Study Questions for Final
First we must find the equilibria of the model.
n1 = n1 (1-m) + n1 r1 (1-n1/K)
n2 = R n2 + m n1
Simplifying both equations:
0 = - m n1 + n1 r1 (1-n1/K)
0 = (R - 1) n2 + m n1
The first equation will be at equilibrium if n1=0 or if
0 = - m + r1 (1-n1/K)
whose solution is:
n1 = K (1 - m/r1)
The first possibility (n1=0) is not of interest because if the source doesn't have any individuals, then certainly the sink would not (besides, plugging this possibility into the second equation indicates that the equilibrium would be an extinct system, with n1 = n2 = 0).
Using the second of these solutions and plugging it into the second equation we get:
0 = (R - 1) n2 + m K (1 - m/r1)
which may be solved for n2:
n2 = m K (1 - m/r1)/(1-R)
This equilibrium will be positive as long as R<1 (true by assumption) and m < r1 (it is reasonable to assume a low migration rate).
Near this equilibrium, we can approximately describe the dynamics using a linear stability matrix (the matrix of partial derivatives evaluated at the equilibrium).
In this case, the local stability matrix is:
{{ 1+m-r1 , 0 }, { m , R }}
Since this is a diagonal matrix, the eigenvalues are given by the diagonal elements, 1+m-r1 and R.
As long as the migration rate is small relative to the intrinsic growth rate of the source popualation (m < r1) and the number of offspring per parent in the sink population is less than 1 (R<1), both eigenvalues will be less than one. We must also check, however, whether the eigenvalues could be negative. This isn't possible for R (which is the number of offspring), but it can be for 1+m-r1. Stability thus requires that 1+m-r1 > -1 or, in other words, 2+m > r1, in which case the source-sink metapopulation would be stable. If this condition isn't met, we expect oscillations away from the equilibrium (as in the logistic model).