Delay-induced Turing instability in reaction-diffusion equations

Abstract

Time delays have been commonly used in modeling biological systems and can significantly change the dynamics of these systems. Quite a few works have been focused on analyzing the effect of small delays on the pattern formation of biological systems. In this paper, we investigate the effect of any delay on the formation of Turing patterns of reaction-diffusion equations. First, for a delay system in a general form, we propose a technique calculating the critical value of the time delay, above which a Turing instability occurs. Then we apply the technique to a predator-prey model and study the pattern formation of the model due to the delay. For the model in question, we find that when the time delay is small it has a uniform steady state or irregular patterns, which are not of Turing type; however, in the presence of a large delay we find spiral patterns of Turing type. For such a model, we also find that the critical delay is a decreasing function of the ratio of carrying capacity to half saturation of the prey density.

Pattern dynamics of a predator–prey reaction–diffusion model with spatiotemporal delay

“…The bifurcation scenarios in reaction-diffusion equations can be classified by three different mechanisms: (1) oscillatory in time but uniform in space induced by spatially homogeneous Hopf bifurcation [5][6][7][8][9][10], (2) oscillatory in space but stationary in time induced by Turing bifurcation [11][12][13], and (3) oscillatory both in space and time induced by wave bifurcation [14,15]. Based on these bifurcation scenarios, some unavoidable factors in the interactions of all kinds of species or elements, such as time delay [16][17][18], predator cannibalism [19], and noise [20], are also considered to be significant ingredients to induce these bifurcation behaviors.…”

Dynamics of a periodic switched predator–prey system with hibernation, birth pulse and impulsive harvesting

“…Li and Wang [10] B Jianjun Jiao jiaojianjun05@126.com considered dynamics of a Ivlev-type predator-prey system with constant rate harvesting. Recently, many literatures [11][12][13][14][15][16][17][18][19][20][21][22] applied impulsive differential equations to investigate predator-prey systems with impulsive effect. Jiao et al [11] showed the harvesting policy for a delayed stage-structured Holling II predator-prey model with impulsive stocking prey.…”

Turing–Hopf bifurcation analysis of a predator–prey model with herd behavior and cross-diffusion

“…We know that functional response function that reflects predator-prey interaction relationships is a crucial component of predator-prey model. In order to describe the features of the predator-prey interaction, many types of usual functional response functions, such as Holling I-IV types, ratio-dependent type, Hassell-Varley type, Beddington-DeAngelis type, Crowley-Martin type and the ones with Allee effect [3][4][5][6], have been proposed and investigated widely.…”

“…The diffusion-driven instability of the equilibrium leads to a spatially inhomogeneous distribution of species concentration, which is the so-called Turing instability. Although Turing instability was first investigated in a morphogenesis, it has quickly spread to ecological systems [3,4,6,[13][14][15][16][17][18][19][20][21][22][23][24], chemical reaction system [25][26][27][28][29][30] and other reaction-diffusion system [31][32][33][34][35][36][37][38]. From [39], we know that the phenomenon of spatial pattern formation in (1) with diffusion can not occur under all possible diffusion rates.…”