Which type of differential equation is this?

[Guest]

Hello, I'm struggling to get started on this problem.
If you could explain which type of differential equation it is, and how you know that for sure, that would be plenty of help as a first step. Thanks

A population r(t) of rabbits (at time t) satisfies

dr/dt = kr (1 − r/r∗) − αfr

where k > 0 is a constant representing the rabbit breeding rate, r∗ > 0 is the (constant) maximum sustainable rabbit population size in the absence of predation, f > 0 is the population of foxes, and α > 0 is the (constant) rate of predation of rabbits by foxes.

Suppose that the fox population, f, is constant. Solve the differential equation above, and determine
(a) the size of the rabbit population as t → ∞;
(b) the maximum predation rate α for which the rabbit population does not die out as t → ∞;
(c) the value of α which maximises αfr (the total number of rabbits caught) as t → ∞, and the corresponding rabbit population.

[HallsofIvy]

That is a first order separable equation. [tex]dr/dt= kr(1- r/r*)- αfr= r(k- kr/r*- αf)= r((k- αf)- kr/r*)[/tex]. We can "separate" it as [tex]\frac{dr}{r((k- αf)- kr/r*)}= dt[/tex] and integrate both sides. On the left, separate it into "partial fractions"- find numbers A and B so that [tex]\frac{1}{r((k- αf)- kr/r*)}= \frac{A}{r}+ \frac{B}{(k- αf)- kr/r*}[/tex]. Those integrals will both give logarithms which you can combine into a single logarithm and so write r in terms of exponentials of t.