We have mentioned 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). Our model was the ``exponential growth'' equation:
(1)
(*) the rate of change of the population
(dy/dt) is always proportional to y.
The constant of proportionality k in the equation represents the net growth rate per unit time -- birth rate minus death rate.
A) For example, suppose you had k = .07, y(0) = 1000, t in years, and you used (2) to calculate a prediction of the population at t = 100 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 our 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 ratio of the rate of change of the population to the population is proportional at each time to the difference between M (the maximum sustainable population) and the population.
Write Verhulst's proposed model as a differential equation of the form dy/dt = f(t,y) for some appropriate function f(t,y).
D) Which of the following slope fields matches your equation from C)?
E) Solve your equation using the separation of variables technique.
Note that
Solutions to these questions and to the lab questions from Friday November, 14 and Monday, November 17 will be due on Wednesday, November 19. One set of solutions per group of four. This assignment will count for two of the parts of the lab/discussion grade for the course.