Singular perturbations of the Holling I predator-prey system with a focus

Zegeling, André; Kooij, Robert E.

doi:10.1016/j.jde.2020.04.011

Cited by 24 publications

(14 citation statements)

References 37 publications

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“…As was shown in [16] system (1.1) is bounded under the conditions which we imposed on h(x), p(x) and δ in Definitions 1.1, 1.2 and 1.3, i.e. any solution starting in the first quadrant of the phase plane will enter (and stay in) a bounded region.…”

Section: Global Propertiesmentioning

confidence: 99%

“…A conclusion from [16] is that the perturbed singular system in the case of a focus can only contain limit cycles coming from the degenerate singularity at x = x g = 1 and from a big singular cycle as indicated in Fig. 16.…”

Section: Perturbation Of the Degenerate Systemmentioning

confidence: 99%

“…The previous bifurcation mechanisms showed the existence of two limit cycles created from a Andronov-Hopf bifurcation and three limit cycles from a center case bifurcation. In [16] the limit of δ ↑ 1 was discussed for the situation of a 0 = a 1 = 0 in (1.2), a system traditionally referred to as the Holling I system. It was proved that when the singularity is a focus, then exactly two limit cycles are created after perturbation of the singular system δ = 1.…”

Section: Singular Bifurcation Near ı =mentioning

confidence: 99%

“…Therefore a natural starting point is to take it to be a constant instead of choosing some specific asymptotic behaviour like in the cases of the Holling II and Holling III types. Moreover, there are some biological arguments to justify this cut-off, see [16]. Another reason for this choice is that it allows for some general extensions by making a small perturbation of the functional response function.…”

Section: Introductionmentioning

confidence: 99%

“…The two parameters a 0 and a 1 do not have an intrinsic biological meaning. As was discussed in [16], there are two kinds of modelling, one is phenomenological where the functional response is chosen in such a way to fit experimental data, the other are the so-called mechanistic models where the function is derived from fundamental biological properties. Our choice is of the phenomenological type similar to what Holling did in his paper [13].…”

Section: Introductionmentioning

confidence: 99%

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Several Bifurcation Mechanisms for Limit Cycles in a Predator–Prey System

Zegeling

Kooij

2021

Qual. Theory Dyn. Syst.

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The research presented in this paper compares the occurrence of limit cycles under different bifurcation mechanisms in a simple system of two-dimensional autonomous predator-prey ODEs. Surprisingly two unconventional approaches, for a singular system and for a system with a center, turn out to produce more limit cycles than the traditional Andronov-Hopf bifurcation. The system has a functional response function which is a monotonically increasing cubic function of x for 0 ≤ x ≤ 1 where x represents the prey density, and which is constant for x > 1. It acts as a proxy for investigating more general systems. The following results are obtained. For the Andronov-Hopf bifurcation the highest order of the weak focus is 2 and at most 2 small-amplitude limit cycles can be created. In the center bifurcation cases are shown to exist with at least 3 limit cycles. In the singular perturbation cases are shown to exist with at least 4 limit cycles and in some cases an exact upper bound of 2 limit cycles is obtained. Finally we indicate how the conclusions can be extended to more general systems. We show how an arbitrary number of limit cycles can be created by choosing an appropriate functional response function and growth function for the prey. One special situation is the system with group defense: the three bifurcation mechanisms typically produce less limit cycles if a group defense element is included.

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Section: Global Propertiesmentioning

confidence: 99%

Section: Perturbation Of the Degenerate Systemmentioning

confidence: 99%

Section: Singular Bifurcation Near ı =mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Several Bifurcation Mechanisms for Limit Cycles in a Predator–Prey System

Zegeling

Kooij

2021

Qual. Theory Dyn. Syst.

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A singularly perturbed vector‐bias malaria model incorporating bed net control

Salman

2022

Math Methods in App Sciences

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A malaria transmission disease model with host selectivity and insecticide‐treated bed nets (ITNs), as an intervention for controlling the disease, is formulated. Since the vector is an insect, the vector time scale is faster than the host time scale. This leads to a singularly perturbed model with two distinctive intrinsic time scales, two‐slow for the host and one‐fast for the vector. The basic reproduction number frakturℜfrakturo$$ {\mathfrak{\Re}}_{\mathfrak{o}} $$ is calculated, and the local stability analysis is performed at equilibria of the model when the perturbation parameter ϵ>0$$ \epsilon >0 $$. The model is analyzed when ϵ→0$$ \epsilon \to 0 $$ using asymptotic expansions technique. The results show that if over 30% of humans use ITNs, then frakturℜfrakturo$$ {\mathfrak{\Re}}_{\mathfrak{o}} $$ can be reduced below 1, and hence, malaria disease can be eliminated. In addition, the dynamics on the slow surface indicate that the infected vectors decay fast when ϵ=0.001$$ \epsilon =0.001 $$ according to the numerical simulations.

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