Narcisa Apreutesei, Nicholas Bessonov, Vitaly Volpert, Vitali Vougalter
Spatial structures and generalized travelling waves for an integro-differential equation
(525K, pdf)
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. 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 and 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). GTWs were introduced 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.