Fixation in a cyclic Lotka-Volterra model

We study a cyclic Lotka-Volterra model of N interacting species populating a d-dimensional lattice. In the realm of a Kirkwood approximation, a critical number of species Nc(d) above which the system fixates is determined analytically. We find Nc = 5, 14, 23 in dimensions d = 1, 2, 3, in remarkably good agreement with simulation results in two dimensions. A cyclic variant of the Lotka-Volterra model of interacting populations, originally introduced by Vito Volterra for description of struggle for existence among species [1,2], has then appeared in a number of apparently unrelated fields ranging from plasma physics [3] to integrable systems [4,5]. Recently, the cyclic Lotka-Volterra model (also known as the cyclic voter model) has attracted a considerable interest as it was realized that introduction of the spatial structure drastically enriches the dynamics [6–11]. Namely, if species live on a one-dimensional (1D) lattice, a homogeneous initial state evolves into a coarsening mosaic of interacting species. This heterogeneous spatial structure spontaneously develops when the number of species is sufficiently small, N < N c , where N c = 5 in one dimension [6,8,11]. For N ≥ N c fixation occurs, i.e. the system approaches a frozen state. Little is known in higher dimensions, even existence of N c has not yet been established theoretically or numerically (in simulations on 2D lattices with N ≤ 10 species, no sign of fixation has been found and instead a reactive steady state has been observed [6–11]). In this work we investigate the cyclic Lotka-Volterra model in the framework of a Kirkwood-like approximation. This approach predicts a finite N c in all spatial dimensions. In the following, we shall use the language of voter model [12]. Consider the cyclic voter model with N possible opinions. Each site of a d-dimensional cubic lattice is occupied by a voter which has an opinion labeled by α, with α = 1,. .. , N. Voters can change their opinions in a cyclic manner, α → α − 1 modulo N , according to the opinions of their neighborhood. Specifically, the following sequential dynamics is implemented: (i) we choose randomly a site (of opinion α, say) and one of its 2d nearest neighbors (of opinion β); (ii) if β = α − 1, then the chosen site changes its opinion from α to β = α − 1; (iii) otherwise, opinion does not change. We set the time …