral of California shows (Fig. 4.39) a somewhat concave curve for denser population. This is due to deer living in the managed area where food supply is high, have a shorter life expectancy than deer living in unmanaged area. In the latter case, there is increased hunting pressure, intraspecific competition etc.

Humans also have greatly increased their own ecological longevity because of greater medical knowledge and facility, increased nutrition and adequate and proper sanitation. Thus, the curve depicting the survivality of human beings approaches the sharp angled type I minimum normality curve.

#### The r-and K-strategies shown by Survivorship Curves:

MacArthur and Wilson (1967) suggested another way of classifying evolutionary strategies, when they applied the terms r-selected and K-selected to populations. The initials r and K are taken from the logistic equation, used for describing the actual rate of growth of populations (R):

where: r is maximum rate of increase of the population.

K is number of organisms that are able to live in the population, when it is in equilibrium or, in other words, it is the carrying capacity of the population.

N is number of organisms in the population at time t.

As can be seen from the above logistic equation, r-selected populations are ones where maximum rate of increase (r) is important. In temperate and arctic regions, populations undergo periodic reduction (irrespective of their genotypes) due to catastrophic weather conditions. These crashes in population are followed by longer period of rapid population increase.

An r-selected population has the ability to take advantage of these favourable situation through increased fecundity and earlier maturity. They have many offsprings which under normal circumstances die before reaching maturity, but survives if circumstances change and are selected. Thus, r-selections are associated with the type III of survivorship curve (Fig. 4.36).

A K-selected population is associated with a steady carrying capacity. For example, in ‘constant’ tropical environments, where populations fluctuate little, populations remain near the limit imposed by resources (K). Adaptations that improve competitive ability and efficiency of resource utilisation are selected.

Thus K-selected populations are less able to take advantage of particular opportunities to expand (than r-selected populations). They are generally more stable and less likely to suffer high mortality. K-selected organisms usually have few and well-cared young. Thus, they are associated with Type I and II of survivorship curves (Fig. 4.36).

r-selected refers to the growth capacity (exponential growth rate) while K-selected denotes the carrying capacity of the environment for the population. The two classes (r- and K-selection) are the extreme ends of a continuum. Each end is associated with a whole group of characteristics of life which fit together into a particular evolutionary strategy. Eric Pianka (1970) listed a variety for traits given in Table 4.11.

In any ecological system populations are constantly undergoing r- or K-selection. Their position on the r-K continuum is dependent upon the strength of selection pressure and where they balance out. Such a case happens in natural situation is difficult to show.

Pianka (1970) placed insects at the r- selected end of the spectrum and mammals at the K-selected end, as insect populations fluctuate more than mammal populations. He reasoned that small organisms move more rapidly relative to their body lengths and use more energy relative to their body weight. They have more rapid development and shorter generations than large animals.

Many ecologists have attempted to contrast genetic responses to r-selected and K- selected spectrums in laboratory populations. Francisco Ayala (1965) showed that when populations of Drosophila were maintained for long periods under crowded conditions, the numbers of adults per cage increased gradually.

This was due to selective of traits that improved fecundity and survival at high densities. In another experiment, Drosophila populations were kept much below the carrying capacity by removing adults. The selective effects of low density with those of high mortality resulted. Similar experiments of bacteria and protozoans on laboratory populations have also given negative results.

### Attribute # 4. Population Growth:

Living things undergo sexual maturity and have the ability to produce young ones of its own type. In other words, natural populations have the ability to grow. The capacity for populations to grow is enormous, particularly when they are introduced in new regions having suitable habitats.

This rapid increase in numbers lead to the development of mathematical techniques to predict the growth of population and its regulation. The study of population is called demography.

Reproduction provides an increase in population growth. But, in nature, populations do not explode. Reproduction continues, but populations do not always grow. Darwin rightly pointed out that the sizes of populations are often regulated by environmental factors.

A population, thus, is a changing entity. This changing characteristic of population is attributed to factors such as density, natality, mortality, survivorship, age structure, growth rate, emigration, immigration and other attributes. These factors are always in flux as species constantly adjust to seasons, to physical forces, and to one another, even when the community and the ecosystem seem to be in an unchanging state. The study of changes in the organism number in a population and the factors related to such changes is termed population dynamics.

