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

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  • 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.

    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 $N_3(t)$

    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 $N_2(t)$

    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 $N_1(t)$

    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

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

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