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Spatial structures and generalized travelling waves for an integro-differential equation. (English) Zbl 1191.35041

Summary: Some models in population dynamics with intra-specific competition lead to integro-differential equations where the integral term corresponds to nonlocal consumption of resources. The principal difference of such equations in comparison with traditional reaction-diffusion equation is that homogeneous in space solutions can lose their stability resulting in emergence of spatial or spatio-temporal structures. We study the existence and global bifurcations of such structures. In the case of unbounded domains, transition between stationary solutions can be observed resulting in propagation of generalized travelling waves (GTW). In this work their existence and properties are studied for the integro-differential equation. Similar to the reaction-diffusion equation in the monostable case, we prove the existence of generalized travelling waves for all values of the speed greater or equal to the minimal one. We illustrate these results by numerical simulations in one and two space dimensions and observe a variety of structures of GTWs.

MSC:

35B32 Bifurcations in context of PDEs
35K57 Reaction-diffusion equations
45K05 Integro-partial differential equations
92D25 Population dynamics (general)
35K58 Semilinear parabolic equations
35R09 Integro-partial differential equations
35C07 Traveling wave solutions
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