# American Institute of Mathematical Sciences

May  2017, 37(5): 2513-2538. doi: 10.3934/dcds.2017108

## 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  February 2017

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.

Citation: Meng Liu, Chuanzhi Bai, Yi Jin. Population dynamical behavior of a two-predator one-prey stochastic model with time delay. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2513-2538. doi: 10.3934/dcds.2017108
##### References:

show all references

##### References:
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 $N_3(t)$
. (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 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 $N_2(t)$
. (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 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
, initial values $N_1(\theta)=0.5+0.1\sin \theta$, $N_2(\theta)=0.1+0.05\sin \theta$, $N_3(\theta)=0.05+0.03\sin \theta$, $M_1(\theta)=0.4+0.1\sin \theta$, $M_2(\theta)=0.3+0.05\sin \theta$, $M_3(\theta)=0.1+0.05\sin \theta$">Figure 5.  Solutions of model (2) with $a_{12}=1.32$, other parameters are taken as Fig.1(a), initial values $N_1(\theta)=0.5+0.1\sin \theta$, $N_2(\theta)=0.1+0.05\sin \theta$, $N_3(\theta)=0.05+0.03\sin \theta$, $M_1(\theta)=0.4+0.1\sin \theta$, $M_2(\theta)=0.3+0.05\sin \theta$, $M_3(\theta)=0.1+0.05\sin \theta$
 $(i')~~1>\rho_1$ $\lim\limits_{t\rightarrow+\infty}N_i(t)=0,~i=1,2,3,~~a.s.$ $(ii')~~\rho_1>1>\rho_2$ $\lim\limits_{t\rightarrow+\infty}\langle N_1(t)\rangle=\frac{b_1}{a_{11}},~\lim\limits_{t\rightarrow+\infty}N_2(t)=\lim\limits_{t\rightarrow+\infty}N_3(t)=0,~~a.s.$ $(iii')~\rho_2>1>\rho_3$ $\lim\limits_{t\rightarrow+\infty}\langle N_1(t)\rangle=\frac{\Delta_1-\tilde{\Delta}_1}{A_{33}},~\lim\limits_{t\rightarrow+\infty}\langle N_2(t)\rangle=\frac{\Delta_2-\tilde{\Delta}_2}{A_{33}},~\lim\limits_{t\rightarrow+\infty}N_3(t)=0,~~a.s.$ $(iv')~~\rho_3>1$ $\lim\limits_{t\rightarrow+\infty}\langle N_i(t)\rangle=\frac{A_i-\tilde{A}_i}{A},~i=1,2,3,~~a.s.$
 $(i')~~1>\rho_1$ $\lim\limits_{t\rightarrow+\infty}N_i(t)=0,~i=1,2,3,~~a.s.$ $(ii')~~\rho_1>1>\rho_2$ $\lim\limits_{t\rightarrow+\infty}\langle N_1(t)\rangle=\frac{b_1}{a_{11}},~\lim\limits_{t\rightarrow+\infty}N_2(t)=\lim\limits_{t\rightarrow+\infty}N_3(t)=0,~~a.s.$ $(iii')~\rho_2>1>\rho_3$ $\lim\limits_{t\rightarrow+\infty}\langle N_1(t)\rangle=\frac{\Delta_1-\tilde{\Delta}_1}{A_{33}},~\lim\limits_{t\rightarrow+\infty}\langle N_2(t)\rangle=\frac{\Delta_2-\tilde{\Delta}_2}{A_{33}},~\lim\limits_{t\rightarrow+\infty}N_3(t)=0,~~a.s.$ $(iv')~~\rho_3>1$ $\lim\limits_{t\rightarrow+\infty}\langle N_i(t)\rangle=\frac{A_i-\tilde{A}_i}{A},~i=1,2,3,~~a.s.$
 [1] Ke Wang, Qi Wang, Feng Yu. Stationary and time-periodic patterns of two-predator and one-prey systems with prey-taxis. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 505-543. doi: 10.3934/dcds.2017021 [2] Sungrim Seirin Lee, Tsuyoshi Kajiwara. The effect of the remains of the carcass in a two-prey, one-predator model. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 353-374. doi: 10.3934/dcdsb.2008.9.353 [3] Miljana JovanoviĆ, Marija KrstiĆ. Extinction in stochastic predator-prey population model with Allee effect on prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2651-2667. doi: 10.3934/dcdsb.2017129 [4] Moitri Sen, Malay Banerjee, Yasuhiro Takeuchi. Influence of Allee effect in prey populations on the dynamics of two-prey-one-predator model. Mathematical Biosciences & Engineering, 2018, 15 (4) : 883-904. doi: 10.3934/mbe.2018040 [5] Feiying Yang, Wantong Li, Renhu Wang. Invasion waves for a nonlocal dispersal predator-prey model with two predators and one prey. Communications on Pure & Applied Analysis, 2021, 20 (12) : 4083-4105. doi: 10.3934/cpaa.2021146 [6] Ming Liu, Dongpo Hu, Fanwei Meng. Stability and bifurcation analysis in a delay-induced predator-prey model with Michaelis-Menten type predator harvesting. