We are interested in PDE models that are used in evolutionary ecology to investigate the effect of climate change on species’ range. More precisely, we will consider the Spatial Infinitesimal Model, a Kinetic model, and the Kirkpatrick-Barton model, a macroscopic model. Our goal will be to provide a rigorous macroscopic limit between those two models. We will also detail a biological study of the effect of pollen dispersion, where the macroscopic limit mentioned is used.
To describe the macroscopic limit from the Spatial Infinitesimal Model to the Kirkpatrick-Barton model, we take advantage of a Tanaka inequality satisfied by the reproduction operator, combined to parabolic estimates to control the spatial dynamics of the solution. As a by-product of this argument, we obtain estimates on a higher moment of solutions of the kinetic model, provided the reproduction rate is large enough.