### Course 1: Arnaud Ducrot

#### Threshold phenomenon for bistable equations with diffusion

**Abstract:**In these lectures we shall discuss some threshold properties of bistable equations with diffusion, both with Laplace operator and nonlocal convolution operator. For homogeneous solutions, bistable equation roughly means that small population densities yield extinction while larger initial densities lead to the persistence of the population. When a spatial variable is considered, namely for non-homogeneous initial data, the threshold phenomenon is much more complex and the aim of these lectures is to shed some light of such phenomenon, both for local and nonlocal diffusion. First, under suitable assumptions on the stability of the homogenous steady states, using travelling wave solutions, we will explain that the solution spreads when the initial data is a step function with a sufficiently large length and height. Next we present an asymptotic analysis of the length when the height of the step initial data is slightly larger than the bistable threshold.