**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.

.

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