Inside dynamics of solutions of integro-differential equations

In this paper, we investigate the inside dynamics of the positive solutions of integro-differential equations {equation*} ∂_t u(t,x)= (J u)(t,x) -u(t,x) + f(u(t,x)), \ t>0 and x, {equation*} with both thin-tailed and fat-tailed dispersal kernels J and a monostable reaction term $f.$ The notion of inside dynamics has been introduced to characterize traveling waves of some reaction-diffusion equations [23]. Assuming that the solution is made of several fractions υⁱ≥ 0 (i∈ I ⊂ N), its inside dynamics is given by the spatio-temporal evolution of υⁱ. According to this dynamics, the traveling waves can be classified in two categories: pushed and pulled waves. For thin-tailed kernels, we observe no qualitative differences between the traveling waves of the above integro-differential equations and the traveling waves of the classical reaction-diffusion equations. In particular, in the KPP case (f(u)≤ f'(0)u for all u∈(0,1)) we prove that all the traveling waves are pulled. On the other hand for fat-tailed kernels, the integro-differential equations do not admit any traveling waves. Therefore, to analyse the inside dynamics of a solution in this case, we introduce new notions of pulled and pushed solutions. Within this new framework, we provide analytical and numerical results showing that the solutions of integro-differential equations involving a fat-tailed dispersal kernel are pushed. Our results have applications in population genetics. They show that the existence of long distance dispersal events during a colonization tend to preserve the genetic diversity.