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5.3. Location of Complex Roots 179

There are values of a for which the fixed point is an unstable population

density. For example, in the case a = 4, the fixed point is 0.75. Calculate

the population density for ten successive years when the initial density is

0.74. Make a diagram to illustrate the situation. How do you account for

the instability? What do you think will happen to the population density

in the long run?

For which values of the parameter a will the nonzero fixed point u for

the function f(x) represent a stable population density?

Consider the possibility of a population density which oscillates between

two values which are the roots other than 0 and u of the equation x =

f(f (x)). Verify that this equation can be written

x[ax - (a - l)][Q”X” - a(a + 1)x + (u + l)] = 0.

Verify that, when a = 3, it has a triple root x = 2/3, that when a > 3, it
has three positive real roots with one on either side of (a - 1)/a, and when
a < 3, it has two nonreal roots. Discuss the stability of the fixed point when
a is less than, equal to and greater than 3. Compute the derivative off at
the fixed point, and discuss the significance of the value of that derivative.
If you have access to a computer, examine the possibility of a population
density which has a k-year cycle, for which, if zi+i = f (xi), we have zk =
xc, but x0,x1, x2,... , z&i all differ.

5.3 Location of Complex Roots

Certain classes of difference (recursion) and differential equations have as-
sociated with them a polynomial whose roots give information about the
properties of the solutions. In Exporation E.50 we will see for instance,
that if the zeros of the related polynomial lie in the interior of the unit disc
in the complex plane, then the nth term of a recursion will tend to 0 as n
grows large. For differential equations, it is often useful to know whether or
not there are any zeros in the right half plane (i.e. for which the real part
is positive). One approach is to adopt the idea used in our second proof of
the Fundamental Theorem of Algebra (Section 4.5), that if a polynomial
has a zero in the region surrounded by a closed curve, then the image of
that curve under the action of the polynomial makes at least one circuit of
the origin.
In the exercises, we will first look at estimates of the size of the zeros,
and then sample techniques for locating zeros within certain regions of the
complex plane.