The role of top predator interference on the dynamics of a food chain model

In this paper, the effects of top predator interference on the dynamics of a food chain model involving an intermediate and a top predator are considered. It is assumed that the interaction between the prey and intermediate predator follows the Volterra scheme, while that between the top predator and its favorite food depends on Beddington–DeAngelis type of functional response. The boundedness of the system, existence of an attracting set, local and global stability of non-negative equilibrium points are established. Number of the bifurcation and Lyapunov exponent bifurcation diagrams is established. It is observed that, the model has different types of attracting sets including chaos. Moreover, increasing the top predator interference stabilizes the system, while increasing the normalization of the residual reduction in the top predator population destabilizes the system.     

Dynamical consequences of predator interference in a tri-trophic model food chain

A model food chain involving a specialist and a generalist predator is proposed and studied. One of the salient features of this model food chain is that it combines both the schemes (Volterra and Leslie) of modeling predator–prey interaction in one system in such a way that the demerits of these individual formulations are suppressed and the resulting model system represents a common unit of real world food webs. The stability analysis of the proposed model is carried out. The Hopf bifurcation conditions of the positive equilibrium point are established. Our numerical computations show that chaotic dynamics is sensitive to changes in values of parameters measuring attributes of either interacting populations or their environments. Two dimensional parameter scans suggest that the model food chain displays short-term recurrent chaos. This can be regarded as a plausible explanation for why it has been so difficult to detect deterministic chaos in natural populations.     

Permanence and extinction of a three-species ratio-dependent food chain model with delay and prey diffusion

In this paper, we study a diffusive three-species ratio-dependent food chain model, using differential inequality, to obtain sufficient conditions that ensure the permanence of the system and the extinction of predator species. Our results reinforce the main result of Sun Wen, Shihua Chen and Huihai Mei [Positive periodic solution of a more realistic three-species Lotka-Volterra model with delay and density regulation, Chaos, Solitons and Fractals, in press].     

Permanence for the Michaelis-Menten type discrete three-species ratio-dependent food chain model with delay

A discrete three trophic level food chain model with ratio-dependent Michaelis-Menten type functional response is investigated. It is shown that under some appropriate conditions the system is permanent. The results indicate that, to make the species coexist in the long run, it is a surefire strategy to keep the death rate of the predator and top predator rather small and the intrinsic growth rate of the prey relatively large.     

The dynamics of a mutual interference age structured predator-prey model with time delay and impulsive perturbations on predators

In this paper, we introduce a mutual interference age structured predator-prey (natural enemy-pest) model with constant maturation time delay for the prey, and then propose a pest management strategy by constant periodic releasing for the predator. We show that there exists a global attractive pest-eradication periodic solution when the periodic releasing amount μ₁ and μ₂ are lager than some critical value. Further, to obtain a more effective pest control strategy, we give the conditions (involving the estimate of μ₁ and μ₂) in which the model is uniformly permanent and the pest population is under the economic threshold level. We believe that the results will provide reliable tactic basis for the practical pest management.     

Analysis of a tritrophic food chain model

We consider a tritrophic food chain model in which the classical assumption of the domino effect is supplemented with an adaptive parameter. Dynamical behaviours such as boundedness, existence of periodic orbits, persistence, as well as stability are analyzed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Disease control in a food chain model supplying alternative food

Necessity to find a non-chemical method of disease control is being increasingly felt due to its eco-friendly nature. In this paper the role of alternative food as a disease controller in a disease induced predator–prey system is studied. Stability criteria and the persistence conditions for the system are derived. Bifurcation analysis is done with respect to rate of infection. The main goal of this study is to show the non-trivial consequences of providing alternative food in a disease induced predator–prey system. Numerical simulation results illustrate that there exists a critical infection rate above which disease free system cannot be reached in absence of alternative food whereas supply of suitable alternative food makes the system disease free up to certain infection level. We have computed the disease free regions in various parametric planes. This study is aimed to introduce a new non-chemical method for controlling disease in a predator–prey system.     

Effect of time-delay on a food chain model

This paper aims to study the effect of discrete time-delay on a tritrophic food chain model with Holling type-II functional responses. Dynamical behaviours such as boundedness, stability, persistence and bifurcation of the model are studied. Our analytical findings are illustrated through computer simulation. Biological implications of our analytical findings are addressed critically.     

Dynamics of a non-autonomous ratio-dependent food chain model

A simple non-autonomous ratio-dependent food chain model is investigated. It is shown that the system is permanence, extinction, ultimate boundedness and globally asymptotic stability under some appropriate conditions. Moreover, by employing Mawhin’s coincidence degree theory, some easily applicable criteria are established for the global existence of positive periodic solution of this model.     

Stability and Hopf bifurcation analysis in a three-level food chain system with delay

A class of three level food chain system is studied. With the theory of delay equations and Hopf bifurcation, the conditions of the positive equilibrium undergoing Hopf bifurcation is given regarding τ as the parameter. The stability and direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold argument, and numerical simulations are performed to illustrate the analytical results.     

Positive periodic solution of a more realistic three-species Lotka-Volterra model with delay and density regulation

In the paper, a more realistic three-species ratio-dependent Lotka-Volterra model with delay and diffusive and density regulation is investigated. By mean of the powerful and effective coincidence degree theory, we establish sufficient conditions for the existence of at least one positive periodic solution of the model. What’s more, the conditions are easily verifiable.     

