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 - A, 2017, 37 (5) : 2513-2538. doi: 10.3934/dcds.2017108
References:
[1]

S. Ahmad and I. M. Stamova, Almost necessary and sufficient conditions for survival of species, Nonlinear Anal. Real World Appl., 5 (2004), 219-229.  doi: 10.1016/S1468-1218(03)00037-3.  Google Scholar

[2]

J. BaoX. MaoG. Yin and C. Yuan, Competitive Lotka-Volterra population dynamics with jumps, Nonlinear Anal., 74 (2011), 6601-6616.  doi: 10.1016/j.na.2011.06.043.  Google Scholar

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I. Barbalat, Systems dequations differentielles d'oscillations nonlineaires, Rev. Roumaine Math. Pures Appl., 4 (1959), 267-270.   Google Scholar

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J. R. Beddington and R. M. May, Harvesting natural populations in a randomly fluctuating environment, Science, 197 (1977), 463-465.   Google Scholar

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C. A. Braumann, Variable effort harvesting models in random environments: Generalization to density-dependent noise intensities, Math. Biosci., 177/178 (2002), 229-245.   Google Scholar

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C. A. Braumann, Itô versus Stratonovich calculus in random population growth, Math. Biosci., 206 (2007), 81-107.   Google Scholar

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Y. CaiY. KangM. Banerjee and W. Wang, A stochastic SIRS epidemic model with infectious force under intervention strategies, J. Differential Equations, 259 (2015), 7463-7502.   Google Scholar

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J. Connell, On the prevalence and relative importance of interspecific competition: evidence from field experiments, Am. Nat., 115 (2011), 351-370.   Google Scholar

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N. H. DangN. H. Du and T. V. Ton, Asymptotic behavior of predator-prey systems perturbed by white noise, Acta Appl. Math., 115 (2011), 351-370.  doi: 10.1007/s10440-011-9628-4.  Google Scholar

[10]

M. Farkas and H. Freedman, Stability conditions for two predator one prey systems, Acta. Appl. Math., 14 (1989), 3-10.   Google Scholar

[11]

K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic, Dordrecht, 1992. doi: 10.1007/978-94-015-7920-9.  Google Scholar

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R. Z. Hasminskii, Stochastic Stability of Differential Equations, Netherlands, 1980. doi: 10.1007/978-94-009-9121-7.  Google Scholar

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A. Hastings and T. Powell, Chaos in a three-species food chain, Ecology, 72 (1991), 896-903.  doi: 10.2307/1940591.  Google Scholar

[14]

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.  doi: 10.1137/S0036144500378302.  Google Scholar

[15]

L. C. Hung, Stochastic delay population systems, Appl. Anal., 88 (2009), 1303-1320.   Google Scholar

[16]

N. Ikeda and S. Watanabe, A comparison theorem for solutions of stochastic differential equations and its applications, Osaka J. Math., 14 (1977), 619-633.   Google Scholar

[17]

C. JiD. Jiang and N. Shi, A note on a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes with stochastic perturbation, J. Math. Anal. Appl., 377 (2011), 435-440.   Google Scholar

[18]

D. Jiang and N. Shi, A note on nonautonomous logistic equation with random perturbation, J. Math. Anal. Appl., 303 (2005), 164-172.   Google Scholar

[19]

A. L. Koch, Competitive coexistence of two predators utilizing the same prey unde rconstant environmental conditions, J. Theor. Biol., 44 (1974), 387-395.   Google Scholar

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[21]

X. Li and X. Mao, Population dynamical behavior of non-autonomous Lotks-Volterra competitive system with random perturbation, Discrete Contin. Dyn. Syst., 24 (2009), 523-545.   Google Scholar

[22]

M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differential Equations, 248 (2010), 1-20.   Google Scholar

[23]

W. LiX. Zhang and C. Zhang, A new method for exponential stability of coupled reaction diffusion systems with mixed delays: Combining Razumikhin method with graph theory, J. Franklin Inst., 352 (2015), 1169-1191.   Google Scholar

[24]

M. Liu, Global asymptotic stability of stochastic Lotka-Volterra systems with infinite delays, IMA J. Appl. Math., 80 (2015), 1431-1453.   Google Scholar

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M. Liu and C. Bai, Analysis of a stochastic tri-trophic food-chain model with harvesting, J. Math. Biol., 73 (2016), 597-625.   Google Scholar

[26]

