## 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.