| Citation: |
Chunyan Ji, Daqing Jiang, Yanan Zhao. QUALITATIVE ANALYSIS OF STOCHASTIC RATIO-DEPENDENT PREDATOR-PREY SYSTEMS[J]. Journal of Applied Analysis & Computation, 2019, 9(2): 475-500. doi: 10.11948/2156-907X.20170230
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QUALITATIVE ANALYSIS OF STOCHASTIC RATIO-DEPENDENT PREDATOR-PREY SYSTEMS
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Abstract
In this paper, two stochastic ratio-dependent predator-prey systems are considered. One is just with white noise, and the other one is taken into both white noise and color noise account. Sufficient criteria for extinction and persistence in time average are established. The critical value between persistence and extinction is obtained. Moreover, we show that there is stationary distribution for the stochastic system with regime-switching. Finally, examples and simulations are carried on to verify these results. -
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References
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Chunyan Ji, Daqing Jiang, Yanan Zhao. QUALITATIVE ANALYSIS OF STOCHASTIC RATIO-DEPENDENT PREDATOR-PREY SYSTEMS[J]. Journal of Applied Analysis & Computation, 2019, 9(2): 475-500. doi: 10.11948/2156-907X.20170230
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Figure 1. Simulation of paths of
$ (x(t), y(t)) $ of system (1.3) (red solid line) and the corresponding deterministic system (blue dotted line), respectively. Population in both systems will die out -
Figure 2. Simulation of paths of
$ (x(t), y(t)) $ of system (1.3) (red solid line) and the corresponding deterministic system (blue dotted line), respectively. In this situation, there is a periodic solution of the corresponding deterministic system, while the large white noise makes both the prey and the predator die out a.s -
Figure 3. Simulation of paths of
$ (x(t), y(t)) $ ,$ \frac{1}{t}\int_0^tb(s)x(s)ds $ of system (1.3) (red solid line) and the corresponding deterministic system (blue dotted line), respectively. In both of two systems, the prey is persistent, but the predator is not, and$ b(t)x(t) $ is stable in time average of system (1.3) -
Figure 4. Simulation of paths of
$ (x(t), y(t)) $ ,$ \frac{1}{t}\int_0^tb(s)x(s)ds $ of system (1.3) (red solid line) and the corresponding deterministic system (blue dotted line), respectively. In both of two systems, the prey is persistent, but the predator is not, and$ b(t)x(t) $ is stable in time average of system (1.3) -
Figure 5. Simulation of paths of
$ x(t), y(t) $ ,$ \frac{1}{t}\int_0^tb(s)x(s)ds $ of system (1.3) (red solid line) and the corresponding deterministic system (blue dotted line), respectively. There is a periodic solution of the corresponding deterministic system, but the large white noise makes the predator extinction and the prey be stable in time average -
Figure 6. Simulation of paths of
$ (x(t), y(t)) $ of system (1.3) (red solid line) and the corresponding deterministic system (blue dotted line), respectively, and the time average of the solution of system (1.3) (green solid line) -
Figure 7. Simulation of paths of
$ x(t), y(t) $ (see the fist column in the second and third lines), and their histogram (see the second column in the second and third lines) of system (1.4) with the Markov chain (see the picture in the first line)