ANALYSIS OF BOGDANOV–TAKENS BIFURCATION OF CODIMENSION 2 IN A GAUSE-TYPE MODEL WITH CONSTANT HARVESTING OF BOTH SPECIES AND DELAY EFFECT

In this paper, we investigate the dynamics of a system in which both prey and predator are harvested with constant rate. Our main objective is to find the effects of harvesting on equilibria, stability, and bifurcations in the system, which may be useful for biological management. The existence and stability of equilibrium points of the model are further investigated. A thorough qualitative analysis has been carried out based on bifurcation theory in dynamical systems and to validate our analytical findings, a large scale numerical simulation has been performed by using plausible values of parameters involved. It is shown that the model can exhibit Hopf bifurcation. The first Lyapunov coefficient is calculated to determine the direction of limit cycle of Hopf bifurcation. Also, it has been proven analytically that the system exhibits Bogdanov–Takens bifurcation of codimension 2. Moreover, discrete-time delay effect has been included due to gestation of the predator species on the same system and observed Hopf bifurcation with respect to the delay parameter. This study renders important tools for investigations of the dynamics of biotic organisms for the management and control of over harvesting. Some phase plane analysis has been carried out to support our analytical results.

HOW DO PARTIALLY HIDING BEHAVIOR AND COMPLETELY HIDING BEHAVIOR IMPACT INTERACTING POPULATIONS?

As the two main behaviors of prey populations in ecological systems, the partially hiding behavior (PHB) and the completely hiding behavior (CHB) play a significant role in determining the dynamics of predator–prey models. This work examines to the dynamical consequences of predator–prey systems with the PHB and the CHB. Previous research has independently studied the two behaviors, and the general conclusions are that the two behaviors can have positive and/or negative impacts on the considered population models. However, to our knowledge, no study has combined and compared the two behaviors in studying the dynamical consequences of predation interactions. Motivated by this, we investigated the dynamical consequences induced by the PHB and the CHB. From a mathematical point of view, the dynamical behaviors are studied and the corresponding sufficient conditions are given. Our findings are general and some published models are special cases of ours. From an ecological point of view, we find that the size of the ecological regions is mainly determined by the two behaviors, and which one is ecologically beneficial for the health coexistence of the interacting populations are primarily determined by the functional response and the attack coefficient of predators. Moreover, we conclude that the evolutionary and optimal choices of prey behavior (PHB or CHB) depend on the predators attack coefficient (large or small attack coefficient) and the resource level (abundant or pool resource level).

A GENERALIZED PREDATOR–PREY SYSTEM WITH HABITAT COMPLEXITY

This paper presents a generalized predator–prey system and considers the effect of habitat complexity on the dynamical consequences. The results show that habitat complexity has a major impact on the dynamical consequences of the considered system. On the one hand, habitat complexity has a stabilizing impact under certain conditions. A numerical simulation in our study and in experiments conducted in the published studies elaborate on this stabilizing effect. On the other hand, the most interesting and open issue is that a destabilizing effect of habitat complexity is found theoretically. All results are explained and illustrated from the ecological viewpoint.

An integrated project of fish and broiler: SIS model with optimal harvesting

The paper analyzes the influence of a susceptible–infectious–susceptible (SIS) infectious disease affecting both fish and broiler species. The paper also considers a joint SIS project of fish and broiler in which the growth rates of both species vary with available nutrients and environmental carrying capacities of biomasses. The nutrients for both species are functions of the biomasses of the two species. The harvesting rates of fish and broiler depend linearly on common effort function. It is assumed that the diseases are transmitted to the susceptible populations by direct contact with the infected populations. Using the medicine, some portion of the infected populations are transmitted to the susceptible populations. The existence of steady states and their stability are investigated analytically. The joint profit of the SIS model is maximized using Pontryagin’s maximum principle and corresponding optimum harvesting rates are also obtained. Using Mathematica software, the models are illustrated and the optimum results are obtained and presented in tabular and graphical forms.