Mathematics 126 -- Calculus for Social Science 2

Discussion 2: Population Growth Models

October 24, 2001

Background

One mathematical model of population growth for a species (that is, a differential equation whose solutions are supposed to exhibit all or some of the same behavior as actual populations, and allow us to make predictions about the real-world population under consideration) is the ``exponential growth'' equation:
(1)

dP/dt = kP
(k > 0), which has general solution
(2)P(t) = P(0)e^kt
As we know, saying that a population satisfies the equation (1) for all t is the same as saying that

(*) the rate of change of the population (dP/dt) is always proportional to P.

The constant of proportionality k in the equation represents the net growth rate per unit time -- birth rate minus death rate.

Discussion Questions

A) For example, suppose you had k = .04, P(0) = 1000, t in years, and you used (2) to calculate a prediction of the population at t = 50 years. What number does (2) give you?

B) What are some of the limitations of the functions (2) as mathematical descriptions of real populations? What are we ``leaving out''? Are we ``leaving out'' too much of the real world to get realistic results?

C) If our population is living in a fixed habitat with limited, but self-renewing, resources (for instance, plants as food for the individuals of an animal species to eat), then one possible additional constraint we might want to build into our population model is that there is a limited sustainable maximum population M that the habitat can support. In biology, M would be called the carrying capacity. One mathematical model along these lines was proposed by the Belgian scientist P.F. Verhulst in the 1830's, and intensively studied by biologists and demographers since. Verhulst's idea was to consider, instead of (*), the proposed relationship:

(**) The rate of change of the natural logarithm of the population is proportional to the difference between M (the maximum sustainable population) and the population.

Write Verhulst's proposed model as a differential equation of the form dP/dt = f(t,P) for the appropriate function f(t,P). This is called the logistic equation.

D) Solve your equation using the separation of variables technique, and the table of integrals as necessary. You should be able to transform your answer to

P(t) = M/(1 + c e^-Mkt)
by algebraic manipulations, where c is an arbitrary constant. Take M = 5000, find the particular solution with P(0) = 200, and plot for t = 0 to t = 10. Describe the behavior of the population function over this time interval.

E) A ``thought question'' -- Is a fixed carrying capacity for human populations realistic? For instance do you think that the carrying capacity for humans on the Earth is the same now as it was 2000 years ago? Why or why not?

Assignment

Solutions will be due in class, Wednesday October 31. One set of solutions per group.