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Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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POPULATION GROWTH MODELS

I.  Non-overlapping generations

When the generations in a population do not overlap (all  parents are dead by the time that offspring mature), like occurs for many insects and annual plants (like garden weeds), we can predict the size of the population over time by knowing two factors:

The number of individuals in the population (represented as Nt); and

The average number of offspring produced per adult (represented as Ro).

If we multiply these two numbers together (Nt x Ro), we will be able to estimate the population size in the next generation.

Let's assume that the net reproductive rate of a population is 1.5, and that we start with 10 individuals.

This model will predict that population size will be the following over the next 5 generations:

Generation  Estimated Population Size   Population increase

1   10   =  10    -

2   10 x 1.5  = 15    5

3   15 x 1.5  = 22.5    7.5

4   22.5 x  1.5   = 33.75    11.25

5   33.75 x 1.5  = 50.625   16.875

Note that the population is growing at an increasing rate over time.  Thus, population growth does not occur in a linear fashion.   This is termed geometric growth.

II.  Overlapping generations

However, many species have overlapping generations (adults still alive when offsping begin having children).  To predict population size at any given time for overlapping generations, we need to know:

Population Size at some time period  (No)

Birth rate in the population at that extact time (b)

Death rate in the population at that exact time (d)

The amount of time you want to predict population size into the future (t)

From b and d, we can calculate the growth rate of the population at that exact moment:

r = b - d

From these variables, we can construct the following equation:

Nt = Noert, where e=2.72

If  No=10, and r=1.5

Time   Estimated Population Size   Population Increase

0   10 x 2.721.5*0 = 10    -

1   10 x 2.721.5*1 = 44.8    34.8

2   10 x 2.721.5*2 = 200.9    156.1

3   10 x 2.721.5*3 = 900.2    699.3

4   10 x 2.721.5*4 = 4034.3   3134.1

Again, you can see that this population is not growing in a linear fashion as it is adding individuals at an increasing rate over time.  This is termed exponential growth.

III.  Density-dependent growth.

Populations cannot grow forever.  Eventually they will run out of resources or space to live.  When that occurs, the death rate will increase.  Eventually, the rate of death and birth will be equal, so that populations will quit growing.  This is termed Logistic Growth, and is graphically displayed as an S-shaped curve.

The population size at which birth and death rates are equal is termed the Carrying Capacity.  This number will change given the amount of resources in the environment.  Regions with more resources can support a larger carrying capacity.  Regions with fewer resources will support a lower carrying capacity.

This type of population regulation is termed 'density dependent' as the death rate is related to the population size.  The larger th population, the greater the death rate.

Why is this so?  If there are few individuals, there are lots of food, and everyone is healthy.   But, if populations are large, there is less  food, and individuals become weak and prone to disease.  Good examples of this can be seen as the death rate of deer in the winter increases as the deer population gets larger.

IV.  Density-independent growth

Populations are not always regulated by changing death rates due to increases in population size.  Some environmental factors effect death rates indepedent of population size.  As such, these factors will influence populations of any size.  Examples of density-independent factors are climate, fire, flood, volcanoes.  These factors may be most important in regulating the size of small animal and insect populations.

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