Pointwise persistence in full chemotaxis models with logistic source on bounded heterogeneous environments

TL;DR

This paper proves that in chemotaxis models with logistic sources on bounded heterogeneous environments, populations persist at every point over time, ensuring the existence of strictly positive solutions.

Contribution

It establishes pointwise persistence for full chemotaxis models with local and nonlocal logistic sources, extending previous mass persistence results to local pointwise persistence.

Findings

Global existence and boundedness of solutions

Pointwise persistence at all locations

Existence of strictly positive entire solutions

Abstract

The current paper is concerned with pointwise persistence in full chemotaxis models with local as well as nonlocal time and space dependent logistic source in bounded domains. We first prove the global existence and boundedness of nonnegative classical solutions under some conditions on the coefficients in the models. Next, under the same conditions on the coefficients, we show that pointwise { persistence} occurs, that is, any globally defined positive solution is bounded below by a positive constant independent of its initial condition when the time is large enough. It should be pointed out that in \cite{TaoWin15}, the authors established the persistence of mass for globally defined positive solutions, which indicates that any extinction phenomenon, if occurring at all, necessarily must be spatially local { in }nature, whereas the population as a whole always persists. The pointwise persistence proved in the current paper implies that not only the population as a whole persists, but also it persists at any location eventually. It also implies the existence of strictly positive entire solutions.