Comments on “L. N. Guin, M. Haque, P. K. Mandal, The spatial patterns through diffusion-driven instability in a predator–prey model, Appl. Math. Model. 36 (2012) 1825–1841.”

Formation of spatial patterns in prey-predator system is a central issue in ecology. In this paper Turing structure through diffusion driven instability in a modified Leslie-Gower and Holling-type II predator-prey model has been investigated. The parametric space for which Turing spatial structure takes place has been found out. Extensive numerical experiments have been performed to show the role of diffusion coefficients and other important parameters of the system in Turing instability that produces some elegant patterns that have not been observed in the earlier findings. Finally it is concluded that the diffusion can lead the prey population to become isolated in the two-dimensional spatial domain.

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Comment on “Existence and stability of periodic solution of a Lotka–Volterra predator–prey model with state dependent impulsive effects” [J. Comput. Appl. Math. 224 (2009) 544–555]

Journal of Computational and Applied Mathematics ◽

10.1016/j.cam.2010.04.001 ◽

2010 ◽

Vol 234 (10) ◽

pp. 2916-2923 ◽

Cited By ~ 6

Author(s):

Yuan Tian ◽

Kaibiao Sun ◽

Lansun Chen

Keyword(s):

Periodic Solution ◽

Impulsive Effects ◽

Predator Prey ◽

Predator Prey Model ◽

Prey Model ◽

State Dependent ◽

Existence And Stability ◽

Appl Math

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Stability and Turing Patterns in a Predator–Prey Model with Hunting Cooperation and Allee Effect in Prey Population

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127420501370 ◽

2020 ◽

Vol 30 (09) ◽

pp. 2050137

Author(s):

Danxia Song ◽

Yongli Song ◽

Chao Li

Keyword(s):

Spatial Patterns ◽

Allee Effect ◽

Turing Instability ◽

Local System ◽

Positive Equilibrium ◽

Predator Population ◽

Predator Prey ◽

Turing Bifurcation ◽

Predator Prey Model ◽

Prey Model

In this paper, we are concerned with a diffusive predator–prey model where the functional response follows the predator cooperation in hunting and the growth of the prey obeys the Allee effect. Firstly, the existence and stability of the positive equilibrium are explicitly determined by the local system parameters. It is shown that the ability of the hunting cooperation can affect the existence of the positive equilibrium and stability, and the intrinsic growth rate of the predator population does not affect the existence of the positive equilibrium, but affects the stability. Then the diffusion-driven Turing instability is investigated and the Turing bifurcation value is obtained, and we conclude that when the ability of the cooperation in hunting is weaker than some critical value, there is no Turing instability. The standard weakly nonlinear analysis method is employed to derive the amplitude equations of the Turing bifurcation, which is used to predict the types of the spatial patterns. And it is found that in the Turing instability region, with the parameter changing from approaching Turing bifurcation value to approaching Hopf bifurcation value, spatial patterns emerge from spot, spot-stripe to stripe. Finally, the numerical simulations are used to support the analytical results.

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