Deterministic extinction by mixing in cyclically competing species

We consider a cyclically competing species model on a ring with global mixing at finite rate, which corresponds to the well-known Lotka-Volterra equation in the limit of infinite mixing rate. Within a perturbation analysis of the model from the infinite mixing rate, we provide analytical evidence that extinction occurs deterministically at sufficiently large but finite values of the mixing rate for any species number $N\ge 3$. Further, by focusing on the cases of rather small species numbers, we discuss numerical results concerning the trajectories toward such deterministic extinction, including global bifurcations caused by changing the mixing rate.

Figure 1

A schematic description of the model. An invasion through the nearest neighbor (NN) occurs along the cyclic competition. Individual species on randomly chosen two sites are exchanged at rate $v/2$.

Figure 2

(a) The maximum real part of the eigenvalues $\lambda$ at the extinction fixed points for $N=3$. The imaginary part is shown in the inset. (b) $P_{\alpha }\left(t\right)$ computed by Eqs. (2) and (4) for $v=1$ from the symmetric initial condition with a perturbation.

Figure 3

(a) The maximum real part of eigenvalues at the extinction fixed point for $N=4$ as a function of $p$. (b) Eigenvalues $\lambda$ at the symmetric fixed point with $N=4$. The top panel shows the two eigenvalues with the largest and the second largest real part, and the bottom panel shows their respective imaginary parts. (c) $P_{\alpha }\left(t\right)$ computed by Eqs. (2) and (4) with $N=4$ from the symmetric initial condition with a perturbation for $v=0.02$. (d) $P_{\alpha }\left(t\right)$ computed by Eqs. (2) and (4) with $N=4$ from the symmetric initial condition with a perturbation for $v=1$. (e), (f) MC simulations: the species densities with $L=9000$ (e), (f) and the spatiotemporal plot with $L=1000$ (g), (h) for $v=0.02$ (e), (g) and $v=1$ (f), (h), where $x$ is site number.