Question - If the fish population at a given time is 240,000, give an estimate of the number of fish born in one week.
Given
- Constant Harvesting Rate $h = 2100$ fish per week
- Per Capita Death Rate $α = 0.2$ fish per day per fish
- Per Capita Birth Rate $β = 0.7$ fish per day per fish
So let the population be $X(t)$ at a given time $t$
The differential equation we get is $$\frac{dX}{dt} = βX - αX - h$$
Putting in the values we get $\frac{dX}{dt} = 0.7X - 0.2X - \frac{2100}{7}$ (keeping it per day)
$\frac{dX}{dt} = 0.5X - 300$
Solving this we get $X(t) = \frac{Ce^{\frac{t}{2}} + 300}{0.5}$
Taking an initial condition of $X(0) = X_0$
Then solving the general solution to get the particular solution we get
$X(t) = (X_0 - 600)e^{\frac{t}{2}} + 600$
Now the question asks us to estimate the number of fish born in one week, so $X(t) = 240000$ But there is no other information so how will we approach this bit, like we don't know the value of $X_0$