Characterization of phase transitions in a model ecosystem of sessile species

We consider a model ecosystem of sessile species competing for space. In particular, we consider the system introduced by Mathiesen et al. [J. Mathiesen, N. Mitarai, K. Sneppen, and A. Trusina, Phys. Rev. Lett. 107, 188101 (2011)] where species compete according to a fixed interaction network with links determined by a Bernoulli process. In the limit of a small introduction rate of new species, the model exhibits a discontinuous transition from a high-diversity state to a low-diversity state as the interaction probability between species, $\gamma$, is increased from zero. Here we explore the effects of finite introduction rates and system size on the phase transition by utilizing efficient parallel computing. We find that the low state appears for $\gamma >{\gamma }_c$. As $\gamma$ is increased further, the high state approaches the low state, suggesting the possibility that the two states merge at a high $\gamma$. We find that the fraction of time spent in the high state becomes longer with higher introduction rates, but the availability of the two states is rather insensitive to the value of the introduction rate. Furthermore, we establish a relation between the introduction rate and the system size, which preserves the probability for the system to remain in the high-diversity state.

Figure 1

(a) Mean diversity and (b) mean length of frozen boundary as a function of interaction frequency $\gamma$ for different introduction rates $\alpha$. $N={320}^2$.

Figure 2

Snapshots of system in (a), (b) low- and (c), (d) high-diversity state. (a) and (c) show magnification of upper left corners of the snapshots (b) and (d), respectively, with marked frozen (black) and active (white) boundaries. $N={192}^2,\gamma =0.7$, and $\alpha =0.0005$.

Figure 3

Time series of diversity and frozen boundary length with introduction rate $\alpha =0.003$ and interaction frequency $\gamma =0.16$ in an $N={192}^2$ system. For higher $\gamma$ and $\alpha$ the diversity (green dashed line) fluctuations are stronger and the frozen boundary length (blue dotted line) is a more reliable criterion. The solid red curve shows whether the system is classified to be in the high or low state, based on the frozen boundary length. The curve shows the frozen boundary length threshold value that was crossed for the current classification.

Figure 4

(a) Distribution of boundary lengths with $N={320}^2,\gamma =0.12$, for various $\alpha$. (b) Location of maximum (mode) as a function of $\gamma$.

Figure 5

Fraction of (a) high- and (b) low-state samples over $\gamma$ and $\alpha$. $N={320}^2$.

Figure 6

Phase diagram. Classification threshold for the frozen boundary is questionable in grey area due to boundary length approaching zero.