Discrete model of periodic pattern formation through a combined autocrine–juxtacrine cell signaling

In this paper, we investigate the spatiotemporal dynamics of a Leslie–Gower predator–prey model incorporating a prey refuge subject to the Neumann boundary conditions. We mainly consider Hopf bifurcation and steady-state bifurcation which bifurcate from the constant positive steady-state of the model. In the case of Hopf bifurcation, by the center manifold theory and the normal form method, we establish the bifurcation direction and stability of bifurcating periodic solutions; in the case of steady-state bifurcation, by the local and global bifurcation theories, we prove the existence of the steady-state bifurcation, and find that there are two typical bifurcations, Turing bifurcation and Turing–Hopf bifurcation. Via numerical simulations, we find that the model exhibits not only stationary Turing pattern induced by diffusion which is dependent on space and independent of time, but also temporal periodic pattern induced by Hopf bifurcation which is dependent on time and independent of space, and spatiotemporal pattern induced by Turing–Hopf bifurcation which is dependent on both time and space. These results may enrich the pattern formation in the predator–prey model.

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Crossover between Spatially Confined Precipitation and Periodic Pattern Formation in Reaction Diffusion Systems

Physical Review Letters ◽

10.1103/physrevlett.77.2834 ◽

1996 ◽

Vol 77 (13) ◽

pp. 2834-2837 ◽

Cited By ~ 10

Author(s):

E. López Cabarcos ◽

Chein-Shiu Kuo ◽

A. Scala ◽

R. Bansil

Keyword(s):

Pattern Formation ◽

Reaction Diffusion ◽

Periodic Pattern ◽

Reaction Diffusion Systems ◽

Diffusion Systems

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Theoretical Aspects of Laser-Induced Periodic Surface Structure Formation

MRS Proceedings ◽

10.1557/proc-29-389 ◽

1983 ◽

Vol 29 ◽

Author(s):

Michael Hutchinson ◽

Ki-Tung Lee ◽

William C. Murphy ◽

A. C. Beri ◽

Thomas F. George

Keyword(s):

Pattern Formation ◽

Maxwell’S Equations ◽

Smooth Surface ◽

Maxwell's Equations ◽

Periodic Pattern ◽

Periodic Surface ◽

Periodic Surface Structure ◽

Surface Grating ◽

Perturbative Solution ◽

Initial Deposition

ABSTRACTLaser-induced periodic pattern formation has been observed on a variety of substances. In particular, low-power lasers have been used to deposit a pattern on a metal surface. For a relatively smooth surface grating, this pattern can be explained in terms of a perturbative solution of Maxwell's equations. However, as the surface grating is enhanced by this initial deposition, the perturbative solution breaks down. An alternate non-perturbative solution of Maxwell's equations for such rough surfaces is considered here.

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Regular and Irregular Vegetation Pattern Formation in Semiarid Regions: A Study on Discrete Klausmeier Model

Complexity ◽

10.1155/2020/2498073 ◽

2020 ◽

Vol 2020 ◽

pp. 1-14

Author(s):

Huayong Zhang ◽

Tousheng Huang ◽

Liming Dai ◽

Ge Pan ◽

Zhao Liu ◽

...

Keyword(s):

Pattern Formation ◽

Discrete Model ◽

Theoretical Models ◽

Vegetation Pattern ◽

Vegetation Patterns ◽

Vegetation Types ◽

Semiarid Regions ◽

Parametric Data ◽

Nonlinear Mechanism ◽

Important Field

The research on regular and irregular vegetation pattern formation in semiarid regions is an important field in ecology. Applying the framework of coupled map lattice, a novel nonlinear space- and time-discrete model is developed based on discretizing the classical Klausmeier model and the vegetation pattern formation in semiarid regions is restudied in this research. Through analysis of Turing-type instability for the discrete model, the conditions for vegetation pattern formation are determined. The discrete model is verified by Klausmeier’s results with the same parametric data, and shows advantages in quantitatively describing diverse vegetation patterns in semiarid regions, such as the patterns of regular mosaicirregular patches, stripes, fractured stripesspots, and stripes-spots, in comparing with former theoretical models. Moreover, the discrete model predicts variations of rainfall and vegetation types can cause transitions of vegetation patterns. This research demonstrates that the nonlinear mechanism of the discrete model better captures the diversity and complexity of vegetation pattern formation in semiarid regions.

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