Abstract
We are interested in the long time behavior of a two-type density-dependent biological population conditioned on non-extinction, in both cases of competition or weak cooperation between the two species. This population is described by a stochastic Lotka–Volterra system, obtained as limit of renormalized interacting birth and death processes. The weak cooperation assumption allows the system not to blow up. We study the existence and uniqueness of a quasi-stationary distribution, that is convergence to equilibrium conditioned on non-extinction. To this aim we generalize in two-dimensions spectral tools developed for one-dimensional generalized Feller diffusion processes. The existence proof of a quasi-stationary distribution is reduced to the one for a d-dimensional Kolmogorov diffusion process under a symmetry assumption. The symmetry we need is satisfied under a local balance condition relying the ecological rates. A novelty is the outlined relation between the uniqueness of the quasi-stationary distribution and the ultracontractivity of the killed semi-group. By a comparison between the killing rates for the populations of each type and the one of the global population, we show that the quasi-stationary distribution can be either supported by individuals of one (the strongest one) type or supported by individuals of the two types. We thus highlight two different long time behaviors depending on the parameters of the model: either the model exhibits an intermediary time scale for which only one type (the dominant trait) is surviving, or there is a positive probability to have coexistence of the two species.
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References
Berezin FA, Shubin MA (1991) The Schrödinger equation. Kluwer Academic Pub, Dordrecht
Cattiaux P (2005) Hypercontractivity for perturbed diffusion semi-groups. Ann Fac des Sc de Toulouse 14(4): 609–628
Cattiaux P, Collet P, Lambert A, Martinez S, Méléard S, San Martin J (2009) Quasi-stationarity distributions and diffusion models in population dynamics. To appear in Ann Prob
Collet P, Martinez S, San Martin J (1995) Asymptotic laws for one dimensional diffusions conditioned on nonabsorption. Ann Prob 23: 1300–1314
Davies EB (1989) Heat kernels and spectral theory. Cambridge University Press
Etheridge AM (2004) Survival and extinction in a locally regulated population. Ann Appl Prob 14: 188–214
Ferrari PA, Kesten H, Martínez S, Picco P (1995) Existence of quasi-stationary distributions. Renew Dyn Approach Ann Probab 23: 501–521
Fukushima M (1980) Dirichlet forms and Markov processes. Kodansha. North-Holland, Amsterdam
Gosselin F (2001) Asymptotic behavior of absorbing Markov chains conditional on nonabsorption for applications in conservation biology. Ann Appl Prob 11: 261–284
Ikeda N, Watanabe S (1988) Stochastic differential equations and diffusion processes, 2nd edn. North- Holland, Amsterdam
Istas J (2005) Mathematical modeling for the life sciences. Universitext, Springer-Verlag
Kavian O, Kerkyacharian G, Roynette B (1993) Some remarks on ultracontractivity. J Func Anal 111: 155–196
Lambert A (2005) The branching process with logistic growth. Ann Appl Prob 15: 1506–1535
Maz’ya V, Shubin M (2005) Discreteness of spectrum and positivity criteria for Schrödinger operators. Ann Math 162: 919–942
Pollett PK, Quasi stationary distributions: a bibliography. http://www.maths.uq.edu.au/~pkp/papers/qsds/qsds.html. Regularly updated
Royer G (1999) Une initiation aux inégalités de Sobolev logarithmiques. S.M.F., Paris
Seneta E, Vere-Jones D (1966) On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J Appl Prob 3: 403–434
Steinsaltz D, Evans SN (2004) Markov mortality models: implications of quasistationarity and varying initial distributions. Theo Pop Bio 65: 319–337
Wang FY (2004) Functional inequalities, Markov processes and Spectral theory. Science Press, Beijing
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Cattiaux, P., Méléard, S. Competitive or weak cooperative stochastic Lotka–Volterra systems conditioned on non-extinction. J. Math. Biol. 60, 797–829 (2010). https://doi.org/10.1007/s00285-009-0285-4
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DOI: https://doi.org/10.1007/s00285-009-0285-4
Keywords
- Stochastic Lotka–Volterra systems
- Multitype population dynamics
- Quasi-stationary distribution
- Yaglom limit
- coexistence