## Tuesday, February 2, 2010

### Harvesting Model

Harvesting a Logistic Population:

When the harvesting term -k is incorporated into into bounded population model we have

.

There are three solution forms for this differential equation, and they correspond to the nature of the stationary solutions ( x(t) = c).

Definition(Stationary Points):

The stationary points of the D. E. are solutions where and are the roots of the characteristic equation

.

The roots are known to be , and the stationary solutions are .

Remark:

Since x(t) is a real function, there are no stationary solutions when .

Case (i) If there is one stationary solution :

When , the differential equation has the form and the solution is

.

The solution with the initial condition is

.

If then .

If then function x(t) has a vertical asymptote at

and the population x(t) becomes extinct at some time (where ), i. e.

.

Case (ii) If there are two stationary solutions and :

When , the differential equation has the form and the solution is

.

The two real roots of the characteristic equation , are .

The solution with the initial condition

If then .

If then the population x(t) becomes extinct at some time , i. e. .

Case (iii) If there are no stationary solutions:

When , the differential equation has the form and the solution is

.

The solution with the initial condition is

The function x(t) has a vertical asymptote at so the population x(t) becomes extinct at some time (where .), i.e.

.