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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. • We explore long transients in time-discrete non-smooth maps. • We analysed transients in 1D and 2D models of population dynamics. • Novel mechanisms of long transients in the considered models are revealed. • The corresponding scaling behaviour of transients is analysed. • Non-smoothness of maps facilitates the emergence of long transients. [ABSTRACT FROM AUTHOR]

Abstract: A minimal reaction–diffusion model for the spatiotemporal spread of an infectious disease is considered. The model is motivated by the Feline Immunodeficiency Virus (FIV) which causes AIDS in cat populations. Because the infected period is long compared with the lifespan, the model incorporates the host population growth. Two different types are considered: logistic growth and growth with a strong Allee effect. In the model with logistic growth, the introduced disease propagates in form of a travelling infection wave with a constant asymptotic rate of spread. In the model with Allee effect the spatiotemporal dynamics are more complicated and the disease has considerable impact on the host population spread. Most importantly, there are waves of extinction, which arise when the disease is introduced in the wake of the invading host population. These waves of extinction destabilize locally stable endemic coexistence states. Moreover, spatially restricted epidemics are possible as well as travelling infection pulses that correspond either to fatal epidemics with succeeding host population extinction or to epidemics with recovery of the host population. Generally, the Allee effect induces minimum viable population sizes and critical spatial lengths of the initial distribution. The local stability analysis yields bistability and the phenomenon of transient epidemics within the regime of disease-induced extinction. Sustained oscillations do not exist. [Copyright &y& Elsevier]