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Coexistence and extinction for stochastic Kolmogorov systems. (English) Zbl 1410.60094
Summary: In recent years there has been a growing interest in the study of the dynamics of stochastic populations. A key question in population biology is to understand the conditions under which populations coexist or go extinct. Theoretical and empirical studies have shown that coexistence can be facilitated or negated by both biotic interactions and environmental fluctuations. We study the dynamics of \(n\) populations that live in a stochastic environment and which can interact nonlinearly (through competition for resources, predator-prey behavior, etc.). Our models are described by \(n\)-dimensional Kolmogorov systems with white noise (stochastic differential equations-SDE). We give sharp conditions under which the populations converge exponentially fast to their unique stationary distribution as well as conditions under which some populations go extinct exponentially fast.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
37H15 Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J05 Discrete-time Markov processes on general state spaces
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References:
[1] Benaïm, M. and Schreiber, S. J. (2009). Persistence of structured populations in random environments. Theor. Popul. Biol.76 19-34. · Zbl 1213.92057
[2] Benaïm, M., Hofbauer, J. and Sandholm, W. H. (2008). Robust permanence and impermanence for stochastic replicator dynamics. J. Biol. Dyn.2 180-195. · Zbl 1140.92025
[3] Benaïm, M. and Lobry, C. (2016). Lotka-Volterra with randomly fluctuating environments or “How switching between beneficial environments can make survival harder.” Ann. Appl. Probab.26 3754-3785. · Zbl 1358.92075
[4] Blath, J., Etheridge, A. and Meredith, M. (2007). Coexistence in locally regulated competing populations and survival of branching annihilating random walk. Ann. Appl. Probab.17 1474-1507. · Zbl 1145.92032
[5] Braumann, C. A. (2002). Variable effort harvesting models in random environments: Generalization to density-dependent noise intensities. Math. Biosci.177 229-245. · Zbl 1003.92027
[6] Caswell, H. (2001). Matrix Population Models. Wiley, New York.
[7] Cattiaux, P. and Méléard, S. (2010). Competitive or weak cooperative stochastic Lotka-Volterra systems conditioned on non-extinction. J. Math. Biol.60 797-829. · Zbl 1202.92082
[8] Cattiaux, P., Collet, P., Lambert, A., Martínez, S., Méléard, S. and San Martín, J. (2009). Quasi-stationary distributions and diffusion models in population dynamics. Ann. Probab.37 1926-1969. · Zbl 1176.92041
[9] Chen, Z. and Kulperger, R. (2005). A stochastic competing-species model and ergodicity. J. Appl. Probab.42 738-753. · Zbl 1085.60030
[10] Chesson, P. (2000). General theory of competitive coexistence in spatially-varying environments. Theor. Popul. Biol.58 211-237. · Zbl 1035.92042
[11] Chesson, P. L. and Ellner, S. (1989). Invasibility and stochastic boundedness in monotonic competition models. J. Math. Biol.27 117-138. · Zbl 0717.92024
[12] Cross, P. C., Lloyd-Smith, J. O., Johnson, P. L. F. and Getz, W. M. (2005). Duelling timescales of host movement and disease recovery determine invasion of disease in structured populations. Ecol. Lett.8 587-595.
[13] Davies, K. F., Chesson, P., Harrison, S., Inouye, B. D., Melbourne, B. and Rice, K. J. (2005). Spatial heterogeneity explains the scale dependence of the native-exotic diversity relationship. Ecology86 1602-1610.
[14] Ethier, S. N. and Kurtz, T. G. (2009). Markov Processes: Characterization and Convergence282. Wiley, New York.
