(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
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
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