Abstract
In this letter, we consider a stochastic generalized logistic equation with Markovian switching. We obtain a critical value which has the property that if the critical value is negative, then the trivial solution of the model is stochastically globally asymptotically stable; if the critical value is positive, then the solution of the model is positive recurrent and has a unique ergodic stationary distribution. We find out that the critical value has a close relationship with the stationary probability distribution of the Markov chain.
Highlights
Logistic equation is one of the most important models in mathematical ecology
Suppose that r and a are affected by white noises, with r → r + σ B (t), a → a – σ B (t), it follows from ( ) that dN(t) = N(t) r – aNθ (t) dt + σ N(t) dB (t) + σ N +θ (t) dB (t), ( )
Where {B (t)}t≥ and {B (t)}t≥ are two independent standard Brownian motions defined on the complete probability space (, F, P) with a filtration {F }t≥, σi stands for the intensity of the white noise, i =
Summary
Logistic equation is one of the most important models in mathematical ecology. A classical generalized logistic equation (or Gilpin-Ayala model [ ]) can be described by the ordinary differential equation dN(t)/dt = N(t) r – aNθ (t) , ( )where N(t) stands for the population size; r, a and θ are positive constants, r represents the growth rate and r/a is the carrying capacity. Several authors have suggested that [ – ] one can use β(t) to model these random changes, where β(t) is a right-continuous Markov chain taking values in a finite-state space S = { , . If the critical value is negative, the trivial equilibrium state of system ( ) is stochastically globally asymptotically stable; if the critical value is positive, the solution of system ( ) is positive recurrent and has a unique ergodic stationary distribution.
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