M. Liu and C. Bai, Optimal harvesting of a stochastic mutualism model with Lévy jumps, Appl. Math. Comput., 276 (2016), 301-309.   Google Scholar

[27]

M. Liu and M. Fan, Permanence of stochastic lotka-volterra systems, J. Nonlinear Sci. , (2016). doi: 10.1007/s00332-016-9337-2.  Google Scholar

[28]

M. Liu and M. Fan, Stability in distribution of a three-species stochastic cascade predator-prey system with time delays, IMA J Appl Math, (2016). doi: 10.1093/imamat/hxw057.  Google Scholar

[29]

M. Liu and P. S. Mandal, Dynamical behavior of a one-prey two-predator model with random perturbations, Commun. Nonlinear Sci. Numer. Simul., 28 (2015), 123-137.   Google Scholar

[30]

M. LiuH. Qiu and K. Wang, A remark on a stochastic predator-prey system with time delays, Appl. Math. Lett., 26 (2013), 318-323.   Google Scholar

[31]

M. LiuK. Wang and Q. Wu, Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle, Bull. Math. Biol., 73 (2011), 1969-2012.   Google Scholar

[32]

Q. Luo and X. Mao, Stochastic population dynamics under regime switching Ⅱ, J. Math. Anal. Appl., 355 (2009), 577-593.   Google Scholar

[33]

X. Mao, Stationary distribution of stochastic population systems, Systems Control Lett., 60 (2011), 398-405.   Google Scholar

[34] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, 2006.   Google Scholar
[35]

R. M. May, Stability and Complexity in Model Ecosystems, Princeton Univ., 1973. Google Scholar

[36]

R. M. May and W. J. Leonard, Nonlinear aspects of competition between three species, SIAM J. Appl. Math., 29 (1975), 243-253.   Google Scholar

[37]

J. W. Moon, Counting Labelled Tress, Canadian Mathematical Congress, Montreal, 1970. Google Scholar

[38]

J. D. Murray, Mathematical Biology I, An introduction, Third edition. Interdisciplinary Applied Mathematics, 17. Springer-Verlag, New York, 2002.  Google Scholar

[39]

B. Øendal, Stochastic Differential Equations: An Introduction with Applications, 5th edition, Springer, Berlin, 1998. Google Scholar

[40]

R. T. Paine, Road maps of interactions or grist for theoretical development, Ecology, 69 (1988), 1648-1654.   Google Scholar

[41]

S. Pasquali, The stochastic logistic equation: stationary solutions and their stability, Rend. Sem. Mat. Univ. Padova, 106 (2001), 165-183.   Google Scholar

[42] D. PratoZabczyk and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, 1996.   Google Scholar
[43]

P. W. PriceC. W. BoutonP. Gross and B. A. McPheron, Interactions among the three trophic levels: Influence of plants on interactions between insect herbivores and natural enemies, Annu. Rev. Ecol. Evol. S., 11 (1980), 41-65.   Google Scholar

[44]

R. Rudnicki, Long-time behaviour of a stochastic prey-predator model, Stoch. Process. Appl., 108 (2003), 93-107.  doi: 10.1016/S0304-4149(03)00090-5.  Google Scholar

[45]

R. Rudnicki and K. Pichór, Influence of stochastic perturbation on prey-predator systems, Math. Biosci., 206 (2007), 108-119.   Google Scholar

[46]

S. J. SchreiberM. Benaïm and K.A.S. Atchadé, Persistence in fluctuating environments, J. Math. Biol., 62 (2011), 655-683.   Google Scholar

[47]

L. Turyn, Persistence in models of 3 interacting predator prey populations-remarks, Math. Biosci., 110 (1992), 125-130.   Google Scholar

[48]

P. Xia, D. Jiang and X. Li, Dynamical behavior of the stochastic delay mutualism system, Abstract and Applied Analysis, 2014 (2014), Art. ID 781394, 19 pp.  Google Scholar

[49]

C. ZhangW. Li and K. Wang, Graph-theoretic method on exponential synchronization of stochastic coupled networks with Markovian switching, Nonlinear Anal. Hybrid Syst., 15 (2015), 37-51.   Google Scholar

[50]

C. ZhangW. Li and K. Wang, Graph-theoretic approach to stability of multi-group models with dispersal, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 259-280.   Google Scholar

[51]

C. ZhangW. Li and K. Wang, Graph theory-based approach for stability analysis of stochastic coupled systems with Lévy noise on networks, IEEE Transactions on Neural Networks and Learning Systems, 26 (2015), 1698-1709.   Google Scholar

