Relaxation oscillations and canard explosion in a predator–prey system of Holling and Leslie types

Atabaigi, Ali; Barati, Ali

doi:10.1016/j.nonrwa.2017.01.006

Cited by 23 publications

(19 citation statements)

References 15 publications

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“…Table 1: Ecological implications of parameters in model (1) Parameter Ecological implications r 1 the intrinsic growth rate of the prey r 2 the natural growth rate of the predator b 1 the strength of competition among individuals for the prey h measure of the food quality Model (1) has been studied by many researchers from various aspects. On the one hand, some researchers have studied bifurcation phenomena with different functional response functions.…”

Section: Introductionmentioning

confidence: 99%

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Dynamics in a slow-fast Leslie-Gower predator-prey model with Beddington-DeAngelis functional response

Wang

2022

Preprint

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This paper deals with a class of Leslie-Gower predator-prey model with Beddington-DeAngelis functional response function. We assume that predator reproduce much slower than prey, which yield a slow-fast system. Using the normal form theory and geometric singular perturbation theory, we provide a detailed mathematical analysis for the existence of supercritical Hopf bifurcation, canard explosion and relaxation oscillation. Numerical simulations are also carried out to substantiate our analytical results.2000 Mathematics Subject Classification: 34C37, 34C07, 37G15.

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Section: Introductionmentioning

confidence: 99%

“…J. Huang et.al [13] and Y. Dai, Y. Zhao, B. Sang [6] investigated the Hopf bifurcation of the model (1) with generalized Holling type III functional response. J. Zhang, J. Su [30] and Y. Dai, Y. Zhao [7] considered predator-prey model with Holling IV functional response.…”

Section: Introductionmentioning

confidence: 99%

Dynamics in a slow-fast Leslie-Gower predator-prey model with Beddington-DeAngelis functional response

Wang

2022

Preprint

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“…Xia et al discussed relaxation oscillations and the mechanism in a periodically excited vector field with pitchfork-Hopf bifurcation [21]. Atabaigi and Barati studied relaxation oscillations and canard explosion in a predatorprey system of Holling and Leslie types [22]. Ai and Sadhu considered the entry-exit theorem and relaxation oscillations in slow-fast planar systems [23].…”

Section: Introductionmentioning

confidence: 99%

Relaxation Oscillations and Dynamical Properties in Two Time-Delay Slow-Fast Modified Leslie-Gower Models

2020

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In this paper, we consider two kinds of time-delay slow-fast modified Leslie-Gower models. For the first system, we prove the existence and uniqueness of relaxation oscillation cycle through the geometric singular perturbation theory and entry-exit function. For the second system, we put forward a conjecture that the relaxation oscillation of the system is unique. Numerical simulation also verifies our results for the systems.

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“…15–18 Such oscillatory behaviors often behave in periodic states characterized by a combination of relatively large amplitude (in spiking state) and nearly harmonic small amplitude oscillations (in rest state). 19–22…”

Section: Introductionmentioning

confidence: 99%

Melnikov-threshold-triggered mixed-mode oscillations in a family of amplitude-modulated forced oscillator

Wang

Zhang

et al. 2019

Journal of Low Frequency Noise, Vibration and Active Control

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This paper deals with the transitions through Melnikov thresholds and the corresponding fast-slow dynamics in a family of bi-parametric mechanical oscillators subjected to an amplitude modulation force both analytically and numerically. Applying Melnikov analytical method, the threshold condition for the occurrence of mixed-mode oscillations is obtained. First, the Melnikov threshold curves are drawn in different parameter space in the unperturbed system. Next, as the control variable crosses above the threshold, two states about mixed-mode oscillations can be identified: the rest state corresponding to local limit cycle from nontrivial equilibrium point and the spiking state corresponding to large amplitude cycle produced by saddle-node bifurcation of limit cycles. Such mixed-mode oscillations are created since the amplitude modulation force slowly and periodically switches between the rest states and spiking states. Furthermore, we elucidate the effect of modulation frequency on phase portraits and dynamical behaviors, and some numerical simulations are also included to illustrate the validity of our study.

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Relaxation oscillations and canard explosion in a predator–prey system of Holling and Leslie types

Cited by 23 publications

References 15 publications

Dynamics in a slow-fast Leslie-Gower predator-prey model with Beddington-DeAngelis functional response

Dynamics in a slow-fast Leslie-Gower predator-prey model with Beddington-DeAngelis functional response

Relaxation Oscillations and Dynamical Properties in Two Time-Delay Slow-Fast Modified Leslie-Gower Models

Melnikov-threshold-triggered mixed-mode oscillations in a family of amplitude-modulated forced oscillator

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