Return to: Equilibria, Stability, and Phase Space
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
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.
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.
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
- 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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