POPULATION GROWTH MODELS

The goal of much of wildlife management is affecting the distribution and abundance of species. May want to increase, stabilize, or decrease populations. To do this, need an understanding of how population numbers change over time and the factors causing those changes. The study of population dynamics is important in understanding these changes and in determining how a management strategy will affect a population.

We will begin by discussing 2 types of population growth models.

Exponential Population Growth and Logistic Population Growth

First have to become familiar with terms and symbols.

lamda = annual rate of change

R = Finite rate of growth (per capita rate of increase)

b = per capita birth (recruitment) rate (young alive/individual after 1 time year)

d = per capita death rate (deaths/individual)

N = population size (N_o = initial pop. size N_t = pop. size at time t)

EXPONENTIAL GROWTH

N_t+1/N_t = =1+R

ex) N_t = 100, N_t+1 = 120, then N_t+1/N_t = 120/100 = 1.2

= 1.2 and R = 0.2

So, if = 1 pop. is not changing size

< 1 pop. is growing smaller

> 1 pop. is growing larger

We can calculate how much a population would grow in a year as:

N_t+1 = N_t or N_t+1 = N_t (1+R)

Draw a graph of exponential pop. growth.

When a population is growing exponentially, the value of does not change with population size. But, every year, more individuals are added to the population than were added the year before. We can calculate this rate of change using the following equation:

N_t+1 - N_t = RN

Another way to think about how the population is changing is to calculate the number of new animals added to the population and subtract the number of animals dying from the population. This can be done as:

Individuals born = bN (young alive at 1 year of age)

Individuals dying = dN (number of animals in the population dying during the year)

so N_t+1 - N_t = bN - dN = (b-d)N

Because N_t+1 - N_t = RN, you can infer that R = b - d.

The rate of pop. change increases linearly w/ pop. size

DRAW GRAPH OF N_t+1 - N_t vs. N

Under what conditions might a species show exponential pop. growth?

1) Placed in a new environment

2) A limiting factor has been removed

3) Cyclic populations may show exponential growth when passing out of their low phase.

4) Show graph of human population OVERHEAD

Exponential population growth is not sustainable.

LOGISTIC GROWTH

There are limits in the environment and all populations will eventually stop growing.

These limits ultimately imposed by resources.

K = Carrying Capacity, the maximum number of animals an area can support on a sustainable basis. This maximum number may be determined by resources (food), space (winter range), or social factors (territoriality).

When there are limits in the environment, the growth rate of the population will approach 0 as the population size approaches K. This will occur because birth rates decrease and death rates increase as more and more animals live in an area.

If we assume that R decreases linearly with population size, then we have a model of population growth called LOGISTIC GROWTH

Draw a graph of R_t vs. N_t

This relationship can be described by the equation R_t = R₀ (1 - N_t/K)

If we substitute this new value of R_t in equation 1 above, we can calculate population growth under the logistic model. This can be done using the equation:

N_t+1 = N_t [1+R₀ (1 - N_t/K)] (3)

Logistic growth produces a sigmoid shaped population growth curve that looks like this.

DRAW GRAPH OF TIME VS. Nt

Now N_t+1 - N_t does this: (draw graph)

What N would support Maximum Sustainable Yield?

This introduces the idea of density dependent regulation.

Birth and death rates may be dependent on the density of the population.

Draw graph of birth and death rates related to pop. size. (also put on level line for density independent birth and death rates).

Where these two lines cross is where population growth stops. This is K, or carrying capacity.

Summary of equations:

Exponential Growth Logistic Growth

Population Size(N_t+1) N_t(1+R)N_t[1+R₀(1-N_t/K)]

Growth rate (N_t+1 - N_t) RN_t N_t[R₀(1-N_t/K)]

Lets work an example

Example of calculating lambda:

We have a population of 100 deer, 50 of which are females. Each female has 2 fawns and 40% of these die before the following spring. What is the birth rate of the population?

50 females * 2 fawns/female = 100 fawns/year

100 * 0.40 = 40 die over winter 60 fawns alive at end of 1st year.

60 fawns / 100 adult deer = 0.60 offspring/ind./yr = b = birth rate

Individuals born = bN

In a population of 100 adult deer, 5 die over the summer, and 20 are shot during the hunting season, and 10 die over the winter, before the next spring. What is the death rate for adult deer?

35/100 = 0.35 deaths/ind.

What is the growth rate of the population?

R = b-d

= 0.60 - 0.35

R = 0.25

LIFE HISTORY STRATEGIES R VS. K SELECTION

In the logistic population growth model, two parameters determined the shape of the curve. These were R (finite rate of growth) and K (carrying capacity).

Some species are often on the lower part of the logistic growth curve, where population growth rate (R) determines the pop. size vs. time.

Other species fill up the habitat and population size is often determined by the carrying capacity (K).

These different categories represent two extremes of what are called life history strategies.

r-selected species and K-selected species.

Realize that r and K-selection is a continuum, and many species will show some characteristics of both r and K-selection.

These different life history strategies have certain characteristics. These are shown on the table I passed out.

OVERHEAD

Discuss parameters from table.