5
Approximation and Location
of Zeros
5.1 Approximation of Roots
The population of,a biological species in a favorable environment will in-
crease steadily from year to year. Often, we can assume that the annual
increase will be given by a factor a, so that if x is this year’s population
and f(z) next year’s, f(x) = az, where a > 1. This equation may model
the situation quite well for small populations, but as the population grows,
factors arise which tend to limit it: nonavailability of food, greater visibil-
ity to predators, conflict arising from crowded conditions. Mathematically,
this can be handled by introducing a “second-order” term in the population
function:
f(x) = ax - bx2 (b > 0).
Thus for large values of x, f(x) will be less than x and the population will
decrease. There will be an intermediate level of population, 20, for which
the factors promoting increase are balanced by those promoting decrease.
This level can be found by solving the quadratic equation x = f(x).
Those familiar with the habits of June bugs, spruce budworm and lem-
mings are aware of another phenonenon. For these, there is no “steady
state” population which persists from year to year. Rather, their numbers
cycle between large and small values. Can we model this using some func-
tion f(z) given above?
If we imagine a two-year cycle of a lean year with population size xl fol-
lowed by a glut year with larger population size x2, then these two numbers
will satisfy
12 = f(C1) Xl = f(x2)
2; # f(x;) for i = 1,2.Thus, x1 and 22 will be roots of the quartic equation
2 = f(f(x)) = a2x - (ab + ba2)x2 + 2ab2x3 - b2x4.Whether one model successfully represents the cycling population turns on
whether Q and b can be chosen so that the quartic has two positive real
roots apart from x = (a - 1)/b which are less than x = a/b (the population
level which leads to extinction).