Consider a population whose number we denote byP

. Suppose thatb

is the average number of births per capita per year andd

is the average number of deaths per capita per year. Then the rate of change of the population is given by the following differential equation,

dP/dt=bP-dP, (1)

wheret

is the time (in years).

(a)Suppose thatb=5

andd=3

. Solve the above ODE subject to the initial conditionP(0)=200

, and enter your expression forP(t)

below:

(b)For your solution above, what happens ast goes to infinity?

so P goes to?

(c)If the following fraction is to be split using partial fractions,

1,000/P(2,000-P)=(A/P)+(B/2,000-P)

find A and B ?

(d)Now suppose that the death rated

is composed of a per capita death rate as before, plus a death rate due to overcrowding and competition for resources of the formP

(it gets worse the bigger the population is), so that

d=3+(P/1,000)

Substitute this into the population differential equation (1) (withb=5

andP(0)=200

as before) and solve forP(t)

.

Hint:you might find the partial fraction in part (c) useful in determining your solution.

Enter your expression forP(t)

below: ?

(e)By examining your solution above, or by considering the differential equation for the populationP

with the modified death rate (part (d)), ast goes to infinity

,so P goes to ?