#### Exponential Population Growth (Growth Without Regulation):

A small population living in a very large and favourable habitat has a growth rate that depends on two factors – the size of the population and the capacity of the population to increase (referred to as biotic potential or intrinsic rate of natural increase). Such a case occurs for aquatic organisms when a new lake is formed or when some deer manage to reach an island where there was no deer earlier.

The important periodical production of offspring results in important differences in the way in which population grows. Young ones may be added to the population only at specific times of the year, that is, during discrete reproductive periods.

Such populations are said to have geometric growth—where the increment of increase is proportional to the number of individuals in the population at the beginning of the breeding season. Geometric growth is the typical pattern of population growth.

There are some organisms which do not have distinct reproductive seasons, but instead add young at any time of the year. Such populations increase more or less continuously and are referred to as exponential growth. Exponential growth rate is the rate at which a population is growing at a particular time, expressed as a proportional increase per unit of time.

In exponential growth, the curve of numbers versus time becomes steeper and steeper (Fig. 4.40). Growth depends on the biotic potential which does not change, and on the size of the population which changes continually—growing larger and larger. As a result, the growth rate of the population increases steadily from a slow rate (when the population is low) to a faster rate (when the population is high).

Examples of exponential growth rate are many in laboratory studies, but in field conditions they are scarce as it requires hard work for accurate censuring. One such example is the ring-necked pheasant population introduced on Protection Island (off the coast of Washington).

The initial population of 8 birds reached to 1,898 in six breeding seasons. Another example is of a herd of tule elk introduced into Grizzly Island (northwest of San Francisco, California). This animal, released in mid-1977, developed from 8 animals to a population of 150 by 1986.

If birth rate equals death rate, the rate of population growth is zero and the population is in a stable condition. If the population grows at a fairly constant rate (say 1%, 5% or less than 1%), the population size will increase exponentially.

However, if the population does not have a stable age distribution the growth rate is faster than predicted from the biotic potential. Subsequently, if growth persists at a constant rate a stable age distribution is quickly established.

**Equation of exponential growth:**

The formula by which exponential growth occurs is

where dN/dt is the population growth rate and refers to the change in numbers (dN) per time interval (dt). Biotic potential, r, is the increase in number of individuals per time period per head (or per individual) and combines birth rate and death rate. N is the number of individuals in the population.

In the above formula, growth rate is higher in a population with a high r compared with one with a low r. Conversely, the growth rate also depend on N, with a slow growth rate when N is small and rapid growth rate when N is large.

The formula given above gives growth rate in a population growing exponentially. If, instead, the population size (Fig. 4.41) at various times during exponential population growth is to be noted then an equivalent expression is the integral equation

where N_{t} is the number of individuals in the population after t units of time N_{0} is the number at time 0 (that is, at the beginning of the period being studied), r is the biotic potential and t is the time period being studied. The constant e is the base of the natural logarithm having a value of approximately 2.718. The term e r is the factor by which the population increases during each time unit and is written as the lower case Greek lambda (λ) that is when t = 1. Then

Suppose a population of 10 duckweeds grows for 4 days and r is 0.20 per day. Then

N_{4} = 10 × e (0.20×4) = 10 × 2.22 = 22.2

Thus, the population size at the end of 4 days is 22 or 23 duckweeds.

#### Logistic Population Growth (Simple Population Regulation):

As has been written earlier in exponential population growth — when a population invades a new area where space and food are in plenty the population undergoes exponential growth. However, the exponential population growth always seems faintly ridiculous because the number of most organisms remain usually constant from year to year.

Thus, the population growth curve shows an exponential, or approximately so, at the beginning having initially a slower growth rate which subsequently gets faster and faster. The population then becomes medium sized and the growth rate begins to slow down until it finally reaches zero, when birth balances death. This type of growth curve looks like a flat S and is called the sigmoid growth curve (Fig. 4.42).

The model depicting this type of growth is the logistic equation, introduced in ecology by Raymond Pearl and L. J. Reed in 1920. The logistic equation is defined as the mathematical expression for a particular sigmoid growth curve in which the percentage rate of increase decreases in linear fashion as population size increases.

