# (Optional, for mathematically trained students) In the three-type evolutionary game of Section 5 and

(Optional, for mathematically trained students) In the
three-type evolutionary game of Section 5 and Figure 12.11, let q3 5 1 2 q1 2
q2 denote the proportion of the orange-throated aggressor types. Then the
dynamics of the population proportions of each type of lizard can be stated as

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And

We did not state this explicitly in the chapter, but a
similar rule for q3 is

(a) Consider the dynamics more explicitly. Let the speed
of change in a variable x in time t be denoted by the derivative dxdt. Then
suppose

Verify that these derivatives conform to the preceding
statements regarding the population dynamics.

(b) Define   Using the chain rule of differentiation,
show that dXdt 5 0, that is, show that X remains constant over time.

(c) From the definitions of the entities, we know that q1
+ q2 + q3 = 1. Combining this fact with the result from
part (b) show that over time, in three-dimensional space, the point (q1,
q2, q3) moves along a circle.

(d) What does the answer to part (c) indicate regarding
the stability of the evolutionary dynamics in the colored-throated lizard
population?