Abstract
In this paper, we propose and investigate a stochastic two-prey one-predator model. Firstly, under some simple assumptions, we show that for each species x i , i=1,2,3, there is a π i which is represented by the coefficients of the model. If π i <1, then x i goes to extinction (i.e., lim t→+∞ x i (t)=0); if π i >1, then x i is stable in the mean (i.e., \(\lim_{t\rightarrow+\infty}t^{-1} \int_{0}^{t}x_{i}(s)\,\mathrm {d}s=\mbox{a positive constant}\)). Secondly, we prove that there is a stationary distribution to this model and it has the ergodic property. Thirdly, we establish the sufficient conditions for global asymptotic stability of the positive solution. Finally, we introduce some numerical simulations to illustrate the theoretical results.
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Acknowledgements
The authors thank the editors and reviewers for their important and valuable comments. Authors were supported by the NSFC of P.R. China (Nos. 11171081 and 11171056).
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Communicated by P.K. Maini.
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Liu, M., Wang, K. Dynamics of a Two-Prey One-Predator System in Random Environments. J Nonlinear Sci 23, 751–775 (2013). https://doi.org/10.1007/s00332-013-9167-4
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DOI: https://doi.org/10.1007/s00332-013-9167-4
Keywords
- Two-prey one-predator model
- Random perturbations
- Extinction
- Stationary distribution
- Global asymptotic stability