Ecological Archives A019-001-A1

Marissa L. Baskett, Steven D. Gaines, and Roger M. Nisbet. 2009. Symbiont diversity may help coral reefs survive moderate climate change. Ecological Applications 19:3–17.

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 ($ g$ ) probability distribution $ p_{im}(g)$ for each symbiont $ i$ in coral $ m$ as well as the symbiont population growth rate $ r_{im}$ in Eq. (3). First, let $ r_f^*(f,t)$ be the asymptotic growth rate of an individual with thermal tolerance phenotype $ f$ at time $ t$ . In addition, let the phenotype $ f$ given a genotype $ g$ be a random normal variable with mean $ g$ and environmental variance $ \sigma_e^2$ , i.e., the phenotype probability distribution given genotype $ g$ is

 
$\displaystyle q(f,g)=\frac{1}{\sqrt{2\pi\sigma_e^2}}e^{-\frac{(f-g)^2}{2\sigma_e^2}}.$
(A.1)

Therefore, the asymptotic growth rate of an individual with genotype $ g$ is

 
$\displaystyle r_g^*(g,t)=\int r_f^*(f,t)q(f,g)df,$
(A.2)

and the total asymptotic population growth rate is

 
$\displaystyle r_{im}(t) = \int r_g^*(g,t)p_{im}(g,t)dg.$
(A.3)

In addition, define $ F(C_m,\mathbf{S_m})$ as the density-dependent component of Eq. (1),

 
$\displaystyle F(C_m,\mathbf{S_m}) = \frac{\hat{r}(t)\sum_jS_{jm}}{K_{Sm}C_m},$
(A.4)

where the vector $ \mathbf{S_m}=(S_{1m},S_{2m})$. Then the realized growth rate given genotype $ g$ is $ r_g(g,t)=r_g^*(g,t)-F(C_m,\mathbf{S_m})$, and the realized growth rate of the entire population is $ \bar{r}_{im}=r_{im}-F(C_m,\mathbf{S_m})$, where $ \frac{dS_{im}}{dt}=\bar{r}_{im}S_{im}$(Lande and Shannon 1996). Given the above definitions, the dynamics of the thermal tolerance genotype probability distribution are

 
$\displaystyle \frac{dp_{im}}{dt} = p_{im}(g,t)(r_g^*(g,t)-r_{im}(t))$
(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 $ r_f^*(f,t)$, for mathematical tractability we use stabilizing selection given an optimal phenotype and selection strength. The phenotype $ f$ is the temperature for which an individual is adapted, and the optimal phenotype $ \theta(t)$ is the actual temperature. Let $ \sigma_{wm}$ determine the width of the fitness function, i.e., increasing $ \sigma_{wm}$means decreasing selection strength. In addition, the maximum possible population growth rate $ \hat{r}_{im}$ depends on temperature such that it follows the empirically-determined relationship of increasing exponentially with the actual temperature $ \theta(t)$ given constants $ a$ and $ b$ , or $ \hat{r}=ae^{b\theta(t)}$ (Norberg 2004). Given these parameters, the asymptotic growth rate for a phenotype $ f$ is

 
$\displaystyle r^*_f(f,t) = \left(1-\frac{(f-\theta(t))^2}{2\sigma_{wm}^2}\right)ae^{b\theta(t)}$
(A.6)

(Lynch et al. 1991; Norberg 2004).

We assume that the genotypic distribution $ p_{im}(g,t)$is a normal distribution with mean $ \bar{g}_{im}$ and variance $ \sigma_{gim}^2$ . Then substituting Eq. (A.6) into Eqs. (A.2)–(A.3) yields the asymptotic growth rate for a population with the genotypic distribution $ p_{im}(g,t)$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 ( $ \int g\frac{dp_{im}(g,t)}{dt}dg$) in Eq. (1) and (given the rate at which mutation increases the genetic variance $ \sigma_M^2$ ) the genetic variance dynamics
( $ \frac{d}{dt}[\int (g-\bar{g}_{im})^2p_{im}(g,t)dg]$) 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 $ \hat{r}$ (rather than the realized population growth rate $ r_{im}$ ). For this analysis, we use a simplified version of the model that ignores symbiont evolutionary dynamics ( $ \frac{d\bar{g}_{im}}{dt}=\frac{d\sigma_{gim}}{dt}=0$) and temperature fluctuations in time (constant $ \theta$ and therefore symbiont average and maximum population growth rates $ r_{im}$ and $ \hat{r}$ ) in order to allow tractable equilibrium analysis. Furthermore, to use the simplest possible model that incorporates symbiont competition, we focus on one coral species ($ C$ ) 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

