# Section 1 - Variables and Parameters

## What is the system we are examining?

The basic idea of this system is that there are two species which are helping each other out in some way. They could be fungus and algae, frogs and toads , or rhinoceroses (rhinoceri?) and those tiny little birds that clean them up. For our purposes, it doesn't matter: we'll call the two species x and y. All that matters is that they've got partnership which makes them both live more procreative lives: in the equations below, this effect is represented by the parameters b and d. We will proceed to examine the equilibria of this system and the interesting bifurcations that occur.

### System

• x' = ax + bxy - ex^2
• y' = cy + dxy - fy^2

### Variables

• x = number of species #1
• y = number of species #2

### Parameters

• a = birth rate of x-population
• b = interaction rate of (effect of interaction on) x-population
• e = death rate of x-population
• c = birth rate of y-population
• d = interaction rate of (effect of interaction on) y-population
• f = death rate of y-population

There are a few variables that are always larger than zero: a and c, because you can't give birth to negative kids; and e and f, because (by definition) dying means that there are less of you around.

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