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We study the price dynamics generated by a stochastic version of a Day–Huang type asset market model with heterogenous, interacting market participants. To facilitate the analysis, we introduce a methodology that allows us to assess the consequences of changes in uncertainty on the dynamics of an asset price process close to stable equilibria. In particular, we focus on noise-induced transitions between bull and bear states of the market under additive as well as parametric noise. Our results are obtained by combining the stochastic sensitivity function (SSF) approach, a mixture of analytical and numerical techniques, due to Mil’shtein and Ryashko (1995) with concepts and techniques from the study of non-smooth 1D maps. We find that the stochastic sensitivity of the respective bull and bear equilibria in the presence of additive noise is higher than under parametric noise. Thus, recurrent transitions are likely to be observed already for relatively low intensities of additive noise.

An eco-epidemiological model of dynamical interaction of susceptible prey, infected prey, and predator is studied in the presence of the Allee effect and random disturbances. A bifurcation analysis of the deterministic variant of the model versus Allee parameter is carried out. Various regimes of tri-rhythmicity with coexistence of three cycles, or two cycles and one chaotic attractor are revealed. The complication of population dynamics due to stochastic transitions between coexisting oscillatory attractors and order-chaos transformations is studied both numerically and analytically, by the method of confidence ellipsoids and tori. The phenomenon of noise-induced extinction of predator with increasing amplitudes of oscillations for susceptible or infected prey is discussed.

A problem of the mathematical modeling and analysis of noise-induced transformations of complex oscillatory regimes in hierarchical population systems is considered. As a key example, we use a three-dimensional food chain dynamical model of the interacting prey, predator, and top predator. We perform a comparative study of the impacts of random fluctuations on three key biological parameters of prey growth, predator mortality, and the top predator growth. A detailed investigation of the stochastic excitement, noise-induced transition from order to chaos, and various scenarios of extinction is carried out. Constructive abilities of the semi-analytical method of confidence domains in the analysis of the noise-induced extinction are demonstrated.