Narrow Results

In this paper we have considered a prey-predator model with Holling type of predation and independent harvesting in either species. The purpose of the work is to offer mathematical analysis of the model and to discuss some significant qualitative results that are expected to arise from the interplay of biological forces. Our study shows that, using the harvesting efforts as controls, it is possible to break the cyclic behaviour of the system and drive it to a required state. Also it is possible to introduce globally stable limit cycle in the system using the above controls.

An analysis is presented for a model of a two species prey-predator system subject to the combined effects of delay and harvesting. Our study shows that, both the delay and harvesting effort may play a significant role on the stability of the system. Computer simulations are carried out to explain some of the mathematical conclusions.

Oscillations exist at all levels of biological systems and are often crucial for their proper functioning. Among the various types of oscillations, limit cycles have received particular attention for more than one hundred years. Specifically, theorems have been established that characterize whether a system might have the capability of exhibiting limit cycles. However, the practical application of these theorems is usually cumbersome and there are hardly any guidelines for devising de novo models that exhibit limit cycles of a desired form. In this paper, we propose a simple method for constructing and customizing stable limit cycles in two-dimensional systems according to desired features, including frequency, amplitude, and phase shift between system variables. The method is based on "inverting" a criterion proposed by Lewis for characterizing oscillations in two-dimensional S-system models. First, we execute comprehensive simulations that result in a set of over 2000 prototype limit cycles. Second, we show with examples how these prototypes can be further customized to adhere to predetermined specifications. This two-step process is fast and efficacious, especially when one considers the paucity of alternative methods. Finally, we illustrate how one may create systems with more complex dynamics by modulating the prototypes with external input signals.

We consider a two predator and one prey model with Holling type II functional response incorporating a constant prey refuge. Depending upon the constant prey refuge m, which provides a criterion for protecting m of prey from predation, sufficient conditions for stability and global stability of equilibria are obtained. We find the critical value of this refuge parameter m for which the dynamical system undergoes a Hopf bifurcation and then makes use of center manifold theorem and normal form methods to find the direction of the Hopf bifurcation as well as the stability of the resulting limit cycle. The influence of the prey refuge parameter is also investigated at the interior equilibrium. Numerical simulations were carried out to illustrate and support the analytical results.

This work deals with the dynamics of a bioeconomic continuous time model, where the combined action of the fishing effort exerted by men (as a predator) and multiple Allee effect or depensation on the growth rate of a self-regenerating resource (the prey) are considered.

It has been recently established that a depensation phenomenon appears by diverse causes and new functions have been proposed to describe multiple Allee effects. One of these formalizations is here incorporated in the well-known Smith's model, one of the simplest models to open access fisheries.

We prove that this new and complex expression is topologically equivalent to a simpler form. Then, we postulate that the parsimony principle must be used to describe this phenomenon.

It is also shown that in the phase plane of biomass-effort on the proposed model, the origin is an attractor equilibrium for all parameters values as a consequence of the Allee effect. Moreover, there is a subset of the parameter values, for which two limit cycles exist surrounding the unique positive equilibrium point of the system, one of them being asymptotically stable (the non damped oscillatory tragedy of the commons); hence, multiestability exists, particularly three-stability.

A mathematical model is developed to assess humoral and cellular immune responses against Trypanosoma cruzi infection. Analysis of the model shows a unique non-trivial equilibrium, which is locally asymptotically stable, except in the case of a strong cellular response. When the proliferation of the activated CD8 T cells is increased, this equilibrium becomes unstable and a limit cycle appears. However, this behavior can be avoided by increasing the action of the humoral response. Therefore, unbalanced humoral and cellular responses can be responsible for long asymptomatic period, and the control of Trypanosoma cruzi infection is a consequence of well coordinated action of both humoral and cellular responses.

This paper investigates the dynamical behavior of the modified May–Holling–Tanner model in the presence of dynamic alternative resources. We study the role of dynamic alternative resources on the survival of the species when there is prey rarity. Detailed mathematical analysis and numerical evaluations, including the situation of ecosystem collapsing, have been presented to discuss the coexistence of species’, stability, occurrence of different bifurcations (saddle-node, transcritical, and Hopf) in three cases in the presence of prey and alternative resources, in the absence of prey and in the absence of alternative resources. It has been obtained that the multiple coexisting states and their stability are outcomes of variations in predation rate for alternative resources. Also, the occurrence of Hopf bifurcation, saddle-node bifurcation, and transcritical bifurcation are due to variations in the parameters of dynamic alternative resources. The impact of dynamic alternative resources on species’ density reveals the fact that if the predation rate for alternative resources increases, then the prey biomass increases (under some restrictions), and variations in the predator’s biomass widely depend upon the quality of food items. This study also points out that the survival of predators is possible in the absence of prey. In the theme of ecological balance, this study suggests some theoretical points of view for the eco-managers.