Multistability and Bifurcations in Coevolutionary Dynamics — Epoche C1
The claim, and the terms it needs A coevolutionary arms race — the reciprocal escalation in which a predator's attack traits and a prey's defence traits each drive change in the other — is commonly pictured as settling into a stable cycle, with the two species chasing one another round a fixed orbit. This essay sets out why that picture is only one of the outcomes such systems admit, and why the alternatives matter. The two alternatives named in the title need their meanings fixed first. Multistability is the property of a dynamical system of possessing more than one stable long-run state, so that where the system ends up depends on where it started; the trait space is then partitioned into basins of attraction separated by boundaries called separatrices. A bifurcation is a qualitative change in the set of long-run states as a parameter is varied smoothly — an equilibrium appearing, vanishing, or losing stability — so that a small change in a rate constant produces a discontinuous change in behaviour. Both are properties of the equations, not intuitions about biology, so the argument has to run through the equations. What follows derives them, identifies exactly which structural feature of an antagonistic interaction permits cycling at all, locates the two bifurcations that matter, and then corrects one claim in the original statement of this case that the analysis shows to be impossible. What the trait equations say, and what the rate constants are Write $\tau_P$ and $\tau_H$ for the mean values in the two populations of the traits that mediate the interaction — say the toxicity of a prey species and the physiological resistance of its predator. The dynamics are usually written as motion up a fitness gradient: $$\frac{d\tau_P}{dt} = k_P\,\frac{\partial W_P(\tau_P,\tau_H)}{\partial \tau_P}, \qquad \frac{d\tau_H}{dt} = k_H\,\frac{\partial W_H(\tau_P,\tau_H)}{\partial \tau_H},$$ where $W_P$ and $W_H$ are fitness functions and $k_P$, $k_H$ are called rates of evolution. Written that way the constants look adjustable at will, and that impression should be dispelled, because two independent derivations fix what they are. The quantitative-genetic derivation is Lande's (1979). For a normally distributed trait under weak selection, the response of the mean per generation is $$\frac{d\bar{z}}{dt} = G\,\frac{\partial \ln \bar{W}}{\partial \bar{z}},$$ with $G$ the additive genetic variance — the part of the trait's variance that is transmitted from parent to offspring — and the gradient taken on the logarithm of mean population fitness. Two things follow. The coefficient is not free: it is a measurable quantity, and one that itself changes as selection erodes variance or mutation and gene flow replenish it. And the gradient is on $\ln \bar{W}$, not on $W$, because a proportional rather than absolute fitness advantage is what determines the change in frequency. The second derivation comes from adaptive dynamics, where the trait changes by successive invasions of rare mutants rather than by shifts in a standing distribution. Dieckmann and Law (1996) obtained what is now called the canonical equation: $$\frac{d\bar{x}_i}{dt} = \tfrac{1}{2}\,\mu_i\,\sigma_i^{2}\,N_i^{*}\;\left.\frac{\partial f_i(x',\bar{x})}{\partial x'}\right|_{x'=\bar{x}_i}.$$ Here $\mu_i$ is the probability of a mutation per birth, $\sigma_i^{2}$ the variance of mutational step sizes, $N_i^{*}$ the equilibrium population size, and $f_i(x',\bar{x})$ the invasion fitness : the long-run growth rate of a rare mutant with trait $x'$ in a population whose residents have traits $\bar{x}$. Each factor earns its place. The product $\mu_i N_i^{*}$ is the rate at which mutants are supplied per unit time, so larger populations and higher mutation rates evolve faster. The factor $\sigma_i^{2}$ appears because the expected displacement per successful invasion scales with the step size and the probability of invasion also scales with it, giving two powers. The factor $\tfrac{1}{2}$ is the one worth pausing on: mutational steps are symmetric about the resident, but only those on the favoured side have any chance of invading, so exactly half the mutation distribution contributes and the other half is discarded. The correction to draw from both derivations is the same. The coefficients in the coevolutionary equations are not tuning knobs. They are composites of genetic variance, mutation supply and population size, all of which vary and some of which are themselves dynamical. Why cycling is possible at all Here is a question the arms-race picture never asks and that turns out to control everything else: why should two species chasing each other produce a cycle rather than simply climbing to a joint peak and stopping? The answer is a structural fact about the vector field, and it is exact. The planar results used here — the gradient criterion, the trace-determinant classification and the Poincaré-Bendixson theorem — are standard, and are taken in the form set out in Strogatz's Nonlinear Dynamics and Chaos (1994). A planar system $\dot{x} = F(x,y)$, $\dot{y} = G(x,y)$ is a gradient system — that is, there exists a single scalar function $V$ with $F = -\partial V/\partial x$ and $G = -\partial V/\partial y$ — if and only if the cross-derivatives match, $\partial F/\partial y = \partial G/\partial x$. Gradient systems cannot oscillate: $V$ strictly decreases along every non-constant trajectory, so no trajectory can return to where it began, and periodic orbits are impossible. If coevolution were motion downhill on one shared surface, cycles would be ruled out by a two-line argument. Applied to the equations above, the condition reads $$\frac{\partial}{\partial \tau_H}\!\left(k_P\,\frac{\partial W_P}{\partial \tau_P}\right) \;=\; \frac{\partial}{\partial \tau_P}\!\left(k_H\,\frac{\partial W_H}{\partial \tau_H}\right).$$ For a mutualism in which both partners gain from matching each other, the two sides can coincide, and such systems are indeed typically n