Abstract
We model the evolution of two populations \(U_t, V_t \) in competition by a two-dimensional size-dependent branching process. The population characteristics are assumed to be close to each other, as in a resident-mutant situation. Given that \(U_t = m\) and \(V_t = n\) the expected values of \(U_{t+1}\) and \(V_{t+1}\) are given by \(me^{r - K(m + bn)} \) and \(ne^{\tilde{r} - \tilde{K} (n + am)}\), respectively, where \(r, \tilde{r}\) model the intrinsic population growth, \(K, \tilde{K} \) model the force of inhibition on the population growth by the present population (such as scarcity of food), and a, b model the interaction between the two populations. For small \(K, \tilde{K} \) the process typically follows the corresponding deterministic Ricker competition model closely, for a very long time. Under some conditions, notably a mutual invasibility condition, the deterministic model has a coexistence fixed point in the open first quadrant. The asymptotic behaviour is studied through the quasi-stationary distribution of the process. We initiate a study of those distributions as the inhibitive force \(K, \tilde{K}\) approach 0.
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Acknowledgments
The author has worked on Ricker competition models with Henrik Fagerholm, Mats Gyllenberg, and Brita Jung. The present paper was presented at the International Conference on Difference Equations and Applications ICDEA 2012 in Barcelona. Many discussions took place in the constructive and inspiring conference atmosphere. In particular, questions and comments by Rafael Luís, Eduardo Cabral Balreira, and Saber Elaydi are gratefully acknowledged. Travel funding was provided by the Magnus Ehrnrooth Foundation.
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Högnäs, G. (2016). On a Stochastic Ricker Competition Model. In: Alsedà i Soler, L., Cushing, J., Elaydi, S., Pinto, A. (eds) Difference Equations, Discrete Dynamical Systems and Applications. ICDEA 2012. Springer Proceedings in Mathematics & Statistics, vol 180. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-52927-0_10
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DOI: https://doi.org/10.1007/978-3-662-52927-0_10
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