 If \(\lambda_{S} = \mu_{S}\), then \(dS/dt = 0\), and we have an equilibrium \(S^*\). Assume that \(S\) species are present on our island. We characterize the total immigration rate as follows. Let \(\lambda_{A}\) be the average immigration rate of new species colonizing the island per unit time, and thus the total immigration rate is \(\lambda_{S} = \lambda_{A}(P-S)\). The total extinction rate can be characterized as well. Let the average extinction rate per species per unit be \(\mu_{A}\). Then \(\mu_{S} = \mu_{A}S.\) Therefore, \(dS/dt = \lambda_{A}(P-S) - \mu_{A}\). Finally, we can derive \(dS/dt\) evaluated at the equilibrium \(S^*\),