Science:Math Exam Resources/Courses/MATH102/December 2013/Question C 02 (c)/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 1: Suppose N is a constant. What is the steady state number of fish $(F_{*})$ in the lake? If the lake has no fish in it initially, at what time $(t_{*})$ does the population size reach half its steady state value $F_{*}/2$? (You do not have to show a derivation of the solution F(t) of the equation for full points - simply stating it is sufficient - but you must show the rest of the calculation.)