Dynamical behaviors of stochastic type K monotone Lotka-Volterra systems

Dejun Fan; Xiaoyu Yi; Ling Xia; Jingliang Lv

doi:10.3934/dcdsb.2018291

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2019, Volume 24, Issue 6: 2901-2922. Doi: 10.3934/dcdsb.2018291

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Dynamical behaviors of stochastic type K monotone Lotka-Volterra systems

Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai 264209, China

* Corresponding author: Jingliang Lv
* Corresponding author: Jingliang Lv

Received: August 2018

Early access: October 2018

Published: June 2019

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Abstract

Two n-species stochastic type K monotone Lotka-Volterra systems are proposed and investigated. For non-autonomous system, we show that there is a unique positive solution to the model for any positive initial value. Moreover, sufficient conditions for stochastic permanence and global attractivity are established. For autonomous system, we prove that for each species, there is a constant which can be represented by the coefficients of the system. If the constant equals 1, then the corresponding species will be nonpersistent on average. To illustrate the theoretical results, the corresponding numerical simulations are also given.

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Mathematics Subject Classification: Primary: 60G15, 60H10; Secondary: 37A50.

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Figure 1. Solutions of system (16) for $r_1 = 0.055,~r_2 = 0.045,~r_3 = 0.035,~a_{11} = 0.01,~a_{12} = -0.05,~a_{13} = 0.01,~a_{21} = -0.05$, $ a_{22} = 0.1,~a_{23} = 0.01,~a_{31} = 0.01,~a_{32} = 0.01,~a_{33} = 0.1.$ The horizontal axis represents the time $t$. (a) is with $\sigma_1 = 0.3,~\sigma_2 = 0.3606,~\sigma_3 = 0.3243$; (b) is with $~\sigma_1 = 0.3,~\sigma_2 = 0.4359,~\sigma_3 = 0.3243$; (c) is with $\sigma_1 = 0.35,~\sigma_2 = 0.4359,~\sigma_3 = 0.2646$.

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Figure 2. Solutions of system (16) for $r_1 = 0.55,~r_2 = 0.24,~r_3 = 0.36,~a_{11} = 0.095,~a_{12} = -0.05,~a_{13} = 0.01,~a_{21} = -0.05,$ $a_{22} = 0.0095,~a_{23} = 0.01,~a_{31} = 0.01,~a_{32} = 0.01,~a_{33} = 0.1,~\sigma_1 = 0.2,~\sigma_2 = 0.1612,~\sigma_3 = 0.1732.$ The horizontal axis represents the time $t$.

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Figure 3. Solutions of system (16) for $r_1 = 0.55,~r_2 = 0.24,~r_3 = 0.36,~a_{11} = 0.095,~a_{12} = -0.05,~a_{13} = 0.01,~a_{21} = -0.05,$ $~a_{22} = 0.0095,~a_{23} = 0.01,~a_{31} = 0.01,~a_{32} = 0.01,~a_{33} = 0.1,~\sigma_1 = $$ 0.2,~\sigma_2 = 0.1612,~\sigma_3 = 0.1732,x_1(0) = 10.3,~y_1(0) = 10.2,~x_2(0) =$ $7.5,~y_2(0) = 7.3,~x_3(0) = 5.2,~y_3(0) = 5.1.$ The horizontal axis represents the time $t$.

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