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the sun rises above the mountains at either end of the narrow twisting valley two armies slowly advance towards one another and to war as strategic adviser to general reddington leader of the red army you have been charged with planning this attack it falls to you to predict whether or not the red arm can successfully defeat general bluford and its blue invaders or if it would be better to fall back and live to fight another day you look down the valley and you can see that the enemy outnumber your own troops two to one however you also know that your red soldiers are much better trained than their blues so the question is could we use maths to predict the outcome of the battle and help us decide what to do mathematical models are useful tools for analyzing real-world situations they help us to simplify complex processes and allow us to calculate the potential outcomes the first step in applying a good mathematical model to any system is to understand the basic processes that make our system tick in this case the overall system is the battle and the underlying interaction is the combat between individual soldiers for the sake of simplicity let's assume that each soldier on the front line only fights with the opposite soldier on the other side when one is victorious the loser is instantly replaced by another soldier from that army this gives us a static front line between the armies stretching from one side of the valley to the other this means that the soldiers further back from the front don't actually do any fighting until they reach the line this may seem like a small detail but it becomes very important later on let's introduce two new parameters alpha and beta to account for how well trained and well armed each army is alpha represents how effective each red soldier is at defeating the enemy where a value of one means each red soldier on the line defeats one blue soldier per unit of time and a value of zero means that the red soldiers are utterly worthless and don't achieve anything ever similarly beta represents the same thing but for the blue soldiers in reality these parameters would probably depend on other factors since soldiers get tired as the battle goes on the parameters would have some dependence on time and if we factor in the morale of the soldiers it would probably also depend on both the size of their army and the enemies army too however to keep things simple we'll for now assume that they are constant this will be one of the main simplifying assumptions for our model we can now use these constants to construct a differential equation that describes how the size of each army changes with time we use a for the number of red soldiers and b for the number of blue soldiers at some time t the rate of...