R

Recipe 11.2 (continued)

If conditions (i) and (ii) are both satisfied, a model is guaranteed to exhibit periodic dynamics. The type of cycle that occurs will have one of three forms:

• At /x = fi* an infinite number of neutrally stable periodic solutions surround the equilibrium point (x^y^).

• For all values of /x slightly larger than ¡x* a single stable limit cycle surrounds the equilibrium point, (x.^ yM). This is referred to as a supercritical bifurcation because it takes place for values of ¡x above ¡x*.

• For all values of ¡x slightly smaller than /x* a single unstable limit cycle surrounds the equilibrium point, (xyM). This is referred to as a subcritical bifurcation because it takes place for values of ¡x below ¡x*.

With this classification, we can now add a third condition:

(iii) If (xyM) is stable at p. = ¡x*, then the cyclic behavior will consist of a unique stable limit cycle for all values of /x in an interval between ¡x* and ¡x* + a where a is a small positive constant (a supercritical Hopf bifurcation). Conversely, if (x^, y^) is unstable at ¡x = ¡x*, the cyclic behavior will consist of a unique unstable limit cycle for all values of fx in an interval between fx* a and ¡x* where a is a small positive constant (a subcritical Hopf bifurcation).

Condition (iii) is the most difficult to check (because a local stability analysis is inconclusive when a(fx*) = 0). We provide an on-line supplementary Mathematica package to automate the calculations.

We now apply the Hopf bifurcation theorem to the sexual selection model (11.7). First we must decide which parameter we will treat as the bifurcation parameter. From a mathematical standpoint it does not matter which parameter we use, but we certainly want to choose one that affects the stability of the equilibrium. An obvious choice would be the composite parameter y = -b Gp - c Gz + a B, because the equilibrium is stable only when y is negative. Therefore, we take y as the bifurcation parameter (represented by fi in Recipe 11.2). The critical value of the parameter, which is where the equilibrium switches from stable to unstable, is then y* = 0 (i.e., the real parts of the eigenvalues are zero at y = 0).

Next we need to identify the real and imaginary parts of the eigenvalues at this equilibrium. The eigenvalues for this equilibrium are

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