Spatial dynamics in a predator-prey model with herd behavior

Huang

et al. 2017

Entropy

Wind-induced vegetation patterns were proposed a long time ago but only recently a dynamic vegetation-sand relationship has been established. In this research, we transformed the continuous vegetation-sand model into a discrete model. Fixed points and stability analyses were then studied. Bifurcation analyses are done around the fixed point, including Neimark-Sacker and Turing bifurcation. Then we simulated the parameter space for both bifurcations. Based on the bifurcation conditions, simulations are carried out around the bifurcation point. Simulation results showed that Neimark-Sacker bifurcation and Turing bifurcation can induce the self-organization of complex vegetation patterns, among which labyrinth and striped patterns are the key results that can be presented by the continuous model. Under the coupled effects of the two bifurcations, simulation results show that vegetation patterns can also be self-organized, but vegetation type changed. The type of the patterns can be Turing type, Neimark-Sacker type, or some other special type. The difference may depend on the relative intensity of each bifurcation. The calculation of entropy may help understand the variance of pattern types.

Section: Bifurcation Analysismentioning

confidence: 99%

Coupled Effects of Turing and Neimark-Sacker Bifurcations on Vegetation Pattern Self-Organization in a Discrete Vegetation-Sand Model

Huang

et al. 2017

Entropy

“…They modeled the poisoning function in terms of the 2/3 power of phytoplankton, and obtained that the dynamics of the plankton population depends on the fraction of the phytoplankton population that aggregates to form patches. We refer readers to references [6,7,8,9,10] as some other related works on predator-prey model with herd behavior.…”

Section: Introductionmentioning

confidence: 99%

Global dynamics of a predator–prey model with defense mechanism for prey

Yuan

Applied Mathematics Letters

2016

Self Cite

Recently, Venturino and Petrovskii proposed a general predator-prey model with group defence for prey species [E. Venturino and S. Petrovskii, Spatiotemporal behavior of a prey-predator system with a group defense for prey, Ecol. Complex. 14, 37-47 (2013)]. The local dynamics had been studied and showed that the model might undergo Hopf bifurcation, and have an extinction domain. In this paper, we dedicate ourselves to the investigation of the global dynamics of the model by establishing the conditions of the nonexistence of periodic orbits, and the existence and uniqueness of limit cycles.

“…The linear mortality rate has been used intensively by researchers [9][10][11][12][13]. When population density is relatively high, the quadratic mortality rate dominates the mortality [14]. When the population density is large, the hyperbolic mortality rate dominates the mortality [15].…”

Section: Introductionmentioning

confidence: 99%

The effect of prey refuge and time delay on a diffusive predator‐prey system with hyperbolic mortality

Yang

2016

Complexity

In this article, we investigate the effect of prey refuge and time delay on a diffusive predator-prey system with Holling II functional response and hyperbolic mortality subject to Neumann boundary condition. More precisely, we study Turing instability of positive equilibrium by using refuge as parameter, instability and Hopf bifurcation induced by time delay. In addition, by the theory of normal form and center manifold, we derive conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution.