Stability Of Equilibrium States For A Stochastically Perturbed Pielou’s Equation [

It is widely recognized that the theory of stochastic difference equations is a mathematical area of great interest. Concrete systems and processes is one of its subareas, which is of some interest nowadays. Specifically, in fields such as systems biology, ecology, biochemistry, genetics, and physiology dynamics, numerous population models are described by studying linear and nonlinear difference equations and systems. These studies have been very productive and helpful to develop the basic theory of the qualitative behaviour of nonlinear rational and exponential difference equations and systems. In this paper, we investigate a type of Pielou’s difference equation that is subject to additive stochastic perturbations modelled as a sequence of independent random variables (ξ_n ) with zero mean and unit variance. These perturbations are proportional to the deviation of the system state \(X_n \) from its equilibrium points, resulting in the following stochastic difference equation. X_(n+1)=(αX_n)/(1+X_(n-1) )+σ(X_n-X^* ) ξ_(n+1) where parameters a and σ have arbitrary values. Utilizing the general method of constructing Lyapunov functional for stochastic difference equations in discrete time, we establish necessary and sufficient conditions for the asymptotic mean square stability of the two equilibrium points (zero and positive). These conditions also serve as sufficient conditions for the stability in probability of the equilibrium points of the original nonlinear equation. To validate the derived results, numerical examples and simulations of equation solutions are presented, accompanied by numerous graphical illustrations of stability trajectories of solutions.