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Harvesting

The removal of a constant number of populations during each time period is known as harvesting or fishing. Ever since primitive humans began hunting and fishing thousands of years ago, there has been a need to know how killing a certain number of animals will affect the population at large. The fact that there are over 750 plants and animals on the endangered species list indicates that humans are not always cognizant of how their actions will affect plants and animals.

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 P² - h.

Question 8

Use the Population Simulator to generate trajectories to the logistic model with harvesting (and simultaneously display the solutions on the phase line).

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 bifurcation: a qualitative change in the dynamics of a system. The critical value of h for which exactly one equilibrium exists is called a bifurcation value. For values of h less than this value, the population dynamics all look qualitatively "the same" as each other, and this is especially apparent if we represent the dynamics on the phase line. At the critical value, the dynamics change, but then for larger values of h, the dynamics on the phase line again look qualitatively "the same" as each other.

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.

Question 9

Since you are now an expert on population modeling, you are asked by your state's Department of Natural Resources (DNR) to help plan how many hunting licenses to issue for an animal that used to be on the endangered species list, but has made a spectacular recovery. The data they provide (see below) indicate that the population, when left to itself, averages 100,000 animals, and experiences yearly variation of about 10% per year, with an occasional harsh winter or drought reducing the population by 33%.

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.

Population Characteristics and Data:

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

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