the equation for the species with vanishing abundance (call it $k$) reads $$0=1  - (C/K)\sum_{j \ne k} c_{k,j} N_j$$ The r.h.s. of this equation is just the growth rate of the $k$th species, so that for $C$ smaller than the critical value, the growth rate of this species is positive, and for $C$ above this value, it is negative. Thus, the $k$th species cannot invade for $C$’s slightly larger than the value at which that species disappears from the community.

In Fig. \[phase\], we present the May line in the $C$-$\sigma$ plane, showing the dependence of $C_1$ on $\sigma$. The data was obtained by calculating the May point for a set of 100 random $c_{i,j}$ matrices for a given sigma, and averaging. We see that $C_1$ increases with decreasing $\sigma$, and appears to approach unity for $\sigma \to 0$. We shall return to this point later.

![The phase boundaries in the $C$, $\sigma$ plane. The leftmost, red, line is the May line where the deterministic solution with all species present disappears. The next, solid black, line to the right is the simulationally determined line where the concave downward part of the species occupancy curve disappears, for the case $K=100$, $\lambda=0.01$. 