Order and chaos in predator to prey ratio-dependent food chain (original) (raw)
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2020
We study of dynamics of a three species food chain model, where the growth rate of middle predator is reduced due to prey population is also consumed by top predator and growth rate of prey is also suppressed due to the same reason. The goal of our study is to demonstrate the presence of chaos in the class of ecological models. The model is analyzed in term of stability like as local and global stability at each equilibrium point including interior equilibrium point. To check the the validate of theoretical formulation by numerical simulation by considering the required set of parameters. Subject class :30C45.
Order and chaos in a food web consisting of a predator and two independent preys
Communications in Nonlinear Science and Numerical …, 2005
A food web consisting of two independent preys and a predator is modeled incorporating modified Holling type-II functional response. The mathematical model has a unique and bounded solution. The necessary and sufficient conditions for persistence of the food web are obtained. Bifurcation diagram has been obtained for selected range of different parameters. The system exhibits chaos for a range of parametric values when long time behavior is studied. The computation of Lyapunov exponents and the existence of strange attractor also indicate the chaotic behavior of the system.
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International Journal of Biomathematics, World Scientific Publications, 2016
Complex dynamics of modified Hastings-Powell(HP)model(phytoplankton-zooplankton-fish)with Holling type IV functional response and density dependent mortality(closure terms)for top predator species is investigated in this article. Closure terms describe the mortality of top predator in plankton food chain models. Modified HP model with Holling type IV functional response gives rise to similar type of chaotic dynamics(inverted ‘teacup attractor’)as observed in original HP model with Holling type II functional response. It is observed that introduction of nonlinear closure terms eliminate chaos and system dynamics becomes stable. Observation of this article support the ‘Steele-Henderson conjecture’ that, nonlinear closure terms eliminate or reduces limit cycles and chaos in plankton food chain models. Chaotic or stable dynamics are numerically verified by Lyapunov exponents method and Sil’nikov eigenvalue analysis and also illustrated graphically by plotting bifurcation diagrams. It is assumed that mortality of fish population, caused by higher order predators(which are not explicitly included in the model)is not constant, rather it exhibits random variation throughout the year. To incorporate the effect of random mortality of fish population, white noise term is introduced into the original deterministic model. It is observed that the corresponding stochastic model is stable in mean square when the intensity of noise is small. Key words : Holling type IV; chaos; closure term; white noise; mean square. 1
Chaos in the three-species Sokol-Howell food chain system with fear
Communications in Mathematical Biology and Neuroscience, 2022
In this paper, the influence of predation fear on the dynamics of the three species food chain system is formulated mathematically and investigated. It is assumed that the food is transferred from the lower level to the upper level according to the Sokol-Howell type of functional response due to the anti-predator property of each prey in the system. The boundedness and persistence conditions are established for the proposed food chain system. The local and global stability analysis is investigated. The occurrence conditions of local bifurcation including the Hopf bifurcation near the equilibrium points are obtained. In the end, numerical simulation is performed to validate the theoretical results and present the dynamical behavior of the system. Different mathematical tools such as strange attractor, bifurcation diagram, and Lyapunov exponents are used to detect chaos in the proposed system. It is observed that the model is capable of exhibiting complex dynamics including chaos. It is also pointed out that a suitable predation fear can control the chaotic dynamics and make the system stable.
Complex Dynamics of a Three Species Food-Chain Model with Holling type IV Functional Response
Nonlinear Analysis: Modelling and Control
In this paper, dynamical complexities of a three species food chain model with Holling type IV predator response is investigated analytically as well as numerically. The local and global stability analysis is carried out. The persistence criterion of the food chain model is obtained. Numerical bifurcation analysis reveals the chaotic behavior in a narrow region of the bifurcation parameter space for biologically realistic parameter values of the model system. Transition to chaotic behavior is established via period-doubling bifurcation and some sequences of distinctive period-halving bifurcation leading to limit cycles are observed.
Chaos and bifurcation of a nonlinear discrete prey-predator system
The discrete-time Prey-predator system obtained by two dimensional map was studied in present study. The fixed points and their stability were analyzed. Bifurcation diagram has been obtained for selected range of different parameters. As some parameters varied, the model exhibited chaos as a long time behavior. Lyapunov exponents and fractal dimension of the chaotic attractor of our map were also calculated. Complex dynamics such as cycles and chaos were observed.
Nonlinear Analysis: Modelling and Control
We study how predator behavior influences community dynamics of predator-prey systems. It turns out that predator behavior plays a dominant role in community dynamics. The hybrid model studied in this paper reveals that period-doubling and period-doubling reversals can generate short-term recurrent chaos (STRC), which mimics chaotic dynamics observed in natural populations. STRC manifests itself when deterministic changes in a system parameter interrupt chaotic behavior at unpredictable intervals. Numerical results reinforce an earlier suggestion that period-doubling reversals could control chaotic dynamics in ecological models. In ecological terms, the prey and intermediate predator populations may go to extinction in the event of a catastrophe. The top predator is always a survivor. In contrast to this, this is not the case when the constituent populations are interacting through Holling type II functional response. Even this top predator can go to extinction in the event of such ...
On the dynamical behavior of three species food web model
Chaos Solitons & Fractals, 2007
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.
Nonlinear Analysis: Modelling and Control
We study how predator behavior influences community dynamics of predatorprey systems. It turns out that predator behavior plays a dominant role in community dynamics. The hybrid model studied in this paper reveals that period-doubling and period-doubling reversals can generate short-term recurrent chaos (STRC), which mimics chaotic dynamics observed in natural populations. STRC manifests itself when deterministic changes in a system parameter interrupt chaotic behavior at unpredictable intervals. Numerical results reinforce an earlier suggestion that period-doubling reversals could control chaotic dynamics in ecological models. In ecological terms, the prey and intermediate predator populations may go to extinction in the event of a catastrophe. The top predator is always a survivor. In contrast to this, this is not the case when the constituent populations are interacting through Holling type II functional response. Even this top predator can go to extinction in the event of such catastrophes.