# American Institute of Mathematical Sciences

September  2021, 41(9): 4145-4183. doi: 10.3934/dcds.2021032

## Coexistence and exclusion of competitive Kolmogorov systems with semi-Markovian switching

 1 School of Mathematical Sciences, Anhui University, Hefei, 230601, China 2 School of Mathematical Science, Huaiyin Normal University, Huai'an, 223300, China 3 Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China

* Corresponding author: Hui Wan

Received  June 2020 Revised  November 2020 Published  September 2021 Early access  February 2021

This work investigates the dynamics of competitive Kolmogorov systems formulated in a semi-Markov regime-switching framework. The conditional holding time of each environmental regime is allowed to follow arbitrary probability distribution on the nonnegative half-line in the sense of approximations. Sharp sufficient conditions of the coexistence and competitive exclusion of species are established, and in the case of species coexistence, the convergence rate of the transition probability to the unique stationary measure is estimated. In weaker conditions, these results extend the existing results to the semi-Markov regime-switching environment. Particularly, the method of proving the exponential convergence of the transition probability to the invariant measure for the population models formulated as random differential equations driven by a semi-Markov process is proposed.

Citation: Dan Li, Hui Wan. Coexistence and exclusion of competitive Kolmogorov systems with semi-Markovian switching. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4145-4183. doi: 10.3934/dcds.2021032
##### References:
 [1] N. Bacaër and M. Khaladi, On the basic reproduction number in a random environment, J. Math. Biol., 67 (2013), 1729-1739.  doi: 10.1007/s00285-012-0611-0. [2] N. Bacaër and A. Ed-Darraz, On linear birth-and-death processes in a random environment, J. Math. Biol., 69 (2014), 73-90.  doi: 10.1007/s00285-013-0696-0. [3] Y. Bakhtin and T. Hurth, Invariant densities for dynamical systems with random switching, Nonlinearity, 25 (2012), 2937-2952.  doi: 10.1088/0951-7715/25/10/2937. [4] J. Bao and J. Shao, Permanence and extinction of regime-switching predator-prey models, SIAM J. Math. Anal., 48 (2016), 725-739.  doi: 10.1137/15M1024512. [5] M. Benaïm and C. Lobry, Lotka-Volterra with randomly fluctuating environments or "how switching between beneficial environments can make survival harder", Ann. Appl. Probab., 26 (2016), 3754-3785.  doi: 10.1214/16-AAP1192. [6] M. Benaïm, S. L. Borgne, F. Malrieu and P. A. Zitt, Qualitative properties of certain piecewise deterministic Markov processes, Annales de I'Institut Henri Poincare-Probabilites et Statistiques, 51 (2015), 1040-1075.  doi: 10.1214/14-AIHP619. [7] M. Benaïm, Stochastic persistence, preprint, arXiv: 1806.08450. [8] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic, New York, 1979. [9] N. Dexter, Stochastic models of foot and mouth disease in feral pigs in the Australian semi-arid rangelands, J. Appl. Ecol., 40 (2003), 293-306.  doi: 10.1046/j.1365-2664.2003.00792.x. [10] N. H. Dang, N. H. Du and G. Yin, Existence of stationary distributions for Kolmogorov systems of competitive type under telegraph noise, J. Differential Equations, 257 (2014), 2078-2101.  doi: 10.1016/j.jde.2014.05.029. [11] S. N. Evans, A. Hening and S. J. Schreiber, Protected polymorphisms and evolutionary stability of patch-selection strategies in stochastic environments, J. Math. Biol., 71 (2015), 325-359.  doi: 10.1007/s00285-014-0824-5. [12] M. Hairer and J. C. Mattingly, Yet another look at Harris' ergodic theorem for Markov chains, In Seminar on Stochastic Analysis, Random Fields and Applications VI, Progress in Probability, Springer, 63 (2011), 109–117. doi: 10.1007/978-3-0348-0021-1_7. [13] Q. He, Fundamentals of Matrix-Analytic Methods, Springer, New York, 2014. doi: 10.1007/978-1-4614-7330-5. [14] A. Hening and D. H. Nguyen, Coexistence and extinction for stochastic Kolmogorov systems, Ann. Appl. Probab., 28 (2018), 1893-1942.  