(a)

The logistic model equation is .

At time , panthers in the preserves.

At time , panthers in the preserve.

The florida preserve capacity is .

Substitute , and in .

.

Substitute and in .

Substitute and in .

Substitute in .

.

Therefore, the equation for the population of the panthers in the preserve is .

(b) Find the population after years.

Substitute in .

Therefore, the population of the preserve after years is .

(c) Find the time to population reach .

Substitute in .

Therefore, the population reaches after .

(d)

The differential equation is in the form of logistic differential equation .

Substitute and in .

The differential equation is .

The initial condition is .

Step size is .

Eulers Method :

Using , , and . 

. 

. 

.

. 

.

Therefore, the population after years is .

(e)

The population of the preserve is .

is increasing most rapidly where , corresponds to .

(a) .

(b) The population of the preserve after years is .

(c) The population reaches panthers after .

(d) ; .

(e)  .