Viewing communities as coupled oscillators: elementary forms from Lotka and Volterra to Kuramoto

Abstract

Ecosystems and their embedded ecological communities are almost always by definition collections of oscillating populations. This is apparent given the qualitative reality that oscillations emerge from consumer-resource interactions, which are the simple building blocks for ecological communities. It is also likely always the case that oscillatory consumer-resource pairs will be connected to one another via trophic cross-feeding with shared resources or via competitive interactions among resources. Thus, one approach to understanding the dynamics of communities conceptualizes them as collections of oscillators coupled in various arrangements. Here we look to the pioneering work of Kuramoto on coupled oscillators and ask to what extent can his insights and approaches be translated to ecological systems. We explore all possible coupling arrangements of the simple case of three oscillator systems with both the Kuramoto model and with the classical Lotka-Volterra equations that are foundational to ecology. Our results show that the six-dimensional analogous Lotka-Volterra systems behave strikingly similarly to that of the corresponding Kuramoto systems across all possible coupling combinations. This qualitative similarity in the results between these two approaches suggests that a vast literature on coupled oscillators that has largely been ignored by ecologists may in fact be relevant in furthering our understanding of ecosystem and community organization.

Instability of oscillations in the Rosenzweig-MacArthur model of one consumer and two resources

“…Our starting point is the Rosenzweig-MacArthur set of equations [15] for two resources R 1 , R 2 and one consumer C…”

“…Here β i is the rate of taking advantage of resource R i , and α ij is the limitation of growth of resource R i , imposed by resource R j . Further, p is the attack rate of the consumer, b is the functional response term of the consumer, and m is the mortality rate of the consumer [15].…”

New forms of structure in ecosystems revealed with the Kuramoto model

“…This arrangement has led to some speculations on its meaning for ecological communities in general [ 19 , 20 ] as well as empirical confirmation in the field [ 23 ]. Elsewhere it has been shown that the pattern of coupling based on the classical consumer resource model of MacArthur [ 28 ] follows precisely the qualitative predictions of coupling patterns from the Kuramoto model [ 29 ].…”

“…[ 30 ] studied the behaviour of the model with γ i,j distributed as a distance related power law in a spatially explicit framework, and Girón and colleagues examined the consequences of allowing some of the γ i , j to be positive and others negative. In an earlier work, Hajian-Forooshani & Vandermeer [ 29 ] compared a six-dimensional predator/prey framework modelled in the Lotka–Volterra style to the same framework in the Kuramoto model showing that the synchronization patterns were qualitatively identical. Numerous studies, not necessarily ecological, have focused on a central feature of the model, the chimeric state that inevitably emerges when the γ i , j collectively are too small to engage all the oscillators in complete collective synchrony but too large to permit complete independence of oscillator behaviour [ 31 ].…”

“…The dynamics of coupled ecological oscillators is complicated and strongly dependent on the nature of the coupling [ 8 , 18 , 29 ], with in-phase synchrony resulting from predators sharing prey and anti-phase synchrony resulting from prey competing with one another. Empirical verification of these theoretical propositions is rare [ 23 ].…”