Migratory effect of middle predator in a tri‐trophic food chain model

A tri‐trophic food chain model in a two‐patch environment is considered. Although tri‐trophic food chain model is well studied, the study considering migration of middle predator is lacking. To the best of our knowledge, the present investigation is the first study in this direction. Both prey and predator density‐dependent migrations are considered to observe the effects on stability and persistence of the system. We observe that migration of middle predator has the ability to control chaos in tri‐trophic food chain model. Our results indicate that the chance of predator extinction enhances for prey density‐dependent middle predator migration. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

Revisited Hastings and Powell model with omnivory and predator switching

The effect of omnivory in predator–prey system is debatable regarding its stabilizing or destabilizing characteristics. Earlier theoretical studies predict that omnivory is stabilizing or destabilizing depending on the condition of the system. The effect of omnivory in the food chain system is not yet properly understood. In the present paper, we study the effect of omnivory in a tri-trophic food chain system on the famous Hastings and Powell model. Omnivory enhances the chance of predator switching between prey and middle predator. The novelty of this paper is to study the effect of predator switching of the top predator which is omnivorous in nature. Our results suggest that in the absence of switching, an increase of omnivory stabilizes the system from chaotic dynamics, however, if we further increase the strength of omnivory, the system becomes unstable and middle predator goes to extinction. It is also observed that the predator switching enhance the stability and persistence of all populations.     

STABILITY ANALYSIS OF A TRITROPHIC FOOD CHAIN MODEL WITH AN ADAPTIVE PARAMETER FOR THE PREDATOR

Abstract The study of three‐species communities have become the focus of considerable attention, and because the studies of ecological communities start with their food web, we consider a tritrophic food chain model comprised of the prey, the predator, and the super‐predator. The classical assumption of the domino effect is supplemented with an adaptive parameter for the predator (in the absence of prey). Thus, the model exhibits an equilibrium with the predator‐top‐predator steady state, which is a saddle point. Dynamical behaviors such as boundedness, existence of periodic orbits, persistence, as well as stability are analyzed. The long‐term coexistence of the three interacting species is addressed, and the stability analysis of the model shows that the biologically most relevant equilibrium point is globally asymptotically stable whenever it satisfies a certain criterion. Practical implications are explored and related to real populations.     

The dynamical behaviors of a food chain model with impulsive effect and Ivlev functional response

In this paper, a food chain model with Ivlev functional response and impulsive effect of top predator is investigated. Conditions for extinction of mid-level predator are given. By using the Floquet theory of linear τ-period impulsive differential equation and small amplitude perturbation skills, we show that the lowest-level prey and the mid-level predator extinction periodic solution is unstable, while the mid-level predator eradication periodic solution is stable, and meanwhile, we prove that the system is permanent if the impulsive period is larger than some critical value. Furthermore, influences of the impulsive perturbation on the inherent oscillation are studied numerically, which displays complicated behavior including a sequence of direct and inverse cascade of period doubling, period halfing as well as chaos.     

Pattern formation and control of spatiotemporal chaos in a reaction diffusion prey–predator system supplying additional food

Spatiotemporal dynamics of a predator–prey system in presence of spatial diffusion is investigated in presence of additional food exists for predators. Conditions for stability of Hopf as well as Turing patterns in a spatial domain are determined by making use of the linear stability analysis. Impact of additional food is clear from these conditions. Numerical simulation results are presented in order to validate the analytical findings. Finally numerical simulations are carried out around the steady state under zero flux boundary conditions. With the help of numerical simulations, the different types of spatial patterns (including stationary spatial pattern, oscillatory pattern, and spatiotemporal chaos) are identified in this diffusive predator–prey system in presence of additional food, depending on the quantity, quality of the additional food and the spatial domain and other parameters of the model. The key observation is that spatiotemporal chaos can be controlled supplying suitable additional food to predator. These investigations may be useful to understand complex spatiotemporal dynamics of population dynamical models in presence of additional food.     

