The diffusion-driven instability and complexity for a single-handed discrete Fisher equation

Abstract

For a reaction diffusion system, it is well known that the diffusion coefficient of the inhibitor must be bigger than that of the activator when the Turing instability is considered. However, the diffusion-driven instability/Turing instability for a single-handed discrete Fisher equation with the Neumann boundary conditions may occur and a series of 2-periodic patterns have been observed. Motivated by these pattern formations, the existence of 2-periodic solutions is established. Naturally, the periodic double and the chaos phenomenon should be considered. To this end, a simplest two elements system will be further discussed, the flip bifurcation theorem will be obtained by computing the center manifold, and the bifurcation diagrams will be simulated by using the shooting method. It proves that the Turing instability and the complexity of dynamical behaviors can be completely driven by the diffusion term. Additionally, those effective methods of numerical simulations are valid for experiments of other patterns, thus, are also beneficial for some application scientists.

Bifurcation analysis for a discrete space-time model of logistic type

“…It follows that Remark 2.3. If m = 2, then system (2.1) is reduced to u t+1 = pu t 1+u t + d(−u t + v t ), v t+1 = pv t 1+v t + d(u t − v t ), whose flip bifurcation was discussed in [30], and bifurcation diagrams were shown. So, we omit the numerical simulations in this work.…”

“…Turing bifurcation, named diffusion driven instability, on discrete space-time systems attracted much attention recently; see, e.g., [12,13,14,23,24] and the references therein. Some authors focus on other bifurcation, such as flip bifurcation; see, e.g., [25,29,30]. In [30], Zhang et al discussed the following model which can generate flip bifurcation u t+1 = pu t 1+u t + d(−u t + v t ), v t+1 = pv t 1+v t + d(u t − v t ), which is the spatial form of system (1.1)-(1.2) when m = 2.…”

“…Some authors focus on other bifurcation, such as flip bifurcation; see, e.g., [25,29,30]. In [30], Zhang et al discussed the following model which can generate flip bifurcation u t+1 = pu t 1+u t + d(−u t + v t ), v t+1 = pv t 1+v t + d(u t − v t ), which is the spatial form of system (1.1)-(1.2) when m = 2. The same analysis can be found in [29] and [25].…”

The Diffusion-Driven Instability for a General Time-Space Discrete Host-Parasitoid Model

“…Since some objects in nature are not only changing with time but also have a difusion phenomenon in space, numerous mathematical models are combined with time and space, and we can call this type of models reaction-difusion models. In recent decades, the reaction-difusion model has been widely used and numerous experts and scholars have used it to study the dynamic behavior of various organisms in nature [1][2][3][4][5][6][7][8][9][10][11], and the results of these studies have played a positive role in environmental protection and governance.…”

“…Referring to the approach in the literature [1,2,16,17], we started to study the stability of the system (2) and (3). Let H n (i, j) � H n (i, j) − H * , P n (i, j) � P n (i, j) − P * , then (2) can be rewritten as follows:…”