Qualitative analysis of a stochastic predator-prey system with disease in the predator. (English) Zbl 1296.92210
Summary: A stochastic predator-prey system with disease in the predator population is proposed, the existence of global positive solution is derived. When the white noise is small, there is a stationary distribution. In addition, conditions of global stability for the deterministic system are also established from the above result. By Lyapunov function, the long time behavior of solution around the disease-free equilibrium of deterministic system is derived. These results mean that stochastic system has the similar property with the corresponding deterministic system. When the white noise is small, however, large environmental noise makes the result different. Finally, numerical simulations are carried out to support our findings.
MSC:
| 92D25 | Population dynamics (general) |
| 34F05 | Ordinary differential equations and systems with randomness |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
References:
| [1] | DOI: 10.1088/0951-7715/18/2/022 · Zbl 1078.34035 · doi:10.1088/0951-7715/18/2/022 |
| [2] | DOI: 10.1016/S0378-4754(97)00106-7 · Zbl 1017.92504 · doi:10.1016/S0378-4754(97)00106-7 |
| [3] | DOI: 10.1016/S0025-5564(01)00089-X · Zbl 0987.92027 · doi:10.1016/S0025-5564(01)00089-X |
| [4] | DOI: 10.1016/S0362-546X(98)00126-6 · Zbl 0922.34036 · doi:10.1016/S0362-546X(98)00126-6 |
| [5] | Gard T. C., Introduction to Stochastic Differential Equations (1988) · Zbl 0628.60064 |
| [6] | DOI: 10.1007/978-94-009-9121-7 · doi:10.1007/978-94-009-9121-7 |
| [7] | DOI: 10.1137/S0036144500378302 · Zbl 0979.65007 · doi:10.1137/S0036144500378302 |
| [8] | DOI: 10.1016/j.jde.2005.06.017 · Zbl 1089.34041 · doi:10.1016/j.jde.2005.06.017 |
| [9] | DOI: 10.1016/j.jmaa.2011.02.037 · Zbl 1232.34072 · doi:10.1016/j.jmaa.2011.02.037 |
| [10] | DOI: 10.1016/j.jmaa.2009.05.039 · Zbl 1190.34064 · doi:10.1016/j.jmaa.2009.05.039 |
| [11] | DOI: 10.1016/j.jmaa.2010.11.008 · Zbl 1216.34040 · doi:10.1016/j.jmaa.2010.11.008 |
| [12] | DOI: 10.1016/j.jmaa.2010.06.003 · Zbl 1194.92053 · doi:10.1016/j.jmaa.2010.06.003 |
| [13] | DOI: 10.1016/j.jmaa.2004.08.027 · Zbl 1076.34062 · doi:10.1016/j.jmaa.2004.08.027 |
| [14] | DOI: 10.1016/j.mcm.2011.02.004 · Zbl 1225.60114 · doi:10.1016/j.mcm.2011.02.004 |
| [15] | DOI: 10.1016/j.cnsns.2010.12.026 · Zbl 1219.92064 · doi:10.1016/j.cnsns.2010.12.026 |
| [16] | DOI: 10.1016/j.cnsns.2010.06.015 · Zbl 1221.34152 · doi:10.1016/j.cnsns.2010.06.015 |
| [17] | Mao X. R., Stochastic Differential Equations and Applications (1997) · Zbl 0892.60057 |
| [18] | DOI: 10.1016/S0304-4149(01)00126-0 · Zbl 1058.60046 · doi:10.1016/S0304-4149(01)00126-0 |
| [19] | DOI: 10.1016/j.jmaa.2009.06.050 · Zbl 1184.34064 · doi:10.1016/j.jmaa.2009.06.050 |
| [20] | DOI: 10.1016/S0025-5564(01)00049-9 · Zbl 0978.92031 · doi:10.1016/S0025-5564(01)00049-9 |
| [21] | DOI: 10.1016/S0096-3003(01)00156-4 · Zbl 1024.92017 · doi:10.1016/S0096-3003(01)00156-4 |
| [22] | DOI: 10.1137/060649343 · Zbl 1140.93045 · doi:10.1137/060649343 |
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