Species extinction occurs regularly and unavoidably in ecological systems. The time scales for extinction can broadly vary and inform on the ecosystem's stability. We study the spatio-temporal extinction dynamics of a paradigmatic population model where three species exhibit cyclic competition. The cyclic dynamics reflects the non-equilibrium nature of the species interactions. While previous work focusses on the coarsening process as a mechanism that drives the system to extinction, we found that unexpectedly the dynamics to extinction is much richer. We observed three different types of dynamics. In addition to coarsening, in the evolutionary relevant limit of large times, oscillating traveling waves and heteroclinic orbits play a dominant role. The weight of the different processes depends on the degree of mixing and the system size. By analytical arguments and extensive numerical simulations we provide the full characteristics of scenarios leading to extinction in one of the most surprising models of ecology.
Deep Dive into Three-fold way to extinction in populations of cyclically competing species.
Species extinction occurs regularly and unavoidably in ecological systems. The time scales for extinction can broadly vary and inform on the ecosystem’s stability. We study the spatio-temporal extinction dynamics of a paradigmatic population model where three species exhibit cyclic competition. The cyclic dynamics reflects the non-equilibrium nature of the species interactions. While previous work focusses on the coarsening process as a mechanism that drives the system to extinction, we found that unexpectedly the dynamics to extinction is much richer. We observed three different types of dynamics. In addition to coarsening, in the evolutionary relevant limit of large times, oscillating traveling waves and heteroclinic orbits play a dominant role. The weight of the different processes depends on the degree of mixing and the system size. By analytical arguments and extensive numerical simulations we provide the full characteristics of scenarios leading to extinction in one of the most s
Stochastic many-particle systems provide a testing ground for non-equilibrium dynamics. In nature, systems frequently evolve away from equilibrium and then relax to an equilibrium steady state. Understanding the relaxation process is a central topic in non-equilibrium physics. Near-equilibrium fluctuations are governed by the same laws that hold in steady state and the transient is typically an exponential decay. Many systems, however, comprise absorbing states, which can be reached but never be left by the dynamics. In this case no fluctuations are present in the steady states. Such systems arise in a broad variety of problems, e.g. physics, chemistry or epidemics [1]. Much effort has been spent on the investigation of simple, diffusion-limited chemical reactions where the decay to equilibrium can obey power laws [2].
Understanding transitions into absorbing states is not only fundamental for nonequilibrium physics, but is also highly relevant for ecology. Here, absorbing states correspond to the extinction of species. Another characteristic feature of ecological systems are cyclic interactions. As a classic example, the work of Lotka and Volterra describes the dynamics of fish populations in the adriatic as persistent oscillations due to predator-prey interactions. Other examples include coral reef invertebrates [3], rodents in the high arctic tundra in Greenland [4], cyclic competition between different mating strategies of lizards [5] and chemical warfare of Escherichia coli bacteria under laboratory conditions [6].
Recent work has investigated cyclic competition in one-dimensional systems with no or only weak diffusion of the reacting agents [7,8,9,10]. Coarse-graining of temporally growing and annihilating domains has been identified as the mechanism that eventually leads to species extinction. However, individual’s mobility may be significant and alter this picture qualitatively.
In this article, we investigate the spatio-temporal dynamics of extinction in a paradigmatic model of three species in cyclic competition. Individuals are positioned on a one-dimensional lattice and are equipped with fast mobility that leads to effective diffusion. The system possesses absorbing states in the form of extinction of two of the three species and, because of fluctuations, the dynamics eventually comes to rest there. However, the time-scales until extinction occurs provide information on the stability of species diversity [11]. We identify three distinct types of dynamics that lead to extinction. These types of dynamics arise from the possible influences that intrinsic fluctuations can have on the coarsening process and on the traveling waves that the cyclic dynamics induces. The different dynamics lead to characteristic dependences of the extinction-time probability on the elapsed time t and the system size N . We provide semi-phenomenological arguments that quantify the functional form and the scaling behaviour of the extinction-time probability. These arguments yield information on the emergence and characteristics of the different types of dynamics.
Consider a stochastic, spatial variant of the May-Leonard model which serves as a prototype for cyclic, rock-paper-scissors-like species interactions. Three species A, B, C compete with each other in a cyclic manner, at rate σ, and reproduce at rate µ upon availability of empty space ∅:
For increasingly large populations intrinsic fluctuations eventually become negligible. If in addition spatial structure is absent, i.e., if every individual can interact with every other in the population at equal probability, the population dynamics is aptly described by deterministic rate equations for the densities s = (a, b, c) of the species A, B and C:
Hereby the indices are understood as modulo 3 and ρ = a + b + c denotes the total density. May and Leonard showed that these equations possess 4 absorbing fixed points, corresponding to the survival of one of the species and to an empty system [12]. Furthermore a reactive fixed point s * = µ σ+3µ (1, 1, 1) exists that represents coexistence of all three species. Linear stability analysis shows that s * is unstable. The absorbing steady states that correspond to extinction, (1, 0, 0), (0, 1, 0) and (0, 0, 1), are heteroclinic points. The Lyapunov function L = abc/ρ 3 demonstrates that the trajectories of the deterministic equations (2), when initially close to the reactive fixed point, spiral outward on an invariant manifold. On this manifold the trajectories then approach the boundary of the phase space and form heteroclinic cycles, converging to the boundary and the absorbing states without ever reaching them.
However, intrinsic noise from finite-system sizes [13,14] and spatial correlations alter the above behaviour [15,16,17,18]. While fluctuations ultimately drive the system into one of the absorbing fixed points [19], the formation of spatial patterns can substantially delay extinction and promote species coexisten
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