[52]

Q. Zhang and D. Jiang, The coexistence of a stochastic Lotka-Volterra model with two predators competing for one prey, Appl. Math. Comput., 269 (2015), 288-300.   Google Scholar

[53]

Y. ZhaoS. Yuan and J. Ma, Survival and stationary distribution analysis of a stochastic competitive model of three species in a polluted environment, Bull. Math. Biol., 77 (2015), 1285-1326.   Google Scholar

[54]

C. Zhu and G. Yin, On hybrid competitive Lotka-Volterra ecosystems, Nonlinear Anal., 71 (2009), e1370-e1379.   Google Scholar

[55]

X. ZouD. Fan and K. Wang, Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1507-1519.   Google Scholar

show all references

References:
[1]

S. Ahmad and I. M. Stamova, Almost necessary and sufficient conditions for survival of species, Nonlinear Anal. Real World Appl., 5 (2004), 219-229.  doi: 10.1016/S1468-1218(03)00037-3.  Google Scholar

[2]

J. BaoX. MaoG. Yin and C. Yuan, Competitive Lotka-Volterra population dynamics with jumps, Nonlinear Anal., 74 (2011), 6601-6616.  doi: 10.1016/j.na.2011.06.043.  Google Scholar

[3]

I. Barbalat, Systems dequations differentielles d'oscillations nonlineaires, Rev. Roumaine Math. Pures Appl., 4 (1959), 267-270.   Google Scholar

[4]

J. R. Beddington and R. M. May, Harvesting natural populations in a randomly fluctuating environment, Science, 197 (1977), 463-465.   Google Scholar

[5]

C. A. Braumann, Variable effort harvesting models in random environments: Generalization to density-dependent noise intensities, Math. Biosci., 177/178 (2002), 229-245.   Google Scholar

[6]

C. A. Braumann, Itô versus Stratonovich calculus in random population growth, Math. Biosci., 206 (2007), 81-107.   Google Scholar

[7]

Y. CaiY. KangM. Banerjee and W. Wang, A stochastic SIRS epidemic model with infectious force under intervention strategies, J. Differential Equations, 259 (2015), 7463-7502.   Google Scholar

[8]

J. Connell, On the prevalence and relative importance of interspecific competition: evidence from field experiments, Am. Nat., 115 (2011), 351-370.   Google Scholar

[9]

N. H. DangN. H. Du and T. V. Ton, Asymptotic behavior of predator-prey systems perturbed by white noise, Acta Appl. Math., 115 (2011), 351-370.  doi: 10.1007/s10440-011-9628-4.  Google Scholar

[10]

M. Farkas and H. Freedman, Stability conditions for two predator one prey systems, Acta. Appl. Math., 14 (1989), 3-10.   Google Scholar

[11]

K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic, Dordrecht, 1992. doi: 10.1007/978-94-015-7920-9.  Google Scholar

[12]

R. Z. Hasminskii, Stochastic Stability of Differential Equations, Netherlands, 1980. doi: 10.1007/978-94-009-9121-7.  Google Scholar

[13]

A. Hastings and T. Powell, Chaos in a three-species food chain, Ecology, 72 (1991), 896-903.  doi: 10.2307/1940591.  Google Scholar

[14]

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.  doi: 10.1137/S0036144500378302.  Google Scholar

[15]

L. C. Hung, Stochastic delay population systems, Appl. Anal., 88 (2009), 1303-1320.   Google Scholar

[16]

N. Ikeda and S. Watanabe, A comparison theorem for solutions of stochastic differential equations and its applications, Osaka J. Math., 14 (1977), 619-633.   Google Scholar

[17]

C. JiD. Jiang and N. Shi, A note on a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes with stochastic perturbation, J. Math. Anal. Appl., 377 (2011), 435-440.   Google Scholar

[18]

D. Jiang and N. Shi, A note on nonautonomous logistic equation with random perturbation, J. Math. Anal. Appl., 303 (2005), 164-172.   Google Scholar

[19]

A. L. Koch, Competitive coexistence of two predators utilizing the same prey unde rconstant environmental conditions, J. Theor. Biol., 44 (1974), 387-395.   Google Scholar

[20] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993.   Google Scholar
[21]

X. Li and X. Mao, Population dynamical behavior of non-autonomous Lotks-Volterra competitive system with random perturbation, Discrete Contin. Dyn. Syst., 24 (2009), 523-545.   Google Scholar

