dS/dt = f(S) = kS(1-S/N)(S/M-1)

Question

of a fox squirrel population from the previous section. Suppose that the parameters
M and k remain relatively constant over the long term but as more people move into
the area. the parameter N (the carrying capacity) decreases.
(a) Assuming that M <= N, sketch the graph of the function f(S) for fixed values
of k and M and several values of N.
(b) At what value of N does a bifurcation occur?
(c) How does the population of fox squirrels behave if the parameter N slowly and
continuously decreases toward the bifurcation value?

Answer 1

what's the trouble? Expand the right side and you have

dS/dt = k/MN (-S^3 + (M+N)S^2 - MNS)

That's just a polynomial, which you can easily integrate to find S(t)

Answer 2

for purposes of integration n k m are constant
then
ds/dt = -ks (s^2/nm - S/n -s/m +1)

= -k [ s^3/nm - s^2/n - s^2/m + s]

=k[ -s^4/4nm +s^3/3n + s^3/3m -s^2/2 + c]

You can play with this perhaps
http://www.wolframalpha.com/input/?lk=3&i=plot++-s^4%2F4n+%2Bs^3%2F3n+%2B+s^3%2F3+-s^2%2F2
I ran it for m = 1
you can try it with various values of n