The growth rate of the population is determined by biotic potential and the size of the population is modified by the environmental resistance (by all the factors that control crowding). These factors may include lowered production due to mothers’ poor nutrition, high rate of death because of predators or parasites, increased emigration etc.

Environmental resistance gradually increases as the size of the population gets closer to the carrying capacity (usually represented by K), which is the number of individuals in a population that the resources of a habitat can support.

Thus, the S-shaped or sigmoid growth curve which has been dealt above, comprises of population that increases slowly at first, then more rapidly, but it subsequently slows down as environmental resistance increases until equilibrium is reached and maintained. This can be represented by a simple logistic equation

K is the maximum carrying capacity.

The logistic growth curve, however, shows another basic pattern of growth, termed as the J-shaped growth curve. In the J-shaped growth curve (Fig. 4.43), the density of the population increases rapidly in exponential manner.

It frequently tends to overshoot the carrying capacity and then drops back rather sharply, as environmental resistance or other limiting factors become effective more or less suddenly. This curve can be presented by the simple model based on the experimental equation (considered earlier)

The equation given above for the J-shaped growth form is same as that of the exponential equation except that a limit is imposed on N. The unrestricted growth is suddenly halted when the population runs out of resources like food or space when frost or heat wave or any other environmental factor intervenes or when the reproductive season suddenly terminates.

When the population reaches the upper limit of N, it remains at this level for a while and then a sudden decline takes place. It thus produces a relaxation-oscillation (boom-and-bust) pattern in density. Such a pattern is the characteristic of many populations in nature, such as algal bloom, annual plants, zooplankton bloom, some insects and, perhaps, lemmings on the Tundra.

The S-shaped or sigmoid pattern of growth shows a gradual increasing action of detrimental factors (environmental resistance or negative feedback) as the density of the population increases. However, in the J-shaped population growth, negative feedback is delayed until right at the end when it goes beyond the carrying capacity.

The equation of the S-shaped curve differs from that of the J-shaped one, in the addition of one expression K – N/K or I – N/K, a measure of the portion of available limiting factors not used by the population.

The S-shaped pattern of growth is followed by a great variety of populations represented by microorganisms, plants and animals, both in natural and laboratory populations. The S-shaped growth form includes two kinds of time lag: (1) the time needed for an organism to start increasing when conditions are favourable, and (2) the time required for organisms to react to unfavorable conditions by altering birth and death rates. The various phases seen in S-shaped curve are the lag, logistic growth, and point of inflection, environmental resistance and carrying capacity, phases (Fig. 4.42).

The lag phase is the time lag necessary for a population to become acclimated to its environment. The point of inflection is the maximum rate of increase. The environmental resistance phase is the slowing of population growth due to limiting resources. The population ultimately reaches the carrying/capacity condition when the rate of population increase is zero and the population density is maximum. Globally the humans have yet to reach the carrying capacity condition.

### Attribute # 5. Age Structure:

Another important attribute of population is age distribution or structure. Age structure of a population refers to the proportion of individuals of various ages. For most animals, the age of an individual is important in specifying its role in the population. Age distribution influences both natality and mortality. The reproductive status of a population is determined by the ratio of the various age groups.

It also indicates what may be expected in the future. A rapidly growing or expanding population generally will contain large number of young individuals stable population will show an even distribution of age classes, while a decline or collapsing population will have large number of old individuals (Fig. 4.44).

The population that is growing or declining at a constant per head rate is called stable age distribution. If the population is not changing and its growth or declining rate is zero, then it is called stationary age distribution. It can be calculated from the l_{x} column of the life table (Table 4.10). It tends to have large number of older individuals than younger ones. However, real populations usually have an age structure quite different than the above ‘ two because of various events in its recent past.

It is evident that populations tend to go to a normal or stable age distribution. Once a stable age distribution is achieved, any unusual increase of natality or mortality will last for a short time and the population would spontaneously return to the stable condition.

As rapidly increasing pioneer population gradually reaches mature and stable condition having slow or zero growth, the percentage of younger age class individuals decreases (Fig. 4.45). It has also been seen that the average age of individuals also increases in a stable population.

