Population dynamical behavior of a two-predator one-prey stochastic model with time delay

Meng Liu; Chuanzhi Bai; Yi Jin

doi:10.3934/dcds.2017108

Article Contents

2017, Volume 37, Issue 5: 2513-2538. Doi: 10.3934/dcds.2017108

This issue Previous Article Traveling waves and entire solutions for an epidemic model with asymmetric dispersal Next Article Fourier spectral approximations to the dynamics of 3D fractional complex Ginzburg-Landau equation

Population dynamical behavior of a two-predator one-prey stochastic model with time delay

1.
School of Mathematical Science, Huaiyin Normal University, Huaian 223300, China
2.
School of Mathematics and Statistics, Northeast Normal University, Jilin 130024, China
3.
Department of Mathematics, Harbin Institute of Technology, Weihai 264209, China

Received: November 2015

Revised: December 2016

Published: May 2017

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Abstract

In this paper, the convergence of the distributions of the solutions (CDS) of a stochastic two-predator one-prey model with time delay is considered. Some traditional methods that are used to study the CDS of stochastic population models without delay can not be applied to investigate the CDS of stochastic population models with delay. In this paper, we use an asymptotic approach to study the problem. By taking advantage of this approach, we show that under some simple conditions, there exist three numbers $ ρ_1>ρ_2>ρ_3$, which are represented by the coefficients of the model, closely related to the CDS of our model. We prove that if $ ρ_1<1$, then $ \lim\limits_{t\to +∞}N_i(t)=0$ almost surely, $ i=1,2,3;$ If $ρ_i>1>ρ_{i+1}$, $ i=1,2$, then $ \lim\limits_{t\to +∞}N_j(t)=0$ almost surely, $ j=i+1,...,3$, and the distributions of $ (N_1(t),...,N_i(t))^\mathrm{T}$ converge to a unique ergodic invariant distribution (UEID); If $ ρ_3>1$, then the distributions of $ (N_1(t),N_2(t),N_3(t))^\mathrm{T}$ converge to a UEID. We also discuss the effects of stochastic noises on the CDS and introduce several numerical examples to illustrate the theoretical results.

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Mathematics Subject Classification: Primary: 34F05, 60H10; Secondary: 92B05, 60J27.

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Figure 1. Model (2) with $\sigma_1^2/2=0.3,~\sigma_2^2/2=0.05$, $~\sigma_3^2/2=0.05$, $r_1=1.2$, $r_2=-0.15$, $r_3=-0.01$, $a_{11}=1.6$, $a_{12}=1.2$, $a_{13}=0.3$, $a_{21}=-0.85$, $a_{22}=1.9$, $a_{23}=0.6$, $a_{31}=-0.4$, $a_{32}=1$, $a_{33}=2.1$, $\tau_{12}=3$, $\tau_{13}=7$, $\tau_{21}=1$, $\tau_{23}=5$, $\tau_{31}=4$, $\tau_{32}=10$, $N_1(\theta)=0.5+0.1\sin \theta$, $N_2(\theta)=0.1+0.05\sin \theta$, $N_3(\theta)=0.05+0.03\sin \theta$. (a) is the paths of $N_1(t)$, $N_2(t)$ and $N_3(t)$ and their time average; (b) is the probability density functions of $N_1(t)$, $N_2(t)$ and $N_3(t)$

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Figure 2. Model (2) with $\sigma_1^2/2=0.3,~\sigma_2^2/2=0.05,~\sigma_3^2/2=0.5$, other parameters are taken as Fig.1. (a) is the paths of $N_1(t)$, $N_2(t)$ and $N_3(t)$ and the time average of $N_1(t)$ and $N_2(t)$; (b) is the probability density functions of $N_1(t)$ and $N_2(t)$

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Figure 3. Model (2) with $\sigma_1^2/2=0.3,~\sigma_2^2/2=0.47$, $~\sigma_3^2/2=0.5$, other parameters are taken as Fig.1. (a) is the paths of $N_1(t)$, $N_2(t)$ and $N_3(t)$ and the time average of $N_1(t)$; (b) is the probability density functions of $N_1(t)$

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Figure 4. The paths of $N_1(t)$, $N_2(t)$ and $N_3(t)$ of model (2) with $\sigma_1^2/2=2,~\sigma_2^2/2=0.47,~\sigma_3^2/2=0.5$, other parameters are taken as Fig.1

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Figure 5. Solutions of model (2) with $a_{12}=1.32$, other parameters are taken as Fig.1(a), initial values $N_1(\theta)=0.5+0.1\sin \theta$, $N_2(\theta)=0.1+0.05\sin \theta$, $N_3(\theta)=0.05+0.03\sin \theta$, $M_1(\theta)=0.4+0.1\sin \theta$, $M_2(\theta)=0.3+0.05\sin \theta$, $M_3(\theta)=0.1+0.05\sin \theta$

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$(i')~~1>\rho_1$	$\lim\limits_{t\rightarrow+\infty}N_i(t)=0,~i=1,2,3,~~a.s.$
$(ii')~~\rho_1>1>\rho_2$	$\lim\limits_{t\rightarrow+\infty}\langle N_1(t)\rangle=\frac{b_1}{a_{11}},~\lim\limits_{t\rightarrow+\infty}N_2(t)=\lim\limits_{t\rightarrow+\infty}N_3(t)=0,~~a.s.$
$(iii')~\rho_2>1>\rho_3$	$\lim\limits_{t\rightarrow+\infty}\langle N_1(t)\rangle=\frac{\Delta_1-\tilde{\Delta}_1}{A_{33}},~\lim\limits_{t\rightarrow+\infty}\langle N_2(t)\rangle=\frac{\Delta_2-\tilde{\Delta}_2}{A_{33}},~\lim\limits_{t\rightarrow+\infty}N_3(t)=0,~~a.s.$
$(iv')~~\rho_3>1$	$\lim\limits_{t\rightarrow+\infty}\langle N_i(t)\rangle=\frac{A_i-\tilde{A}_i}{A},~i=1,2,3,~~a.s.$

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