On the dynamical behavior of three species food web model

In this paper, a mathematical model consisting of two preys one predator with Beddington–DeAngelis functional response is proposed and analyzed. The local stability analysis of the system is carried out. The necessary and sufficient conditions for the persistence of three species food web model are obtained. For the biologically reasonable range of parameter values, the global dynamics of the system has been investigated numerically. Number of bifurcation diagrams has been obtained; Lyapunov exponents have been computed for different attractor sets. It is observed that the model has different types of attractors including chaos.     

A Holling II functional response food chain model with impulsive perturbations

In this paper, we investigate a three trophic level food chain system with Holling II functional responses and periodic constant impulsive perturbations of top predator. Conditions for extinction of predator as a pest are given. By using the Floquet theory of impulsive equation and small amplitude perturbation skills, we consider the local stability of predator eradication periodic solution. Further, influences of the impulsive perturbation on the inherent oscillation are studied numerically, which shows the rich dynamics (for example: period doubling, period halfing, chaos crisis) in the positive octant. The dynamics behavior is found to be very sensitive to the parameter values and initial value.     

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.     

Evolving to the edge of chaos: Chance or necessity?

We show that ecological systems evolve to edges of chaos (EOC). This has been demonstrated by analyzing three diverse model ecosystems using numerical simulations in combination with analytical procedures. It has been found that all these systems reside on EOC and display short-term recurrent chaos (strc). The first two are non-linear food chains and the third one is a linear food chain. The dynamics of first two is dictated by deterministic changes in system parameters. In contrast to this, dynamics of the third model system (the linear food chain) is governed by both deterministic changes in system parameters as well as exogenous stochastic perturbations (unforeseen changes in initial conditions) of these dynamical systems.     

Evolving to the edge of chaos: Chance or necessity?

We show that ecological systems evolve to edges of chaos (EOC). This has been demonstrated by analyzing three diverse model ecosystems using numerical simulations in combination with analytical procedures. It has been found that all these systems reside on EOC and display short-term recurrent chaos (strc). The first two are non-linear food chains and the third one is a linear food chain. The dynamics of first two is dictated by deterministic changes in system parameters. In contrast to this, dynamics of the third model system (the linear food chain) is governed by both deterministic changes in system parameters as well as exogenous stochastic perturbations (unforeseen changes in initial conditions) of these dynamical systems.     

Spread of a disease and its effect on population dynamics in an eco-epidemiological system

In this paper, an eco-epidemiological model with simple law of mass action and modified Holling type II functional response has been proposed and analyzed to understand how a disease may spread among natural populations. The proposed model is a modification of the model presented by Upadhyay et al. (2008) [1]. Existence of the equilibria and their stability analysis (linear and nonlinear) has been studied. The dynamical transitions in the model have been studied by identifying the existence of backward Hopf-bifurcations and demonstrated the period-doubling route to chaos when the death rate of predator (μ₁) and the growth rate of susceptible prey population (r) are treated as bifurcation parameters. Our studies show that the system exhibits deterministic chaos when some control parameters attain their critical values. Chaotic dynamics is depicted using the 2D parameter scans and bifurcation analysis. Possible implications of the results for disease eradication or its control are discussed.     

Top-predator interference and gestation delay as determinants of the dynamics of a realistic model food chain

An attempt has been made to understand the role of top predator interference and gestation delay on the dynamics of a three species food chain model involving intermediate and top predator populations. Interaction between the prey and an intermediate predator follows the Volterra scheme (with Holling type IV functional response), while that between the top predator and its prey depends on Beddington–DeAngelis type functional response. Stability switches and Hopf-bifurcation occurs when the delay crosses some critical value. Model system exhibits irregular behavior when the interference is high or gestation period is larger than its critical value. Furthermore, the direction of Hopf-bifurcation and the stability of the bifurcating periodic solutions are determined using the center manifold theorem and normal form theory. Computer simulations have been carried out to illustrate the analytical findings. Different diagnostic tests, like, initial sensitivity, Lyapunov exponent, recurrence plot tests ensure the complex dynamical behavior of the model system. Finally, we observed the subcritical Hopf-bifurcation phenomena in the designed model system and the bifurcating periodic solution is unstable for the considered set of parameter values.     

