# Simple Combat Model

#### grandy

##### Member
The academics and the technicians of a university faculty are planning a paintball battle and the mathematicians are trying to predict their chances of winning with a continuous model. They are representing the numbers in the two teams at any time t as and respectively and have decided that the system to model the encounter should be dA/dT= - k2T and dT/dt= - K1A, where k1 and k2 are positive constants.
The muscles and eyesight of the academics have of course suffered from too much ‘book learning’ over the years, but at least they are aware of their limitations as good soldiers. They estimate that although both sides can fire paintballs at the same rate as each other, denoted by shots per minute, the technicians have a probability of 0.035 of hitting their target with a single paintball shot whilst the academics have only a 0.01 probability of hitting theirs.
Based on the initial conditions A(0)=100 ,T(0)=80, use Maple to produce a numerical approximation andgraph of the populations over a 20 minutes battle then comment on the outcomeafter that time.

Last edited by a moderator:

#### CaptainBlack

##### Well-known member
Re: Two Species Population Model

The academics and the technicians of a university faculty are planning a paintball battle and the mathematicians are trying to predict their chances of winning with a continuous model. They are representing the numbers in the two teams at any time t as and respectively and have decided that the system to model the encounter should be dA/dT= - k2T and dT/dt= - K1A, where k1 and k2 are positive constants.
The muscles and eyesight of the academics have of course suffered from too much ‘book learning’ over the years, but at least they are aware of their limitations as good soldiers. They estimate that although both sides can fire paintballs at the same rate as each other, denoted by shots per minute, the technicians have a probability of 0.035 of hitting their target with a single paintball shot whilst the academics have only a 0.01 probability of hitting theirs.
Based on the initial conditions A(0)=100 ,T(0)=80, use Maple to produce a numerical approximation andgraph of the populations over a 20 minutes battle then comment on the outcomeafter that time.
These are one of the Lanchester models of combat, in this case you can solve it by differentiating one of the equations again to get:
$\frac{d^2A}{dt^2}=+k_1k_2A$
Which is a linear constant coefficient ODE and so the general solution can be found by the usual methods. The given initial condition and the condition that $$A \not\to \infty$$ will allow the solution to be found.

However you question is ill posed, the question asked is the chance of winning, but the model presented is a continuous model, and you are not given sufficient information to find $$k_1$$ and $$k_1$$. The first of these problems can be overcome by dividing time up into small slices and modelling casualties in a time slice as a Poisson RV. But that leaves the second problem of the finding the rate constants.

CB