Abstract
Integrodifference equations are discrete-time analogues of reaction-diffusion 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.
Highlights
Dynamical stabilization is a concept that was first described by [22] in the context of a predator-prey reaction-diffusion model
Integrodifference equations (IDEs) describe local reaction and spatial spread in discrete time and share many close connections with reaction-diffusion equations, yet the phenomenon of dynamical stabilization has not been formally
In our previous work [2], we focused on generalized spreading speeds and found the phenomenon of dynamical stabilization numerically, but did not study in detail
Summary
Dynamical stabilization is a concept that was first described by [22] in the context of a predator-prey reaction-diffusion model. Equation 11 will have one positive and one negative real root, independent of c This result, in combination with the analysis around (0, 0), shows that the necessary condition for the existence of a monotone connection between (0, 0) and (1, 0) is satisfied for c ≥ c∗. A traveling profile that increases from the unstable state N ∗ = 1 may exist in the second-iterate IDE defined by the operator Q ◦ Q We summarize these results and illustrate how they relate to dynamical stabilization . Oscillations arise near the positive fixed point of the delay system so that non-monotone waves connecting 0 to 1 may exist for r∗ < r < r∗∗ with r∗∗ > 2 In both cases, the speed of the traveling waves is c∗ as given by 5
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