Abstract
This paper is concerned with a delay Lotka-Volterra model under regime switching diffusion in random environment. Permanence and asymptotic estimations of solutions are investigated by virtue of V-function technique, M-matrix method, and Chebyshev's inequality. Finally, an example is given to illustrate the main results.
Highlights
The delay differential equation dx (t) dt = x (t) (a − bx (t) + cx (t τ)) (1)has been used to model the population growth of certain species and is known as the delay Lotka-Volterra model or the delay logistic equation
We assume in this paper that the Markov chain r(t) is irreducible. This is a very reasonable assumption as it means that the system can switch from any regime to any other regime
Assumption 6 implies that there exists a constant θ > 0 such that the matrix A(θ) is a nonsingular Mmatrix
Summary
Has been used to model the population growth of certain species and is known as the delay Lotka-Volterra model or the delay logistic equation. The switching between different environments is memoryless and the waiting time for the switch has an exponential distribution This indicates that we may model the random environments and other random factors in the system by a continuoustime Markov chain r(t), t ≥ 0 with a finite state space S = {1, 2, . The stochastic permanence and extinction of a logistic model under regime switching were considered in [20, 24], asymptotic results of a competitive Lotka-Volterra model in random environment are obtained in [25], a new single-species model disturbed by both white noise and colored noise in a polluted environment was developed and analyzed in [26], and a general stochastic logistic system under regime switching was proposed and was treated in [27]. An example is given to illustrate our main results
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