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3197-3222. doi: 10.3934/dcdss.2020259 [7] Tomás Caraballo, Renato Colucci, Luca Guerrini. On a predator prey model with nonlinear harvesting and distributed delay. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2703-2727. doi: 10.3934/cpaa.2018128 [8] Mary Ballyk, Ross Staffeldt, Ibrahim Jawarneh. A nutrient-prey-predator model: Stability and bifurcations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 2975-3004. doi: 10.3934/dcdss.2020192 [9] Yu Ma, Chunlai Mu, Shuyan Qiu. Boundedness and asymptotic stability in a two-species predator-prey chemotaxis model. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021218 [10] Wei Feng, Jody Hinson. Stability and pattern in two-patch predator-prey population dynamics. Conference Publications, 2005, 2005 (Special) : 268-279. doi: 10.3934/proc.2005.2005.268 [11] Guihong Fan, Gail S. K. Wolkowicz. Chaotic dynamics in a simple predator-prey model with discrete delay. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 191-216. doi: 10.3934/dcdsb.2020263 [12] Xiaoying Wang, Xingfu Zou. On a two-patch predator-prey model with adaptive habitancy of predators. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 677-697. doi: 10.3934/dcdsb.2016.21.677 [13] Sílvia Cuadrado. Stability of equilibria of a predator-prey model of phenotype evolution. Mathematical Biosciences & Engineering, 2009, 6 (4) : 701-718. doi: 10.3934/mbe.2009.6.701 [14] Yinshu Wu, Wenzhang Huang. Global stability of the predator-prey model with a sigmoid functional response. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1159-1167. doi: 10.3934/dcdsb.2019214 [15] Antoni Leon Dawidowicz, Anna Poskrobko. Stability problem for the age-dependent predator-prey model. Evolution Equations & Control Theory, 2018, 7 (1) : 79-93. doi: 10.3934/eect.2018005 [16] Pankaj Kumar, Shiv Raj. Modelling and analysis of prey-predator model involving predation of mature prey using delay differential equations. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021035 [17] Sungrim Seirin Lee. Dependence of propagation speed on invader species: The effect of the predatory commensalism in two-prey, one-predator system with diffusion. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 797-825. doi: 10.3934/dcdsb.2009.12.797 [18] Komi Messan, Yun Kang. A two patch prey-predator model with multiple foraging strategies in predator: Applications to insects. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 947-976. doi: 10.3934/dcdsb.2017048 [19] Yun Kang, Sourav Kumar Sasmal, Komi Messan. A two-patch prey-predator model with predator dispersal driven by the predation strength. Mathematical Biosciences & Engineering, 2017, 14 (4) : 843-880. doi: 10.3934/mbe.2017046 [20] S. Nakaoka, Y. Saito, Y. Takeuchi. Stability, delay, and chaotic behavior in a Lotka-Volterra predator-prey system. Mathematical Biosciences & Engineering, 2006, 3 (1) : 173-187. doi: 10.3934/mbe.2006.3.173

2020 Impact Factor: 1.392