Keywords
bifurcation analysis
limit cycles predatorprey systems
cicly Canard
Sistemas planares de Filippov
análisis de bifurcación
ciclo límite
momodelo depredador - presa
ciclo de Canard
How to Cite
Abstract
This paper presents the necessary conditions to guarantee the existence of a stable limit cycle in a predator - prey model and some geometrical aspects to perform a qualitative analysis in two - dimensional Filippov dynamic systems. With these defined guidelines, the dynamics of a predator - prey model are studied when exploitation in predators is restricted if the number of prey is lower than a critical value. The study is carried out by the bifurcation analysis in relation to two parameters: exploitation and protection of the populations to interact.
References
V.I. Arnold, Ordinary Differential Equations, The MIT Press, translated from Russian and edited by Richard A. Silverman, Cambridge MA, London, 1973. https://mitpress.mit.edu/books/ordinary-differential-equations
C.A. Buzzi , T. de Carvalho, P.R. da Silva, Closed poly-trajectories and Poincaré index of non-smooth vector fields on the plane, Journal of Dynamical and Control Systems, 19(2013), no. 2, 173–193. Doi: 10.1007/s10883-013-9169
L. Edelstein-Keshet, Mathematical Models in Biology, Society for Industrial and Applied Mathematics, Philadelphia PA, 2005. Doi:10.1137/1.9780898719147
A.F. Filippov, Differential Equations with Discontinuous Righthand Sides, Mathematics and Its Applications 18 (Soviet Series), Kluwer Academic Publishers Group, Dordrecht, 1988. Doi: 10.1007/978-94-015-7793-9
H.I. Freedman, Deterministic Mathematical Models in Population Ecology. Marcel Dekker, New York, 1980. Doi: 10.2307/3556198
M. Guardia, T.M. Seara, M.A. Teixeira, Generic bifurcations of low codimension of planar Filippov Systems, Journal of Differential Equations, 250(2010), no. 4, 1967–2023. Doi: 10.1016/j.jde.2010.11.016
Y. Kuang, H.I. Freedman, Uniqueness of limit cycles in Gause-type models of predator-prey systems, Mathematical Biosciences, 88(1988), no. 1, 67–84. Doi: 10.1016/0025-5564(88)90049-1
Y. Kuznetsov, Elements of Applied Bifurcation Theory, Applied Mathematical Sciences 112, Springer, New York,} 1995. Doi: 10.1007/978-1-4757-3978-7
Y. Kuznetsov, S. Rinaldi, A. Gragnani, One-parameter bifurcations in planar Filippov systems, International Journal of Bifurcation and Chaos,13(2003), no. 8, 2157–2188. Doi: 10.1142/S0218127403007874
J.D. Murray, Mathematical Biology: I. An Introduction, 3rd Edition, Springer, New York, 2002. Doi: 10.1007/b98868
J. Sotomayor, Lições de equações diferenciais ordinárias, Projeto Euclides, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, 11(1979).
J. Yang, S. Tang, R.A. Cheke, Global stability and sliding bifurcations of a non-smooth Gause predator-prey system, Applied Mathematics and Computation, 224(2013), no. 1, 9–20. Doi: 10.1016/j.amc.2013.08.024
Comments
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
Copyright (c) 2021 Revista de Matemática: Teoría y Aplicaciones