doi: 10.1214/17-AAP1347. [15] Z. Hou, J. A. Filar and A. Chen, Markov Processes and Controlled Markov Chains, Kluwer Academic Publishers, 2002. doi: 10.1007/978-1-4613-0265-0. [16] Z. Hou, J. Luo, P. Shi and S. K. Nguang, Stochastic stability of Ito differential equations with semi-Markovian jump parameters, IEEE Trans. Autom. Control, 51 (2006), 1383-1387.  doi: 10.1109/TAC.2006.878746. [17] L. Hu, M. Tang, Z. Wu, Z. Xi and J. Yu, The threshold infection level for Wolbachia invasion in random environments, J. Differential Equations, 266 (2019), 4377-4393.  doi: 10.1016/j.jde.2018.09.035. [18] V. Jurdjevic, Geometric Control Theory, Cambridge Stud. Adv. Math., vol.52, Cambridge University Press, 1997. [19] D. Li, S. Liu and J. Cui, Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching, J. Differential Equations, 263 (2017), 8873-8915.  doi: 10.1016/j.jde.2017.08.066. [20] D. Li, S. Liu and J. Cui, Threshold dynamics and ergodicity of an SIRS epidemic model with semi-Markov switching, J. Differential Equations, 266 (2019), 3973-4017.  doi: 10.1016/j.jde.2018.09.026. [21] M. Liu and K. Wang, Persistence and extinction of a stochastic single-specie model under regime switching in a polluted environment, J. Theor. Biol., 264 (2010), 934-944.  doi: 10.1016/j.jtbi.2010.03.008. [22] Q. Liu, D. Jiang, T. Hayat and A. Alsaedi, Dynamical behavior of a multigroup SIRS epidemic model with standard incidence rates and Markovian switching, Discrete Cont. Dyn. Syst., 39 (2019), 5683-5706.  doi: 10.3934/dcds.2019249. [23] S. P. Meyn and R. L. Tweedie, Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes, Adv. Appl. Prob., 25 (1993), 518-548.  doi: 10.2307/1427522. [24] J. D. Murray, Mathematical Biology I: An Introduction, Springer, Berlin, 2002. [25] P. I. Ndiaye, D. J. Bicout, B. Mondet and P. Sabatier, Rainfall triggered dynamics of Aedes mosquito aggressiveness, J. Theor. Biol., 243 (2006), 222-229.  doi: 10.1016/j.jtbi.2006.06.005. [26] D. H. Nguyen and G. Yin, Coexistence and exclusion of stochastic competitive Lotka-Volterra models, J. Differential Equations, 262 (2017), 1192-1225.  doi: 10.1016/j.jde.2016.10.005. [27] R. Rudnicki, K. Pichór and M. Tyran-Kamińska, Markov semigroups and their applications, In: P. Garbaczewski, R. Olkiewicz (Eds.), Lecture Notes in Physics: Dynamics of Dissipation, Springer-Verlag, Berlin, 2002,215–238. [28] M. Sankaran, N. P. Hanan and R. J. Scholes, Determinants of woody cover in African savannas, Nature, 438 (2005), 846-849.  doi: 10.1038/nature04070. [29] C. Serra, M. D. Martínez, X. Lana and A. Burgueño, European dry spell length distributions, years 1951–2000, Theor. Appl. Climatol., 114 (2013), 531-551.  doi: 10.1007/s00704-013-0857-5. [30] M. J. Small and D. J. Morgan, The relationship between a continuous-time renewal model and a discrete Markov chain model of precipitation occurrence, Water Resources Research, 22 (1986), 1422-1430.  doi: 10.1029/WR022i010p01422. [31] L. Stettner, On the Existence and Uniqueness of Invariant Measure for Continuous Time Markov Processes, LCDS Report No. 86-18, Brown University, Providence, April 1986. [32] A. D. Synodinos, B. Tietjen, D. Lohmann and F. Jeltsch, The impact of inter-annual rainfall variability on African savannas changes with mean rainfall, J. Theor. Biol., 437 (2018), 92-100.  doi: 10.1016/j.jtbi.2017.10.019. [33] M. Turelli, Does environmental variability limit niche overlap?, Proc. Natl. Acad. Sci. USA, 75 (1978), 5085-5089.  doi: 10.1073/pnas.75.10.5085. [34] B. Wang and Q. Zhu, Asymptotic stability in distribution of stochastic systems with semi-Markovian switching, Int. J. Control, 92 (2019), 1314-1324.  doi: 10.1080/00207179.2017.1392042. [35] H. Yang and X. Li, Explicit approximations for nonlinear switching diffusion systems in finite and infinite horizons, J. Differential Equations, 265 (2018), 2921-2967.  doi: 10.1016/j.jde.2018.04.052.

