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May  2010, 13(3): 537-557. doi: 10.3934/dcdsb.2010.13.537

Spatial structures and generalized travelling waves for an integro-differential equation

1. 

Department of Mathematics, Technical University of Iasi, Iasi

2. 

Institute of Mechanical Engineering Problems, 199178 Saint Petersburg, Russian Federation

3. 

Institute of Mathematics, University Lyon 1, 69622 Villeurbann

4. 

University of Toronto, Department of Mathematics, Toronto, ON, M5S 2E4, Canada

Received  May 2009 Revised  September 2009 Published  February 2010

Some models in population dynamics with intra-specific competition lead to integro-differential equations where the integral term corresponds to nonlocal consumption of resources [8][9]. 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 [4]. 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). GTWs are introduced in [18] for reaction-diffusion systems as global in time propagating solutions. 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.
Citation: Narcisa Apreutesei, Nikolai Bessonov, Vitaly Volpert, Vitali Vougalter. Spatial structures and generalized travelling waves for an integro-differential equation. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 537-557. doi: 10.3934/dcdsb.2010.13.537
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