  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.
