Regime shifts, extinctions and long transients in models of population dynamics with density-dependent dispersal

Predicting extinctions resulting from ecosystems' regime shifts has long been a focus of biological conservation and ecological management. Mathematical modelling plays a key role in assessing the possibility of such events. Traditionally, however, models focused on long-term, asymptotic behaviour of ecosystems. Meanwhile, the environment is usually non-stationary, which may mean that the long-term behaviour is never observed. Correspondingly, over the last two decades there has been a growing appreciation of the role of transients both in empirical ecology and theoretical studies, in particular in the context of species extinctions. In this paper, we theoretically explore long transients and extinctions occurring in several paradigmatic models of increasing complexity, such as single species, two-species and three-species systems. We consider the population dynamics in a local 'patch', the patch being connected to the rest of the population via density-dependent dispersal. We consider both deterministic and stochastic scenarios. We discover many different patterns of long transient dynamics with quick regime shifts between 'safe' (persistent) dynamics and unsafe ones resulting in extinctions. Remarkably, delayed extinction can occur after thousands of generations of apparently safe population dynamics. We classify transient regimes and reveal their underlying mechanisms. Environmental noise can either shorten transients or can create a new type of a long transient. Our study suggests that not only are long transients ubiquitous, but there is also a great variety of them. The omnipresence of long transients emphasizes the need to account for them in nature conservation programs as well as future theoretical research.