Appendix A. Further model derivation.
In this supplement, we provide two model explorations: (1) a full derivation of the symbiont genetic model and (2) an analysis that explains the structure of the density dependence in the symbiont population model.
A.1 Genetic model derivation
First, we derive the symbiont genetic dynamics in Eqn. (1)–(3) based on Lynch et al. (1991) (see also Lynch and Lande 1993; Lynch 1996). In particular, we derive the dynamics of the tolerance genotype ( ) probability distribution for each symbiont in coral as well as the symbiont population growth rate in Eq. (3). First, let
be the asymptotic growth rate of an individual with thermal tolerance phenotype at time . In addition, let the phenotype given a genotype be a random normal variable with mean and environmental variance
, i.e., the phenotype probability distribution given genotype is
(A.1) |
Therefore, the asymptotic growth rate of an individual with genotype is
(A.2) |
and the total asymptotic population growth rate is
(A.3) |
In addition, define as the density-dependent component of Eq. (1),
(A.4) |
where the vector . Then the realized growth rate given genotype is , and the realized growth rate of the entire population is , where (Lande and Shannon 1996). Given the above definitions, the dynamics of the thermal tolerance genotype probability distribution are
(A.5) |
(Fig. 1a; Lynch et al. 1991). Note that the functional form of density dependence we use leads to genotype distribution dynamics unaffected by density dependence.
To define the phenotype-dependent instantaneous growth rate , for mathematical tractability we use stabilizing selection given an optimal phenotype and selection strength. The phenotype is the temperature for which an individual is adapted, and the optimal phenotype is the actual temperature. Let determine the width of the fitness function, i.e., increasing means decreasing selection strength. In addition, the maximum possible population growth rate depends on temperature such that it follows the empirically-determined relationship of increasing exponentially with the actual temperature given constants and , or (Norberg 2004). Given these parameters, the asymptotic growth rate for a phenotype is
(A.6) |
(Lynch et al. 1991; Norberg 2004).
We assume that the genotypic distribution
is
a normal
distribution with mean
and
variance
. Then substituting Eq. (A.6)
into Eqs. (A.2)–(A.3) yields the asymptotic growth
rate for a population with the genotypic distribution
in
Eq. (3) (Lynch
et al. 1991).
In addition, the genotypic distribution dynamics in Eq. (A.5) and the asymptotic growth
rate function in Eq. (A.6)
yield the dynamics of the mean genotype dynamics (
)
in Eq. (1) and (given the rate at which mutation increases the genetic
variance
) the genetic variance dynamics
(
)
in Eq. (2) (Lynch
et al. 1991). Although constant genetic variance is
often a simplifying assumption in quantitative genetic models,
accounting for changes in genetic variance can be important to the
dynamics of models with strong selection (Barton
1999) and
coevolution (Kopp and
Gavrilets 2006). Note that the constant rate at which
mutation increases the genetic variance used here limits the potential
rate of evolution but does not limit the absolute amount of evolution
possible.
A.2 Symbiont density-dependence
Second, we analyze the symbiont competitive dynamics to explain the model structure with symbiont density dependence scaled by the maximum growth rate (rather than the realized population growth rate ). For this analysis, we use a simplified version of the model that ignores symbiont evolutionary dynamics ( ) and temperature fluctuations in time (constant and therefore symbiont average and maximum population growth rates and ) in order to allow tractable equilibrium analysis. Furthermore, to use the simplest possible model that incorporates symbiont competition, we focus on one coral species ( ) and ignore coral dynamics under the assumption that coral dynamics are much slower than symbiont dynamics such that coral cover is relatively constant on the time scale of symbiont competitive dynamics. Therefore, we follow two symbiont types with the dynamics
(A.7) |
|
(A.8) |
where all of parameters are the same as in the main text (note that,
for these parameters, we lose the subscripts because we do not need to
differentiate between multiple coral species).
We analyze Eqn. (A.7)–(A.8) by determining the local stability of each biologically relevant (i.e., non-negative) equilibrium based on the leading eigenvalue of the Jacobian matrix evaluated at that equilibrium. From this analysis, we find:
Therefore, assuming at least one symbiont has a positive realized population growth rate for the temperature at a given point in time, we expect the system to approach the equilibrium with only the symbiont type that has the greater realized population growth rate for that temperature. Thus, given the realized population growth rate in eq. (3), the symbiont with the thermal tolerance closest to the temperature is the competitive dominant. We employ this model structure in order to have the symbiont competitive outcome depend on the temperature without the use of additional parameters (e.g., temperature- or symbiont-type-dependent parameters and/or carrying capacities if we were to use a more traditional Lotka-Volterra competition model structure).
Note that the equilibrium symbiont density ( ) is lower than the total carrying capacity ( ) unless the symbiont is optimally adapted to the temperature ( ). Therefore, the model structure allows for lower symbiont densities given low levels of thermal stress (temperatures deviating from the symbiont genotype such that ). If we had instead chosen to scale intraspecific density dependence by rather than , then this intermediate outcome would not have been possible; rather, the model would result in an all-or-nothing equilibrium outcome of either zero symbiont density in a bleaching event ( , ) or symbionts at their carrying capacity otherwise ( , ). The temperature-dependent continuum of possible symbiont densities in our model structure (Fig. A1) better represents the biological reality (e.g., Fitt et al. 2000).
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