PapersNo Access

A Mechanistic Stochastic Ricker Model: Analytical and Numerical Investigations

Tamar Gadrich

Department of Industrial Engineering and Management, ORT Braude College, Karmiel 2161002, Israel

E-mail Address: tamarg@braude.ac.il

Search for more papers by this author

and

Guy Katriel

Department of Mathematics, ORT Braude College, Karmiel 2161002, Israel

E-mail Address: katriel@braude.ac.il

Search for more papers by this author

https://doi.org/10.1142/S021812741650067XCited by:5 (Source: Crossref)

Abstract

The Ricker model is one of the simplest and most widely-used ecological models displaying complex nonlinear dynamics. We study a discrete-time population model, which is derived from simple assumptions concerning individual organisms’ behavior, using the “site-based” approach, developed by Brännström, Broomhead, Johansson and Sumpter. In the large-population limit the model converges to the Ricker model, and can thus be considered a mechanistic version of the Ricker model, derived from basic ecological principles, and taking into account the demographic stochasticity inherent to finite populations. We employ several analytical and precise numerical methods to study the model, showing how each approach contributes to understanding the model’s dynamics. Expressing the model as a Markov chain, we employ the concept of quasi-stationary distributions, which are computed numerically, and used to examine the interaction between complex deterministic dynamics and demographic stochasticity, as well as to calculate mean times to extinction. A Gaussian Markov chain approximation is used to obtain quantitative asymptotic approximations for the size of fluctuations of the stochastic model’s time series around the deterministic trajectory, and for the correlations between successive fluctuations. Results of these approximations are compared to results obtained from quasi-stationary distributions and from direct simulations, and are shown to be in good agreement.

Keywords:

References

Akcakaya, R. H. [1991] “ A method for simulating demographic stochasticity,” Ecol. Model. 54, 133–136. Web of Science, Google Scholar
Akcakaya, R. H. [2000] “ Population viability analyses with demographically and spatially structured models,” Ecol. Bull. 48, 23–38. Google Scholar
Anazawa, M. [2010] “ The mechanistic basis of discrete-time population models: The role of resource partitioning and spatial aggregation,” Theor. Popul. Biol. 77, 213–218. Web of Science, Google Scholar
Black, A. J. & McKane, A. J. [2012] “ Stochastic formulation of ecological models and their applications,” TREE 27, 337–345. Web of Science, Google Scholar
Brännström, Å. & Sumpter, D. J. T. [2005a] “ The role of competition and clustering in population dynamics,” Proc. Roy. Soc. B 272, 2065–2072. Web of Science, Google Scholar
Brännström, Å. & Sumpter, D. J. T. [2005b] “ Coupled map lattice approximations for spatially explicit individual-based models of ecology,” Bull. Math. Biol. 67, 663–682. Web of Science, Google Scholar
Brännström, Å. & Sumpter, D. J. T. [2006] “ Stochastic analogues of deterministic single-species population models,” Theor. Popul. Biol. 69, 442–451. Web of Science, Google Scholar
Buckley, F. M. & Pollett, P. K. [2010] “ Limit theorems for discrete-time metapopulation models,” Prob. Surveys 7, 53–83. Google Scholar
Darroch, J. N. & Seneta, E. [1965] “ On quasi-stationary distributions in absorbing discrete-time finite Markov chains,” J. Appl. Prob. 2, 88–100. Google Scholar
Fagerholm, H. & Högnäs, G. [2002] “ Stability classification of a Ricker model with two random parameters,” Adv. Appl. Prob. 34, 112–127. Web of Science, Google Scholar
Geritz, S. A. H. & Kisdi, É. [2004] “ On the mechanistic underpinning of discrete-time population models with complex dynamics,” J. Theoret. Biol. 228, 261–269. Web of Science, Google Scholar
Geritz, S. A. H. & Kisdi, É. [2012] “ Mathematical ecology: Why mechanistic models?” J. Math. Biol. 65, 1411–1415. Web of Science, Google Scholar
Gotzen, B., Liebscher, S. & Wlacher, S. [2011] “ On a class of deterministic population models with stochastic foundation,” Bull. Math. Biol. 73, 1559–1582. Web of Science, Google Scholar
Grimm, V. & Railsback, S. F. [2013] Individual-Based Modeling and Ecology (Princeton University Press, USA). Google Scholar
Gurney, W. S. & Nisbet, R. M. [1998] Ecological Dynamics (Oxford University Press, UK). Google Scholar
Gyllenberg, M., Högnäs, G. & Koski, T. [1994] “ Population models with environmental stochasticity,” J. Math. Biol. 32, 93–108. Web of Science, Google Scholar
Hastings, A. [1997] Population Biology: Concepts and Models (Springer-Verlag, NY). Google Scholar
Hritonenko, N., Rodkina, A. & Yatsenko, Y. [2006] “ Stability analysis of stochastic Ricker population model,” Discr. Dyn. Nat. Soc. 2006, 64590-1–13. Web of Science, Google Scholar
Johansson, A. & Sumpter, D. J. T. [2003] “ From local interactions to population dynamics in site-based models of ecology,” Theor. Popul. Biol. 64, 419–517. Web of Science, Google Scholar
Klok, C. & De Roos, A. M. [1998] “ Effects of habitat size and quality on equilibrium density and extinction time of Sorex araneus populations,” J. Animal Ecol. 67, 195–209. Web of Science, Google Scholar
May, R. M. [1974] “ Biological populations with nonoverlapping generations: Stable points, stable cycles, and chaos,” Science 186, 645–647. Web of Science, Google Scholar
Melbourne, B. A. & Hastings, A. [2008] “ Extinction risk depends strongly on factors contributing to stochasticity,” Nature 454, 100–103. Web of Science, Google Scholar
Melbourne, B. A. [2012] “ Demographic stochasticity,” in Encyclopedia of Theoretical Ecology, eds. Hastings, A. & Gross, L. J. (Univ. of California Press, Berkeley). Google Scholar
Ovaskainen, O. & Meerson, B. [2010] “ Stochastic models of population extinction,” Trends Ecol. Evol. 25, 643–652. Web of Science, Google Scholar
Parzen, E. [1962] “ Spectral analysis of asymptotically stationary time series,” Bull. Inst. Int. Statist. 39, 87–103. Web of Science, Google Scholar
Ricker, W. E. [1954] “ Stock and recruitment,” J. Fisheries Res. Board of Canada 11, 559–623. Google Scholar
Royama, T. [1992] Analytical Population Dynamics (Springer-Verlag, NY). Google Scholar
Sumpter, D. J. T. & Broomhead, D. S. [2001] “ Relating individual behavior to population dynamics,” Proc. Roy. Soc. Lond. B 268, 925–932. Web of Science, Google Scholar
Thieme, H. R. [2003] Mathematics in Population Biology (Princeton University Press). Google Scholar
Turchin, P. [2003] Complex Population Dynamics: A Theoretical/Empirical Synthesis (Princeton University Press). Google Scholar
van Doorn, E. A. & Pollett, P. K. [2013] “ Quasi-stationary distributions for discrete-state models,” Eur. J. Oper. Res. 230, 1–14. Web of Science, Google Scholar