1 Appendix S1 We derive here some asymptotic properties of model (5) but we will use Verhulst’s r-α formulation of the logistic model because it makes mathematical derivation lighter. The usual r-K parameterization is obtained with K= r /α. Firstly, we prove that under perfect mixing (β → ∞) we have the following equality: N1 γ1 = N2 γ2 . For that, we use Eq. (5) to consider the following difference: d dt N1 γ1 − N2 γ2 ⎛ ⎝⎜ ⎞ ⎠⎟ = 1 γ1 dN1 dt − 1 γ2 dN2 dt = r1 γ1 N1− α1 γ1 N1 2− r2 γ2 N2+ α2 γ2 N2 2−β 1 γ1 + 1 γ2 ⎛ ⎝⎜ ⎞ ⎠⎟ N1 γ1 −N2 γ2 ⎛ ⎝⎜ ⎞ ⎠⎟ . In the limit β → ∞, the last term in the RHS becomes predominant and the equation tends to: d dt N1 γ1 − N2 γ2 ⎛ ⎝⎜ ⎞ ⎠⎟ = −β 1 γ1 + 1 γ2 ⎛ ⎝⎜ ⎞ ⎠⎟ N1 γ1 −N2 γ2 ⎛ ⎝⎜ ⎞ ⎠⎟. Consequently N1 γ1 − N2 γ2 tends to zero, that is N1 γ1 = N2 γ2 . From this equality, we can also deduce that NT γ1+γ2 = N1 + N2 γ1+γ2 = N1 γ1 γ1 + N2 γ2 γ2 γ1+γ2 = N1 γ1 = N2 γ2 . (A1) Secondly, we calculate the dynamics of total abundance. Equations (5) modified in the r-α formulation give: dNT dt = dN1 dt + dN2 dt = r1N1−α1N1 2+ r 2N2−α2N2 2 and using (A1), we have:
2 dNT dt = r1N1 γ1 γ1 −α1N1 2 γ1 γ1 ⎛ ⎝⎜ ⎞ ⎠⎟ 2 + r2N2 γ2 γ2 −α2N2 2 γ2 γ2 ⎛ ⎝⎜ ⎞ ⎠⎟ 2 = r
