  We consider a model ecosystem of sessile species competing for space. In
particular, we consider the system introduced in [Mathiesen et al. 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 to 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.
