Dynamical stabilization and traveling waves in integrodifference equations

Abstract

Integrodifference equations are discrete-time analogues of reactiondiffusion equations and can be used to model the spatial spread and invasion of non-native species. They support solutions in the form of traveling waves, and the speed of these waves gives important insights about the speed of biological invasions. Typically, a traveling wave leaves in its wake a stable state of the system. Dynamical stabilization is the phenomenon that an unstable state arises in the wake of such a wave and appears stable for potentially long periods of time, before it is replaced with a stable state via another transition wave. While dynamical stabilization has been studied in systems of reaction-diffusion equations, we here present the first such study for integrodifference equations. We use linear stability analysis of traveling-wave profiles to determine necessary conditions for the emergence of dynamical stabilization and relate it to the theory of stacked fronts. We find that the phenomenon is the norm rather than the exception when the non-spatial dynamics exhibit a stable two-cycle.

Transient Propagation of the Invasion Front in the Homogeneous Landscape and in the Presence of a Road

“…Although the growth function ( 9 ) presents a wealth of spatio-temporal dynamics regimes for various values of the growth parameter , (Andersen 1991 ; Bourgeois et al. 2020 ; Kot and Schaffer 1986 ), here we restrict our consideration by the simplest case providing monotone solutions converging to a travelling wave of the constant height (Lutscher 2019 ). Furthermore, in our model, the growth function ( 9 ) can be approximated by a linear growth function with the same growth factor A , as we assume that the initial population density is low and there is no significant increase in the population size at the beginning of invasion.…”

Rotational Symmetry and Rotating Waves in Planar Integro-Difference Equations

“…Systems such as (1.2) have been of interest to mathematicians and scientists for many years, see for example [6,7,24,25,40,41]. For example, in many applications, N = 1 and u represents the distribution in space Ω of a generation of a certain population, and v represents the subsequent distribution in space of the next generation after the population has reproduced locally (under a reproduction law modelled through the function F (x, •)), and then spread out in space according to the distribution kernel κ [24,25,40].…”

“…For example there may exist a period-2 cycle (or more generally a period-p cycle) in the dynamics of (1.8). This question was analyzed in some detail [6,7] by studying the seconditerate map N 2 in (1.5) and looking for travelling waves of the resulting operator connecting two fixed points of the second iterate f 2 in (1.8). See also [12,17].…”