Abstract
In this paper, we deal with a discrete Lotka-Volterra predator-prey model with time-varying delays. For the general non-autonomous case, sufficient conditions which ensure the permanence and global stability of the system are obtained by using differential inequality theory. For the periodic case, sufficient conditions which guarantee the existence of a unique globally stable positive periodic solution are established. The paper ends with some interesting numerical simulations that illustrate our analytical predictions.MSC:34K20, 34C25, 92D25.
Highlights
After the pioneering work of Berryman [ ] in, the dynamic relationship between predators and their preys has become one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance
6 Conclusions In this paper, we have investigated the dynamic behavior of a discrete Lotka-Volterra predator-prey model with time-varying delays
Sufficient conditions which ensure the permanence of the system are established
Summary
After the pioneering work of Berryman [ ] in , the dynamic relationship between predators and their preys has become one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. [ ] Assume that {x(k)} satisfies x(k) > and x(k + ) ≤ x(k) exp a(k) – b(k)x(k) for k ∈ N , where a(k) and b(k) are non-negative sequences bounded above and below by positive constants. [ ] Assume that {x(k)} satisfies x(k + ) ≥ x(k) exp a(k) – b(k)x(k) , k ≥ N , limk→+∞ sup x(k) ≤ x∗ and x(N ) > , where a(k) and b(k) are non-negative sequences bounded above and below by positive constants and N ∈ N. ) with the initial condition xi( ) ≥ (i = , , ), one has mi ≤ lim inf xi(k) ≤ lim sup xi(k) ≤ Mi. Proof Let (x (k), x (k), x (k)) be any positive solution of system
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