Master stability functions reveal diffusion-driven pattern formation in networks

Andreas Brechtel, Philipp Gramlich, Daniel Ritterskamp, Barbara Drossel, and Thilo Gross

Phys. Rev. E 97, 032307 – Published 19 March 2018

Abstract

We study diffusion-driven pattern formation in networks of networks, a class of multilayer systems, where different layers have the same topology, but different internal dynamics. Agents are assumed to disperse within a layer by undergoing random walks, while they can be created or destroyed by reactions between or within a layer. We show that the stability of homogeneous steady states can be analyzed with a master stability function approach that reveals a deep analogy between pattern formation in networks and pattern formation in continuous space. For illustration, we consider a generalized model of ecological meta-food webs. This fairly complex model describes the dispersal of many different species across a region consisting of a network of individual habitats while subject to realistic, nonlinear predator-prey interactions. In this example, the method reveals the intricate dependence of the dynamics on the spatial structure. The ability of the proposed approach to deal with this fairly complex system highlights it as a promising tool for ecology and other applications.

Received 4 April 2017
Revised 8 October 2017

DOI:https://doi.org/10.1103/PhysRevE.97.032307

Physics Subject Headings (PhySH)

Dynamics of networks Ecological pattern formation Emergence of patterns Network diffusion Network stability Population dynamics

NetworksPhysics of Living SystemsNonlinear Dynamics

Authors & Affiliations

Andreas Brechtel¹, Philipp Gramlich¹, Daniel Ritterskamp², Barbara Drossel¹, and Thilo Gross²

¹Institute of Condensed Matter Physics, Technische Universität Darmstadt, 64289 Darmstadt, Germany
²Department of Engineering Mathematics, Merchant Venturers School of Engineering, University of Bristol, Woodland Road, Bristol BS8 1UB, United Kingdom

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Vol. 97, Iss. 3 — March 2018

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Images

Figure 1
Example meta-food web with four species (blue bubbles and black arrows) on five habitats or patches (gray lines and circles). The stability of the system is described by the $20 \times 20$ Jacobian matrix $J$ , which can be written as $5 \times 5$ matrix of $4 \times 4$ blocks. The blocks contain the intrapatch Jacobian matrix $P$ , describing the dynamics within one patch, and the coupling matrix $C$ , describing the dependence of migration rates on the population sizes, which is given here for a simple diffusion process. While the intrapatch Jacobian only appears on the diagonal of $J$ , the coupling blocks $C$ occur in a pattern given by the Laplacian matrix $L$ that encodes the structure of the patch network and is shown here for a coupling strength of 1.
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Figure 2
Analogy between diffusion in continuous space and in networks. To emphasize the similarity, we have written the network reaction-diffusion system [Eq. (12)] in matrix form and also allowed a coupling matrix $C$ in the instability condition for the continuous space systems. While our derivation considered the simpler case where $C$ is proportional to the identity matrix, such matrices appear in the case of cross diffusion where diffusion of one species depends on the concentration of other species [46]. While the concentrations $X$ are position dependent in continuous space, the state of the system on a network is captured by a discrete set of variables $X_{i}$ , with $i$ being the node index. See text for detailed derivations.
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Figure 3
Master stability function (MSF) for meta-food webs. Shown are MSFs (a)–(c) and spatial geometries (d)–(f) for the same local food web (that of Fig. 1) and coupling matrix $C$ [see Eq. (36) and Appendix for details] but three different geometries (represented by circles, indicating patches, and lines, indicating migration links, in a two-dimensional $x, y$ landscape). The MSF [identical line in (a)–(c)] relates the Jacobian eigenvalue of the meta-food web to the Laplacian eigenvalue of the spatial network. The meta-food web is stable if none of the Laplacian eigenvalues (arrows) fall into ranges where the master stability function [line in (a)–(c)] is positive (blue shaded area). The food web under consideration is unstable on one geometry (a), (d) but stable on another (b), (e). Tightening the coupling, i.e., increasing the weights in the adjacency matrix [indicated by thicker lines in (f)], can destabilize the stable system by stretching the Laplacian spectrum [cf. (b) and (c)].
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Figure 4
Complex master stability functions (MSFs). Shown are two examples of food webs (a), (b) of 20 species (blue bubbles) connected by predator-prey interactions (arrows). The coupling matrix $C$ was constructed such that predators emigrate preferentially from patches with scarce prey and prey emigrates preferentially from patches with abundant predators (see Appendix). The corresponding MSFs [(c) for (a), (d) for (b)] have many forbidden (blue) ranges.
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Figure 5
Localization of eigenvectors. Shown is a system where on a given topology (c) one eigenvalue of the Laplacian lies in one of the forbidden ranges where the MSF of the food web is positive. Hence, the system departs from the homogeneous steady state and approaches a nonhomogeneous state. Color coding the distance of the density of the apex predator population to the homogeneous value (see methods) shows that the impact of the instability is confined to a relatively small area. In cases where the system approaches a state in the vicinity of the homogeneous state, the eigenvector corresponding to the destabilizing eigenvector is indicative of the pattern in the final state. A comparison (a) of the respective eigenvector (open circles) and the observed deviation from the homogeneous value in the final state (dots) shows good agreement in the example system (c), but similar accuracy cannot be guaranteed in general. The localization of many eigenvectors is a generic feature of geographical networks. For the 500-node example geometry in (b) the number of nodes on which eigenvectors have a significant amplitude can be quantified by the participation number (see methods). This reveals that the majority of eigenvectors only extend to relatively few nodes [histogram in (d)], while only four eigenvalues (arrows) have significant amplitude on a large fraction of the nodes. The example shown here is a case of real eigenvalues causing a Turing instability. When the leading eigenvalue is complex, the associated instability is a wave instability (see Appendix).
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Figure 6
Five-species food web, used in Figs. 4 and 4 (a) and for the animated oscillations (b). The shown numbers are the indices of the species. The arrows indicate the direction of biomass flow. In food web (a), species 1 and 2 are the primary producers and species 5 is the apex predator. In network (b) species 1 is the primary producer and species 3 is the observed top predator.
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