[22]

M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differential Equations, 248 (2010), 1-20.   Google Scholar

[23]

W. LiX. Zhang and C. Zhang, A new method for exponential stability of coupled reaction diffusion systems with mixed delays: Combining Razumikhin method with graph theory, J. Franklin Inst., 352 (2015), 1169-1191.   Google Scholar

[24]

M. Liu, Global asymptotic stability of stochastic Lotka-Volterra systems with infinite delays, IMA J. Appl. Math., 80 (2015), 1431-1453.   Google Scholar

[25]

M. Liu and C. Bai, Analysis of a stochastic tri-trophic food-chain model with harvesting, J. Math. Biol., 73 (2016), 597-625.   Google Scholar

[26]

M. Liu and C. Bai, Optimal harvesting of a stochastic mutualism model with Lévy jumps, Appl. Math. Comput., 276 (2016), 301-309.   Google Scholar

[27]

M. Liu and M. Fan, Permanence of stochastic lotka-volterra systems, J. Nonlinear Sci. , (2016). doi: 10.1007/s00332-016-9337-2.  Google Scholar

[28]

M. Liu and M. Fan, Stability in distribution of a three-species stochastic cascade predator-prey system with time delays, IMA J Appl Math, (2016). doi: 10.1093/imamat/hxw057.  Google Scholar

[29]

M. Liu and P. S. Mandal, Dynamical behavior of a one-prey two-predator model with random perturbations, Commun. Nonlinear Sci. Numer. Simul., 28 (2015), 123-137.   Google Scholar

[30]

M. LiuH. Qiu and K. Wang, A remark on a stochastic predator-prey system with time delays, Appl. Math. Lett., 26 (2013), 318-323.   Google Scholar

[31]

M. LiuK. Wang and Q. Wu, Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle, Bull. Math. Biol., 73 (2011), 1969-2012.   Google Scholar

[32]

Q. Luo and X. Mao, Stochastic population dynamics under regime switching Ⅱ, J. Math. Anal. Appl., 355 (2009), 577-593.   Google Scholar

[33]

X. Mao, Stationary distribution of stochastic population systems, Systems Control Lett., 60 (2011), 398-405.   Google Scholar

[34] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, 2006.   Google Scholar
[35]

R. M. May, Stability and Complexity in Model Ecosystems, Princeton Univ., 1973. Google Scholar

[36]

R. M. May and W. J. Leonard, Nonlinear aspects of competition between three species, SIAM J. Appl. Math., 29 (1975), 243-253.   Google Scholar

[37]

J. W. Moon, Counting Labelled Tress, Canadian Mathematical Congress, Montreal, 1970. Google Scholar

[38]

J. D. Murray, Mathematical Biology I, An introduction, Third edition. Interdisciplinary Applied Mathematics, 17. Springer-Verlag, New York, 2002.  Google Scholar

[39]

B. Øendal, Stochastic Differential Equations: An Introduction with Applications, 5th edition, Springer, Berlin, 1998. Google Scholar

[40]

R. T. Paine, Road maps of interactions or grist for theoretical development, Ecology, 69 (1988), 1648-1654.   Google Scholar

[41]

S. Pasquali, The stochastic logistic equation: stationary solutions and their stability, Rend. Sem. Mat. Univ. Padova, 106 (2001), 165-183.   Google Scholar

[42] D. PratoZabczyk and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, 1996.   Google Scholar
[43]

P. W. PriceC. W. BoutonP. Gross and B. A. McPheron, Interactions among the three trophic levels: Influence of plants on interactions between insect herbivores and natural enemies, Annu. Rev. Ecol. Evol. S., 11 (1980), 41-65.   Google Scholar

[44]

R. Rudnicki, Long-time behaviour of a stochastic prey-predator model, Stoch. Process. Appl., 108 (2003), 93-107.  doi: 10.1016/S0304-4149(03)00090-5.  Google Scholar

[45]

R. Rudnicki and K. Pichór, Influence of stochastic perturbation on prey-predator systems, Math. Biosci., 206 (2007), 108-119.   Google Scholar

[46]

S. J. SchreiberM. Benaïm and K.A.S. Atchadé, Persistence in fluctuating environments, J. Math. Biol., 62 (2011), 655-683.   Google Scholar

[47]

L. Turyn, Persistence in models of 3 interacting predator prey populations-remarks, Math. Biosci., 110 (1992), 125-130.   Google Scholar

