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Inside dynamics of solutions of integro-differential equations

Abstract : In this paper, we investigate the inside dynamics of the positive solutions of integro-differential equations partial derivative(t)u(t, x) =(J star u)(t,x) - u(t, x) + f(u(t,x)), t >0 and x is an element of R, 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 v(i) >= 0 (i is an element of I subset of N), its inside dynamics is given by the spatio-temporal evolution of v(i). 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 is an element of (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.
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Olivier Bonnefon, Jerome Coville, Jimmy Garnier, Lionel Roques. Inside dynamics of solutions of integro-differential equations. Discrete and Continuous Dynamical Systems - Series B, American Institute of Mathematical Sciences, 2014, 19 (10), pp.3057-3085. ⟨10.3934/dcdsb.2014.19.3057⟩. ⟨hal-02632900⟩



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