Population dynamics of a two‐stage species with recruitment

Abstract

This work models and analyzes the qualitative dynamics of a two‐stage species with the factor of recruitment. It shows how the dynamics is determined by a basic threshold parameter scriptR. As a result, it is proved that if scriptRMathClass-rel≤1, then the extinction equilibrium point is globally asymptotically stable, assuring that the species would disappear under this condition. Numerical simulations, produced by varying the parameters obtained from empirical data, show different situations regarding the evolution of the population and allow us to validate the model. Copyright © 2012 John Wiley & Sons, Ltd.

General functional response and recruitment in a predator–prey system with capture on both species

“…Consequently, the fraction of the population surviving to recruitment is very small . In , the authors set out an approach to these kinds of models, providing one that considers recruitment in a two‐stage species.…”

Modelo para el control óptimo del VIH con tasa de infección dependiente de la densidad del virus

“…Algunos autores utilizan una tasa de infección saturada (dependiente de la densidad del virus) para eliminar esa característica (en particular se recomienda ver [5,25,29,30,36] y las referencias allí); es decir, que se describe mediante una tasa de infección funcional, en particular la respuesta funcional Holling tipo II, la respuesta funcional de BeddingtonDeAngelis, y otras más sofisticadas que pueden encontrarse en [5,61,60]. Suponga entonces que la tasa de infección entre células T y el virus V crece rápidamente cuando la concentración promedio del VIH es pequeña, y luego crece más lentamente cuando V es más grande, hasta alcanzar cierta tasa máxima de infección β [1,6,22,25,28,29,34,37], lo que puede describirse por la expresión,…”

Border Collision Bifurcations in a Generalized Model of Population Dynamics

“…In a previous work [12], we studied a discrete time continuous and differentiable dynamical system in biology, which models the population dynamics of a two-stage species with recruitment and capture. In such work, to be coherent with the biological meaning of the model, the possible values of the essential parameters are determined by two nonnegative constraints under which the dynamics of the discrete model considered coincide exactly with its continuous counterpart analyzed in Ladino and Valverde [13]. In both the discrete and continuous models, the system exhibits a transcritical bifurcation while one of the equilibria is a global attractor of the system (alternatively), depending on the value of a threshold parameter which is a function of the key parameters.…”

“…To better explain, in the present contribution we consider the model proposed in Ladino and Valverde [13] (which is a two-dimensional model in continuous time) and we obtain its discrete time formulation by considering the variation of each state variable in a unit time. Even if this is a simplified manner to obtain the discrete counterpart of the initial model, we proceeded in such a way for the following main reasons.…”

“…In fact, for the numerical simulations, we will consider actual data related to the population parameters on the state of fisheries for two fish species, that is, Prochilodus mariae and Prochilodus magdalenae, which inhabit in the Orinoco and Magdalena rivers of Colombia (CCI and INCODER [14]). As far as the other parameters of the model are concerned, because of the difficulty of finding related serious research publications, we consider the values theoretically estimated in Ladino and Valverde [13]. Although using real data for the parameters would be of great interest, the numerical analysis we perform has the advantage of allowing us to simulate and analyze different scenarios of the feasible biological parametric space.…”