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##### References:
 [1] N. Bacaër and M. Khaladi, On the basic reproduction number in a random environment, J. Math. Biol., 67 (2013), 1729-1739.  doi: 10.1007/s00285-012-0611-0. [2] N. Bacaër and A. Ed-Darraz, On linear birth-and-death processes in a random environment, J. Math. Biol., 69 (2014), 73-90.  doi: 10.1007/s00285-013-0696-0. [3] Y. Bakhtin and T. Hurth, Invariant densities for dynamical systems with random switching, Nonlinearity, 25 (2012), 2937-2952.  doi: 10.1088/0951-7715/25/10/2937. [4] J. Bao and J. Shao, Permanence and extinction of regime-switching predator-prey models, SIAM J. Math. Anal., 48 (2016), 725-739.  doi: 10.1137/15M1024512. [5] M. Benaïm and C. Lobry, Lotka-Volterra with randomly fluctuating environments or "how switching between beneficial environments can make survival harder", Ann. Appl. Probab., 26 (2016), 3754-3785.  doi: 10.1214/16-AAP1192. [6] M. Benaïm, S. L. Borgne, F. Malrieu and P. A. Zitt, Qualitative properties of certain piecewise deterministic Markov processes, Annales de I'Institut Henri Poincare-Probabilites et Statistiques, 51 (2015), 1040-1075.  doi: 10.1214/14-AIHP619. [7] M. Benaïm, Stochastic persistence, preprint, arXiv: 1806.08450. [8] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic, New York, 1979. [9] N. Dexter, Stochastic models of foot and mouth disease in feral pigs in the Australian semi-arid rangelands, J. Appl. Ecol., 40 (2003), 293-306.  doi: 10.1046/j.1365-2664.2003.00792.x. [10] N. H. Dang, N. H. Du and G. Yin, Existence of stationary distributions for Kolmogorov systems of competitive type under telegraph noise, J. Differential Equations, 257 (2014), 2078-2101.  doi: 10.1016/j.jde.2014.05.029. [11] S. N. Evans, A. Hening and S. J. Schreiber, Protected polymorphisms and evolutionary stability of patch-selection strategies in stochastic environments, J. Math. Biol., 71 (2015), 325-359.  doi: 10.1007/s00285-014-0824-5. [12] M. Hairer and J. C. Mattingly, Yet another look at Harris' ergodic theorem for Markov chains, In Seminar on Stochastic Analysis, Random Fields and Applications VI, Progress in Probability, Springer, 63 (2011), 109–117. doi: 10.1007/978-3-0348-0021-1_7. [13] Q. He, Fundamentals of Matrix-Analytic Methods, Springer, New York, 2014. doi: 10.1007/978-1-4614-7330-5. [14] A. Hening and D. H. Nguyen, Coexistence and extinction for stochastic Kolmogorov systems, Ann. Appl. Probab., 28 (2018), 1893-1942.  doi: 10.1214/17-AAP1347. [15] Z. Hou, J. A. Filar and A. Chen, Markov Processes and Controlled Markov Chains, Kluwer Academic Publishers, 2002. doi: 10.1007/978-1-4613-0265-0. [16] Z. Hou, J. Luo, P. Shi and S. K. Nguang, Stochastic stability of Ito differential equations with semi-Markovian jump parameters, IEEE Trans. Autom. Control, 51 (2006), 1383-1387.  doi: 10.1109/TAC.2006.878746. [17] L. Hu, M. Tang, Z. Wu, Z. Xi and J. Yu, The threshold infection level for Wolbachia invasion in random environments, J. Differential Equations, 266 (2019), 4377-4393.  doi: 10.1016/j.jde.2018.09.035. [18] V. Jurdjevic, Geometric Control Theory, Cambridge Stud. Adv. Math., vol.52, Cambridge University Press, 1997. [19] D. Li, S. Liu and J. Cui, Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching, J. Differential Equations, 263 (2017), 8873-8915.  