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Discrete Dynamics in Nature and Society
Volume 2016 (2016), Article ID 9724139, 13 pages
http://dx.doi.org/10.1155/2016/9724139
Research Article

Border Collision Bifurcations in a Generalized Model of Population Dynamics

1Department of Mathematics and Physics, University of Los Llanos, 500001 Villavicencio, Colombia
2Department of Economics and Law, University of Macerata, 62100 Macerata, Italy
3Department of Mathematics, University of Castilla-La Mancha, 02071 Albacete, Spain

Received 21 December 2015; Revised 24 February 2016; Accepted 22 March 2016

Academic Editor: Xiaohua Ding

Copyright © 2016 Lilia M. Ladino et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We analyze the dynamics of a generalized discrete time population model of a two-stage species with recruitment and capture. This generalization, which is inspired by other approaches and real data that one can find in literature, consists in considering no restriction for the value of the two key parameters appearing in the model, that is, the natural death rate and the mortality rate due to fishing activity. In the more general case the feasibility of the system has been preserved by posing opportune formulas for the piecewise map defining the model. The resulting two-dimensional nonlinear map is not smooth, though continuous, as its definition changes as any border is crossed in the phase plane. Hence, techniques from the mathematical theory of piecewise smooth dynamical systems must be applied to show that, due to the existence of borders, abrupt changes in the dynamic behavior of population sizes and multistability emerge. The main novelty of the present contribution with respect to the previous ones is that, while using real data, richer dynamics are produced, such as fluctuations and multistability. Such new evidences are of great interest in biology since new strategies to preserve the survival of the species can be suggested.