### 4.5.3 Model analysis

It is possible to obtain long term harvesting rates easily for the logistic model by making two assumptions.

1. The harvesting occurring during the exponential growth phase follows the results obtained previously;
2. As the population approaches equilibrium, we can take theevalue of to be zero.

From the logistic equation, the last assumption leads immediately to the result

This is simply a quadratic equation in P, the population. Clearly, the roots of this equation must be real for the population to actually exist. We have:

cP2 - McP + Mh =0 whose solution is given by

For real fish, we require real roots so that

This leads immediately to the condition

Hence the admissible harvesting values are given by the inequality .

Notice that as the population , which is half that of the totally unharvested population.

1. Case

Evidently, P1 is unstable equilibrium point whereas P2 is stable.

2. Case

3. Case