Hybrid competitive Lotka–Volterra ecosystems: countable switching states and two-time-scale models

References

• Lotka, A. J. (1925). Elements of Physical Biology. Baltimore: William and Wilkins.
   Google Scholar
• Volterra, V. (1926). Variazioni e fluttuazioni del numero d’individui in specie d’animali conviventi. Mem. Acad. Lincei. 2:31–113.
   Google Scholar
• Dieu, N. T., Du, N. H., Nguyen, D. H., Yin, G. (2016). Protection zones for survival of species in random environment. SIAM J. Appl. Math. 76(4):1382–1402. DOI:10.1137/15M1032004.
   Web of Science ®Google Scholar
• Hening, A., Nguyen, D., Yin, G. (2018). Stochastic population growth in spatially heterogeneous environments: The density-dependent case. J. Math. Biol. 76(3):697–754. DOI:10.1007/s00285-017-1153-2.
   PubMed Web of Science ®Google Scholar
• Nguyen, D. H., Yin, G. (2017). Coexistence and exclusion of stochastic competitive Lotka–Volterra models. J. Differ. Equations 262(3):1192–1225. DOI:10.1016/j.jde.2016.10.005.
   Web of Science ®Google Scholar
• Yin, G., Zhu, C. (2010). Hybrid Switching Diffusions: Properties and Applications. New York: Springer
   Google Scholar
• Li, X. Z., Tang, C. L., Ji, X. H. (1999). The criteria for globally stable equilibrium in n-dimensional Lotka–Volterra systems. J. Math. Anal. Appl. 240(2):600–606. DOI:10.1006/jmaa.1999.6612.
   Web of Science ®Google Scholar
• Xiao, D., Li, W. (2000). Limit cycles for the competitive Lotka–Volterra systems. J. Differ. Equations 164(1):1–15. DOI:10.1006/jdeq.1999.3729.
   Web of Science ®Google Scholar
• Arnold, L., Horsthemeke, W., Stucki, J. W. (1979). The influence of external real and white noise on the lotka volterra model. Biomed. J. 21:451–471. DOI:10.1002/bimj.4710210507.
   Google Scholar
• Khasminskii, R. S., Klebaner, F. C. (2001). Long term behavior of solutions of the Lotka–Volterra systems under small perturbations. Ann. Appl. Probab. 11:952–963. DOI:10.1214/aoap/1015345354.
   Web of Science ®Google Scholar
• Imhof, L. A. (2005). The long-run behavior of stochastic replicator dynamics. Ann. Appl. Probab. 15(1B):1019–1045. DOI:10.1214/105051604000000837.
   Web of Science ®Google Scholar
• Khasminskii, R. Z., Potsepun, N. (2006). On the replicator dynamics behavior under Stratonovich type random perturbations. Stoch. Dyn. 6(2):197–211. DOI:10.1142/S0219493706001712.
   Web of Science ®Google Scholar
• Du, N. H., Kon, R., Sato, K., Takeuchi, Y. (2004). Dynamical behavior of Lotka–Volterra competition systems: Non-automous bistable case and the effect of telegraph noise. J. Comp. Appl. Math. 170(2):399–422. DOI:10.1016/j.cam.2004.02.001.
   Web of Science ®Google Scholar
• Luo, Q., Mao, X. (2007). Stochastic population dynamics under regime switching. J. Math. Anal. Appl. 334(1):69–84. DOI:10.1016/j.jmaa.2006.12.032.
   Web of Science ®Google Scholar
• Tran, K., Yin, G. (2014). Stochastic competitive Lotka–Volterra ecosystems under partial observation: Feedback controls for permanence and extinction. J. Franklin Inst. 351(8):4039–4064. DOI:10.1016/j.jfranklin.2014.04.015.
   Web of Science ®Google Scholar
• Zhu, C., Yin, G. (2009). On hybrid competitive Lotka–Volterra ecosystems. Nonlinear Analysis 71(12):e1370–e1379. DOI:10.1016/j.na.2009.01.166.
   Web of Science ®Google Scholar
• Zhu, C., Yin, G. (2009). On competitive Lotka–Volterra model in random environments. J. Math. Anal. Appl. 357(1):154–170. DOI:10.1016/j.jmaa.2009.03.066.
   Web of Science ®Google Scholar
• Blankenship, G. B., Papanicolaou, G. C. (1978). Stability and control of stochastic systems with wide band noise. SIAM J. Appl. Math. 34(3):437–476. DOI:10.1137/0134036.
   Web of Science ®Google Scholar
• Kushner, H. J. (1984). Approximation and Weak Convergence Methods for Random Processes, with Applications to Stochastic System Theory. Cambridge, MA: MIT Press
   Google Scholar
• Yin, G., Zhang, Q. (2013). Continuous-Time Markov Chains and Applications: A Singular Perturbation Approach. New York: Springer
   Google Scholar
• Nguyen, D. H., Yin, G. (2016). Modeling and analysis of switching diffusion systems: Past dependent switching with a countable state space. SIAM J. Control Optim. 54(5):2450–2477. DOI:10.1137/16M1059357.
   Web of Science ®Google Scholar
• Anderson, W. J. (2012). Continuous-time Markov Chains: An Applications-Oriented Approach. Springer
   Google Scholar
• Mao, X., Marion, G., Renshaw, E. (2002). Environmental brownian noise suppresses explosions in population dynamics. Stochastic Process. Appl. 97(1):95–110. DOI:10.1016/S0304-4149(01)00126-0.
   Web of Science ®Google Scholar
• Badowski, G., Yin, G. (2002). Stability of hybrid dynamic systems containing singularly perturbed random processes. IEEE Trans. Autom. Control 47(12):2021–2032. DOI:10.1109/TAC.2002.805682.
   Web of Science ®Google Scholar
• Mao, X., Sabanis, S., Renshaw, R. (2003). Asymptotic behavior of the stochastic Lotka–Volterra model. J. Math. Anal. Appl. 287(1):141–156. DOI:10.1016/S0022-247X(03)00539-0.
   Web of Science ®Google Scholar
• Khasminskii, R. Z. (1980). Stochastic Stability of Differential Equations. Alphen aan den Rijn, Netherlands: Sijthoff and Noordhoff.
   Google Scholar
• Mao, X. (1997). Stochastic Differential Equations and Applications. Horwood Publishing,
   Google Scholar
• Yin, G., Zhang, H. Q. (2004). Two-time-scale Markov chains and applications to quasi-birth-death queues. SIAM J. Appl. Math. 65(2):567–586. DOI:10.1137/S003613990139756X.
   Web of Science ®Google Scholar
• Tran, K., Yin, G. (2015). Optimal harvesting strategies for stochastic competitive Lotka–Volterra ecosystems. Automatica 55:236–246. DOI:10.1016/j.automatica.2015.03.017.
   Web of Science ®Google Scholar
• Mao, X., Yuan, C. (2006). Stochastic Differential Equations with Markovian Switching. London: Imperial College Press
   Google Scholar
• Applebaum, D. (2009). Lévy Processes and Stochastic Calculus. Cambridge, UK: Cambridge Univ. Press.
   Google Scholar
• Skorohod, A. V. (1989). Asymptotic Methods in the Theory of Stochastic Differential Equations. Providence, RI: Amer. Math. Soc.
   Google Scholar