 
$\displaystyle \frac{dS_1}{dt}$$\displaystyle =$$\displaystyle \frac{S_1}{K_SC}\left(r_1K_SC-\hat{r}(S_1+S_2)\right)$
(A.7)
   
$\displaystyle \frac{dS_2}{dt}$$\displaystyle =$$\displaystyle \frac{S_2}{K_SC}\left(r_2K_SC-\hat{r}(S_1+S_2)\right),$
(A.8)


where all of parameters are the same as in the main text (note that, for these parameters, we lose the $ m$ 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:

In summary, we find that if both symbionts have negative realized population growth rates ($ r_1,r_2<0$ ), then the zero equilibrium ( $ \bar{S}_1= \bar{S}_2=0$) is the only locally stable equilibrium, as we would expect intuitively. If at least one symbiont has a positive realized population growth rate, then the only locally stable equilibrium is the edge equilibrium for the symbiont with the greater realized population growth rate (i.e., if $ r_2<r_1$ , the only locally stable equilibrium is $ \bar{S}_1=K_SCr_1/\hat{r}$and $ \bar{S}_2=0$, and vice versa).

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 $ \alpha_{ij}$parameters and/or carrying capacities if we were to use a more traditional Lotka-Volterra competition model structure).

Note that the equilibrium symbiont density ( $ \bar{S}_i=K_SCr_i/\hat{r}$) is lower than the total carrying capacity ($ K_SC$ ) unless the symbiont is optimally adapted to the temperature ( $ r_i=\hat{r}$). Therefore, the model structure allows for lower symbiont densities given low levels of thermal stress (temperatures deviating from the symbiont genotype such that $ 0<r_i<\hat{r}$ ). If we had instead chosen to scale intraspecific density dependence by $ r_i$ rather than $ \hat{r}$, then this intermediate outcome would not have been possible; rather, the model would result in an all-or-nothing equilibrium outcome of either zero symbiont density in a bleaching event ($ r_i<0$ , $ \bar{S}_i=0$ ) or symbionts at their carrying capacity otherwise ($ r_i>0$ , $ \bar{S}_i=K_SC$ ). 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).


 

 
   FIG. A1. Intra-annual fluctuations in symbiont density (green broken line) and temperature (blue solid line): five years from the simulation with one evolving symbiont and past (ISST) temperature data for Curaçao (excerpt from data presented in Fig. 2).

 

LITERATURE CITED

Barton, N. H. 1999. Clines in polygenic traits. Genet. Res. 74:223–236.

Fitt, W. K., F. K. McFarland, M. E. Warner, and G. C. Chilcoat. 2000. Seasonal patterns of tissue biomass and densities of symbiotic dinoflagellates in reef corals and relation to coral bleaching. Limnology and Oceanography 45:677–685.

Kopp, M., and S. Gavrilets. 2006. Multilocus genetics and the coevolution of quantitative traits. Evolution 60:1321–1336.

Lande, R., and S. Shannon. 1996. The role of genetic variation in adaptation and population persistence in a changing environment. Evolution 50 :434–437.

Lynch, M. 1996. A quantitative-genetic perspective on conservation issues. Pages 471–501 in J. C. Avise and J. L. Hamrick, editors. Conservation Genetics. Chapman and Hall, New York, New York, USA.

Lynch, M., W. Gabriel, and A. M. Wood. 1991. Adaptive and demographic responses of plankton populations to environmental-change. Limnology and Oceanography 36:1301–1312.

Lynch, M., and R. Lande. 1993. Evolution and extinction in response to environmental change. Pages 234–250 in P. M. Kareiva, J. G. Kingsolver, and R. B. Huey, editors. Biotic interactions and global change. Sinauer Associates, Sunderland, Massachusetts, USA.

Norberg, J. 2004. Biodiversity and ecosystem functioning: A complex adaptive systems approach. Limnology and Oceanography 49:1269–1277.


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