Long transients in discontinuous time-discrete models of population dynamics

Abstract

Traditionally, mathematical modelling of population dynamics was focused on long-term, asymptotic behaviour (systems attractors), whereas the effects of transient regimes were largely disregarded. However, recently there has been a growing appreciation of the role of transients both in empirical ecology and theoretical studies. Among the main challenges are identification of the mechanisms triggering transients in various dynamical systems and understanding of the corresponding scaling law of the transient’s lifetime; the latter is of a vital practical importance for long-term ecological forecasting and regime shifts anticipation. In this study, we reveal and investigate various patterns of long transients occurring in two generic time-discrete population models which are mathematically described by discontinuous (piece-wise) maps. In particular, we consider a single-species population model and a predator–prey system, in each model we assume that the dispersal of species at the end of each season is density dependent. For both models, we demonstrate transients due to crawl-by dynamics, chaotic repellers, chaotic saddles, ghost attractors, and a rich variety of intermittent regimes. For each type of transient, we investigate the corresponding scaling law of the transient’s lifetime. We explore the space of key model parameters, to find where particular types of long transients can be expected, and we show that long transients are omnipresent since they can be observed within a wide range of model parameters. We also reveal the possibility of complex patterns occurring as a cascade of transients of different types. We also considered a stochastic version of the model where some parameters exhibit random fluctuations. We show that stochasticity can reduce, extend or produce new patterns of long transients. We conclude that the discontinuity in population models significantly facilitates the emergence of long transients by creating new types and increasing parameter domains of the corresponding transient dynamics. Another important conclusion is that the asymptotic regime of population dynamics is hardly possible to predict based on a finite time course of species densities, which is crucial for ecosystem management and decision making.