fish population density

Question

We consider a fish population density (number of individuals per space unit) N growing following the logistic model, i.e. with growth rate R(N) = rN(1 − (N/K)) where r and K are positive constants

This fish population is harvested, with a harvesting rate h(N)=qEN where E(effort of fishermen) and q(catchability of considered type of fish) are positive constants
 

a)At which fish population density NM does the gowth rate R(N) eaxctly balances (i.e. is equal to) the harvesting rate h(N)? The value NM is called the Maximal Sustainable Yeld, MSY

b)Assume now that N = NM, i.e. N is constant, and consider h as a function of E, the fishing effort: h(E) = qENM. What level of fishing effort E will maximizes the harvesting rate?

Answer 1

(a)

The fish population density is N .

Growth rate or Change population density with respect to time = dN/dt = R(N) = rN(1 − (N/K)) .

Harvesting rate of fishes is h(N)=qEN .

Growth rate R(N) is balances with the harvesting rate h(N) ⇒ R(N) = h(N) .

rN(1 − (N/K)) = qEN 

N[r(1 − (N/K))] = N[qE] 

N = 0 or r(1 − (N/K)) = qE

N = 0 or r − r(N/K)) = qE

N = 0  or r(N/K)) = r - qE ⇒ N = (r - qE)K /r

N = 0  or N = (1 - qE/r)K

N = 0 is corresponds to the extinction of fish population .

So N = (1 - qE/r)K .

Growth rate R(N) is balances with the harvesting rate h(N) when N = (1 - qE/r)K .