Ricker model: Influence of periodic and stochastic parametric modulation.

Abstract

Linear and nonlinear properties of the response of the discrete Ricker system ${\mathit{x}}_{\mathit{n}+1}$=${\mathit{x}}_{\mathit{n}}$exp[r(1-${\mathit{x}}_{\mathit{n}}$)] against modulating the control parameter r with periodic, aperiodic, and stochastic perturbations ${\ensuremath{\xi}}_{\mathit{n}}$ via r(1+\ensuremath{\Delta}${\ensuremath{\xi}}_{\mathit{n}}$) are investigated analytically and numerically in the range where the unmodulated map undergoes its first pitchfork bifurcation from the fixed point ${\mathit{x}}^{\mathrm{*}}$=1 into a period-2 limit cycle. Parametric modulation does not destroy the fixed point. On the contrary, it is stabilized in a universal manner for small modulation amplitudes \ensuremath{\Delta}, irrespective of the kind of perturbations destroyed. However, modulation dramatically changes the period-2 limit cycle of the unmodulated map. A formula for the stability boundary ${\mathit{r}}_{\mathit{c}}$(\ensuremath{\Delta}) of the fixed point is derived and evaluated for various periodic and stochastic modulation types as a function of modulation amplitude \ensuremath{\Delta}. For periodic modulation linear conditions are elucidated for the bifurcation of limit cycles at ${\mathit{r}}_{\mathit{c}}$(\ensuremath{\Delta}) that are harmonic or subharmonic with respect to the modulation. Structure and dynamics of the time-dependent nonlinear solutions that bifurcate out of the fixed point and that depend sensitively on details of the modulation dynamics are investigated analytically in comparison with numerical simulations. The difference in the response against periodic modulation with even and odd periods N is elucidated in particular for N=2 and 3, where the difference is most striking. The peculiarities of the temporal behavior of the map under modulation with large periods are explained. The influence of stochastic perturbations on the period-2 limit cycle of the unmodulated system is determined analytically and numerically by evaluating various averaged quantities that seem appropriate for a statistical description of the stochastic response. Conditions are evaluated under which the flip behavior of successive sign changes, sgn(1-${\mathit{x}}_{\mathit{n}+1}$)=-sgn(1-${\mathit{x}}_{\mathit{n}}$), of the unmodulated map is preserved under modulation.