Question: We apply Lanchester's model to the Battle of Trafalgar (1805), when a fleet of 40 British ships expected to face a combined French and Spanish fleet of 46 ships. Suppose that there were x British ships and y opposing ships at time t. We assume that all the ships are identical so that constants in the differential equations in Lanchester's model are equal:
(dx/dt) = -ay
(dy/dt) = -ax
(a) Write a differential equation involving dy/dx, and solve it using the initial sizes of the two fleets.
(b) If the battle were fought until all the British ships were put out of action, how many French/Spanish ships does this model predict would be left at the end of the battle? Admiral Nelson, who was in command of the British fleet, did not in fact send his 40 ships against the 46 French and Spanish ships. Instead he split the battle into two parts, sending 32 of his ships against 23 of the French/Spanish ships and his other 8 ships against their other 23.
(c) Analyze each of these two sub-battles using Lanchester's model. Find the solution trajectory for each sub-battle. Which side is predicted to win each one? How many ships from each fleet are expected to be left at the end?
(d) Suppose, as in fact happened, that the remaining ships from each sub-battle then fought each other. Which side is predicted to win, and with how many ships remaining?