Science:Math Exam Resources/Courses/MATH102/December 2013/Question C 02 (d)/Statement

The population of fish in a particular lake is given by the function F(t) where F is measured in number of fish and t is measured in days. A company that manages fish stocks is hired to restock the lake, adding fish at a constant rate. Only N fishers are allowed to fish in the lake at a time. A simple model for this scenario is given by the equation:

${\frac {dF}{dt}}=I-\alpha NF$

Where I and $\alpha$ are constant and two cases for N are considered.

Case 2: The agency that administers fishing permits decides that the number of fishers should be proportional to the number of fish currently in the lake, $N=F/3$. What is the population size $F_{*}$ to which F(t) approaches as $t\to \infty$ in this scenario?