The changing age structure, with an increasing percentage of old individuals, has some strong impacts on life style and economical and sociological consequences. A greater proportion of our resources will have to be used for helping the elderly and a small proportion used for education and other child welfare services. However, the economic burden may not be greater as the dependency ratio (the number of workers compared to the number of non-workers) will not be too different.

Age structure also can be expressed in terms of three categories: pre-reproductive, reproductive and post-reproductive. In accordance to their lifespan, the relative duration of these ages varies greatly with different organisms. In case of humans, during recent times, the three age categories are relatively equal with a third of the human life falling in each class.

However, early humans had a much shorter post-reproductive period. Insects have extremely long pre-reproductive period, a very short reproductive period and no post-reproductive period. For example, mayflies require from one to several years to develop during the larval stage and adults emerge to live for only a few days.

In fish population that have a very high potential natality rate, a phenomena called dominant age class has been observed. When in one year large survival of eggs and larval fish takes place, then in subsequent years reproduction is suppressed.

### Attribute # 6. Patterns of Distribution:

Distribution or dispersion of individuals within a population describes their spacing with respect to one another.

**In a population, individuals may be distributed according to four types of pattern (Fig. 4.46):**

All the above four types of distribution are found in nature.

(a) Random distribution occurs in individuals that are distributed throughout a homogeneous area without regard to the presence of others. It takes place when the environment is uniform and there is no tendency to aggregate.

Lone parasites or predators show a random distribution as they are often engaged in random searching behaviour for their host or prey.

(b) In regular or uniform or spaced distribution, each individual maintains a minimum distance between itself and its neighbour. It may occur when competition between individuals is severe or when there is strong hostility—which eventually promotes even spacing.

Trees present in forests that have reached sufficient height to form a forest canopy show a regular uniform distribution due to competition for sunlight. Other examples are monoculture crops, orchard or pine plantation, desert shrubs etc. A similar regular pattern of distribution is seen in territorial animals.

(c) Clumped distribution takes place in individuals who maintain discrete groups. Clumping or aggregation, by far, is the most common pattern of distribution.

**It may occur due to:**

(1) The social predisposition of individuals to form groups,

(2) Clumped distribution of resources (the most common cause), or

(3) Tendency of progeny to remain close to their parents.

Salamanders prefer to live in clumps under logs. Birds travel in large flocks. Trees form clumps of individuals through vegetative reproduction.

(d) In regular clumped distribution individuals are clumped and are spaced out evenly from other similar clumps.

Herds of animals or vegetative clones in plants show either random or are clumped in a regular pattern. In the absence of hostility and mutual attraction, individuals may distribute themselves at random. Thus, in a population, the position of an individual is not influenced by the positions of other individuals. A random distribution pattern implies that spacing is not related to a biological process. It is often used as a model to compare it with an observed distribution.

**To determine the type of spacing and the degree of clumping, several methods have been suggested of which two are mentioned:**

1. To compare the actual frequency of occurrence of different sized groups obtained in a series of samples. If the occurrence of small sized and large sized groups is more frequent and the occurrence of mid-sized groups less frequent than expected, then the distribution is clumped. The reverse is seen in uniform distribution.

2. The distance between individuals are measured and the square root of the distance is plotted against frequency. The shape of the resulting polygon indicates the pattern of distribution. A symmetrical polygon (bell-shaped) indicates random distribution, a slanted polygon to the right indicates a uniform distribution, and one slanted to the left indicates a clumped distribution.

The pattern of dispersion for many species reflects the arrangement of habitat patches in the environment. For example, apple leaves are habitat patches for the mite. The pattern of dispersion for other species may be due to an interaction between the spatial arrangement of habitat patches and other ecological or behavioral processes.

For example, kangaroo rats, in order to construct their burrows, require certain soil characteristics. It may be assumed that individual kangaroo rats would simply aggregate within suitable habitat patches when they can easily construct burrows.

However, the aggregated dispersion in the population was not entirely due to habitat patchiness as banner tailed kangaroo rats have a tendency to leave the place of their birth. The movement of individuals may also influence the pattern of dispersion. For example, in case of plants, dispersal of seeds often depends on the action of other organisms.

