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doi: 10.3934/dcds.2021032

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

1. 

School of Mathematical Science, Huaiyin Normal University, Huai'an, 223300, China

2. 

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  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 & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2021032
References:
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Q. LiuD. JiangT. 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.  Google Scholar

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P. I. NdiayeD. J. BicoutB. 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.  Google Scholar

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[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. Google Scholar

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[29]

C. SerraM. D. MartínezX. 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.  Google Scholar

[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.  Google Scholar

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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. Google Scholar

[32]

A. D. SynodinosB. TietjenD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

M. BenaïmS. L. BorgneF. 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.  Google Scholar

[7]

M. Benaïm, Stochastic persistence, preprint, arXiv: 1806.08450. Google Scholar

[8]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic, New York, 1979.  Google Scholar

[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.  Google Scholar

[10]

N. H. DangN. 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.  Google Scholar

[11]

S. N. EvansA. 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.  Google Scholar

[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.  Google Scholar

[13]

Q. He, Fundamentals of Matrix-Analytic Methods, Springer, New York, 2014. doi: 10.1007/978-1-4614-7330-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

Z. HouJ. LuoP. 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.  Google Scholar

[17]

L. HuM. TangZ. WuZ. 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.  Google Scholar

[18]

V. Jurdjevic, Geometric Control Theory, Cambridge Stud. Adv. Math., vol.52, Cambridge University Press, 1997.  Google Scholar

[19]

D. LiS. 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.  Google Scholar

[20]

D. LiS. 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.  Google Scholar

[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.  Google Scholar

[22]

Q. LiuD. JiangT. 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.  Google Scholar

[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.  Google Scholar

[24]

J. D. Murray, Mathematical Biology I: An Introduction, Springer, Berlin, 2002.  Google Scholar

[25]

P. I. NdiayeD. J. BicoutB. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[28]

M. SankaranN. P. Hanan and R. J. Scholes, Determinants of woody cover in African savannas, Nature, 438 (2005), 846-849.  doi: 10.1038/nature04070.  Google Scholar

[29]

C. SerraM. D. MartínezX. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[32]

A. D. SynodinosB. TietjenD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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