Long-term transients and complex dynamics of a stage-structured population with time delay and the Allee effect

Abstract

Traditionally, mathematical modelling in population ecology is mainly focused on asymptotical behaviour of the model, i.e. as given by the system attractors. Recently, however, transient regimes and especially long-term transients have been recognized as playing a crucial role in the dynamics of ecosystems. In particular, long-term transients are a potential explanation of ecological regime shifts, when an apparently healthy population suddenly collapses and goes extinct.In this paper, we show that the interplay between delay in maturation and a strong Allee effect can result in long-term transients in a single species system. We first derive a simple 'conceptual' model of the population dynamics that incorporates both a strong Allee effect and maturation delay. Unlike much of the previous work, our approach is not empirical since our model is derived from basic principles. We show that the model exhibits a high complexity in its asymptotic dynamics including multi-periodic and chaotic attractors. We then show the existence of long-term transient dynamics in the system, when the population size oscillates for a long time between locally stable stationary states before it eventually settles either at the persistence equilibrium or goes extinct. The parametric space of the model is found to have a complex structure with the basins of attraction corresponding to the persistence and extinction states being of a complicated shape. This impedes the prediction of the eventual fate of the population, as a small variation in the maturation delay or the initial population size can either bring the population to extinction or ensure its persistence.3

Pattern Formation in a Reaction‐Diffusion Predator‐Prey Model with Weak Allee Effect and Delay

“…en, considering that the predator-prey model with Allee effect is more realistic, people begin to introduce delay into the predator-prey model and discuss the effects of Allee effect and delay on the dynamics of the model [2,[17][18][19][20][21][22]. We try to introduce weak Allee effect and searching delay into model (1), and then we get…”

Global Hopf bifurcation of a population model with stage structure and strong Allee effect

“…Let C(x 0 , τ 0 , T 0 ) denote the connected component of (x 0 , τ 0 , T 0 ) in Σ(F ) and the set of stationary solution of (15) by…”

“…Since then, a fair amount of work [23,22,11] has been devote to it, such as the study the Hopf bifurcation and global Hopf bifurcation [23,22], chaos, peak-adding and period-doubling bifurcations [11]. Morozov et al [15] considered a age-structured single-species population, in which the author derived the model detailedly using the general framework of agestructured populations, by taking the reproduction function r(y) = αy n−1 1+βy m , they arrived atẏ (t) = αe −δτ y n (t − τ ) 1 + βy m (t − τ ) − Dy(t).…”

“…In [15], Morozov et al indicated that the growth of adult population in system (1) has the property of strong Allee effect when m > n > 1, where the 'Allee effect' named after Allee [1]. Literatures [10,21] as well as the references therein showed that the phenomenon of the growth rate per capita achieves its peak at a positive density is called an Allee effect.…”

“…We would like to mention that the situation in Figure 1(b)(classical Mackey-Glass case) is not Allee effect since the growth rate per capita r(y) always decreases as y increases. A large amount of simulations has been done in [15] to reveal the rich phenomenon of strong Allee effect (n = 2, m = 4) model, such as long-term transients, peakadding bifurcations, and period-doubling bifurcations. Morozov et al have done a lot of work to exhibit them in numerical simulation by taking A = αe −δτ as a constant.…”