The varying degree of aggregation of individuals is characteristic of the internal structure of most population. Aggregation will subsequently increase competition between individuals for resources such as nutrients, food, space etc.

This often is counter-balanced by the increased survival of the group due to its ability to defend itself, or to find resource, or to modify microhabitat conditions. Thus, both under-crowding (lack of aggregation) and over-crowding may be limiting. This view was put forward by W. C. Allee, a Quaker and V. E. Shelford, and was termed as the Allee effect or Allee principle of aggregation.

Allee effect stresses that any optimal function (faster body growth, increased reproduction, or longer life) takes place at an intermediate rather than at minimal density. For instance, at low density, a drop in reproductive rate takes place as some females may go unmated because they were not found by males or because of an unbalanced sex ratio.

### Attribute # 7. Population Genetics:

To understand how populations evolve and ecosystems change over time, an understanding of population genetics and natural selection is necessary. Population genetics may be defined as the study of changes in the frequencies of genes and genotypes within a population.

Natural selection is an evolutionary process where the frequencies of genetic traits borne by individuals in a population change as a result of differential survival and reproductive success. Populations bear genetic variations and the amount of variation changes substantially from species to species and from place to place.

**Genetic variation is important to a population because:**

(1) It is the capacity of populations to respond to environmental change through evolution, and

(2) Variation in progeny may increase the likelihood of some individuals to be well-adapted to particular habitat patches or to changed conditions. Genetic variation is caused primarily by mutation and by gene flow where different genes have a selective advantage.

**Measurement of genetic variation in population:**

**To understand the nature of genetic variation in populations, ecologists have developed a number of techniques:**

1. In general cases, for a particular trait, through phenotypic observation, ecologists can tell whether an individual is homozygous or heterozygous for that trait. Then, by observing different members of that population, he may be able to determine the extent of heterozygocity in the population.

2. Genetic variation can be determined by electrophoresis, that is, by examining the protein products of genes. This process deals with the fact that proteins produced by a heterozygote have slightly different forms called allozymes. Homozygotes, on the other hand, produce only a single type of protein.

3. Recently, to examine genetic variation, ecologists have employed recombinant DNA technology. Here a special class of enzyme, called restriction enzymes (that recognizes a specific base sequence on the DNA molecule), are used to cut both the strands of DNA at the site of specific base sequence.

By the use of different restriction enzymes the DNA molecule can be cut at two places to produce a DNA fragment called a restriction fragment. Restriction fragments of different lengths are referred to as restriction fragment length polymorphism or RFLP. RFLPs are heritable and used for determining genetic variation or as genetic markers for studying genetic disorders.

4. Another recent approach for examining DNA directly is DNA fingerprinting. It is an enhancement of RFLP technology. It is used to detect variations in short segments of DNA, having repeated sequences of nucleotides, referred to as variable number tandem repeats (VNTR). Such VNTRs may be found at different loci of different chromosomes. Restriction enzymes are used to cut these repeating sequences. These may be used as a measure of the level of genetic variation in populations.

**Genetic variation and population size:**

As different alleles may cause differences in form and function, they are liable to come under the influence of natural selection. Natural selection may favour an increase of one allele over another in a population. This may lead to genetic drift (changes in gene frequency).

The rate of fixation of alleles is inversely proportional to the size of the population. Thus, in a small population, genetic variation decreases more rapidly than in the case of larger populations. When small populations continue to exist for long, then loss of genetic variation takes place due to genetic drift and close inbreeding, a situation referred to as population bottleneck.

Such a condition had occurred in East Africa in populations of cheetahs which exhibited practically no genetic variation. Small subpopulations often restrict the evolutionary responsiveness to selective pressures of changing environment, thus making these subpopulations vulnerable to extinction.

#### Population Regulation:

Individuals are subjected to a number of environmental hazards which affects its growth and proliferation. Populations are made up of individuals and the size of the population depends upon the reproductive fitness and lifespan of those individuals. Thus, population size is affected by factors such as nutrients, flood, drought, predators, diseases etc.

