Generalized logistic equation on Networks - Elbetch, Bilel (Comptes Rendus. Mathématique)
In this paper, we consider a general single species model in a heterogeneous environment of $n$ patches ($n\ge 2$), where each patch follows a generalized logistic law. First, we prove the global stability of the model. Second, in the case of perfect mixing, i.e. when the migration rate tends to infinity, the total population follows a generalized logistic law with a carrying capacity which in general is different from the sum of the $n$ carrying capacities. Next, we give some properties of the total equilibrium population and we compute its derivative at no dispersal. In some particular cases, 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 the $n $ carrying capacities. Finally, we study an example of two-patch model where the first patch follows a logistic law and the second a Richard’s law, we give a complete classification of the model parameter space as to whether dispersal is beneficial or detrimental to the sum of two carrying capacities.