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Phase portraits, Hopf bifurcations and limit cycles of Leslie-Gower predator-prey systems with harvesting rates
Stochastic Lotka-Volterra system with unbounded distributed delay
| 1. | School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China, China |
$\dot{x}(t)=\mbox{diag}(x_1(t), \cdots, x_n(t))(r+Ax(t)+B\int_{-\infty}^0x(t+\theta)d\mu(\theta))$
into the corresponding stochastic system
$dx(t)=\mbox{diag}(x_1(t), \cdots, x_n(t))[(r+Ax(t)+B\int_{-\infty}^0x(t+\theta)d\mu(\theta))dt+\beta dw(t)].$
This paper obtains one condition under which the above stochastic system has a global almost surely positive solution and gives the asymptotic pathwise estimation of this solution. This paper also shows that when the noise is sufficiently large, the solution of this stochastic system will converge to zero with probability one. This reveals that the sufficiently large noise may make the population extinct.
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