**The various limiting factors are:**

**(a) Factors those are constant:**

There are factors that are relatively constant and limits the population to a fairly constant size, as individuals has to compete for the resources. Plants, for example, compete for space and light, birds for nesting territories, heterotrophs for food etc. However, large changes in population are not produced.

**(b) Factors those are variable:**

Although certain factors like seasonal drought or cold are variable they are, however, predictable. Their presences are felt for some months or few days and may sometimes result in population crash. Evasive actions like migration or dropping of leaves (deciduous trees) may be taken to avoid such predictable factors.

**(c) Factors that are unpredictable (density independent and density dependent****):**

Ecosystems are subjected to irregular or unpredictable extrinsic disturbances like weather, water currents, pollution etc. These physical factors often influence the population size. When there is low probability of physical stress such as storms or fire, populations tends to be biologically controlled and their density is self-regulatory.

**Factors favourable or limiting to a population are either:**

(i) Density-independent, that is its effect on the population is independent of the population size, or

(ii) Density-dependent, if its effect is a function of population density, climactic factors often acts in a density-independent manner, while biotic factors act in a density-dependent manner. The J-shaped growth curve occurs in case of density-independent population whose growth slows down or stops. On the other hand, sigmoid growth curve occurs in density-dependent population where self-crowding and other factors regulate the population growth.

**The primary differences between density- independent and density-dependent factors are:**

1. Density-independent or extrinsic factors of the environment cause variations (sometimes drastic) in population density. This may cause shifting of carrying capacity levels (asymptotic).

Density-dependent or intrinsic factors (such as competition) tend to maintain a stable population density.

2. In physically stressed ecosystem, density-independent environmental factors play a greater role.

In favourable environment, density-dependent natality and mortality play an important role.

3. Density-independent factors involve interaction with the rest of the community.

Density-dependent factors are the population’s own response to density.

4. The main density-independent factors are predation, parasitism, disease and interspecific competition.

Density-dependent factors include intra- specific competition, immigration, emigration and physical and behavioural changes that affects reproduction and survival.

For many organisms, intraspecific competition is one of the most important density dependent factors. Like animals, plants exhibit density-dependent population regulation mechanisms. At very high density, plant populations undergo a process termed self-thinning. When in an area seeds are sown at a high density, the emerging young plants compete with one another. As the seedlings grow, competition among them becomes tougher and tougher.

Many die leading to a gradual decline in the number of surviving plants. A similar condition occurs in over-populated caterpillars that tends to overshoot the carrying capacity conditions. Holling (1966), has emphasised the importance of behavioural characteristics, where a given insect parasite can effectively control the insect host at different densities.

Population studies generally depend upon the type of ecosystem of which it is a part. Physically controlled and self-regulatory ecosystems are arbitrary. It presents an oversimplified model. However, it is a relevant approach, as human efforts have been directed towards replacing self-maintained ecosystems with monocultures and stressed systems. At the same time, the cost of physical and chemical control has risen due to the resistance of pest to pesticides and the toxic chemical by-products in food, water and air, have become a potential threat to mankind.

This has led to the increased implementation of integrative pest management (IPM). Evidences of the above is the generation of increased interest in a new frontier termed ecologically based pest management. This involves efforts to reestablish natural, density-dependent, ecosystem-level controls in agricultural and forest ecosystems.

## Acknowledgments

We thank P. Visconti for developing and providing the crosswalk between the IUCN habitat classification scheme and the ESA CCI land cover maps. We also thank 3 anonymous reviewers for providing useful comments that improved the quality of the manuscript and acknowledge the many thousands of individuals and organizations that contribute to the IUCN Red List assessments of birds and mammals. L.S. and M.A.J.H. were supported by an ERC consolidation grant (ERC – CoG SIZE 647224).

