This paper is a follow-up to a previous work where we considered a multi-patch model, each patch following a logistic law, the patches being coupled by symmetric migration terms. In this paper we drop the symmetry hypothesis. First, in the case of perfect mixing, i.e when the migration rate tends to infinity, the total population follows a logistic law with a carrying capacity which in general is different from the sum of the n carrying capacities, and depends on the migration terms. Second, we determine, in some particular cases, the conditions under which fragmentation and asymmetrical migration can lead to a total equilibrium population greater or smaller than the sum of the carrying capacities. Finally, for the three-patch model, we show numerically the existence of at least three critical values of the migration rate for which the total equilibrium population equals the sum of the carrying capacities.

In this paper, we consider a two-patch model coupled by migration terms, where each patch follows a Richards law. First, we prove the global stability of the model. Second, in the case when the migration rate tends to infinity, the total carrying capacity is given, which in general is different from the sum of the two carrying capacities and depends on the parameters of the growth rate and also on the migration terms. Using the theory of singular perturbations, we give an approximation of the solutions of the system in this case. Finally, we determine the conditions under which fragmentation and migration can lead to a total equilibrium population which might be greater or smaller than the sum of two carrying capacities and we give a complete classification for all possible cases. The total equilibrium population formula for a large migration rate plays an important role in this classification. We show that this choice of local dynamics has an influence on the effect of dispersal. Comparing the dynamics of the total equilibrium population as a function of the migration rate with that of the logistic model, we obtain the same behavior. In particular, we have only three situations that the total equilibrium population can occur: it is always greater than the sum of two carrying capacities, always smaller, and a third case, where the effect of dispersal is beneficial for lower values of the migration rate and detrimental for the higher values. We end by examining the two-patch model where one growth rate is much larger than the second one, we compare the total equilibrium population with the sum of the two carrying capacities.

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