# American Institute of Mathematical Sciences

June 2019, 24(6): 2901-2922. doi: 10.3934/dcdsb.2018291

## 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

Received  August 2018 Published  October 2018

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.

Citation: Dejun Fan, Xiaoyu Yi, Ling Xia, Jingliang Lv. Dynamical behaviors of stochastic type K monotone Lotka-Volterra systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2901-2922. doi: 10.3934/dcdsb.2018291
##### References:
 [1] X. Bai and J. Jiang, Comparison theorems for neutral stochastic functional differential equations, J. Diff. Equ., 260 (2016), 7250-7277. doi: 10.1016/j.jde.2016.01.027. [2] X. Bai and J. Jiang, Comparison theorem for stochastic functional differential equations and applications, J. Dyn. Diff. Equ., 29 (2017), 1-24. doi: 10.1007/s10884-014-9406-x. [3] I. Barbalat, Systems dequations differentielles d'osci d'oscillations nonlineaires, Rev. Math Pures. Appl., 4 (1959), 267-270. [4] A. Berman and R. J. Plemmons, Nonnegative Matrices in Mathematical Sciences, Academic Press, New York, 1979. [5] L. Chen and J. Jiang, Stochastic epidemic models driven by stochastic algorithms with constant step, Discrete Contin. Dyn. Syst. Ser. B, 21 (2017), 721-736. doi: 10.3934/dcdsb.2016.21.721. [6] S. Cheng, Stochastic population systems, Stoch. Anal. Appl., 27 (2009), 854-874. doi: 10.1080/07362990902844348. [7] T. G. Hallam and Z. Ma, Persistence in population models with demographic fluctuations, J. Math. Biol., 24 (1986), 327-339. doi: 10.1007/BF00275641. [8] M. W. Hirsch, Systems of differential equations which are competitive or cooperative Ⅰ: Limit sets, SIAM J. Math. Anal., 13 (1982), 167-179. doi: 10.1137/0513013. [9] M. W. Hirsch, Systems of differential equations that are competitive or cooperative Ⅱ: Convergence almost everywhere, SIAM J. Math. Anal., 16 (1985), 423-439. doi: 10.1137/0516030. [10] M. W. Hirsch, Systems of differential equations which are competitive or cooperative Ⅲ: Competing species, Nonlinearity, 1 (1988), 51-71. doi: 10.1088/0951-7715/1/1/003. [11] M. W. Hirsch, System of differential equations that are competitive or cooperative. Ⅳ: Structural stability in three-dimensional systems, SIAM J. Math. Anal., 21 (1990), 1225-1234. doi: 10.1137/0521067. [12] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus: 2nd Edition, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-1-4612-0949-2. [13] X. Li, A. Gray, D. Jiang and X. Mao, Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching, J. Math. Anal. Appl., 376 (2011), 11-28. doi: 10.1016/j.jmaa.2010.10.053. [14] X. Li, D. Jiang and X. Mao, Population dynamical behavior of Lotka-Volterra system under regime switching, J. Comput. Appl. Math., 232 (2009), 427-448. doi: 10.1016/j.cam.2009.06.021. [15] X. Li and X. Mao, Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation, Discrete. Contin. Dyn. Syst., 24 (2009), 523-545. doi: 10.3934/dcds.2009.24.523. [16] X. Liang and J. Jiang, On the finite dimensional dynamical systems with limited competition, Trans. Am. Math. Soc., 354 (2002), 3535-3554. doi: 10.1090/S0002-9947-02-03032-5. [17] X. Liang and J. Jiang, The dynamical behaviour of type-K competitive Kolmogorov systems and its application to three-dimensional type-K competitive Lotka-Volterra systems, Nonlinearity, 16 (2003), 785-801. doi: 10.1088/0951-7715/16/3/301. [18] M. Liu and K. Wang, Persistence and extinction in stochastic non-autonomous logistic systems, J. Math. Anal. Appl., 375 (2011), 443-457. doi: 10.1016/j.jmaa.2010.09.058. [19] M. Liu and K. Wang, Population dynamical behavior of Lotka-Volterra cooperative systems with random perturbations, Discrete Contin. Dyn. Syst., 33 (2013), 2495-2522. doi: 10.3934/dcds.2013.33.2495. [20] M. Liu, K. 