Narrow Results

The Ricker model is one of the simplest and most widely-used ecological models displaying complex nonlinear dynamics. We study a discrete-time population model, which is derived from simple assumptions concerning individual organisms’ behavior, using the “site-based” approach, developed by Brännström, Broomhead, Johansson and Sumpter. In the large-population limit the model converges to the Ricker model, and can thus be considered a mechanistic version of the Ricker model, derived from basic ecological principles, and taking into account the demographic stochasticity inherent to finite populations. We employ several analytical and precise numerical methods to study the model, showing how each approach contributes to understanding the model’s dynamics. Expressing the model as a Markov chain, we employ the concept of quasi-stationary distributions, which are computed numerically, and used to examine the interaction between complex deterministic dynamics and demographic stochasticity, as well as to calculate mean times to extinction. A Gaussian Markov chain approximation is used to obtain quantitative asymptotic approximations for the size of fluctuations of the stochastic model’s time series around the deterministic trajectory, and for the correlations between successive fluctuations. Results of these approximations are compared to results obtained from quasi-stationary distributions and from direct simulations, and are shown to be in good agreement.

We study the collective behavior of populations, coupling the equilibrium and chaotic subsystems by mutual migration. It is assumed that the dynamics of an isolated subsystem is modeled by the Ricker map, and the intensity of migrations within the metapopulation is subject to random perturbations. In the deterministic case, we specify parameter zones of mono- and birhythmicity with regular and chaotic attractors. Noise-induced multistage transitions from order to chaos and vice versa are investigated from an approach that combines direct numerical simulations, studies of chaotic transients, stochastic sensitivity, and confidence domains.

We study a discrete-time model of host–parasitoid interactions, where the host is subject to a strong Allee effect and the parasitoid is aggregated. The system may have multiple coexisting steady states and there are two host population thresholds. The hosts become extinct if it is below the Allee threshold. The other threshold depends on the Allee threshold beyond which the host also becomes extinct due to overcompensated density dependence. When the initial host population size is between the two thresholds, we derive a critical parasitoid population size above which both populations become extinct. The critical size depends on the degree of aggregation of parasitoids. It is shown that both populations are more likely to become extinct if parasitoid aggregation is increased. Numerical simulations reveal that a strong Allee effect on the host can stabilize the host–parasitoid interactions on one hand but may drive both populations to extinction on the other hand. Further, aggregation of the parasitoid can promote population persistence when the host is subject to a strong Allee effect with a large Allee threshold. However, a more aggregated parasitoid population is more vulnerable to extinction if the growth rate of hosts is large.

In recent years, the stability of the predator–prey model subject to the Allee effect has become an interesting issue. This study investigates the effect of Allee effect on the stability of a discrete-time predator–prey model with Holling type-II functional response. Using equilibrium analysis, stability analysis and bifurcation theory, the mathematical characteristics of the proposed model are examined. Model experiences flip bifurcation and Neimark–Sacker bifurcation based on the center manifold theorem and bifurcation theory. Our analytical results are demonstrated by numerical simulations.

We study cycles of the discrete-time nonlinear dynamical system under parametric random disturbances. A dynamics of deviations of the random solutions from the deterministic cycle is analyzed with the help of first approximation linear systems. A closed system for second moments of these deviations is obtained and explicit formulas for the stable periodic solution are given. Constructive abilities of our mathematical technique are demonstrated for parametrical analysis of stochastically forced periodic regimes in the Ricker population model with Allee effect. A threshold noise intensity corresponding to the onset of stochastic extinction is estimated.