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2010, Volume 13, Issue 3: 537-557. Doi: 10.3934/dcdsb.2010.13.537
This issue Previous Article Global solutions for an age-dependent model of nucleation, growth and ageing with hysteresis Next Article Asymptotic stability of traveling wavefronts in a delayed population model with stage structure on a two-dimensional spatial lattice

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

  • 3.

    Institute of Mathematics, University Lyon 1, 69622 Villeurbann

  • 4.

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

Received:  April 30, 2009
Revised:  August 31, 2009
Published:  September 2010
Abstract Related Papers Cited by
    Abstract
  • 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.
    Mathematics Subject Classification: Primary: 35K57; Secondary: 45K05, 92D25.

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