3 differential equations, including the logistic function
I would like to derive formally a particular result, that I, intuitively, suspect to be true. Therefore, I want to use the following differential equations:
There are n groups in a particular population, each with its own rate of reproduction r1, r2, r3, ..., rn.
(1) Say that, when the total number of individuals in the population, for all groups, P=P1+P2+P3+...+Pn is below the carrying capacity, the number of individuals grows over time according to the logistic function, with each group growing at their own reproduction rate r1, r2, r3, ...,rn.
(2) Say that, when the total number of individuals the population, for all groups, is above its carrying capacity, the number of individuals shrinks over time at the same decline rate d; which is the same for all groups.
(3) Say that, the carrying capacity K=K(t) randomly fluctuates over time ("random walk"), between Kmin and Kmax.
I want to model this set of differential equations, in order to demonstrate that when reproduction rate rk=max(r1,r2,r3,...,rn), that in the limit for time t going to infinite, P=Pk, while Pi=0, for all i != k.
Meaning: When the carrying capacity of a population fluctuates, the entire population will tend to consist exclusively of individuals from the group with the highest reproduction rate, while all the other groups will tend to disappear completely.
If anybody could help me to derive this result formally from a set of differential equations, I would be most grateful.