[48]

P. Xia, D. Jiang and X. Li, Dynamical behavior of the stochastic delay mutualism system, Abstract and Applied Analysis, 2014 (2014), Art. ID 781394, 19 pp.  Google Scholar

[49]

C. ZhangW. Li and K. Wang, Graph-theoretic method on exponential synchronization of stochastic coupled networks with Markovian switching, Nonlinear Anal. Hybrid Syst., 15 (2015), 37-51.   Google Scholar

[50]

C. ZhangW. Li and K. Wang, Graph-theoretic approach to stability of multi-group models with dispersal, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 259-280.   Google Scholar

[51]

C. ZhangW. Li and K. Wang, Graph theory-based approach for stability analysis of stochastic coupled systems with Lévy noise on networks, IEEE Transactions on Neural Networks and Learning Systems, 26 (2015), 1698-1709.   Google Scholar

[52]

Q. Zhang and D. Jiang, The coexistence of a stochastic Lotka-Volterra model with two predators competing for one prey, Appl. Math. Comput., 269 (2015), 288-300.   Google Scholar

[53]

Y. ZhaoS. Yuan and J. Ma, Survival and stationary distribution analysis of a stochastic competitive model of three species in a polluted environment, Bull. Math. Biol., 77 (2015), 1285-1326.   Google Scholar

[54]

C. Zhu and G. Yin, On hybrid competitive Lotka-Volterra ecosystems, Nonlinear Anal., 71 (2009), e1370-e1379.   Google Scholar

[55]

X. ZouD. Fan and K. Wang, Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1507-1519.   Google Scholar

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 $N_1(\theta)=0.5+0.1\sin \theta$, $N_2(\theta)=0.1+0.05\sin \theta$, $N_3(\theta)=0.05+0.03\sin \theta$, $M_1(\theta)=0.4+0.1\sin \theta$, $M_2(\theta)=0.3+0.05\sin \theta$, $M_3(\theta)=0.1+0.05\sin \theta$
$(i')~~1>\rho_1$ $\lim\limits_{t\rightarrow+\infty}N_i(t)=0,~i=1,2,3,~~a.s.$
$(ii')~~\rho_1>1>\rho_2$ $\lim\limits_{t\rightarrow+\infty}\langle N_1(t)\rangle=\frac{b_1}{a_{11}},~\lim\limits_{t\rightarrow+\infty}N_2(t)=\lim\limits_{t\rightarrow+\infty}N_3(t)=0,~~a.s.$
$(iii')~\rho_2>1>\rho_3$ $\lim\limits_{t\rightarrow+\infty}\langle N_1(t)\rangle=\frac{\Delta_1-\tilde{\Delta}_1}{A_{33}},~\lim\limits_{t\rightarrow+\infty}\langle N_2(t)\rangle=\frac{\Delta_2-\tilde{\Delta}_2}{A_{33}},~\lim\limits_{t\rightarrow+\infty}N_3(t)=0,~~a.s.$
$(iv')~~\rho_3>1$ $\lim\limits_{t\rightarrow+\infty}\langle N_i(t)\rangle=\frac{A_i-\tilde{A}_i}{A},~i=1,2,3,~~a.s.$
$(i')~~1>\rho_1$ $\lim\limits_{t\rightarrow+\infty}N_i(t)=0,~i=1,2,3,~~a.s.$
$(ii')~~\rho_1>1>\rho_2$ $\lim\limits_{t\rightarrow+\infty}\langle N_1(t)\rangle=\frac{b_1}{a_{11}},~\lim\limits_{t\rightarrow+\infty}N_2(t)=\lim\limits_{t\rightarrow+\infty}N_3(t)=0,~~a.s.$
$(iii')~\rho_2>1>\rho_3$ $\lim\limits_{t\rightarrow+\infty}\langle N_1(t)\rangle=\frac{\Delta_1-\tilde{\Delta}_1}{A_{33}},~\lim\limits_{t\rightarrow+\infty}\langle N_2(t)\rangle=\frac{\Delta_2-\tilde{\Delta}_2}{A_{33}},~\lim\limits_{t\rightarrow+\infty}N_3(t)=0,~~a.s.$
$(iv')~~\rho_3>1$ $\lim\limits_{t\rightarrow+\infty}\langle N_i(t)\rangle=\frac{A_i-\tilde{A}_i}{A},~i=1,2,3,~~a.s.$
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