# Background

The modified Predator-Prey model is simply a more realistic version of the predator-prey model. The equations are

```
u' = u(1 - u) - (auv)/(u + d)
```

v' = bv(1 - v/u)

```
```

####
Where u represents the population of the prey and v represents the population of the predators. We also know that a, b, and d must all be positive.

With this model comes several ideal conditions:

1) The predator v and its prey u are the only relevant species in the ecosystem. Also, u is the only food source for v, there is no predation on v, and v is u's only predator.

2) The prey u has an unlimited food supply.

3) Both populations live in a fixed region.

There are several limiting circumstances we know from a biological standpoint.

1) If the population of both the predator and the prey are extinct, then they both stay extinct.

2) If the prey population becomes extinct, then since u is the only food source for v then the predator population eventually becomes extinct.

3) If the predator population becomes extinct then the prey population will grow unchecked.

The constants a, b, and d could be environmental conditions, which keeps the growth rates in check if one population dies out.

Matthew Hirsch igel@wpi.edu

Matthew Shaw shaw@wpi.edu