Bifurcation and Stability of Prey-Predator Model with Beddington-DeAngelis Functional Response

In this paper we discuss the harvesting of the prey species making a fraction of them to be accessed by the predator while both the prey and predator are being subjected to Beddington-DeAngelis functional response. It is observed that a Hopf-bifurcation may occur around the interior equilibrium taking the environmental carrying capacity of the prey species as the parameter. Some numerical examples and the corresponding curves are studied using Maple to explain the results of the proposed model

Behavior of the Solutions for Predator-Prey Dynamic Systems with Beddington-DeAngelis Type Functional Response on Periodic Time Scales in Shifts

We consider two-dimensional predator-prey system with Beddington-DeAngelis type functional response on periodic time scales in shifts. For this special case we try to find under which conditions the system has δ±-periodic solution

Seasonally Perturbed Prey-Predator Ecological System with the Beddington-DeAngelis Functional Response

On the basis of the theories and methods of ecology and ordinary differential equation, a seasonally perturbed prey-predator system with the Beddington-DeAngelis functional response is studied analytically and numerically. Mathematical theoretical works have been pursuing the investigation of uniformly persistent, which depicts the threshold expression of some critical parameters. Numerical analysis indicates that the seasonality has a strong effect on the dynamical complexity and species biomass using bifurcation diagrams and Poincaré sections. The results show that the seasonality in three different parameters can give rise to rich and complex dynamical behaviors. In addition, the largest Lyapunov exponents are computed. This computation further confirms the existence of chaotic behavior and the accuracy of numerical simulation. All these results are expected to be of use in the study of the dynamic complexity of ecosystems

Extinction and permanence of the predator-prey system with general functional response and impulsive control

Traditional approach for modelling the evolution of populations in the predator-prey ecosystem has commonly been undertaken using specific impulsive response function, and this kind of modelling is applicable only for a specific ecosystem under certain environ- mental situations only. This paper attempts to fill the gap by modelling the predator-prey ecosystem using a ‘generalized’ impulsive response function for the first time. Different from previous research, the present work develops the modelling for an integrated pest management (IPM) especially when the stocking of predator (natural enemy) and the har- vesting of prey (pest) occur impulsively and at different instances of time. The paper firstly establishes the sufficient conditions for the local and the global stabilities of prey eradica- tion periodic solution by applying the Floquet theorem of the Impulsive different equation and small amplitude perturbation under a ‘generalized’ impulsive response function. Sub- sequently the sufficient condition for the permanence of the system is given through the comparison techniques. The corollaries of the theorems that are established by using the ‘general impulsive response function’ under the locally asymptotically stable condition are found to be in excellent agreement with those reported previously. Theoretical results that are obtained in this work is then validated by using a typical impulsive response func- tion (Holling type-II) as an example, and the outcome is shown to be consistent with the previously reported results. Finally, the implication of the developed theories for practical pest management is illustrated through numerical simulation. It is shown that the elim- ination of either the preys or the pest can be effectively deployed by making use of the theoretical model established in this work. The developed model is capable to predict the population evolutions of the predator-prey ecosystem to accommodate requirements such as: the combinations of the biological control, chemical control, any functional response function, the moderate impulsive period, the harvest rate for the prey and predator pa- rameter and the incremental stocking of the predator paramete

Analysis of an Impulsive One-Predator and Two-Prey System with Stage-Structure and Generalized Functional Response

An impulsive one-predator and two-prey system with stage-structure and generalized functional response is proposed and analyzed. By reasonable assumption and theoretical analysis, we obtain conditions for the existence and global attractivity of the predator-extinction periodic solution. Sufficient conditions for the permanence of this system are established via impulsive differential comparison theorem. Furthermore, abundant results of numerical simulations are given by choosing two different and concrete functional responses, which indicate that impulsive effects, stage-structure, and functional responses are vital to the dynamical properties of this system. Finally, the biological meanings of the main results and some control strategies are given

Dynamical Complexity of a Spatial Phytoplankton-Zooplankton Model with an Alternative Prey and Refuge Effect

The spatiotemporal dynamics of a phytoplankton-zooplankton model with an alternative prey and refuge effect is investigated mathematically and numerically. The stability of the equilibrium point and the traveling wave solution of the phytoplankton-zooplankton model are described based on theoretical mathematical work, which provides the basis of the numerical simulation. The numerical analysis shows that refuges have a strong effect on the spatiotemporal dynamics of the model according to the pattern formation. These results may help us to understand prey-predator interactions in water ecosystems. They are also relevant to research into phytoplankton-zooplankton ecosystems

Globally exponentially stable periodic solution in a general delayed predator-prey model under discontinuous prey control strategy

This paper studies the solution behaviour of a general delayed predator-prey model with discontinuous prey control strategy. The positiveness and boundeness of the solution of the system is firstly investigated using the comparison theorem. Then the sufficient conditions are derived for the existence of positive periodic solutions using the differential inclusion theory and the topological degree theory. Furthermore, the positive periodic solution is proved to be globally exponentially stable by employing the generalized Lyapunov approach. The global finite-time convergence is also discussed for the system state. Finally, the numerical simulations of four examples are given to validate the correctness of the theoretical results

Mathematical and Dynamic Analysis of a Prey-Predator Model in the Presence of Alternative Prey with Impulsive State Feedback Control

The dynamic complexities of a prey-predator system in the presence of alternative prey with impulsive state feedback control are studied analytically and numerically. By using the analogue of the Poincaré criterion, sufficient conditions for the existence and stability of semitrivial periodic solutions can be obtained. Furthermore, the corresponding bifurcation diagrams and phase diagrams are investigated by means of numerical simulations which illustrate the feasibility of the main results