# A macroscopic limit for an ecology model.

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.