Phase separation explains a new class of self-organized spatial patterns in ecological systems

Abstract

The origin of regular spatial patterns in ecological systems has long fascinated researchers. Turing's activator-inhibitor principle is considered the central paradigm to explain such patterns. According to this principle, local activation combined with longrange inhibition of growth and survival is an essential prerequisite for pattern formation. Here, we show that the physical principle of phase separation, solely based on density-dependent movement by organisms, represents an alternative class of self-organized pattern formation in ecology. Using experiments with self-organizing mussel beds, we derive an empirical relation between the speed of animal movement and local animal density. By incorporating this relation in a partial differential equation, we demonstrate that this model corresponds mathematically to the wellknown Cahn-Hilliard equation for phase separation in physics. Finally, we show that the predicted patterns match those found both in field observations and in our experiments. Our results reveal a principle for ecological self-organization, where phase separation rather than activation and inhibition processes drives spatial pattern formation.he activator-inhibitor principle, originally conceived by Turing in 1952 (1), provides a potential theoretical mechanism for the occurrence of regular patterns in biology (2-6) and chemistry (7-9), although experimental evidence in particular for biological systems has remained scarce (3, 4, 10). In the past decades, this principle has been applied to a wide range of ecological systems, including arid bush lands (11-15), mussel beds (16, 17), and boreal peat lands (18,19). The principle, in which a local positive activating feedback interacts with large-scale inhibitory feedback to generate spatial differentiation in growth, birth, mortality, respiration, or decay, explains the spontaneous emergence of regular spatial patterns in ecosystems even under near-homogeneous starting conditions. Physical theory offers an alternative mechanism for pattern formation, proposed by Cahn and Hilliard in 1958 (20). They identified that density-dependent rates of dispersal can lead to separation of a mixed fluid into two phases that are separated in distinct spatial regions, subsequently leading to pattern formation. The principle of density-dependent dispersal, switching between dispersion and aggregation as local density increases, has become a central mathematical explanation for phase separation in many fields (21) such as multiphase fluid flow (22), mineral exsolution and growth (23), and biological applications (24-28). Although aggregation due to individual motion is a commonly observed phenomenon within ecology, application of the principles of phase separation to explain pattern formation in ecological systems is absent both in terms of theory and experiments (25,26).Here, we apply the concept of phase separation to the formation of spatial patterns in the distribution of aggregating mussels. On intertidal flats, establishing mussel beds exhibit spatial self-org...

Pattern formation at multiple spatial scales drives the resilience of mussel bed ecosystems

“…We therefore based our analysis on estimates that are, for as much as possible, based on experiments in case of mussel movement, and literature evidence in case of growth and mortality processes. Moreover, we provide extensive bifurcation analyses of the models of small-scale and large-scale self-organization in previous papers 13,14,20,25,34,35 .…”

“…Yet, because research questions and techniques often vary among different scales, there is a strong tendency to study the processes at these different scales separately [10][11][12] . For instance, some studies focus on the spatial patterns that result from the movement of individuals, for example, for shelter, improved mate choice or the sharing of information 5,[13][14][15][16][17] . Other studies focus on large-scale ecosystem processes that create spatial variation in predation pressure, resource availability 8,12,[18][19][20] and other environmental conditions [21][22][23][24] .…”

“…This interplay between facilitation and competition explains the banded patterns occurring at the 5-10-metre scale 20 . At the smallest scale, individual mussels aggregate by means of densitydependent movement, actively moving into small clusters and out of larger ones, to form a reticulate network of clusters at B15-cm intervals in which mussels attach to each other by means of byssal threads, thereby protecting each other against predation and dislodgement 13,14,26,27 . Both the large-scale banded patterns and the reticulate network of small-scale clusters result from a selforganization process, meaning that the patterns emerge from the interactions between the mussels, rather than being imposed by the landscape 1 .…”

Active Matter Clusters at Interfaces

“…Hydrodynamic continuum models, on the other hand, do not treat each agent as an individual, but instead study an average alignment and density profile within the system (Toner et al, 2005). Hydrodynamic swarming models have established active matter as a type of nonequilibrium complex fluid (Toner and Tu, 1995;Marchetti et al, 2013) and provided a unified framework to study phase transitions (Levine et al, 2000;Toner et al, 2005), instabilities (Bertin et al, 2009), and pattern formation (Liu et al, 2013) in active systems.…”

Production efficiency of mussel bottom culture

“…A reduction of losses through a reduction in within plot-scale mussel density by using better spreading at seeding has been mentioned as a major improvement in Newell et al (2007). Higher losses at higher densities might be caused by the smothering of mussels in the under layers, in multi-layered mussels at high densities or as an effect of the re-organisation process (Liu et al 2013). This type of loss acts at the scale of the area occupied by the mussels and is probably also related to the speed and time at which the mussels are flushed out of the vessel.…”