Differential and difference equation

*how to solve this problem?*

Fishery A uses the following model for the absolute growth rate

dP/dt= r P(1-P/K)-h

where h is the constant absolute harvest rate.

(a)If r = 1.04 and K = 100 calculate the harvest rate h which gives

an equilibrium at Peq = 55.

(b)Using the above values of r, K and h show that the model can be

rewritten in the factorised form

dP/dt=r/K(P − a)(b − P).

and find the values of a and b.

(c)Thus, use separation of variables and partial fractions to show

that

ln |P − 45| − ln |P − 55| = kAt + C

where C is an arbitrary constant. Determine the value of kA.

(d) [2 Marks] Next, calculate the value of the arbitrary constant if

P = 46 when t = 0.

Then, find the time taken for the population to increase from 46 to

54 kilotons.

Fishery B uses the following model for the relative growth rate

1/P dP/dt= r(1 −P/K) − e

where e is the constant relative harvest rate (also called the

effort).

(a)If r = 1.04 and K = 100 calculate the effort e which gives an

equilibrium

at Peq = 55.

(b)Using the above values of r, K and e show that the model can be

rewritten in the factorised form

dP/dt=r/KP(c − P).

and give the value of c.

(c)Thus, use separation of variables and partial fractions to show

that

ln |P| − ln |P − 55| = kBt + C

where C is an arbitrary constant. Determine the value of kB.

(d)Next, calculate the value of the arbitrary constant if P = 46

when t = 0.

Then, find the time taken for the population to increase from 46 to

54 kilotons.