The purpose of the following game is to introduce a study of quadratic equations by considering a simple population problem.

Alien worms have invaded earth and threaten all our gardens. We want to know how they will propagate and have collected the following information.

These worms produce little baby worms at the end of a summer (August). The the worms grow old and die, leaving their progeny in a dormant form of worms that must survive a winter to give rise to a new generation (in May). The following spring a certain fraction of these worms come to life. Some worms might remain dormant for a year or more before reviving. Others might be lost due to squishing, predation, disease, or weather. But in order for the worms to survive as a species, a sufficiently large population must be renewed from year to year.

From this description we have the following parameters and variables:

1. The number of worms produced per worm in August = a

2. The fraction of one-year old worms that spring to life in May = b

3. The fraction of worms that survive a given winter = c

4. Assume that worms older than two years are dead.

We can figure out what is going on with these growing worms using words, but it saves writing if we use symbols such as a, b, c, etc.!

pn = number of worms in the n^th generation.

sn = the number of one-year old worms in April before reviving.

<sn> = the number of one-year old worms left in May after some reviving.

s^(o)n = the number of new worms produced in August.

In May, a fraction, a, of recent worms and b of one-year old worms produce the next batch of worms.

The next generation is produced by the previous generation population times the number of worm offspring per worm plus the surviving worms from the winter.

pn + 1 = (worms from recent worms) + (worms from worms that survive the winter)

pn + 1 = a pn + b sn

In August, new (0-year) worms are produced at the rate of a per worm.

s^(o)n = a pn

Over the winter the worm supply changes due to mortality and aging. Worms that were new in generation n, s^(o)n, will be one year old in the next generation, n + 1.

sn+1 = c s^(o)n

Combining equations

sn+1 = ac pn

That is, the number of dormant worms in the n + 1 generation depends on the preceding n generation and the number of one-year old worms in April. While the number of one-year old worms in April for the n + 1 generation depends on the number of worms in generation n. This is an example of two coupled equations.

Of all the possible relationships let us require that the population of the next generation be proportional to the previous generation and that the next generation of worms be proportional to the previous generation of worms. Another way of stating this is to note that the dormant worm population depends on the worm population and therefore the proportionality constant for the two relations must be the same.

pn+1 = a pn + b sn = k pn

sn+1 = a c pn = k sn

Solve for k.

a pn + abc/k pn = k pn

ak + abc = k^{^2}

Relate the next generation to the previous generation:

pn+1 = (a +/- Sqrt(a^{^2} + 4abc))pn

What does this mean? For the upper sign there is exponential growth, but for the lower sign there is exponential growth if the root is less than a.

pn can never be negative and a, b, c => 0. The positive sign in front of the square root is OK, but what about the negative sign? If the negative square root is greater than a then pn+1 would become negative meaning a negative population which we have not allowed! If a - Sqrt(a^{^2} + 4abc) > 0 then abc = 0.

Now relax these real life criteria and play with the values and signs of the coefficients.