A food chain system with Holling type IV functional response and impulsive perturbations

AbstractIn this paper, a three-trophic-level food chain system with Holling type IV functional response and impulsive perturbations is established. We show that this system is uniformly bounded. Using the Floquet theory of impulsive equations and small perturbation skills, we find conditions for the local and global stabilities of the prey and top predator-free periodic solution. Moreover, we obtain sufficient conditions for the system to be permanent via the comparison theorem. We display some numerical examples to substantiate our theoretical results

An Impulsive Two-Prey One-Predator System with Seasonal Effects

In recent years, the impulsive population systems have been studied by many researchers. However, seasonal effects on prey are rarely discussed. Thus, in this paper, the dynamics of the Holling-type IV two-competitive-prey one-predator system with impulsive perturbations and seasonal effects are analyzed using the Floquet theory and comparison techniques. It is assumed that the impulsive perturbations act in a periodic fashion, the proportional impulses (the chemical controls) for all species and the constant impulse (the biological control) for the predator at different fixed time but, the same period. In addition, the intrinsic growth rates of prey population are regarded as a periodically varying function of time due to seasonal variations. Sufficient conditions for the local and global stabilities of the two-prey-free periodic solution are established. It is proven that the system is permanent under some conditions. Moreover, sufficient conditions, under which one of the two preys is extinct and the remaining two species are permanent, are also found. Finally, numerical examples and conclusion are given

Complex Dynamics of a Discrete-Time Predator-Prey System with Ivlev Functional Response

The dynamics of a discrete-time predator-prey system with Ivlev functional response is investigated in this paper. The conditions of existence for flip bifurcation and Hopf bifurcation in the interior of R+2 are derived by using the center manifold theorem and bifurcation theory. Numerical simulations are presented not only to substantiate our theoretical results but also to illustrate the complex dynamical behaviors of the system such as attracting invariant circles, periodic-doubling bifurcation leading to chaos, and periodic-halving phenomena. In addition, the maximum Lyapunov exponents are numerically calculated to confirm the dynamical complexity of the system. Finally, we compare the system to discrete systems with Holling-type functional response with respect to dynamical behaviors

The Dynamics of a Predator-Prey System with State-Dependent Feedback Control

A Lotka-Volterra-type predator-prey system with state-dependent feedback control is investigated in both theoretical and numerical ways. Using the Poincaré map and the analogue of the Poincaré criterion, the sufficient conditions for the existence and stability of semitrivial periodic solutions and positive periodic solutions are obtained. In addition, we show that there is no positive periodic solution with period greater than and equal to three under some conditions. The qualitative analysis shows that the positive period-one solution bifurcates from the semitrivial solution through a fold bifurcation. Numerical simulations to substantiate our theoretical results are provided. Also, the bifurcation diagrams of solutions are illustrated by using the Poincaré map, and it is shown that the chaotic solutions take place via a cascade of period-doubling bifurcations

Seasonal Effects on a Beddington-DeAngelis Type Predator-Prey System with Impulsive Perturbations

We study a Beddington-DeAngelis type predator-prey system with impulsive perturbation and seasonal effects. First, we numerically observe the influence of seasonal effects on the system without impulsive perturbations. Next, we find the conditions for the local and global stabilities of prey-free periodic solutions by using Floquet theory for the impulsive equation and small amplitude perturbation skills, and for the permanence of the system via comparison theorem. Finally, we show that seasonal effects and impulsive perturbation can give birth to various kinds of dynamical behavior of the system including chaotic phenomena by numerical simulations

Dynamics of a Beddington-DeAngelis-type predator-prey system with constant rate harvesting

In the paper, a predator-prey system with Beddington-DeAngelis functional response and constant rate harvesting is considered. Various dynamical behaviors of the system including saddle-node points and a cusp of codimension 2 are investigated by using the analysis of qualitative method and bifurcation theory. Also, it is shown that the system undergoes several kinds of bifurcation such as the saddle-node bifurcation, the subcritical and supercritical Hopf bifurcation, Bogdanov-Takens bifurcation by choosing the death rate of the predator and the harvesting rate of the prey as the bifurcation parameters. Some numerical examples are illustrated in order to substantiate our theoretical results. These results unveil far richer dynamics compared to the system without harvesting

Spatiotemporal Dynamics of a Predator-Prey System with Linear Harvesting Rate

We investigate a spatial ratio-dependent predator-prey system with linear harvesting rate. By using linear stability and bifurcation analysis, we obtain the conditions for Hopf and Turing bifurcation in the spatial domain. In addition, we classify spatial pattern formations of the system by making use of numerical simulations. In fact, the numerical simulations unveil that the typical Turing patterns, such as spotted, spot-stripelike mixtures and stripelike patterns, can be observed even if the system has the linear harvesting rate. In order to analyze these patterns via the spatial frequency, the discrete Fourier transform is used. Moreover, we discuss that the linear harvesting system is more realistic than a predator-prey system with constant harvesting. Our results disclose that the spatially extended system with linear harvesting rate has more complex dynamic patterns in the two-dimensional space. It may help to understand the effects of harvesting on species in the real world

Research Article Extinction and Permanence of a Three-Species Lotka-Volterra System with Impulsive Control Strategies

A three-species Lotka-Volterra system with impulsive control strategies containing the biological control �the constant impulse � and the chemical control �the proportional impulse � with the same period, but not simultaneously, is investigated. By applying the Floquet theory of impulsive differential equation and small amplitude perturbation techniques to the system, we find conditions for local and global stabilities of a lower-level prey and top-predator free periodic solution of the system. In addition, it is shown that the system is permanent under some conditions by using comparison results of impulsive differential inequalities. We also give a numerical example that seems to indicate the existence of chaotic behavior. Copyright q 2008 Hunki Baek. 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. 1

Dynamics of a Predator-Prey System with Mixed Functional Responses

A predator-prey system with two preys and one predator is considered. Especially, two different types of functional responses, Holling type and Beddington-DeAngelis type, are adopted. First, the boundedness of system is showed. Stabilities analysis of system is investigated via some properties about equilibrium points and stabilities of two subsystems without one of the preys of system. Also, persistence conditions of system are found out and some numerical examples are illustrated to substantiate our theoretical results