Global attractors and extinction dynamics of cyclically competing species

Abstract

Transitions to absorbing states are of fundamental importance in nonequilibrium physics as well as ecology. In ecology, absorbing states correspond to the extinction of species. We here study the spatial population dynamics of three cyclically interacting species. The interaction scheme comprises both direct competition between species as in the cyclic Lotka-Volterra model, and separated selection and reproduction processes as in the May-Leonard model. We show that the dynamic processes leading to the transient maintenance of biodiversity are closely linked to attractors of the nonlinear dynamics for the overall species' concentrations. The characteristics of these global attractors change qualitatively at certain threshold values of the mobility and depend on the relative strength of the different types of competition between species. They give information about the scaling of extinction times with the system size and thereby the stability of biodiversity. We define an effective free energy as the negative logarithm of the probability to find the system in a specific global state before reaching one of the absorbing states. The global attractors then correspond to minima of this effective energy landscape and determine the most probable values for the species' global concentrations. As in equilibrium thermodynamics, qualitative changes in the effective free energy landscape indicate and characterize the underlying nonequilibrium phase transitions. We provide the complete phase diagrams for the population dynamics and give a comprehensive analysis of the spatio-temporal dynamics and routes to extinction in the respective phases.

Characterization of spiraling patterns in spatial rock-paper-scissors games

“…[15,17,18,23,38]. In particular, the functional dependence of the CGLE parameter (7) differs from that used in Refs.…”

“…[15] for a variant of the model considered here with only dominance-removal competition (ζ = μ = 0 and δ D = δ E ). The treatment was then extended to also include dominance-replacement competition (with μ = 0 and δ D = δ E ) [ 17,23] and has recently been generalized to more than three species [38]. In all these works, the derivation of the CGLE relies on the fact that the underlying mean-field dynamics quickly settles on a two-dimensional manifold on which the flows approach the absorbing boundaries forming heteroclinic cycles [13,22].…”

“…4). It is worth noting that in a number of earlier works with μ = 0, the quantities λ and v were considered to not vary with the CGLE parameter c, see, e.g., [15,17,23]. The prediction (9) can be used to theoretically estimate the spiral wavelength, see, e.g., Fig.…”

“…In recent years, many models for the RPS cyclic competition have been considered. In particular, various two-dimensional versions of the model introduced by May and Leonard [22] have been studied [15,[17][18][19]21,23]. In spatial variants of the May-Leonard model, it was found that mobility implemented by pair-exchange among neighbors can significantly influence species diversity: Below a certain mobility threshold species coexist over long periods of time and selforganize by forming fascinating spiraling patterns, whereas biodiversity is lost when that threshold is exceeded [15].…”

“…[15,17,18,21,23]. The model that we consider accounts for cyclic competition with dominanceremoval [15,18,21,23] and dominance-replacement [16], also including reproduction, mutation, and mobility in the form of hopping and pair exchange between nearest neighbors. Our approach is inspired by the experiments of Ref.…”