Permanency in predator–prey models of Leslie type with ratio-dependent simplified Holling type-IV functional response

Amirabad, H. Qolizadeh; RabieiMotlagh, Omid; Mohammadinejad, H.M.

doi:10.1016/j.matcom.2018.09.023

Cited by 9 publications

(9 citation statements)

References 19 publications

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“…They give the sufficient condition for Hopf bifurcation by calculating the first‐order Lyapunov quantity. Theorem 4.1 and conjecture 4.9 in Amirabad et al indicate that the multiplicity of the weak focus is greater than 1 and there are at least two limit cycles arising from the Hopf bifurcation. However, after above scaling, we know that system without the harvesting takes the form

({, \begin{matrix} left \\ array \frac{d x}{d t} = x (1 - x) - \frac{α x y^{2}}{x^{2} + β y^{2}}, \\ array \frac{d y}{d t} = σ y (1 - \frac{y}{x}) \end{matrix})

…”

Section: The System Without Harvestingmentioning

confidence: 99%

“…Zhou considered system with the ratio‐dependent Holling type III functional response and studied the stability of the positive equilibrium and the existence of periodic solutions by analyzing the Hopf bifurcation. Then, based on the Holling type IV functional response, Amirabad et al considered system with the ratio‐dependent Holling type IV functional response

((), P ((), \frac{x}{y}) = \frac{m x y}{x^{2} + a^{2} y^{2}})

as follows:

({, \begin{matrix} left \\ array \frac{d x}{d t} = r x (1 - \frac{x}{K}) - \frac{m x y^{2}}{x^{2} + a y^{2}}, \\ array \frac{d y}{d t} = s y (1 - \frac{y}{b x}) \end{matrix})

with m >0 is the conversion rate of predator and a >0 is the half saturation constant, which simulates the group defense of prey. They investigated the permanency of the system and derived the sufficient conditions for the global stability of positive equilibrium by constructing a Lyapunov function and Hopf bifurcation by calculating the first‐order Lyapunov quantity.…”

Section: Introductionmentioning

confidence: 99%

“…21,22 Zhou 22 considered system (1.1) with the ratio-dependent Holling type III functional response and studied the stability of the positive equilibrium and the existence of periodic solutions by analyzing the Hopf bifurcation. Then, based on the Holling type IV functional response, Amirabad et al 23 considered system (1.1) with the ratio-dependent Holling type IV functional response…”

Section: Introductionmentioning

confidence: 99%

“…The rest of this paper is organized as follows. In Section , we first consider system without the harvesting and answer the unsolved problem about the Hopf bifurcation in Amirabad et al In Section , the existence and the qualitative properties of the equilibria of system are discussed. In Section , we present a detailed investigation of the bifurcations at the degenerate equilibria including two saddle‐node bifurcations and a Bogdanov‐Takens bifurcation of codimension 2.…”

Section: Introductionmentioning

confidence: 99%

“…Let = 0.58, = 0.2, = 1 3 , and h = 4477 405000 , then E 22 = (0.4943, 0.4943), and system (1.4) has two limit cycles enclosing unstable focus E 22 , the inner limit cycle is stable, and the outer one is unstable (see Figure 4B). 23 ∶= − 16( + 1) 9 ( 3 + 3 2 + 3 + 3 − + )…”

mentioning

confidence: 99%

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Bifurcations of a Leslie‐Gower prey‐predator system with ratio‐dependent Holling IV functional response and prey harvesting

Yao

2019

Math Methods in App Sciences

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The dynamics of a Leslie‐Gower prey‐predator system with ratio‐dependent Holling IV functional response and constant harvesting rate of prey are taken into account. The results developed in this article reveal far richer dynamics compared with the system without harvesting. We first make qualitative and bifurcation analysis of the system without harvesting and show that the system has a weak focus of multiplicity at most 2, at which a Hopf bifurcation occurs. However, the system with harvesting has four nonhyperbolic equilibria for some parameter values, such as two saddle‐node, a cusp, and a weak focus of multiplicity at most 4, and exhibits two saddle‐node bifurcations, a Bogdanov‐Takens bifurcation of codimension 2, and a Hopf bifurcation. It reveals that there exist some critical harvesting values such that the species are in danger of extinction when the harvesting rate is greater than the critical values, which indicates that the dynamics of the system are sensitive to the constant prey harvesting. Moreover, numerical simulations are presented to illustrate our theoretical results.

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