Göran Högnäs:
On the quasi-stationary distribution of a stochastic Ricker model
Åbo Akademi. Reports on Computer Science & Mathematics, Ser. A.
No. 177, September 1996. 16 pp. - Revised version to be published in
Stochastic Processes and their Applications.
Abstract:
We model the evolution of a single-species
population by a size-dependent branching
process $Z_t$ in discrete time. Given that $Z_t = n $ the expected value of
$Z_{t+1} $ may be written $ n \exp(r - \gamma n)$ where $r > 0$ is a growth parameter
and $\gamma > 0 $ is an
(inhibitive) environmental parameter. For small values of $\gamma$ the short-term
evolution of the normed
process $\gamma Z_t$ follows the deterministic Ricker model closely. As long as
the parameter $r$ remains below the critical value $r_c \approx 2.6924 $ where
the chaotic region for the Ricker model starts, the quasi-stationary distribution
of $\gamma Z_t$ is shown to converge weakly to the uniform distribution
on the unique stable
periodic orbit. The long-term behavior of $\gamma Z_t$ differs from that of the
Ricker model, however: $\gamma Z_t$ has a finite lifetime a.\ s.
-
The methods used rely on the central limit theorem and Markov's inequality
as well as elementary dynamical systems theory.
Keywords : Size-dependent branching process; Quasi-stationary distribution;
Weak convergence; Ricker model; Stable period; Markov's inequality; Entropy function
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