Applications of Systems of Equations 451for a conventional force, x, versus a guerrilla force, y, on the interval [O, 5] forthe initial conditions (i) x(O) = 0, y(O) = 0 and (ii) x(O) = 4, y(O) = l. In
each case, decide which force wins the battle, when victory occurs, and the
number of combatants in the winning force at the time of victory.
- The Lanchesterian model for two guerrilla forces engaged in battle is
(11)dxdt = f(t) - Ax - Bxy
dy =g(t)-Cy-Dxy
dtwhere A, B, C, and Dare nonnegative constants and where f(t) and g(t) are
the reinforcement rates, Ax and Cy are the operational loss rates, and Bxy
and Dxy are the combat loss rates.
a. Write the system of equations for two guerrilla forces engaged in combat
when there are no reinforcements and no operational losses.b. Divide one equation of your system by the other equation to obtain a
differential equation in x and y.c. Find the general solution to the differential equation of part b. That is,
find the trajectories of the system.d. Sketch a phase-plane portrait for the system in the first quadrant.
e. For each force determine conditions which ensure victory and determine the
number of combatants in the winning force at the end of the battle. Specify
conditions under which there is a tie.- Use SOLVESYS or your computer software to solve the Lanchester combat
model
dxdt = 2 - .5x - .5xy
(12)
dy
dt = 1 - .2y - .25xy
for two guerrilla forces engaged in battle on the interval [O, 5] for the initial
conditions:
(i) x(O) = 0, y(O) = 0 (ii) x(O) = 0, y(O) = 6(iii) x(O) = 5, y(O) = 0 and (iv) x(O) = 5, y(O) = 6
In each case, display a phase-plane graph of y versus x. Do you think either
force will ever win?