The simple population growth equation (L? E 2 has solution P : 137140 for some constant C. If 'r' :> [ii this leads to unrestricted growth, which is not realistic in man}:r situations. rP :I A more nuanced variant is the Logistic equation where . the population at time t is P : P[t) _ . the maximum sustainable population is the constant K . the maximum rate of population growth is the constant 3 _ 4 This is a separable differential equation which can be written as dP Integration of the LHS of this equation can be done using the partial fraction expansion : rdt. (1) 44+ B 13.0% P 1s where -A= so .3: gm. We now integrate both sides of equation [1) and nd lufP]1n(1) :'rt+C K Where C is the constant of integration. It know that initially, P = 3: then we nd that C = ln[K). We can rearrange this equation to express P as a function of t: P: gm