Dispersal-induced destabilization of metapopulations and oscillatory Turing patterns in ecological networks

Abstract

As shown by Alan Turing in 1952, differential diffusion may destabilize uniform distributions of reacting species and lead to emergence of patterns. While stationary Turing patterns are broadly known, the oscillatory instability, leading to traveling waves in continuous media and sometimes called the wave bifurcation, remains less investigated. Here, we extend the original analysis by Turing to networks and apply it to ecological metapopulations with dispersal connections between habitats. Remarkably, the oscillatory Turing instability does not lead to wave patterns in networks, but to spontaneous development of heterogeneous oscillations and possible extinction of species. We find such oscillatory instabilities for all possible food webs with three predator or prey species, under various assumptions about the mobility of individual species and nonlinear interactions between them. Hence, the oscillatory Turing instability should be generic and must play a fundamental role in metapopulation dynamics, providing a common mechanism for dispersal-induced destabilization of ecosystems.

Beyond connectivity: how the structure of dispersal influences metacommunity dynamics

“…We further illustrate that differences in the pattern of connections and initial conditions among communities lead to the emergence of a wide array of stable asynchronous dynamics, leading to substantial differences in the potential persistence of an unstable predator-prey interaction at both the regional metacommunity and local patch level, despite the otherwise strict homogeneity of our metacommunities in which each community is identical and connected to exactly four other community patches. Our study is the first to produce this range of dynamical variation as a result of such subtle changes in spatial structure; this contrasts with earlier work that has only produced comparable variation through heterogeneity in connectivity or degree distribution (Holland and Hastings 2008;Gilarranz and Bascompte 2012), differences in dispersal among species (de Roos et al 1998), or alteration of community dynamics in more complex models (Marleau et al 2014;Hata et al 2014). The appearance of this striking range of dynamical variation, both in the frequency and and quality of asynchrony among metacommunities shows the important role of the pattern of connections among local communities in shaping the regional dynamics of metacommunities.…”

“…While these are typically studied in the context of Turing instabilities, wherein spatial processes also destabilize the synchronized homogeneous state, alternative mechanisms for the formation of spatial patterns have been described (Cahn and Hilliard 1958;Liu et al 2013), and they can occur in systems with locally stable homogeneous states (Wolfrum 2012). The patterns we observe are surprising because they are equivalent to the special case of traveling wave patterns as their equilibria are limit cycles rather than fixed points, which typically require at least three species to occur (Turing 1952;Hata et al 2014). Here they can arise with only two species because the interaction between the predator and prey is already prone to oscillation (Yang et al 2004).…”

Controlling Synchronization Patterns in Complex Networks

“…Travelling waves have recently been investigated on networks with a certain degree of regularity, e.g., regular rings, tree networks, and small-world networks [Isele, 2014;Kouvaris et al, 2014;; Turing patterns, which have been originally described in reaction diffusion systems, have been generalized to networks [Nakao and Mikhailov, 2010;Hata et al, 2012;Wolfrum, 2012;Hata et al, 2014]; Chimera states, where the nodes in a regular network with homogeneous local dynamics separate into two groups with distinctly different dynamical behavior, gained a lot of attention recently [Kuramoto and Battogtokh, 2002;Abrams and Strogatz, 2004;Sethia et al, 2008;Omelchenko et al, 2011Omelchenko et al, , 2012Hagerstrom et al, 2012;Omelchenko et al, 2013;Vüllings et al, 2014;Zakharova et al, 2014;Omelchenko et al, 2015]. In principle, all of these dynamical states could be controlled with an adaptive control scheme based on the SG method.…”

Self-organization of network dynamics into local quantized states

“…The basic mechanisms of pattern formation by local self-activation and lateral inhibition, or short-range positive feedback and long-range negative feedback [4,5] are ubiquitous in ecological and biological spatial systems, from morphogenesis and developmental biology [1,6] to adaptive strategies in living organisms [7,8] and spatial heterogeneity in predator-prey systems [9]. Heterogeneity and patchiness in vegetation dynamics, associated with Turing patterns in vegetation dyanmics have been proposed as a connection between pattern and process in ecosystems [10,11], suggesting a link between spatial vegetation patterns and vulnerability to catastrophic shifts in water-stressed ecosystems [12][13][14].The theory of non-equilibrium self-organization and Turing patterns has been recently extended to network-organized natural and socio-technical systems [15][16][17][18], including complex topological structures such as multiplex [19,20], directed [21] and cartesian product networks [22]. Self-organization is rapidly emerging as a central paradigm to understand neural computation [23][24][25].…”

“…The theory of non-equilibrium self-organization and Turing patterns has been recently extended to network-organized natural and socio-technical systems [15][16][17][18], including complex topological structures such as multiplex [19,20], directed [21] and cartesian product networks [22]. Self-organization is rapidly emerging as a central paradigm to understand neural computation [23][24][25].…”