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.     

Complex dynamics in the Leslie–Gower type of the food chain system with multiple delays

In this paper, we present a Leslie–Gower type of food chain system composed of three species, which are resource, consumer, and predator, respectively. The digestion time delays corresponding to consumer-eat-resource and predator-eat-consumer are introduced for more realistic consideration. It is called the resource digestion delay (RDD) and consumer digestion delay (CDD) for simplicity. Analyzing the corresponding characteristic equation, the stabilities of the boundary and interior equilibrium points are studied. The food chain system exhibits the species coexistence for the small values of digestion delays. Large RDD/CDD may destabilize the species coexistence and induce the system dynamic into recurrent bloom or system collapse. Further, the present of multiple delays can control species population into the stable coexistence. To investigate the effect of time delays on the recurrent bloom of species population, the Hopf bifurcation and periodic solution are investigated in detail in terms of the central manifold reduction and normal form method. Finally, numerical simulations are performed to display some complex dynamics, which include multiple periodic solution and chaos motion for the different values of system parameters. The system dynamic behavior evolves into the chaos motion by employing the period-doubling bifurcation.     

A generalization of Lotka–Volterra,GLV, systems with some dynamical and topological properties

It is shown that local asymptotic instability is related to the existence of a positive Lyapunov exponent which is a necessary condition for chaos. Also it is proved that linear transformations do not affect the dynamical behaviour of the system. A generalized Lotka–Volterra (GLV) model is introduced and proved that for specific choices of parameters it exhibits chaos. Knots and links which arise from the system which describe the behaviour of a typical nuclear spin are studied. We conjecture that knots and links associated GLV is much more general than Lorenz knots, and the one predator – two preys LV model exhibits chaos for general parameters.     

Waves analysis and spatiotemporal pattern formation of an ecosystem model

In this paper, we consider a predator–prey model given by a reaction–diffusion system. This model incorporates Holling-type-II (Michaelis–Menten) and modified Leslie-Gower functional responses. We show the existence of qualitatively different types of system behaviors realized for various parameter values. Our model is investigated with methods of the qualitative theory and the theory of bifurcations. We generalize the traveling waves existence method for populations dynamics with positive derivative densities, to the predator–prey system in which growth densities may change sign. Parallel to this is a discussion and an analysis of alternative model outcomes such as complex pattern formation and spatio-temporal chaos behavior.     

Study on autonomous and nonautonomous version of a food chain model with intraspecific competition in top predator

In this article, a three-species food chain model with intraspecific competition in top predators has been considered. Ecological and mathematical well posedness of the model system has been established by showing that all the solutions of the model are positive and bounded. The extinction scenarios of intermediate and top predator species along with the existence and stability of all equilibrium points have been discussed. The effects of competition and conversion efficiency of top predators in the dynamics of the system have been discussed with great thrust, and it is observed that the conversion efficiency of top predators deteriorates the stability of the system, whereas intraspecific competition in top predators enhances the stable coexistence of all the populations of the system. Further, nonautonomous version of the model system has been taken into consideration to study the impact of seasonal variation in the dynamics of the model system. Sufficient conditions for the existence of a globally attractive positive periodic have been established in a periodic environment. Finally, numerical simulations have been carried out to validate our analytical findings.     

Aggregation methods in food chains

The aim of this paper is to apply aggregation methods to food chains under batch and chemostat conditions. These predator-prey systems are modelled using ODEs, one for each trophic level. Because the models are based on mass conservation laws, they are conservative and this allows perfect aggregation. Furthermore, it is assumed that the ingestion rate of the predator is smaller than that of the prey. On this assumption, approximate aggregation can be performed, yielding further reduction of the dimension of the system. We will study a food chain often found in wastewater treatment plants. This food chain consists of sewage, bacteria, and worms. In order to show the feasibility of the aggregation methods, we will compare simulated results for the reduced and the full model of this food chain under chemostat conditions.     

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.