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.

中文翻译：

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积分差分方程中的动力稳定和行波

积分差分方程是反应扩散方程的离散时间类似物，可用于模拟非本地物种的空间扩散和入侵。它们以行波的形式支持解决方案，这些波的速度提供了有关生物入侵速度的重要见解。通常，行波会在其唤醒后使系统保持稳定状态。动态稳定是一种现象，在这种波动之后，会出现不稳定状态，并且可能会在很长一段时间内保持稳定状态，然后再通过另一个过渡波将其替换为稳定状态。虽然已经在反应扩散方程组中研究了动力学稳定，但我们在此提出了积分微分方程的第一个此类研究。我们使用行波剖面的线性稳定性分析来确定动力稳定出现的必要条件，并将其与堆积前沿理论联系起来。我们发现，当非空间动力学表现出稳定的两个周期时，该现象是正常现象，而不是例外情况。