cycles

Topic. The subject of the article is the expansion of the author’s research series in the direction of mathematical modeling of specific ecological situations and transitional regimes that arise in nonlinear population processes with complex internal regulation.

The response problem of equilibrium and cycles for stochastically forced Verhulst population model is considered. Theoretical and empirical approaches are used for stochastically sensitivity analysis. The theoretical approach is based on the firth approximation method and the empirical approach is based on direct numerical simulation. The correspondence between the two approaches for Verhulst population model is demonstrated. The increase of discrete system sensitivity to external noise in the period­doubling bifurcation zone under transition to chaos is shown.

We consider the Hopf system as a classical model of a stiff birth of a cycle. In the presence of parametrical and additive random disturbances, various types of the stochastic attractors are observed. The solution of the corresponding Fokker–Planck–Kolmogorov equation is found. The qualitative changes of the form for stochastic attractors under multiplicative noise are shown. The phenomenon of backward stochastic bifurcations is described in details.