Periodic, permanent, and extinct solutions to population models

Abstract

The existence of a critical parameter λc>0 is proven for some population models, that splits the set of parameters into two parts where the existence, resp. nonexistence, of a positive periodic solution is guaranteed. Moreover, it is shown that in a quite wide class of population models, all the positive solutions are permanent, resp. extinct ones, provided there exists, resp. does not exist, a positive periodic solution. The results are based on a theoretical research dealing with a boundary value problem for functional differential equation with a real parameteru′(t)=ℓ(u)(t)+λF(u)(t)for a.e. t∈[a,b],h(u)=0, where ℓ and F:C([a,b];R)→L([a,b];R) are, respectively, linear and nonlinear operators, h:C([a,b];R)→R is a linear functional, and λ∈R is a real parameter. The results are illustrated by numerical simulations.