doi: 10.1016/j.jde.2017.08.066. [20] D. Li, S. Liu and J. Cui, Threshold dynamics and ergodicity of an SIRS epidemic model with semi-Markov switching, J. Differential Equations, 266 (2019), 3973-4017.  doi: 10.1016/j.jde.2018.09.026. [21] M. Liu and K. Wang, Persistence and extinction of a stochastic single-specie model under regime switching in a polluted environment, J. Theor. Biol., 264 (2010), 934-944.  doi: 10.1016/j.jtbi.2010.03.008. [22] Q. Liu, D. Jiang, T. Hayat and A. Alsaedi, Dynamical behavior of a multigroup SIRS epidemic model with standard incidence rates and Markovian switching, Discrete Cont. Dyn. Syst., 39 (2019), 5683-5706.  doi: 10.3934/dcds.2019249. [23] S. P. Meyn and R. L. Tweedie, Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes, Adv. Appl. Prob., 25 (1993), 518-548.  doi: 10.2307/1427522. [24] J. D. Murray, Mathematical Biology I: An Introduction, Springer, Berlin, 2002. [25] P. I. Ndiaye, D. J. Bicout, B. Mondet and P. Sabatier, Rainfall triggered dynamics of Aedes mosquito aggressiveness, J. Theor. Biol., 243 (2006), 222-229.  doi: 10.1016/j.jtbi.2006.06.005. [26] D. H. Nguyen and G. Yin, Coexistence and exclusion of stochastic competitive Lotka-Volterra models, J. Differential Equations, 262 (2017), 1192-1225.  doi: 10.1016/j.jde.2016.10.005. [27] R. Rudnicki, K. Pichór and M. Tyran-Kamińska, Markov semigroups and their applications, In: P. Garbaczewski, R. Olkiewicz (Eds.), Lecture Notes in Physics: Dynamics of Dissipation, Springer-Verlag, Berlin, 2002,215–238. [28] M. Sankaran, N. P. Hanan and R. J. Scholes, Determinants of woody cover in African savannas, Nature, 438 (2005), 846-849.  doi: 10.1038/nature04070. [29] C. Serra, M. D. Martínez, X. Lana and A. Burgueño, European dry spell length distributions, years 1951–2000, Theor. Appl. Climatol., 114 (2013), 531-551.  doi: 10.1007/s00704-013-0857-5. [30] M. J. Small and D. J. Morgan, The relationship between a continuous-time renewal model and a discrete Markov chain model of precipitation occurrence, Water Resources Research, 22 (1986), 1422-1430.  doi: 10.1029/WR022i010p01422. [31] L. Stettner, On the Existence and Uniqueness of Invariant Measure for Continuous Time Markov Processes, LCDS Report No. 86-18, Brown University, Providence, April 1986. [32] A. D. Synodinos, B. Tietjen, D. Lohmann and F. Jeltsch, The impact of inter-annual rainfall variability on African savannas changes with mean rainfall, J. Theor. Biol., 437 (2018), 92-100.  doi: 10.1016/j.jtbi.2017.10.019. [33] M. Turelli, Does environmental variability limit niche overlap?, Proc. Natl. Acad. Sci. USA, 75 (1978), 5085-5089.  doi: 10.1073/pnas.75.10.5085. [34] B. Wang and Q. Zhu, Asymptotic stability in distribution of stochastic systems with semi-Markovian switching, Int. J. Control, 92 (2019), 1314-1324.  doi: 10.1080/00207179.2017.1392042. [35] H. Yang and X. Li, Explicit approximations for nonlinear switching diffusion systems in finite and infinite horizons, J. Differential Equations, 265 (2018), 2921-2967.  doi: 10.1016/j.jde.2018.04.052.
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