The effect of climate shift on a species submitted to dispersion, evolution, growth and nonlocal competition
Résumé
We consider a population structured by a space
variable and a phenotypical trait, submitted to dispersion,
mutations, growth and nonlocal competition. We introduce the
climate shift due to {\it Global Warming} and discuss the dynamics
of the population by studying the long time behavior of the
solution of the Cauchy problem. We consider three sets of
assumptions on the growth function. In the so-called {\it confined
case} we determine a critical climate change speed for the
extinction or survival of the population, the latter case taking place by \lq\lq strictly following the climate shift''. In the so-called {\it
environmental gradient case}, or {\it unconfined case}, we additionally determine the propagation speed
of the population when it survives: thanks to a combination of migration and evolution, it can here be different from the speed of the climate shift. Finally, we consider {\it mixed scenarios}, that are complex situations, where the
growth function satisfies the conditions of the confined case on the right, and the conditions of the unconfined case on the left.
The main difficulty comes from the nonlocal competition term that prevents
the use of classical methods based on comparison arguments. This difficulty
is overcome thanks to estimates on the tails of the solution, and a careful application of the parabolic Harnack inequality.
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