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## How should species density be calculated for a clumped distribution? - Biology

**SAMPLING SEDENTARY ORGANISMS**

**I. Objectives: ** 1. Introduce basic approaches for sampling sedentary organisms

2. Provide experience with application of transect and quadrat sampling

Ecologists have the challenge of evaluating the nature and function of natural systems. However, they are often faced with the problem that they cannot accurately determine many important parameters strictly through qualitative observations. Problems with spatial scale often make it difficult to directly estimate average densities in habitats such as large forested tracts or in aquatic habitats where one cannot directly see the animals or plants of interest. To measure functions in natural system, it is necessary to representatively sample both the biotic and abiotic components. **Since it is usually impossible to sample an entire population within a study area, one must take smaller samples of the community or population of interest.** The problem then becomes how to best estimate the parameters of some ecological system taking into account limited time and resources. **The problem of sampling can be thought of as several issues: 1) what sampling method can be used to collect the needed data, 2) how do we select sampling locations to limit potential bias and obtain best estimates of target parameters, and 3) how can we minimize the number of samples we take while still providing an accurate estimate of parameters?**

Ecologists have developed several basic sampling approaches to deal with sedentary (non-moving) organisms. **The most common method is plot sampling.** This method is relatively straightforward in that it simply involves counting the number of organisms of interest in a defined area (often, but not necessarily, a quadrat). **Plot sampling is a highly versatile approach, providing information on densities, associations, dispersion patterns, and indirect evidence on a variety of community or population processes.** However, **three problems with this method** that will be explored in this lab are **the size of the sampling plot, how plots will be placed, and the number of plots to be sampled.** The size of the plot must obviously vary with the size of the organisms to be sampled, but also may need to be varied depending on distributions of organisms or the size of habitat patches in the environment. It is often useful to have a plot smaller than the average patch size to get information of dispersion patterns, mean numbers within patches, and densities between patches. Also, **larger plots tend to give more accurate estimates of overall densities, but are more cumbersome and time consuming to place and sample.** The placement of plots needs to be done so that there is no bias in the data collected. This is usually achieved through random or haphazard placement (what is the difference?), but may also be limited by time and resources.

**When organisms are known to vary across some defined gradient, a variation of the plot sampling method that is often used is a belt transect. This involves sampling replicate rectangular belts that straddle a known environmental gradient.** Often these belts are broken into subsections. **The problems of size (and shape), placement, and number of replicates also apply to belt transects** **.**

An alternative to plot sampling methods are a variety of methods collectively known as **plotless sampling approaches.** These approaches are generally **quicker to apply** than plot sampling, but usually **provide much more limited information.** **The most common plotless sampling method is point-quarter sampling.** Point-quarter sampling **can provide density measures** if applied correctly and has the advantage over plot sampling that it is **less cumbersome and time-consuming than sampling of larger quadrats**. However, it has the **disadvantages that density estimates may not be accurate if organisms have severely clumped or uniform distributions and the technique does not provide as versatile information as obtained with quadrat sampling**. The basic methodology for point-quarter sampling is: 1) randomly (or haphazardly) select a center point, 2) divide the area around the center point into 4 quadrants (pie sections), with the quadrants crossing at the center point, and 3) record the distance to the nearest tree (or species of interest) in each quadrant.

This procedure is then repeated for additional replicates. Density of trees for Point-quarter sampling can be determined with the following formula:

D=density in distance units squared A=area occupied by trees

d_{i}=distance from the center point to the nearest tree in the ith quadrant

Q=number of quadrants taken for each center point (usually 4)

PQ=number of center points times the number of quadrants per point

**Another method used to obtain qualitative information on the distribution and abundance of sedentary organisms is the line transect.** This method simply involves placing a line of pre-determined length over an area and then recording the number of organisms intercepted by that line (or within a defined distance of the line). This method has the **advantage of being quick and easy, but has the disadvantage of providing information on only relative abundances and not giving quantitative information on actual densities.**

**III. Methodology - Computer Sampling Simulation** .

We will use the Ecobeaker program to investigate several aspects of quadrat sampling, especially how size, number and shape of quadrats taken affect accuracy of quadrat results.

*A. The Ecobeaker Universe*

The Ecobeaker universe presents a 50m X 50m landscape with 3 different types of plants on it, each a different color. The green plants are growing in completely random places around the landscape. The blue plants grow much better when they are near other blue plants, so they are clumped together (a patchy distribution). The red plants compete strongly with each other, so they do not like to grow nearby other red plants. This makes the red plants space themselves out in a somewhat even pattern. There are exactly 80 individuals of each plant type in the whole area.

