Periodic Solutions of the Generalised Logistic Equation

Abstract

We study the periodic generalised logistic equation which is parameterised and includes a degenerate potential. Such an equation can be used to model a population density within a habitat that has refuges in which the population experiences no mortality. It is assumed that all parameters vary periodically - for example to take account for temperature changes of seasons. Under no assumption on the boundary of the domain, and the form of the degenerate potential, we show the existence of a periodic solution. It is necessary for the parameter to fall within a predetermined range for a periodic solution exist and so we are also able to examine properties of a family of periodic solutions, determined by the parameter range. We show that this family of periodic solutions bifurcate and under additional assumptions we are able to show they tend to a blow-up solution of an equivalent logistic equation. In contrast to the existing literature, we have greatly relaxed assumptions on the domain and the degenerate potential. Additionally, we explore how the parabolic maximum principle affects where periodic solutions remain bounded in the domain. We provide examples showing that the solutions remain bounded even in some places where the potential is zero, in contrast to the corresponding elliptic problem.