Delay-driven spatial patterns in a network-organized semiarid vegetation model
Introduction
A variety of vegetation patterns have been observed particularly in arid regions, including spotted vegetation, labyrinths, gap patterns, and regular bands [1], [2], [3], [4], [5]. Apart from the well-recognized mechanisms (i.e. habitat heterogeneity) that underly the formation of vegetation patterns [6], great attention has been paid to the positive feedback between plant growth and water availability that triggers the emergence of self-organized pattern formation [7], and self-organized vegetation patterns have been widely investigated [8], [9], [10], [11], [12], [13], [14], [15]. The widely employed modeling approach is reaction–diffusion systems. As an application, Klausmeier [14] developed a reaction–advection–diffusion system to describe the interaction between plant and water, where the motions of plant and water are, respectively, described by a reaction–diffusion equation and an advection equation. He found a close agreement between theoretical predictions and filed observations. Under the framework of reaction–diffusion systems, the classic mechanisms behind the formation of spatial pattern is Turing bifurcation, which means that a locally stable equilibrium becomes unstable due to certain spatially varying perturbations, leading to spatially heterogeneous steady state [16], [17]. Turing–Hopf bifurcation, a time and space breaking symmetry, is a well-known mechanism for generating spatio-temporal patterns [18], [19], [20], which is also known as oscillatory Turing, or wave instability [21], [22].
However, most of the aforementioned studies were carried out on a spatially continuous domain, that is, the entire landscape is spatially continuous. The frequent anthropogenic activities have been breaking landscape into many fine-scale habitats. These fine-scale habitats form a network connected by dispersal of seeds. Recent studies of pattern formation on network-organized systems in physics show that Turing patterns can take place in large network [23]. A further recent study shows that Turing-like waves can emerge in a one-species delayed reaction–diffusion model on a network [24], differing from the convectional knowledge that Turing patterns can only occur in two-component reaction–diffusion systems [17], [18]. Motivated by these observations, we are interested in understanding whether spatial pattern can emerge in semiarid vegetation models on networks standing for fragmented habitats, which remains unexplored.
To this aim, we propose the following delayed reaction–diffusion vegetation system on a networkHere u and v are densities of water and plant at time t, respectively. Parameters a and b are, respectively, the rates of rainfall and plant mortality. Plant takes up water at a rate . The time delay τ means that some time lag is required for plant to consume water. This semiarid vegetation model is defined on an undirected network with n nodes but no self-loops. is an adjacency matrix, whose entry gij equals 1, if there exists an edge from node j to node i, and 0, otherwise. It is obvious that for . We assume that G is strongly connected, i.e., for any pair of distinct nodes, there exists a path from one node to the other. It is known that the adjacency matrix G is irreducible [25]. The Laplacian matrix L is defined asInteraction between water and plant occurs in each node and their time-dependent concentrations in the ith node are denoted by ui(t) and vi(t), respectively. The water and plant move across the network links and the process is characterized by two diffusion coefficients Du > 0 and Dv > 0, and the links represent dispersal of plant seed and water flow from one habitat to another.
Time delay was first considered in a prey–predator system by Volterra [26] and found to be able to drive the emergence of oscillatory behavior. Since then, many studies about delay-driven pattern formation followed (e.g., [27], [28], [29], [30], [31], [32], [33]). In addition, time delay was also introduced in experiments to induce spiral wave [34]. Time delay in a network-organized model with discrete domain was recently introduced in [24]. It was found that travelling waves can occur following a symmetry-breaking instability of homogeneous stationary stable solution. This Turing-like patterns emerge in a single species system. A follow-up work on a two-component delayed reaction–diffusion system on a complex network shows that the system can evolve towards a stationary Turing pattern or to a wave pattern associated with a diffusion induced symmetry-breaking instability [35].
The paper is structured as follows. In Section 2, we perform analytical analysis to determine the local stability of the coexistence equilibrium between plant and water, derive the conditions for Hopf bifurcation to occur, and finally apply the center manifold theory to determine the direction as well as stability of the Hopf bifurcation. In Section 3, we carry out numerical analysis to demonstrate the dynamic behavior of plant on a network domain. The paper is closed with a brief discussion.
Section snippets
Delay-driven spatial patterns
In this section we first carry out a linear stability analysis around the positive equilibrium (Section 2.1), and (then explore the condition and stability of the emerging Hopf bifurcation by means of the normal form theory and the center manifold theory (Section 2.2). We find that the system (1.1) can generate spatial patterns on the discrete domain when time delay goes beyond a threshold value, which is, however, impossible in the absence of time delay.
The system (1.1) can possibly have three
Numerical simulations
In this section, we carry out numerical simulations to demonstrate our analytical findings. To this end, we consider the following set of parameters based on the real data reported in [14]
Recall that the conditions (2.6) and (2.8) basically determine when Hopf bifurcation induced instability occurs around the positive equilibrium E2. Fig. 1 shows how the critical time delay τ0 varies with parameters a and b. As expected, large a and b promote the emergence of Hopf
Discussion
We have developed a theoretical framework for studying spatiotemporal dynamics of a semiarid vegetation model on a network domain. The network stands for fragmented habitats connected by dispersal of plant seed. Applying standard stability analysis and the center manifold theory, we investigated the direction and stability of delay induced Hopf bifurcation, which leads to the emergence of oscillations that are periodic in time. An interesting finding is that small time delay can easily induce
Acknowledgements
The work is supported by the PRC Grant of NSFC (61877052, 11571301, and 11871065). Zhi Ling and Lai Zhang are additionally supported by the NSF of Jiangsu Province (BK20151305 and BK20181450). Lai Zhang further acknowledges the financial support by the Jiangsu Distinguished Professor Program, and the Yangzhou Talent Program of ‘LvYangJinFeng’.
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