Abstract
The classic Ricker equation \(x_{t+1}=bx_{t}\exp \left( -cx_{t}\right) \) has positive equilibria for \(b>1\) that destabilize when \(b>e^{2}\) after which its asymptotic dynamics are oscillatory and complex. We study an evolutionary version of the Ricker equation in which coefficients depend on a phenotypic trait subject to Darwinian evolution. We are interested in the question of whether evolution will select against or will promote complex dynamics. Toward this end, we study the existence and stability of its positive equilibria and focus on equilibrium destabilization as an indicator of the onset of complex dynamics. We find that the answer relies crucially on the speed of evolution and on how the intra-specific competition coefficient c depends on the evolving trait. In the case of a hierarchical dependence, equilibrium destabilization generally occurs after \(e^{2}\) when the speed of evolution is sufficiently slow (in which case we say evolution selects against complex dynamics). When evolution proceeds at a faster pace, destabilization can occur before \(e^{2}\) (in which case we say evolution promotes complex dynamics) provided the competition coefficient is highly sensitive to changes in the trait v. We also show that destabilization does not always result in a period doubling bifurcation, as in the non-evolutionary Ricker equation, but under certain circumstances can result in a Neimark-Sacker bifurcation.
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References
Cushing, J.M.: A strong ergodic theorem for some nonlinear matrix models for structured population growth. Nat. Resourc. Model. 3(3), 331–357 (1989)
Cushing, J.M.: Difference equations as models of evolutionary population dynamics. J. Biol. Dyn. 13, 103–127 (2019)
Elaydi, S.N.: Discrete Chaos, with Applications in Science and Engineering, 2nd edn. Chapman & Hall/CRC, New York (2008)
Ferriere, R., Fox, G.A.: Chaos and evolution. Trends Ecol. Evol. 10, 480–483 (1995)
May, R.M.: Simple mathematical models with very complicated dynamics. Nature 261, 459–467 (1976)
Mokni, K., Elaydi, S., CH-Chaoui, M., Eladdadi, A.: Discrete evolutionary population models: a new approach. To appear in the J Biol Dyn (2020)
Neutel, A.M., Heesterbeek, J.A.P., de Ruiter, P.C.: Stability in real food webs: weak links in long loops. Science 296, 1120–1123 (2002)
Sugihara, G., May, R.M.: Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series. Nature 344, 734–741 (1990)
Vincent, T., Brown, J.: Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics. Cambridge University Press, Cambridge, UK (2005)
Zimmer, C.: Life after chaos. Science 284, 83–86 (1999)
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Cushing, J.M. (2020). A Darwinian Ricker Equation. In: Baigent, S., Bohner, M., Elaydi, S. (eds) Progress on Difference Equations and Discrete Dynamical Systems. ICDEA 2019. Springer Proceedings in Mathematics & Statistics, vol 341. Springer, Cham. https://doi.org/10.1007/978-3-030-60107-2_10
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DOI: https://doi.org/10.1007/978-3-030-60107-2_10
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