Modify the model in Problem 3 for net rate at which the population P(t) of a cer

Question

Modify the model in Problem 3 for net rate at which the population P(t) of a certain kind of fish changes by also assuming that the fish are harvested at a constant rate h > 0. Problem 3: Using the concept of net rate introduced in Problem 2, determine a model for a population P(t) if the birth rate is proportional to the population present at time t but the death rate is proportional to the square of the population present at time t. Dp/dt = k1P ? K2P^2 P(t) = population of fish at time t b=birth rate d=death rate b infinity P(t) d infinity P^2(t) b = k1P(t) d = k2P^2(t)

Explanation / Answer

Yes, it is correct

It can be solved by seperation of variables:

For h>0 (harvesting rate)

(k1*P - k2*P^2)*h = dt/dP

dP/(k1*P - k2*P^2)*h = dt

=>dP/k1P*h - dP/k2*P^2*h = dt

taking integration on both sides:

=>h(1/K1[lnP]+1/K2[1/P])=t+C h>0 and have constant value

=>lnP+K1/K2[1/P]=(t+C)/h

=>lnP+lne^K1/K2[1/P]=(lne^t+lnC')/lne^h ;; lnC'=C

=>Pe^K1/K2[1/P]=C'e^t-e^h Ans

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