Preventing Noise-Induced Extinction in Discrete Population Models

A problem of the analysis and prevention of noise-induced extinction in nonlinear population models is considered. For the solution of this problem, we suggest a general approach based on the stochastic sensitivity analysis. To prevent the noise-induced extinction, we construct feedback regulators which provide a low stochastic sensitivity and keep the system close to the safe equilibrium regime. For the demonstration of this approach, we apply our mathematical technique to the conceptual but quite representative Ricker-type models. A variant of the Ricker model with delay is studied along with the classic widely used one-dimensional system

Analysis of the Noise-Induced Regimes in Ricker Population Model with Allee Effect via Confidence Domains Technique

We consider a discrete-time Ricker population model with the Allee effect under the random disturbances. It is shown that noise can cause various dynamic regimes, such as stable stochastic oscillations around the equilibrium, noise-induced extinction, and a stochastic trigger. For the parametric analysis of these regimes, we develop a method based on the investigation of the dispersions and arrangement of confidence domains. Using this method, we estimate threshold values of the noise generating such regimes

Controlling the Stochastic Sensitivity in Nonlinear Discrete-Time Systems with Incomplete Information

For stochastic nonlinear discrete-time system with incomplete information, a problem of the stabilization of equilibrium is considered. Our approach uses a regulator which synthesizes the required stochastic sensitivity. Mathematically, this problem is reduced to the solution of some quadratic matrix equations. A description of attainability sets and algorithms for regulators design is given. The general results are applied to the suppression of unwanted large-amplitude oscillations around the equilibria of the stochastically forced Verhulst model with noisy observations

Analysis of noise-induced transitions from regular to chaotic oscillations in the Chen system

The stochastically perturbed Chen system is studied within the parameter region which permits both regular and chaotic oscillations. As noise intensity increases and passes some threshold value, noise-induced hopping between close portions of the stochastic cycle can be observed. Through these transitions, the stochastic cycle is deformed to be a stochastic attractor that looks like chaotic. In this paper for investigation of these transitions, a constructive method based on the stochastic sensitivity function technique with confidence ellipses is suggested and discussed in detail. Analyzing a mutual arrangement of these ellipses, we estimate the threshold noise intensity corresponding to chaotization of the stochastic attractor. Capabilities of this geometric method for detailed analysis of the noise-induced hopping which generates chaos are demonstrated on the stochastic Chen system. © 2012 American Institute of Physics

Noise-induced transitions and shifts in a climate-vegetation feedback model

Motivated by the extremely important role of the Earth’s vegetation dynamics in climate changes, we study the stochastic variability of a simple climate–vegetation system. In the case of deterministic dynamics, the system has one stable equilibrium and limit cycle or two stable equilibria corresponding to two opposite (cold and warm) climate–vegetation states. These states are divided by a separatrix going across a point of unstable equilibrium. Some possible stochastic scenarios caused by different externally induced natural and anthropogenic processes inherit properties of deterministic behaviour and drastically change the system dynamics. We demonstrate that the system transitions across its separatrix occur with increasing noise intensity. The climate–vegetation system therewith fluctuates, transits and localizes in the vicinity of its attractor. We show that this phenomenon occurs within some critical range of noise intensities. A noise-induced shift into the range of smaller global average temperatures corresponding to substantial oscillations of the Earth’s vegetation cover is revealed. Our analysis demonstrates that the climate–vegetation interactions essentially contribute to climate dynamics and should be taken into account in more precise and complex models of climate variability.</jats:p

Stochastic Analysis of Subcritical Amplification of Magnetic Energy in a Turbulent Dynamo

We present and analyze a simplified stochastic $\alpha \Omega -$dynamo model which is designed to assess the influence of additive and multiplicative noises, non-normality of dynamo equation, and nonlinearity of the $\alpha -$% effect and turbulent diffusivity, on the generation of a large-scale magnetic field in the subcritical case. Our model incorporates random fluctuations in the $\alpha -$parameter and additive noise arising from the small-scale fluctuations of magnetic and turbulent velocity fields. We show that the noise effects along with non-normality can lead to the stochastic amplification of the magnetic field even in the subcritical case. The criteria for the stochastic instability during the early kinematic stage are established and the critical value for the intensity of multiplicative noise due to $\alpha -$fluctuations is found. We obtain numerical solutions of non-linear stochastic differential equations and find the series of phase transitions induced by random fluctuations in the $\alpha -$parameter.Comment: 21pages,7 figure