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:
- 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.
- 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, rδ ), the lower the frequency of mimicry. The
greater the benefits of mimicry (due to reductions in
competition, cβ ), 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.