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. doi: 10.1007/s11538-010-9569-5. [21] X. Mao, Exponential Stability of Stochastic Differential Equations, Marcel Dekker, New York, 1994. [22] X. Mao and C. Yuan, Stochastical Differential Equations with Markovian Switching, Imperial College Press, London, 2006. doi: 10.1142/p473. [23] S. Smale, On the differential equations of species in competition, J. Math. Biol., 3 (1976), 5-7. doi: 10.1007/BF00307854. [24] H. L. Smith, Competing subcommunities of mutualists and a generalized Kamke theorem, SIAM J. Appl. Math., 46 (1986), 856-874. doi: 10.1137/0146052. [25] H. L. Smith and H. R. Thieme, Stable coexistence and bi-stability for competitive systems on ordered Banach spaces, J. Diff. Eqns., 176 (2001), 195-222. doi: 10.1006/jdeq.2001.3981. [26] P. Tak$\acute{a}\check{c}$, Convergence to equilibrium on invariant $d$-hypersurfaces for strongly increasing discrete-time semigroups, J. Math. Anal. Appl., 148 (1990), 223-244. doi: 10.1016/0022-247X(90)90040-M. [27] P. Tak$\acute{a}\check{c}$, Domains of attraction of generic omega-limit sets for strongly monotone discrete-time semigroups, J. Reine. Angew. Math., 423 (1992), 101-173. doi: 10.1515/crll.1992.423.101. [28] Y. Takeuchi and N. Adachi, The existence of globally stable equilibria of ecosystems of the generalized Volterra type, J. Math. Biol., 10 (1980), 401-415. doi: 10.1007/BF00276098. [29] Y. Takeuchi, N. Adachi and H. Tokumaru, Global stability of ecosystems of the generalized Volterra type, Math. Biosci., 42 (1978), 119-136. doi: 10.1016/0025-5564(78)90010-X. [30] C. C. Travis and W. M. Post, Dynamics and comparative statics of mutualistic communities, J. Theor. Biol., 78 (1979), 553-571. doi: 10.1016/0022-5193(79)90190-5. [31] C. Tu and J. Jiang, The coexistence of a community of species with limited competition, J. Math. Anal. Appl., 217 (1998), 233-245. doi: 10.1006/jmaa.1997.5711. [32] C. Tu and J. Jiang, The necessary and sufficient conditions for the global stability of type-K Lotka-Volterra system, Proc. Am. Math. Soc., 127 (1999), 3181-3186. doi: 10.1090/S0002-9939-99-05077-7. [33] C. Tu and J. Jiang, Global stability and permanence for a class of type-K monotone systems, SIAM J.Math. Anal., 30 (1999), 360-378. doi: 10.1137/S0036141097325290. [34] Y. Wang and J. Jiang, The long-run behavior of periodic competitive Kolmogorov systems, Nonlinear Anal.: Real World Appl., 3 (2002), 471-485. doi: 10.1016/S1468-1218(01)00034-7. [35] Y. Wang and J. Jiang, Uniqueness and attractivity of the carrying simplices for the discrete-time competitive dynamical systems, J. Differential Equations, 186 (2002), 611-632. doi: 10.1016/S0022-0396(02)00025-6. [36] F. Wu and Y. Hu, Stochastic Lotka-Volterra system with unbounded distributed delay, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 275-288. doi: 10.3934/dcdsb.2010.14.275. [37] M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dyn. Stability Systems, 8 (1993), 189-217. doi: 10.1080/02681119308806158.

