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Gonzalez-Olivares, E., Gonzalez-Yanez, B., Becerra-Klix, R., & Ramos-Jiliberto, R. (2017). Multiple stable states in a model based on predator-induced defenses. Ecol. Complex., 32, 111–120.
Abstract: A large variety of antipredator defenses are exhibited by plants, animals and microbes in nature. A deep understanding of the dynamic consequences of prey responses to predation risk is essential for building a comprehensive theory of food webs. Here we present a simple classification of prey defenses based on the sensitivity of prey immunity to predation respect to abundances of prey and predators. Only three out of six defense types have been analytically studied in the context of predator-prey dynamics, which reveals a serious gap in our current knowledge of ecological interactions. In this study we present a mathematical analysis on a widely occurring type of prey defense whose behavior has not been established in exact terms. The study model considers prey whose average immunity to predators is enhanced by predator abundance. This case, known as inducible defenses, has been reported for a wide array of species. Our results reveal a rich dynamic behavior, in which the predator-prey system exhibits either one, two or three positive equilibrium points, with up to two attractors. Thus, inducible defenses constitute a mechanism that could drive alternative stable states even in very simple food web models. (C) 2017 Elsevier B.V. All rights reserved.
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Vera-Damian, Y., Vidal, C., & Gonzalez-Olivares, E. (2019). Dynamics and bifurcations of a modified Leslie-Gower-type model considering a Beddington-DeAngelis functional response. Math. Meth. Appl. Sci., 42(9), 3179–3210.
Abstract: In this paper, a planar system of ordinary differential equations is considered, which is a modified Leslie-Gower model, considering a Beddington-DeAngelis functional response. It generates a complex dynamics of the predator-prey interactions according to the associated parameters. From the system obtained, we characterize all the equilibria and its local behavior, and the existence of a trapping set is proved. We describe different types of bifurcations (such as Hopf, Bogdanov-Takens, and homoclinic bifurcation), and the existence of limit cycles is shown. Analytic proofs are provided for all results. Ecological implications and a set of numerical simulations supporting the mathematical results are also presented.
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