Box 1A: The Sickly Defender Hypothesis: Proof of Concept Model
To demonstrate the plausibility of constitutive dishonest signals of infection, we use a simple model of their evolution. Our model is based on the Sickly Defender Hypothesis , in which a mimicking male accrues fitness benefits through conflict-avoidance but incurs costs due to foregone mating opportunities. We represent male fitness as the difference between fitness gains (benefits) from reproduction and fitness losses (costs) from competition with other males. The general expression of this, for a male of any type i , is:
\(W_{i}=R_{i}-C_{i}\).
We consider three male phenotypes: Males may be sick (infected, I ), or healthy. Healthy males may either mimic (M ) a diseased state, or not (N ). We assume that the choice of mimicry is made either at birth or early on in development, and is consistent throughout an animal’s lifetime (e.g., a healthy individual either mimics or does not, but does not switch between these strategies).
Both reproductive benefits and competition costs are affected by a male’s phenotype. For a healthy male, reproductive benefits occur at a rate r , whereas sick or mimicking males have a reduced benefit r(1−δ) due to female avoidance of obviously unhealthy individuals. Here, δ  is the proportion of reproductive fitness lost due to illness. Male-male conflict also depends on phenotype. Conflicts are the most intense (and most costly) between two healthy males, which incur costs at a rate c  per encounter. Conflicts between sick (or mimicking) and healthy males incur reduced costs for the sick (or mimicking) individuals because healthy males choose to avoid contact that may transmit infection. Because mimics are healthy and wish to avoid infection, they also reduce contact (and conflict) with sick or mimicking individuals. Thus, the cost to a healthy male of conflict with a sick or mimicking male is c(1−α), and the cost to a mimicking male of conflict with a non-mimicking healthy male is c(1−β), where α and β are the proportionate reductions in conflict costs for avoidance of sick individuals, and avoidance by healthy individuals, respectively. Infected males experience the same landscape of intraspecific interactions as mimicking males, except that they incur an additional fitness cost s  from being sick.
The fitness functions for each type of male are therefore:
\({W_{N}=r-c\left[N+\left(1-\alpha\right)\left(M+I\right)\right]}\)
\({W_{m}=r-c\left[N+\left(1-\alpha\right)\left(M+I\right)\right]}\)
\({W_{I}=r-c\left[N+\left(1-\alpha\right)\left(M+I\right)\right]-s}\)
We can express these fitness functions in terms of the frequency of each phenotype in the environment (such that N +M +I = 1), where I = p , the proportion of individuals who carry the disease,N = x(1-p) , the proportion of healthy individuals who fake sickness, and M = (1-x)(1-p) , the proportion of healthy non-mimics:
\begin{equation} W_{N}=r-c\left[1-\alpha\left(x-xp+p\right)\right]\nonumber \\ \end{equation}\begin{equation} W_{M}=r\left(1-\delta\right)-c\left[1-\beta\left(1-x\right)\left(1-p\right)-\alpha\left(x-xp+p\right)\right]\nonumber \\ \end{equation}\begin{equation} W_{I}=r\left(1-\delta\right)-c\left[1-\beta\left(1-x\right)\left(1-p\right)-\alpha\left(x-xp+p\right)\right]-s\nonumber \\ \end{equation}
We focus on the evolution of x , the frequency of mimicry among healthy individuals. To determine the equilibrium frequency of mimicry, we must study the dynamics of the differential equation
\begin{equation} \dot{x}=x\left(1-x\right)\left[W_{M}\left(x\right)-W_{N}\left(x\right)\right]\nonumber \\ \end{equation}
which describes how the frequency of mimicry changes over time (Nowak 2006). This equation has three possible equilibria: x* = 0 (no mimicry), x* = 1 (all healthy individuals mimic), and x* =\(\hat{x}\) where\(\hat{x}\) satisfies \(W_{M}\left(\hat{x}\right)=W_{N}\left(\hat{x}\right)\)(and mimicry persists in the system). If all healthy males mimic (x = 1), the value of the mimicked signal should quickly erode; therefore we are most interested in the latter equilibrium, which implies that mimicry can evolve, and that its evolved frequency is 0 > \(\hat{x}\) > 1.
For mimicry and non-mimicry to co-occur, two conditions must be met:
  1. The mimicry-free (x* = 0) equilibrium must be unstable, which implies that a mutant, mimicking male in a population of non-mimics must have a higher fitness than the non-mimics, or, mathematically, that \(W_{M}\left(0\right)>\ W_{N}(0)\) so that\(c\beta(1-p)\ >\ r\delta\). Biologically, this means that if the benefits from conflict avoidance exceed the costs of lost mating opportunities when mimicry is rare, mimicry should invade.
  2. The mimicry-exclusive (x* = 1) equilibria must be unstable, meaning that a mutant non-mimicking male in a population of mimics should have a higher fitness. Mathematically:\(W_{M}\left(1\right)<\ W_{N}(1)\) so that \(r\delta>0\). Biologically, this means that, so long as there is a fitness cost to mimicry (in this case due to reduced reproductive opportunity), mimicry will never be the sole evolutionarily stable strategy and will coexist with non-mimicry.
If these conditions are met, the evolved frequency of mimicry (which satisfies \(W_{M}\left(x\right)=W_{N}\left(x\right)\)) is:
\begin{equation} \hat{x}=1-r\delta c\beta\left(1-p\right).\nonumber \\ \end{equation}
The larger the costs of mimicry (due to reductions in mating opportunities,  ), the lower the frequency of mimicry. The greater the benefits of mimicry (due to reductions in competition,  ), the greater the frequency of mimicry.
The frequency of mimicry is also affected by the prevalence of disease in the population (Figure 2): The more frequent the disease, the less effective mimicry becomes until, ultimately, a mimic in a population of non-mimics has a lower fitness than the non-mimics and thus mimicry is lost from the population. This is because, in high disease environments, intense conflicts between healthy males are relatively rare, and individual fitness is primarily driven by differences in attractiveness to females.