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##### References:
 [1] X. Bai and J. Jiang, Comparison theorems for neutral stochastic functional differential equations, J. Diff. Equ., 260 (2016), 7250-7277. doi: 10.1016/j.jde.2016.01.027. [2] X. Bai and J. Jiang, Comparison theorem for stochastic functional differential equations and applications, J. Dyn. Diff. Equ., 29 (2017), 1-24. doi: 10.1007/s10884-014-9406-x. [3] I. Barbalat, Systems dequations differentielles d'osci d'oscillations nonlineaires, Rev. Math Pures. Appl., 4 (1959), 267-270. [4] A. Berman and R. J. Plemmons, Nonnegative Matrices in Mathematical Sciences, Academic Press, New York, 1979. [5] L. Chen and J. Jiang, Stochastic epidemic models driven by stochastic algorithms with constant step, Discrete Contin. Dyn. Syst. Ser. B, 21 (2017), 721-736. doi: 10.3934/dcdsb.2016.21.721. [6] S. Cheng, Stochastic population systems, Stoch. Anal. Appl., 27 (2009), 854-874. doi: 10.1080/07362990902844348. [7] T. G. Hallam and Z. Ma, Persistence in population models with demographic fluctuations, J. Math. Biol., 24 (1986), 327-339. doi: 10.1007/BF00275641. [8] M. W. Hirsch, Systems of differential equations which are competitive or cooperative Ⅰ: Limit sets, SIAM J. Math. Anal., 13 (1982), 167-179. doi: 10.1137/0513013. [9] M. W. Hirsch, Systems of differential equations that are competitive or cooperative Ⅱ: Convergence almost everywhere, SIAM J. Math. Anal., 16 (1985), 423-439. doi: 10.1137/0516030. [10] M. W. Hirsch, Systems of differential equations which are competitive or cooperative Ⅲ: Competing species, Nonlinearity, 1 (1988), 51-71. doi: 10.1088/0951-7715/1/1/003. [11] M. W. Hirsch, System of differential equations that are competitive or cooperative. Ⅳ: Structural stability in three-dimensional systems, SIAM J. Math. Anal., 21 (1990), 1225-1234. doi: 10.1137/0521067. [12] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus: 2nd Edition, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-1-4612-0949-2. [13] X. Li, A. Gray, D. Jiang and X. Mao, Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching, J. Math. Anal. Appl., 376 (2011), 11-28. doi: 10.1016/j.jmaa.2010.10.053. [14] X. Li, D. Jiang and X. Mao, Population dynamical behavior of Lotka-Volterra system under regime switching, J. Comput. Appl. Math., 232 (2009), 427-448. doi: 10.1016/j.cam.2009.06.021. [15] X. Li and X. Mao, Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation, Discrete. Contin. Dyn. Syst., 24 (2009), 523-545. doi: 10.3934/dcds.2009.24.523. [16] X. Liang and J. Jiang, On the finite dimensional dynamical systems with limited competition, Trans. Am. Math. Soc., 354 (2002), 3535-3554. doi: 10.1090/S0002-9947-02-03032-5. [17] X. Liang and J. Jiang, The dynamical behaviour of type-K competitive Kolmogorov systems and its application to three-dimensional type-K competitive Lotka-Volterra systems, Nonlinearity, 16 (2003), 785-801. doi: 10.1088/0951-7715/16/3/301. [18] M. Liu and K. Wang, Persistence and extinction in stochastic non-autonomous logistic systems, J. Math. Anal. Appl., 375 (2011), 443-457. doi: 10.1016/j.jmaa.2010.09.058. [19] M. Liu and K. Wang, Population dynamical behavior of Lotka-Volterra cooperative systems with random perturbations, Discrete Contin. Dyn. Syst., 33 (2013), 2495-2522. doi: 10.3934/dcds.2013.33.2495. [20] M. Liu, K. 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. doi: 10.1007/s11538-010-9569-5. [21] X. Mao, Exponential Stability of Stochastic Differential Equations, Marcel Dekker, New York, 1994. [22] X. Mao and C. Yuan, Stochastical Differential Equations with Markovian Switching, Imperial College Press, London, 2006. doi: 10.1142/p473. [23] S. Smale, On the differential equations of species in competition, J. Math. Biol., 3 (1976), 5-7. doi: 10.1007/BF00307854. [24] H. L. Smith, Competing subcommunities of mutualists and a generalized Kamke theorem, SIAM J. Appl. Math., 46 (1986), 856-874. doi: 10.1137/0146052. [25] H. L. Smith and H. R. Thieme, Stable coexistence and bi-stability for competitive systems on ordered Banach spaces, J. Diff. Eqns., 176 (2001), 195-222. doi: 10.1006/jdeq.2001.3981. [26] P. Tak$\acute{a}\check{c}$, Convergence to equilibrium on invariant $d$-hypersurfaces for strongly increasing discrete-time semigroups, J. Math. Anal. Appl., 148 (1990), 223-244. doi: 10.1016/0022-247X(90)90040-M. [27] P. Tak$\acute{a}\check{c}$, Domains of attraction of generic omega-limit sets for strongly monotone discrete-time semigroups, J. Reine. Angew. Math., 423 (1992), 101-173. doi: 10.1515/crll.1992.423.101. [28] Y. Takeuchi and N. Adachi, The existence of globally stable equilibria of ecosystems of the generalized Volterra type, J. Math. Biol., 10 (1980), 401-415. doi: 10.1007/BF00276098. [29] Y. Takeuchi, N. Adachi and H. Tokumaru, Global stability of ecosystems of the generalized Volterra type, Math. Biosci., 42 (1978), 119-136. doi: 10.1016/0025-5564(78)90010-X. [30] C. C. Travis and W. M. Post, Dynamics and comparative statics of mutualistic communities, J. Theor. Biol., 78 (1979), 553-571. doi: 10.1016/0022-5193(79)90190-5. [31] C. Tu and J. Jiang, The coexistence of a community of species with limited competition, J. Math. Anal. Appl., 217 (1998), 233-245. doi: 10.1006/jmaa.1997.5711. [32] C. Tu and J. Jiang, The necessary and sufficient conditions for the global stability of type-K Lotka-Volterra system, Proc. Am. Math. Soc., 127 (1999), 3181-3186. doi: 10.1090/S0002-9939-99-05077-7. [33] C. Tu and J. Jiang, Global stability and permanence for a class of type-K monotone systems, SIAM J.Math. Anal., 30 (1999), 360-378. doi: 10.1137/S0036141097325290. [34] Y. Wang and J. Jiang, The long-run behavior of periodic competitive Kolmogorov systems, Nonlinear Anal.: Real World Appl., 3 (2002), 471-485. doi: 10.1016/S1468-1218(01)00034-7. [35] Y. Wang and J. Jiang, Uniqueness and attractivity of the carrying simplices for the discrete-time competitive dynamical systems, J. Differential Equations, 186 (2002), 611-632. doi: 10.1016/S0022-0396(02)00025-6. [36] F. Wu and Y. Hu, Stochastic Lotka-Volterra system with unbounded distributed delay, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 275-288. doi: 10.3934/dcdsb.2010.14.275. [37] M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dyn. Stability Systems, 8 (1993), 189-217. doi: 10.1080/02681119308806158.
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 $\sigma_1 = 0.35,~\sigma_2 = 0.4359,~\sigma_3 = 0.2646$.
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 $t$.
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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