*B. Getting into the program*

1. Double click on the Ecobeaker icon

2. Open the file "Random Sampling" using the open command in the file menu.

3. When it is open, you should get a screen that shows the landscape to be sampled in the upper left of the screen. To the right of the landscape are the Species and Population size windows, which show the color and total number for each species. The Sampling parameters window lets you change the width, height and number of quadrats which you use to sample the populations. The Control panel window lets you actually take a sample (by clicking on "sample").

** C. Effect of size, keeping replication constant ** 1. The program has an initial, default quadrat size of 5m x 5m.

2. Click on the sample button in the Control panel to take a sample. A square will be drawn on the Landscape window, showing the area that is being sampled. Just below the Landscape window a dialog box will appear, telling you how many individuals of each species were found within this square. Write down both the number of individuals of each species and the width and height of the area sampled.

3. Repeat this procedure at least 4 more times (5 total), so that you get a range of different estimates that tell you whether you can expect this scheme to consistently give you accurate and precise results.

4. Now increase the size of the sampling quadrat to a larger size such as 10m x 10m. To do this, go to the Sampling parameters window and type "10" for both the quadrat height and quadrat width items. Then click on the change button in the sampling parameters window to make the change in size you specified go into effect.

5. Repeat the sampling procedure for the larger quadrat size.

6. Repeat the entire procedure for at least two more quadrat sizes (e.g. 15m x 15m, 20m x 20m).

7. When you have finished your sequence, estimate the true population size (number in the entire 50m x 50m landscape) from each sample you took for each species. Present the results in a table form. Next, make a plot of population estimate (y-axis) versus quadrat area (x-axis) for each species. This will show you visually what happened to your estimate as you increased quadrat size. You will have several pionts plotted for each quadrat size, from each of the samples you took at that size, and this will show you the range of accuracy you might expect from a quadrat of that size.

8. What are the shapes of your estimated population size versus quadrat area graphs for each species? Is there some point where increasing the size of the quadrat seems to make little difference in accuracy? Is this point the same for all the species?

** D. Effects of quadrat shape ** 1. Change your quadrat size to 10m x 10m and do sampling as described above.

2. Repeat this procedure with 3 other plot shapes that have the same area (e.g., 5m X 20m, 4m X 25m, 2m x 50m).

3. Estimate the true population sizes and construct graphs as described in #7 above (in this case, the y-axis would be estimated population size and the x-axis would be quadrat shape ordered from most narrow shape to exact square).

4. How did changing quadrat shape affect population estimates? Is the affect the same for all species?

** E. Influence of replicate sampling ** 1. Set the quadrat size back to 5m x 5m and take 5 sets of samples as described above.

2. Now go to the Sampling parameters window and change the number of quadrats to 2. This time, when you take a sample, first one 5m x 5m quadrat will appear, as before, and a dialog box will tell you how many individuals of each species were found in that quadrat. When you click on the continue button, a second 5m x 5m quadrat will be laid down, and again you’ll receive a report of how many individuals were found in the second quadrat. Finally, a third dialog box will appear giving you the total number of individuals found within both quadrats. For the purposes of this lab, you should calculate the mean number per quadrat (total number divided by number of replicate quadrats per sample). Repeat this sampling procedure 5 times (5 pairs of quadrats).

3. Repeat the sampling procedure described in step 2 of this section for 5 replicates and 7 replicates per sampling period, recording means for each.

4. Estimate the true population sizes and construct graphs as described in #7 above (under quadrat size). The x-axis in this case will be number of replicate quadrats used and the y-axis will be population estimates for each sampling set.

5. Do these graphs look the same as those you got with a single quadrat? Does accuracy of your estimate change with increasing replication?

**IV. Laboratory Assignment ** Turn in graphs and answers to questions in sections C8, D4 and E5.

**V. Key Points From The Homework:**

1. As you increase the plot size, accuracy increases but the sampling becomes more cumbersome and time-consuming.

You may use the smallest plot sizes for species that are distributed evenly you need to use larger plot sizes for species that are clumped.

2. When sampling a clumped species, it is best to use belt transects.

3. As you increasing the number of replicates, accuracy increases but the sampling also becomes

more cumbersome.