# Section 6 - The Meanings of the Equilibria

## Do these equilibria make sense?

The population of adults stabilizes at n/u, which means that the number of adults (a) is directly proportional to the death rate of the predators and inversely proportional to the predation rate. Thus, the faster the predators are dying, the more adults there are, and the slower the predators are eating adults, the more adults there are. Which makes sense.

The population of children stabilizes at:

```           bn
--------
u (g + d)
```

So, c is directly proportional to the birth rate of the prey and the death rate of the predators and inversely proportional to the predation rate and the (growing up + death) rate. Thus, the faster the adults are having babies and the faster the predators are dying (and so not eating the adults), the more children there are. And the slower the predators are eating adults and the slower the child population is losing members (by growing up and death, hence the sum), the more children there are.

The population of children can also be expressed as:

```     ab
-----
g + d
```

Here, it is again directly proportional to the prey's birth rate and inversely proportional the sum of the two "death" rates (growing up and real death), and also directly proportional the to the number of adults. The terms were merely consolidated.

The population of predators stabilizes at:

```   bg - d(g + d)
--------------
u (g + d)
```

This is more easily seen as gc/ua - d/u or (gc - da) / (ua) when it is close to the equilibrium. It is directly proportional to the rate of change of the adults, not including the predation factor. Thus, the faster the adult prey population is growing, aside from the predation issue, the bigger the predator population.

It is inversely proportional, however, to the size of the adult prey population and the predation rate -- the faster it eats the adults (and the more there are to eat), the smaller the predator population is. This happens because it eats the prey so fast that there are barely any left, and it takes a while to replenish the population --- meanwhile, the predator population shrinks so far that it can never reach its former peak, and it goes to a small limit rather than a large one.

It should be noted that the death rate of the predators (n) has no effect on the limit of the predator population; it changes how it gets there, but the end result is the same.

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