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Dynamic behaviors of the Ricker population model under a set of randomized perturbations. (English) Zbl 0952.92025

Summary: We studied the dynamics of the Ricker population model under perturbations by the discrete random variable $\epsilon$ which follows distribution $P\left\{\epsilon ={a}_{i}\right\}={p}_{i}$, $i=1,\cdots ,n$, $0<{a}_{i}\ll 1$, $n\ge 1$. Under the perturbations, $n+1$ blurred orbits appeared in the bifurcation diagram. Each of the $n+1$ blurred orbits consisted of $n$ sub-orbits. The asymptotes of the $n$ sub-orbits in one of the $n+1$ blurred orbits were ${N}_{t}={a}_{i}$ for $i=1,\cdots ,n$. For other $n$ blurred orbits, the asymptotes of the $n$ sub-orbits were ${N}_{t}={a}_{i}exp\left[r\left(1-{a}_{i}\right)\right]+{a}_{j}$, $j=1,2,\cdots ,n$, for $i=1,\cdots ,n$, respectively.

The effects of variances of the random variable $\epsilon$ on the bifurcation diagrams were examined. As the variance value increased, the bifurcation diagram became more blurred. Perturbation effects of the approximate continuous uniform random variable and random error were compared. The effects of the two perturbations on dynamics of the Ricker model were similar, but with differences. Under different perturbations, the attracting equilibrium points and two-cycle periods in the Ricker model were relatively stable. However, some dynamic properties, such as the periodic windows and the $n$-cycle periods $\left(n>4\right)$, could not be observed even when the variance of a perturbation variable was very small. The process of reversal of the period-doubling, an important feature of the Ricker and other population models observed under constant perturbations, was relatively unstable under random perturbations.

MSC:
 92D25 Population dynamics (general) 37N25 Dynamical systems in biology 37H99 Random dynamical systems