Harvesting

We can have constant harvesting, where a fixed amount is harvested, regardless of population size, and harvesting where a fixed fraction of the population is removed.Constant Harvesting with the Exponential Model - This is done by adding a fixed constant H to the exponential model. The equation is still separable and can be solved in the usual way. In order to find the maximum harvesting, one can use equilibrium points of the model and the discriminant. For real solutions, the discriminant has to be greater than zero; this inequality can be used to find restrictions for the constant harvesting. Constant Harvesting with the Logistic Model - We may have to adjust the carrying capacity to include constant harvesting in the Logistic Model. The same approach of considering the discriminant and solving the inequality for the discriminant greater than zero can be used to find the critical harvesting rate. For constant harvesting, the critical harvesting rate for the exponential model, Hc, is the max (with N>0) of f(N).For constant harvesting with the logistic model, the critical harvesting rate, Hc, is 1/4 a n. For other models, consider the max value (with N>0) of dN/dt. Calculate f'(N) = 0 to find a value for N, then substitute this into f(N) to find Hc. Considering harvesting that is a fixed fraction of the population, B, we can also find a critical rate of Bc above which the population would die out. For this, we have f(N) - BcN = 0, so we can solve f'(N) - Bc = 0 to find the critcal rate. In general, Bc is the maximum growth rate.