Extinction, periodicity and multistability in a Ricker model of stage-structured populations

Abstract

We study the dynamics of a second-order difference equation that is derived from a planar Ricker model of two-stage biological populations. We obtain sufficient conditions for global convergence to zero in the non-autonomous case. This gives general conditions for extinction in the biological context. We also study the dynamics of an autonomous special case of the equation that generates multistable periodic and non-periodic orbits in the positive quadrant of the plane.

Periodic and non-periodic solutions in a Ricker-type second-order equation with periodic parameters

“…Theorem 18 indicates a completely different dynamics where globally stable limit cycles occur when a n is restricted to the interval (0,2). Another of our main results is Theorem 8 that extends previous special cases in [2] and [3]. Further, Corollary 9 is a consequence of Theorem 8.…”

“…it has mimimal period 1 not 2. The behavior described in the corollary is indeed that which is observed for the constant parameter case; see [3] for a discussion of the stability of the variety of solutions mentioned above, which is of the same type as noted in Remark 10. We used a semiconjugate factorization of (1) to investigate its dynamics.…”

“…We use numerical simulations to highlight the rich variety of coexisting solutions that Theorem 13 allows. As noted above, and explained in greater detail in [3], these solutions are attracting (though not locally stable) so they are observable and may be recorded numerically.…”

“…In the case p = 1, i.e. when ( 1) is autonomous with a n = a for all n, the conditions stated in Theorem 13 were examined in [3]. In particular, it was verified that if a ≥ 3.13 then (1) has chaotic solutions from certain initial conditions.…”

“…We study the behavior of solutions of the second-order difference equation x n+1 = x n−1 e an−x n−1 −xn (1) where the parameter {a n } is a periodic sequence of real numbers. This equation is a special case of a stage-structured population model with a Ricker-type recruitment function; see [3]. For more information and additional Ricker-type models see, e.g.…”