Oscillations and chaos behind predator–prey invasion: mathematical artifact or ecological reality?

TLDR

Theoretical ecology struggles to distinguish whether model predictions reflect real phenomena or are artifacts, especially due to scarce detailed ecological data. This study investigates whether invasion dynamics in oscillatory predator–prey systems can be inferred from only two qualitative features: oscillatory kinetics and local random movement. Four unrelated mathematical frameworks—reaction–diffusion equations, coupled map lattices, deterministic cellular automata, and integrodifference equations—were employed to model these features. Simulations across all models consistently revealed either irregular spatiotemporal oscillations or regular travelling‑wave oscillations behind the invasion, indicating that such patterns are an ecological reality arising from the basic assumptions.

Abstract

A constant dilemma in theoretical ecology is knowing whether model predictions corrspond to real phenomena or whether they are artifacts of the modelling framework. The frequent absence of detailed ecological data against which models can be tested gives this issue particular importance. We address this question in the specific case of invasion in a predator–prey system with oscillatory population kinetics, in which both species exhibit local random movement. Given only these two basic qualitative features, we consider whether we can deduce any properties of the behaviour following invasion. To do this we study four different types of mathematical model, which have no formal relationship, but which all reflect our two qualitative ingredients. The models are: reaction–diffusion equations, coupled map lattices, deterministic cellular automata, and integrodifference equations. We present results of numerical simulations of the invasion of prey by predators for each model, and show that although there are certain differences, the main qualitative features of the behaviour behind invasion are the same for all the models. Specifically, there are either irregular spatiotemporal oscillations behind the invasion, or regular spatiotemporal oscillations with the form of a periodic travelling ‘wake’, depending on parameter values. The observation of this behaviour in all types of model strongly suggests that it is a direct consequence of our basic qualitative assumptions, and as such is an ecological reality which will always occur behind invasion in actual oscillatory predator–prey systems.