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

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