Complex dynamics in a food chain with slow and fast processes

This paper is devoted to the analysis of the dynamic behavior of a three-species food chain model, in which two predators compete for the same prey while one of the predators feeds on the other. Under the assumption that the time responses of the three trophic levels are extremely diversified, the model is proved to have homoclinic orbit. We firstly use geometric singular perturbation method to detect singular homoclinic orbits as well as parameter combinations for which these orbits exist. Then, we show, numerically, that there exist also nonsingular homoclinic orbits that tend toward the singular ones for slightly different parameter values. This analysis is particularly helpful to understanding the chaotic behavior of the food chains.     

Global dynamics of a general brucellosis model with discrete delay

For the prevention and control of brucellosis, it is important to investigate the mechanism of brucellosis transmission. Based on the characteristics of the spread of brucellosis, a susceptible-exposed-infectious-brucella (SEIB) delay dynamic model is proposed with the general incidence, elimination rate and shedding rate of pathogen. Under biologically motivated assumptions, it shows the uniqueness of the endemic equilibrium, and investigates the global asymptotically stability of the disease-free equilibrium and the endemic equilibrium. The results suggest that the global stability of equilibria depends entirely on the basic reproduction number $R_0$ and time delay is harmless for the stability of equilibria. Finally, some specific examples and numerical simulations are used to illustrate the utilization of research results and reveal the biological significance of hypothesis $(H_7)$, which implies that the dynamics of brucellosis transmission depend largely on the development of the prevention and control strategies.     

Analysis of ratio-dependent food chain model

In this paper, a food chain model with ratio-dependent functional response is studied under homogeneous Neumann boundary conditions. The large time behavior of all non-negative equilibria in the time-dependent system is investigated, i.e., conditions for the stability at equilibria are found. Moreover, non-constant positive steady-states are studied in terms of diffusion effects, namely, Turing patterns arising from diffusion-driven instability (Turing instability) are demonstrated. The employed methods are comparison principle for parabolic problems and Leray-Schauder Theorem.     

The chaos and control of a food chain model supplying additional food to top-predator

The control and management of chaotic population is one of the main objectives for constructing mathematical model in ecology today. In this paper, we apply a technique of controlling chaotic predator–prey population dynamics by supplying additional food to top-predator. We formulate a three species predator–prey model supplying additional food to top-predator. Existence conditions and local stability criteria of equilibrium points are determined analytically. Persistence conditions for the system are derived. Global stability conditions of interior equilibrium point is calculated. Theoretical results are verified through numerical simulations. Phase diagram is presented for various quality and quantity of additional food. One parameter bifurcation analysis is done with respect to quality and quantity of additional food separately keeping one of them fixed. Using MATCONT package, we derive the bifurcation scenarios when both the parameters quality and quantity of additional food vary together. We predict the existence of Hopf point (H), limit point (LP) and branch point (BP) in the model for suitable supply of additional food. We have computed the regions of different dynamical behaviour in the quantity–quality parametric plane. From our study we conclude that chaotic population dynamics of predator prey system can be controlled to obtain regular population dynamics only by supplying additional food to top predator. This study is aimed to introduce a new non-chemical chaos control mechanism in a predator–prey system with the applications in fishery management and biological conservation of prey predator species.     

Global dynamics of a discretized SIRS epidemic model with time delay

We derive a discretized SIRS epidemic model with time delay by applying a nonstandard finite difference scheme. Sufficient conditions for the global dynamics of the solution are obtained by improvements in discretization and applying proofs for continuous epidemic models. These conditions for our discretized model are the same as for the original continuous model.     

Dynamics of a three-species food chain model with adaptive traits

A three-species food chain model is proposed with dynamically variable adaptive traits in the intermediate consumer. We prove that its solutions are non-negative and bounded, and we analyze the existence and stability of its equilibria. By applying Li and Muldowney’s [Li MY, Muldowney J. On Bendixson’s criterion. J Differ Equ 1993;106:27–39] high-dimensional Bendixson criterion, we show that the positive equilibrium is globally stable under specific conditions. We support our analytical findings with numerical simulations.     

The dynamic complexity of a three species food chain model

In this paper, a three-species food chain model is analytically investigated on theories of ecology and using numerical simulation. Bifurcation diagrams are obtained for biologically feasible parameters. The results show that the system exhibits rich complexity features such as stable, periodic and chaotic dynamics.     

Computer-assisted analysis of chaos in a three-species food chain model

In this paper, a three-species food chain model with Holling type IV and Beddington–DeAngelis functional responses is formulated. Numerical simulations show that this system can generate chaos for some parameter values. But the mechanism behind chaos is still unclear only through numerical simulations. Then, using the topological horseshoe theories and Conley–Moser conditions, we present a computer-assisted analysis to show the chaoticity of this system in the topological sense, that is, it has positive topological entropy. We prove that the Poincaré map of this model possesses a closed uniformly hyperbolic chaotic invariant set, and it is topologically conjugate to a 2-shift map. At last, we consider the impact of fear on this three-species model. It is an important factor in controlling chaos in biological models, which has been validated in other models.