The effect of climate shift on a species submitted to dispersion, evolution, growth, and nonlocal competition

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 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 confined case we determine a critical climate change speed for the extinction or survival of the population, the latter case taking place by "strictly following the climate shift." In the so-called environmental gradient case, or unconfined case, we additionally determine the propagation speed of the population when it survives: thanks to a combination of migration and evolution, it can be different here than the speed of the climate shift. Finally, we consider 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.