Figure 1: Sea lice on wild juvenile salmon 87 6 - 4 2- Nov Jan Mar May Jul Motile louse abundance (ind. fish-1) 87 b 2 Nov Jan Mar May Month Figure 2: Number of motile sea lice per fish during the winter of 2005-2006 on two salmon farms in the Broughton Archipelago: (a) Sargeaunt Pass and (b) Humphrey Rock. The dashed line indicates the time of parasiticide treatment.Sea lice are a parasite that feed on salmon. They occur naturally in low densities on wild salmon, but crowded conditions on salmon farms are ideal for sea louse outbreaks. Sea lice from salmon farms can spread onto wild migrating salmon. To help protect wild salmon, the Canadian federal government requires salmon farms to treat their fish with a parasiticide (or else to harvest the fish) when sea lice counts on a salmon farm reach a threshold of three motile lice per fish. A model for sea louse population on a salmon farm before and after treatment is given by p (t) = Peat if t T' where p(t) is the number of motile lice per fish at time t, 7 is the time of treatment, and P, Q, a, b > 0 are constant parameters. 2. Find the value of Q such that p(t) is continuous for all t. In the winter of 2005-2006, the sea louse populations for two salmon farms in the Broughton Archipelago were fitted to the model above. The population graphs are shown in Figure 2 at the bottom of this document. 3. (a) Use the limit definition of derivative to show that p(t) satisfies the differential equation p'(t) = Jap(t) if t T' (You may use without proof the fact that lim h = log(c) for all c > 0.) 140 (b) Which of the two farms, Sargeaunt Pass or Humphrey Rock, has a larger parameter a? Justify your answer carefully. 4. On the Sargeaunt Pass farm, the parameter b was measured to have the value b = 0.057. Juvenile wild salmon begin migrating past the farm on March 1. On what date should treatment occur so that the sea lice population on the farm is reduced by 95% by the time of migration? (Assume it is not a leap year.)