Logistic growth in seasonally changing environments

TL;DR

This paper studies a mathematical model of population growth with seasonal changes, proving the existence and stability of periodic solutions, and analyzing their behavior near critical parameter values.

Contribution

It introduces a new analysis of periodic solutions in a logistic model with time-dependent refuges, including bifurcation and blow-up phenomena.

Findings

Existence of stable time-periodic solutions under minimal boundary assumptions.

Bifurcation of solutions as the parameter varies.

Solutions can blow up in parts of the domain near critical parameters.

Abstract

We consider a parameter dependent periodic-logistic problem with a logistic term involving a degeneracy that replicates time dependent refuges in the habitat of a population. Working under no or very minimal assumptions on the boundary regularity of the domain we show the existence of a time-periodic solution which bifurcates with respect to the parameter and show their stability. We show that under suitable assumptions that the periodic solution blows up on part of the domain and remains finite on other parts when the parameter approaches a critical value.