few uninvadable equilibria control the system and the transitions are noise induced.[]{data-label="fig1"}](phasesillustration){width="7cm"}

The Wilson-MacArthur line, $C=0$
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When $C=0$, species do not interact with each other and the dynamics of each species is logistic with carrying capacity $K$. Deterministically, an infinitesimally small population grows exponentially and then saturates to the steady state value, $K$. With demographic noise the first immigrant may fail to establish a community (e.g., it may die before reproduction, even if the birth rate is larger than the death rate), so not every immigration results in successful colonization. The chance of a successful colonization depends on the details of the stochastic process (in particular, the variance in the number of offspring per individual, see the analysis of [@DAKSIS] for the SIS model). After a successful recolonization the population still fluctuates around $K$, and in the long run it must go extinct as well since (without migration) the zero population is an absorbing state. The rate of these long-term extinctions (as opposed to colonization failures that take place at short times) depends, again, on the details of the process and the value of $K$ [@OIK19991].

Accordingly, in the $C=0$ limit of our model the history of every species 