Question

In logistic growth, we assume that a population grows at a rate
proportional to its size but that the larger it gets, the more likely it becomes that competition between individuals in the
population for resources (food, living space, etc.) can cause the
death of some individuals. The growth constant is commonly called k and the death constant is called . If the population is P(t) then we have this differential equation:dP/dt= kP - P^2
Assume a population of fish in a lake follows logistic growth; initially, 40 fish are introduced to the lake. Solve the differential equation if k = 0.5 and a

Answer 1

and = :001.
(a) The equation is both separable and Bernoulli. I found it
easier to treat it as a Bernoulli equation but opinions may
vary. Find P(t) and try to predict what will happen to the
population in the long run.
(b) If you solve this by separating variables, it will be easier to
nd the two singular solutions to this dierential equation.
(A hint: they are both constant functions, P = c where c is
some number.) Do the singular solutions have any physical
interpretation?

Answer 2

=.001