Suppose that the population of a species of fish in a certain lake is hypothesized to grow according to a logistic model of population growth with growth rate $r=0.3$ and carrying capacity $k=3000$. Assume initially there are 2500 fish of the species in the lake. Determine the correct differential equation for each of the scenarios below.

(a) Each year 150 fish are harvested from the lake.
(b) Each year 25% of the fish are harvested from the lake.
(c) What is a maximum (within 50) safe fixed amount to harvest each year in order to assure that there will always be some fish in the lake

Question

Suppose that the population of a species of fish in a certain lake is hypothesized to grow according to a logistic model of population growth with growth rate $r=0.3$ and carrying capacity $k=3000$. Assume initially there are 2500 fish of the species in the lake. Determine the correct differential equation for each of the scenarios below.

(a) Each year 150 fish are harvested from the lake.
(b) Each year 25% of the fish are harvested from the lake.
(c) What is a maximum (within 50) safe fixed amount to harvest each year in order to assure that there will always be some fish in the lake

wwe123 · Answer

If P is the population at time t, r is the growth rate, and k is the carrying capacity, then we may model the population with the IVP:

\displaystyle \frac{dP}{dt}=rP\left(1-\frac{P}{k} \right) where \displaystyle P(0)=P_0

This model does not account for any harvesting. So what you need to do is introduce a term that accounts for the harvesting in each case. Beginning with part (a), what should you add to the ODE to account for 150 fish being harvested each year?

austinl · Answer

You literally copied this from another site I asked this on didn't you? @wwe123