The inclusion of harvesting into our mathematical model is easy:
during each time period we assume that a constant number, *h*,
of individuals are removed from the population, therefore the
differential equation that models the population becomes

dP/dt = k P - A P^{2}- h.

- Geometrically describe
what happens to the location of equilibria for small, medium,
and large values of harvesting.
Can you verify this
result by analytically solving for the location of equilibria
of the vector field on the phase line? For what values of
*h*(in terms of*k*and*A*) will there be- two equilibria?
- one equilibrium?
- no equilibria?

- Sketch the vector field for a value of harvesting that
gives rise to a single equilibrium.
Describe the stability of the equilibrium.
Is it a stable or unstable equilibrium?
- Now set the coefficient of overcrowding to zero. Can harvesting alone restrict the growth of a population? In other words, can removing a constant number of individuals from an exponentially growing population result in a bounded population over the long term? Analyze the stability of any equilibria that result and describe in biological terms what is happening in this model.

The appearance or disappearance of equilibria is one example of a

In real populations, both the birth rate and the coefficient of overcrowding may not be constant, but may vary according to environmental factors. For example, stress brought on by a drought may temporarily decrease a population's birth rate, and it can also decrease the carrying capacity of the land (thus increasing the coefficient of overcrowding).

Furthermore, in many populations subject to harvesting, the harvesting is not applied uniformly to all of the population, but is restricted to adult (sexually mature) individuals, which could affect the birth rate. In an effort to lessen harvesting's effect on birthrate, sometimes the harvesting is biased towards harvesting more males than females, since a small number of males can be used to impregnate all of the mature females. Models of population that are more realistic than the logistic model try to account for factors such as these.

The DNR asks you if it is possible to eventually reduce the population to 60,000 individuals by harvesting adult members of the population. They want you to mathematically model this request and recommend a course of action.

- First assume that the population may be modeled by
a logistic law. Use the data below to help you estimate values
of
*k*and*A*for this population, and to estimate a value of*h*that would satisfy the DNR request. Explain how you arrived at your estimates. You may want to numerically explore your model using the Population Simulator. - Is the logistic growth law (with harvesting)
adequate to model the DNR's request?
If so, defend the model. If not, invent a more sophisticated model
that incorporates additional features that you believe are important.
- Write a one page memo to the DNR that presents your analysis to this question, and recommends a course of action.

- Females become sexually mature in their third year; males in their fourth year.
- Dominant males will mate with multiple females.
- On average, a mature female gives birth to 1.1 offspring.
- There appears to be 30% infant mortality in the first two years.
- There appears to be 10-33% adult mortality per year due to natural causes.

Current Population Estimates (in Thousands) Age Males Females ----- ----- ------- 1 15 15 2 11 11 3 8 10 4 7 8 5 or older 8 7

Comments to:
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Created: May 15 1996 ---
Last modified: May 15 1996

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The Geometry Center
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