# American Institute of Mathematical Sciences

October  2018, 23(8): 3275-3296. doi: 10.3934/dcdsb.2018244

## Stochastic non-autonomous Holling type-Ⅲ prey-predator model with predator's intra-specific competition

 1 Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, West Bengal 711103, India 2 Department of Mathematics, Vivekananda College, Thakurpukur, Kolkata-700063, India

Received  December 2017 Revised  February 2018 Published  October 2018 Early access  August 2018

The objective of this article is to study the significance of dynamical properties of non-autonomous deterministic as well as stochastic prey-predator model with Holling type-Ⅲ functional response. Firstly, uniform persistence of the deterministic model has been demonstrated. Secondly, stochastic non-autonomous prey-predator system with Holling type-Ⅲ functional response is proposed. The existence of a global positive solution has been derived. Sufficient conditions for non-persistence in mean, weakly persistence in mean, extinction have been derived. Moreover the sufficient conditions for permanence of the system have been established. The analytical results are verified by numerical simulation.

Citation: Sampurna Sengupta, Pritha Das, Debasis Mukherjee. Stochastic non-autonomous Holling type-Ⅲ prey-predator model with predator's intra-specific competition. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3275-3296. doi: 10.3934/dcdsb.2018244
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##### References:
Numerical simulation for the deterministic system (1) with initial condition (0.2, 0.3) by $a_1(t) = 0.1+0.01 \sin t, \ a_2(t) = 0.02+0.01\sin t$ shows the stable behavior of prey and predator
Numerical simulation for the deterministic system (1) with initial condition (0.2, 0.3) by $a_1(t) = 2+0.1 \sin t,\ a_2(t) = 1+0.1\sin t$ shows the unstable behavior of prey and predator
Numerical simulation for the deterministic system (1) with (0.2, 0.3) by $r(t) = 5 + 2.5 \sin t,~b(t) = 0.22+0.02\sin t,~c(t) = 0.01+0.005\sin t,~d(t)$ $= 0.2+0.01\sin t,~a_1(t) = 0.1+0.1\sin t,~a_2(t) = 1+0.1\sin t$ shows that system is persistent
Numerical simulation for the system (5) with $\frac{\sigma_1^2}{2} = \frac{\sigma_2^2}{2} = 0.21+0.02\sin t$ shows that both prey and predator population goes to extinction
Numerical simulation for the system (5) with $\frac{\sigma_1^2}{2} = 0.19+0.02\sin t,~\frac{\sigma_2^2}{2} = 0.09+0.02\sin t$ shows weakly persistence in the mean of prey and extinction of predator
Numerical simulation for the system (5) with $r(t) = 2.2+0.01 \sin t , \ \sigma_1 = \sigma_2 = 0.02+0.01\sin t$ shows permanence of both prey and predator
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