Complex dynamics of a three species food-chain model with Holling type IV functional response

Upadhyay, Ranjit Kumar; Raw, S. N.

doi:10.15388/na.16.3.14098

Cited by 15 publications

(16 citation statements)

References 30 publications

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“…They discovered an interesting variety of dynamical chaos which they termed as short-term recurrent chaos (strc); a form of chaotic behavior, wherein it is interrupted by other kinds of dynamics at irregular and unpredictable intervals. We found the presence of this phenomenon in several other predator-prey models [Upadhyay & Rai, 2001;Rai & Upadhyay, 2004, 2006Upadhyay, 2000Upadhyay, , 2008. However, whether this phenomenon is common in ecological systems remains to be investigated.…”

Section: Introductionmentioning

confidence: 71%

“…Since the important work of Sir Robert May [1975May [ , 1976, deterministic chaos has been studied in models by many authors theoretically [May & Oster, 1976;Beddington et al, 1976;Hastings & Powell, 1991;Rai & Sreenivasan, 1993;Upadhyay & Rai, 1997Upadhyay et al, 1998;Stone & He, 2007;Upadhyay, 2000Upadhyay, , 2008, in the laboratory [Costantino et al, 1997;Fussmann et al, 2000;King et al, 2004] and in the field [Hanski et al, 1993;Ellner & Turchin, 1995;Turchin & Hanski, 1997;Turchin & Ellner, 2000]. Although chaos was observed by many authors in the models they had considered, it was not observed either in the laboratory studies [Cushing et al, 2001 or from the field studies.…”

Section: Introductionmentioning

confidence: 99%

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Observability of Chaos and Cycles in Ecological Systems: Lessons From Predator–prey Models

Upadhyay

2009

Int. J. Bifurcation Chaos

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We examine and assess deterministic chaos as an observable. First, we present the development of model ecological systems. We illustrate how to apply the Kolmogorov theorem to obtain limits on the parameters in the system, which assure the existence of either stable equilibrium point or stable limit cycle behavior in the phase space of two-dimensional (2D) dynamical systems. We also illustrate the method of deriving conditions using the linear stability analysis. We apply these procedures on some basic existing model ecological systems. Then, we propose four model ecological systems to study the dynamical chaos (chaos and intermittent chaos) and cycles. Dynamics of two predation and two competition models have been explored. The predation models have been designed by linking two predator-prey communities, which differ from one another in one essential way: the predator in the first is specialist and that in the second is generalist. The two competition models pertain to two distinct competition processes: interference and exploitative competition. The first competition model was designed by linking two predator-prey communities through inter-specific competition. The other competition model assumes that a cycling predator-prey community is successfully invaded by a predator with linear functional response and coexists with the community as a result of differences in the functional responses of the two predators. The main criterion behind the selection of these two model systems for the present study was that they represent diversity of ecological interactions in the real world in a manner which preserves mathematical tractability. For investigating the dynamic behavior of the model systems, the following tools are used: (i) calculation of the basin boundary structures, (ii) performing two-dimensional parameter scans using two of the parameters in the system as base variables, (iii) drawing the bifurcation diagrams, and (iv) performing time series analysis and drawing the phase space diagrams. The results of numerical simulation are used to distinguish between chaotic and cyclic behaviors of the systems.The conclusion that we obtain from the first two model systems (predation models) is that it would be difficult to capture chaos in the wild because ecological systems appear to change their attractors in response to changes in the system parameters quite frequently. The detection of chaos in the real data does not seem to be a possibility as what is present in ecological systems is not robust chaos but short-term recurrent chaos. The first competition model (interference competition) shares this conclusion with those of predation ones. The model with exploitative competition suggests that deterministic chaos may be robust in certain systems, but it would not be observed as the constituent populations frequently execute excursions to extinctionsized densities. Thus, no matter how good the data characteristics and analysis techniques are, dynamical chaos may continue to elude ecologists. On the other hand, the model...

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Section: Introductionmentioning

confidence: 71%

Section: Introductionmentioning

confidence: 99%

Observability of Chaos and Cycles in Ecological Systems: Lessons From Predator–prey Models

Upadhyay

2009

Int. J. Bifurcation Chaos

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“…Motivated by Jana, Ghorai, and Kar [5] who carried out the research on the dynamics of prey-predator with stage-structure for the predator. Then Wang [12] did the same in analyzing prey-predator models with stage-structure for predators. There are several results of the research on the dynamics of preypredator and how predator predation patterns.…”

Section: Introductionmentioning

confidence: 99%

“…Dubei [2] developed a general mathematical Gauss model consisting of two prey and one predator populations with Holling type II the functional responses. Upadhyay and Raw [12] examined the dynamical behavior of a three species food chain system with Holling type IV of the predator response. Kar and Chaudury [8] describe a two-prey one-predator model based on Lotka-Volterra dynamics which are affected not only by harvesting but also by the presence of a predator.…”

Section: Introductionmentioning

confidence: 99%

Dynamics and Numerical Simulation of Stage Structure Prey-Predator Models

Savitri¹,

Abadi²

2018

Proceedings of the International Conference on Science and Technology (ICST 2018)

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In the present paper, we studied a prey-predator model with stage-structure for prey. Prey populations which are divided into subpopulations, one is immature prey and the other is mature prey. We investigate the stability of prey-predator models as well as local analysis of four equilibrium obtained and analyzed how the dynamical behavior if the solution is stable or unstable. Numerical simulations are performed from the system around the equilibrium point to show dynamic behavior around that point with several different conditions andinitial values.

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“…Generally, Holling type-I,II and type-III responses have been applied to many theoretical studies. Huang and Xiao [8,6 ] and many other researchers [7,10,13,17,23,25,26,29,34 ] investigated the bifurcation analysis and stability of a Prey-Predator model with Type-II & Type-IV responses.…”

mentioning

confidence: 99%

Three Species Epidemiological Model with Holling Type Functional Responses

2020

International Journal of Mathematical Models and Methods in Applied Sciences

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In the analytical and numerical study, the interaction between three species is modeled, where in the three species are identified as a prey , which is susceptible, the infected prey and the predator with type-II and type-Iv functional responses, which represents a mathematical model of eco-epidemiology. This study is carried out in both analytically and numerically. The boundedness of the model is studied and the stability analysis of the model is carried out at the positive equilibrium point in terms of locally and globally. The conditions for the occurrence of Hopf bifurcation with fixed biological parameter values are investigated and also it is noticed that the bifurcation occurs by sensitive changes in the parameter values of 3 0 l l, , and  which represents the growth rate of a predator, transmission rate from infected prey to susceptible prey and half-saturation constant of predator respectively. Further, the stochastic nature of the model is analyzed both analytically and numerically. It is observed that the system exhibits chaotic behavior with the sensitive parameter values which causes large environmental fluctuations.

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Complex dynamics of a three species food-chain model with Holling type IV functional response

Cited by 15 publications

References 30 publications

Observability of Chaos and Cycles in Ecological Systems: Lessons From Predator–prey Models

Observability of Chaos and Cycles in Ecological Systems: Lessons From Predator–prey Models

Dynamics and Numerical Simulation of Stage Structure Prey-Predator Models

Three Species Epidemiological Model with Holling Type Functional Responses

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