| Citation: |
Nan Cao, Xianlong Fu. STATIONARY DISTRIBUTION OF A LOTKA-VOLTERRA MODEL WITH STOCHASTIC PERTURBATIONS AND DISTRIBUTED DELAY[J]. Journal of Applied Analysis & Computation, 2022, 12(5): 1713-1726. doi: 10.11948/20210175
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STATIONARY DISTRIBUTION OF A LOTKA-VOLTERRA MODEL WITH STOCHASTIC PERTURBATIONS AND DISTRIBUTED DELAY
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Abstract
This paper devotes to the existence of a stationary distribution for a one-prey and two-predator Lotka-Volterra model with stochastic nonlinear perturbations and distributed delay. The studied autonomous system is first proved having a unique global and positive solution. Then, through constructing appropriate Lyapunov function and using Itô formula, sufficient conditions guaranteeing the existence of a stationary distribution of the stochastic system are obtained. Some numerical simulations are also provided in the end to illustrate the main results.
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Keywords:
- Lotka-Volterra model /
- Itô formula /
- global solution /
- stationary distribution /
- distributed delay
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References
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Nan Cao, Xianlong Fu. STATIONARY DISTRIBUTION OF A LOTKA-VOLTERRA MODEL WITH STOCHASTIC PERTURBATIONS AND DISTRIBUTED DELAY[J]. Journal of Applied Analysis & Computation, 2022, 12(5): 1713-1726. doi: 10.11948/20210175
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Figure 1. The left is the solution of the stochastic system (1.3) and the corresponding deterministic system. The right is the density function of
$ x(t) $ ,$ y_{1}(t) $ and$ y_{2}(t) $ for the system (1.3) obtained by 10, 000 simulations running at$ t=200 $ .