## The Annals of Applied Probability

### Coexistence and extinction for stochastic Kolmogorov systems

#### Abstract

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.

The analysis is done by a careful study of the properties of the invariant measures of the process that are supported on the boundary of the domain. To our knowledge this is one of the first general results describing the asymptotic behavior of stochastic Kolmogorov systems in non-compact domains.

We are able to fully describe the properties of many of the SDE that appear in the literature. In particular, we extend results on two dimensional Lotka-Volterra models, two dimensional predator–prey models, $n$ dimensional simple food chains, and two predator and one prey models. We also show how one can use our methods to classify the dynamics of any two-dimensional stochastic Kolmogorov system satisfying some mild assumptions.

#### Article information

Source
Ann. Appl. Probab., Volume 28, Number 3 (2018), 1893-1942.

Dates
Revised: August 2017
First available in Project Euclid: 1 June 2018

https://projecteuclid.org/euclid.aoap/1527840035

Digital Object Identifier
doi:10.1214/17-AAP1347

Mathematical Reviews number (MathSciNet)
MR3809480

Zentralblatt MATH identifier
06919741

#### Citation

Hening, Alexandru; Nguyen, Dang H. Coexistence and extinction for stochastic Kolmogorov systems. Ann. Appl. Probab. 28 (2018), no. 3, 1893--1942. doi:10.1214/17-AAP1347. https://projecteuclid.org/euclid.aoap/1527840035

#### References

• Benaïm, M. (2014). Stochastic persistence. Preprint.
• Benaïm, M. and Schreiber, S. J. (2009). Persistence of structured populations in random environments. Theor. Popul. Biol. 76 19–34.
• Benaïm, M., Hofbauer, J. and Sandholm, W. H. (2008). Robust permanence and impermanence for stochastic replicator dynamics. J. Biol. Dyn. 2 180–195.
• 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.
• 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.
• Braumann, C. A. (2002). Variable effort harvesting models in random environments: Generalization to density-dependent noise intensities. Math. Biosci. 177 229–245.
• Caswell, H. (2001). Matrix Population Models. Wiley, New York.
• Cattiaux, P. and Méléard, S. (2010). Competitive or weak cooperative stochastic Lotka–Volterra systems conditioned on non-extinction. J. Math. Biol. 60 797–829.
• 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.
• Chen, Z. and Kulperger, R. (2005). A stochastic competing-species model and ergodicity. J. Appl. Probab. 42 738–753.
• Chesson, P. (2000). General theory of competitive coexistence in spatially-varying environments. Theor. Popul. Biol. 58 211–237.
• Chesson, P. L. and Ellner, S. (1989). Invasibility and stochastic boundedness in monotonic competition models. J. Math. Biol. 27 117–138.
• 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.
• 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. Ecology 86 1602–1610.
• Ethier, S. N. and Kurtz, T. G. (2009). Markov Processes: Characterization and Convergence 282. Wiley, New York.
• 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.
• 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.
• Friedman, A. (2008). Partial Differential Equations of Parabolic Type. Dover Publications, Mineola, NY.
• Gard, T. C. (1988). Introduction to Stochastic Differential Equations. Monographs and Textbooks in Pure and Applied Mathematics 114. Dekker, New York.
• Hening, A. and Nguyen, D. (2017a). Stochastic Lotka–Volterra food chains. J. Math. Biol. To appear.
• Hening, A. and Nguyen, D. (2017b). Persistence in stochastic Lotka–Volterra food chains with intraspecific competition. Preprint.
• 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.
• Hofbauer, J. (1981). A general cooperation theorem for hypercycles. Monatsh. Math. 91 233–240.
• Hofbauer, J. and So, J. W.-H. (1989). Uniform persistence and repellors for maps. Proc. Amer. Math. Soc. 107 1137–1142.
• Hutson, V. (1984). A theorem on average Liapunov functions. Monatsh. Math. 98 267–275.
• 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.
• Khasminskii, R. (2012). Stochastic Stability of Differential Equations, 2nd ed. Stochastic Modelling and Applied Probability 66. Springer, Heidelberg.
• Lande, R., Engen, S. and Saether, B.-E. (2003). Stochastic Population Dynamics in Ecology and Conservation. Oxford Univ. Press, Oxford.
• Law, R. and Morton, R. D. (1996). Permanence and the assembly of ecological communities. Ecology 77 762–775.
• Liu, M. and Bai, C. (2016). Analysis of a stochastic tri-trophic food-chain model with harvesting. J. Math. Biol. 73 597–625.
• Mao, X. (1997). Stochastic Differential Equations and Their Applications. Horwood Publishing Limited, Chichester.
• 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.
• Nummelin, E. (1984). General Irreducible Markov Chains and Nonnegative Operators. Cambridge Tracts in Mathematics 83. Cambridge Univ. Press, Cambridge.
• Pyšek, P. and Hulme, P. E. (2005). Spatio-temporal dynamics of plant invasions: Linking pattern to process. Ecoscience 12 302–315.
• Rudnicki, R. (2003). Long-time behaviour of a stochastic prey-predator model. Stochastic Process. Appl. 108 93–107.
• Rudnicki, R. and Pichór, K. (2007). Influence of stochastic perturbation on prey–predator systems. Math. Biosci. 206 108–119.
• Schreiber, S. J., Benaïm, M. and Atchadé, K. A. S. (2011). Persistence in fluctuating environments. J. Math. Biol. 62 655–683.
• Schreiber, S. J. and Lloyd-Smith, J. O. (2009). Invasion dynamics in spatially heterogeneous environments. Amer. Nat. 174 490–505.
• Turelli, M. (1977). Random environments and stochastic calculus. Theor. Popul. Biol. 12 140–178.