[15] Evans, S. N., Hening, A. and Schreiber, S. J. (2015). Protected polymorphisms and evolutionary stability of patch-selection strategies in stochastic environments. J. Math. Biol.71 325-359. · Zbl 1322.92057
[16] Evans, S. N., Ralph, P. L., Schreiber, S. J. and Sen, A. (2013). Stochastic population growth in spatially heterogeneous environments. J. Math. Biol.66 423-476. · Zbl 1402.92341
[17] Friedman, A. (2008). Partial Differential Equations of Parabolic Type. Dover Publications, Mineola, NY.
[18] Gard, T. C. (1988). Introduction to Stochastic Differential Equations. Monographs and Textbooks in Pure and Applied Mathematics114. Dekker, New York. · Zbl 0628.60064
[19] Hening, A. and Nguyen, D. (2017a). Stochastic Lotka-Volterra food chains. J. Math. Biol. To appear. · Zbl 1392.92075
[20] Hening, A., Nguyen, D. H. and Yin, G. (2018). Stochastic population growth in spatially heterogeneous environments: The density-dependent case. J. Math. Biol. To appear. · Zbl 1392.92076
[21] Hofbauer, J. (1981). A general cooperation theorem for hypercycles. Monatsh. Math.91 233-240. · Zbl 0449.34039
[22] Hofbauer, J. and So, J. W.-H. (1989). Uniform persistence and repellors for maps. Proc. Amer. Math. Soc.107 1137-1142. · Zbl 0678.58024
[23] Hutson, V. (1984). A theorem on average Liapunov functions. Monatsh. Math.98 267-275. · Zbl 0542.34043
[24] Khasminskii, R. Z. (1960). Ergodic properties of recurrent diffusion processes and stabilization of the solution to the Cauchy problem for parabolic equations. Theory Probab. Appl.5 179-196.
[25] Khasminskii, R. (2012). Stochastic Stability of Differential Equations, 2nd ed. Stochastic Modelling and Applied Probability66. Springer, Heidelberg. · Zbl 1259.60058
[26] Lande, R., Engen, S. and Saether, B.-E. (2003). Stochastic Population Dynamics in Ecology and Conservation. Oxford Univ. Press, Oxford. · Zbl 1087.92064
[27] Law, R. and Morton, R. D. (1996). Permanence and the assembly of ecological communities. Ecology77 762-775.
[28] Liu, M. and Bai, C. (2016). Analysis of a stochastic tri-trophic food-chain model with harvesting. J. Math. Biol.73 597-625. · Zbl 1347.92067
[29] Mao, X. (1997). Stochastic Differential Equations and Their Applications. Horwood Publishing Limited, Chichester. · Zbl 0892.60057
[30] Meyn, S. P. and Tweedie, R. L. (1992). Stability of Markovian processes. I. Criteria for discrete-time chains. Adv. in Appl. Probab.24 542-574. · Zbl 0757.60061
[31] Nummelin, E. (1984). General Irreducible Markov Chains and Nonnegative Operators. Cambridge Tracts in Mathematics83. Cambridge Univ. Press, Cambridge. · Zbl 0551.60066
[32] Pyšek, P. and Hulme, P. E. (2005). Spatio-temporal dynamics of plant invasions: Linking pattern to process. Ecoscience12 302-315.
[33] Rudnicki, R. (2003). Long-time behaviour of a stochastic prey-predator model. Stochastic Process. Appl.108 93-107. · Zbl 1075.60539
[34] Rudnicki, R. and Pichór, K. (2007). Influence of stochastic perturbation on prey – predator systems. Math. Biosci.206 108-119. · Zbl 1124.92055
[35] Schreiber, S. J., Benaïm, M. and Atchadé, K. A. S. (2011). Persistence in fluctuating environments. J. Math. Biol.62 655-683. · Zbl 1232.92075
[36] Schreiber, S. J. and Lloyd-Smith, J. O. (2009). Invasion dynamics in spatially heterogeneous environments. Amer. Nat.174 490-505.
[37] Turelli, M. (1977). Random environments and stochastic calculus. Theor. Popul. Biol.12 140-178. · Zbl 0444.92013
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