Chaos in a Three‐Species Food Chain

Hastings, Alan; Powell, Thomas M.

doi:10.2307/1940591

Cited by 941 publications

(717 citation statements)

References 31 publications

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“…Finally, our results have implications for the rich body of literature that studies the population dynamics of small modules of (usually two to four) interacting populations (McCann et al 1998;Hastings & Powell 1991;Polis & Holt 1992;Gjata et al 2012). Due to the allometric relationship between body mass and metabolic rates (Brown et al 2004), the dynamics of interacting populations depends on their relative niche positions (i.e., predatorÀ-prey body-mass ratios, Yodzis and Innes 1992) and stable configurations that allow for the persistence of all species are usually found if predators are larger than their prey (Otto et al 2007;Kartascheff et al 2010;Heckmann et al 2012).…”

Section: Discussionmentioning

confidence: 72%

Prohibition rules for three-node substructures in ordered food webs with cannibalistic species

Guill

Paulau

2015

Israel J Ecol Evol

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We evaluate the spectrum of ordered three-node substructures in food webs taking self-links (cannibalism) into account. If the order of nodes in the network cannot be neglected, 512 substructures can be distinguished. Simple statistical models of networks impose constraints on the structure that prohibit a large number of substructures completely. We analyse two variants of the widely used niche model, the original niche model and the generalised niche model, and show analytically and numerically that they exclude 344 and 320 substructures, respectively. The prohibition rules for three-node substructures in the two niche-model variants are further contrasted with a large set of empirical food webs, which reveals that up to about 30% of the three-node substructures that occur in empirical food webs are prohibited by the model algorithms.

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Section: Discussionmentioning

confidence: 72%

Prohibition rules for three-node substructures in ordered food webs with cannibalistic species

Guill

Paulau

2015

Israel J Ecol Evol

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“…For the vector field X η (x, 0, w, z), we consider the following data set: Since the corresponding parameters in three-dimensional hyperspace 4 and 5 are identical for data sets (17) and (18), the bifurcation diagram with respect to w 2 ∈ (3.5, 5.5) is identical to Figure 3. For the vector field X η (x, y, w, z), the bifurcation diagram with respect to w 2 ∈ (3.5, 5.5) is identical to Figure 3 for the following set of data: …”

Section: Numerical Simulationsmentioning

confidence: 99%

“…These studies indicate the complexity in the dynamics of such systems. The coexistence of species may not be only in terms of stability of singularities and orbits, it may happen in more complex forms such as quasi-periodic or strange attractors [17].…”

Section: Introductionmentioning

confidence: 99%

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Complex behaviour in four species food-web model

Gakkhar¹,

Priyadarshi²,

Banerjee³

2012

Journal of Biological Dynamics

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A four-dimensional food-web system consisting of a bottom prey, two middle predators and a generalist predator has been developed with modified functional response. The system is well posed and dissipative. Some results on uniform persistence have been developed. The dynamics of the system is found to be chaotic for certain choice of parameters. The coexistence of all four species is possible in the form of periodic orbits/strange attractors for suitably chosen set of parameters.

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“…The classical food chain models with only two trophic levels are shown to be insufficient to produce realistic dynamics [12][13][14][15][16]. Therefore we consider the following three trophic levels food chain model with ratio-dependence which is a simple relation between the populations of the three species: z prey on y and only y, and y prey on x and nutrient recycling is not accounted for.…”

Section: Introductionmentioning

confidence: 99%

Hopf Point Analysis for Ratio-Dependent Food Chain Models

Kara

Can

2008

MCA

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-In this paper periodic and quasi-periodic behavior of a food chain model with three trophic levels are studied. Michaelis-Menten type ratio-dependent functional response is considered. There are two equilibrium points of the system. It is found out that at most one of these equilibrium points is stable at a time. In the parameter space, there are passages from instability to stability, which are called Hopf bifurcation points. For the first equilibrium point, it is possible to find bifurcation points analytically and to prove that the system has periodic solutions around these points. However for the second equilibrium point the computation is more tedious and bifurcation points can only be found by numerical experiments. It has been found that around these points there are periodic solutions and when this point is unstable, the solution is an enlarging spiral from inside and approaches to a limit cycle.

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Chaos in a Three‐Species Food Chain

Cited by 941 publications

References 31 publications

Prohibition rules for three-node substructures in ordered food webs with cannibalistic species

Prohibition rules for three-node substructures in ordered food webs with cannibalistic species

Complex behaviour in four species food-web model

Hopf Point Analysis for Ratio-Dependent Food Chain Models

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