After an initial trait where (eta_{mathrm{NG}}) grows as (sqrt{nu psi_{0})}), the relation between the two figures of merit becomes approximately linear, showing the direct connection between nonlinearity and nonclassical character of the state of the oscillator.
A numerical nonlinear fit gives the functional relation eta_{mathrm{NG}}=0.002+0.207 sqrt{nu psi_{0})}+2.731nu( psi_{0}), (11) showing that after an initial trait, the link between the two quantities becomes linear, ensuring the proportionality of the two figures of merit under scrutiny.
In contrast, the traits evolution can make multiple equilibria and can stabilize or destabilize the system, not depending on the parameter values but on the initial trait values.
The coevolutionary consequences can differ dramatically depending on the initial trait value and the timing of the alternative prey's invasion into the original system.
In this case, destabilization occurs even when the initial trait value of the second prey is very low (c in Fig. 5).
In this case, regardless of the invading prey's initial trait value, the second prey always evolves its defense to higher values and destabilizes the system (Fig. 6).
This result holds for many more conditions of covariance matrices and initial trait values (BJR and MK unpublished results) as long as none of the three trait values evolved to zero prior to achieving equilibrium, a condition causing convergence problems.
The effect of the invasion depends mainly on the initial trait value of the invading prey, the ease with which the prey can evolve (cost constraints and speed), and the stability of the resident system.
The present results show that whether coevolution in a predator prey system is likely to be disturbed by an invading prey species can depend on the initial trait value of the invading prey.
In the opposite case, when the invading prey species is adaptively superior, the invading prey is likely to evolve its defense to lower values regardless of the initial trait value, resulting in destabilization of the population dynamics.
