In atredennick/community_synchrony: Calculates syunchrony and stability metrics from time series data

Appendix X for TITLE: Mathematical Support

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Andrew T. Tredennick\footnote{Correspondance: [e-mail]}\textsuperscript{1}, Claire de Mazancourt\textsuperscript{2}, Michel Loreau\textsuperscript{2}, and Peter B. Adler\textsuperscript{1}

\textit{\small{\textsuperscript{1}Department of Wildland Resources and the Ecology Center, 5230 Old Main Hill, Utah State University, Logan, Utah 84322 USA}}

\textit{\small{\textsuperscript{2}Centre for Biodiversity Theory and Modelling, Experimental Ecology Station, Centre National de la Recherche Scientifique, Moulis, 09200, France}}

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Derivation of Expected Synchrony

Following @Loreau2013 and @DeMazancourt2013a, we define population growth, ignoring observation error, as

\begin{align} \tilde{r_i}(t) = r_{mi} \left[ 1- \frac{\tilde{N_i}(t)+\sum_{j \neq i} \alpha_{ij}\tilde{N_j}(t)} {K_i} + \sigma_{ei}\mu_{ei}(t) + \frac{\sigma_{di}\mu_{di}(t)}{\sqrt{\tilde{N_i}(t)}} \right] \end{align}

\noindent where $\tilde{N_i}(t)$ is the biomass of species i in year t, and $\tilde{r_i}(t)$ is its instantaneous growth rate in year t (tildes denote latent quantities that cannot be known due to observation error). $r_{mi}$ is species i's intrinsic rate of increase, $K_i$ is its carrying capacity, and $\alpha_{ij}$ is the interspecific competition coefficient representing the effect of species j on species i. Just as in our vital rate statistical model (Appendix X), environmental stochasticity is incorporated as an additive year effect on a species' growth rate. In the theoretical population model, environmental stochasticity is incorporated as $\sigma_{ei}\mu_{ei}(t)$, where $\sigma_{ei}^2$ is the environmental variance and $\mu_{ei}$ are normal random variables with zero mean and unit variance that are independent through time but may be correlated. Demographic stochasticity arises from variations in births and deaths among individuals (e.g., same states, different fates), and is included in the model as a first order, normal approximation [@Lande2003; @DeMazancourt2013a]. $\sigma_{di}^2$ is the demographic variance and $\mu_{di}(t)$ are independent normal variables with zero mean and unit variance.

In the general case, C is the variance-covariance matrix for population biomasses at steady state. It solves the Lyapunov equation $\textbf{C}^\infty = \textbf{A} \textbf{C}^\infty \textbf{A}^T + \textbf{B}$, where

\begin{align} A_{ij} = \begin{cases} 1-r_{mi} \frac{N_{i}^}{K_i}, & i=j \ -r_{mi} \frac{N_{i}^}{K_i} \alpha_{ij}, & i \neq j \end{cases} \end{align}

\noindent and

\begin{equation} B_{ij} = N_{i}^ N_{j}^ \sigma_{ei} \sigma_{ej} \text{cov}(\mu_{ei},\mu_{ej}) + \sqrt{N_{i}^ N_{j}^} \sigma_{di}\sigma_{dj} \text{cov}(\mu_{di},\mu_{dj}) \end{equation}

\noindent Similarly, R is the variance-covariance matrix for population growth rates at steady state

\begin{equation} R_{ij} = \frac{r_{mi}r_{mj}}{K_i K_j} \sum_{k,l} \alpha_{ik} \alpha_{jl} C_{kl} + \sigma_{ei} \sigma_{ej} \text{cov}(\mu_{ei},\mu_{ej}) + \frac{\sigma_{di}\sigma_{dj} \text{cov}(\mu_{di},\mu_{dj})}{\sqrt{N_{i}^ N_{j}^}}. \end{equation}

\noindent Then, following the synchrony metric of @Loreau2008a, the sychrony of population biomasses is

\begin{equation} \phi_N = \frac{\sum_{i,j}C_{ij}}{\left( \sum_i \sqrt{C_{ii}} \right)^2} \end{equation}

\noindent and the synchrony of per capita growth rates is

\begin{equation} \phi_R = \frac{\sum_{i,j}R_{ij}}{\left( \sum_i \sqrt{R_{ii}} \right)^2}. \end{equation}

Equations A# and A# show that synchrony population biomasses and growth rates emerge from complex interactions among species' intrinsic growth rates, interspecific interactions, environmental stochasticity, and demographic stochasticity. Given these complexities, it is impossible to determine expected effects of different parameters in a multi-species case. Thus, we analyze a simplified case where interspecific interactions are zero. The variance-covariance matrix of population biomasses at steady state then becomes

\begin{equation} c_{ij} = \frac{B_{ij}}{1-a_i a_j} = \frac{N_{i}^ N_{j}^ \sigma_{ei} \sigma_{ej} \text{cov}(\mu_{ei},\mu_{ej}) + \sqrt{N_{i}^ N_{j}^} \sigma_{di}\sigma_{dj} \text{cov}(\mu_{di},\mu_{dj})}{1 - (1-r_{mi})(1-r_{mj})} \end{equation}

and the variance-covariance matrix of population growth rates becomes

\begin{equation} R_{ij} = \frac{r_{mi}r_{mj}}{K_i K_j} C_{ij} + \sigma_{ei} \sigma_{ej} \text{cov}(\mu_{ei},\mu_{ej}) + \frac{\sigma_{di}\sigma_{dj} \text{cov}(\mu_{di},\mu_{dj})}{\sqrt{N_{i}^ N_{j}^}}, \end{equation}

which simplifies to

\begin{equation} R_{ij} = \frac{1}{1- \frac{r_{mi}r_{mj}}{r_{mi}+r_{mj}}} \left( \sigma_{ei} \sigma_{ej} \text{cov}(\mu_{ei},\mu_{ej}) + \frac{\sigma_{di}\sigma_{dj} \text{cov}(\mu_{di},\mu_{dj})}{\sqrt{N_{i}^ N_{j}^}} \right) \end{equation}

\noindent This is still not a general result, however, because synchrony in population sizes and synchrony in growth rates use weighted factors of the environmental variances and covariances with species-specific parameters. So, we further assume that, along with interspecific interactions being zero, all species have identical growth rates, environmental stochasticity is absent, and all species have identical demographic variance. This represents a theoretical limiting case where the community consists of identical species coexisting in a constant environment where only demographic stochasticity causes temporal fluctuations. Under such conditions, synchrony of population biomasses is

\begin{equation} \phi_N = \frac{1}{\left(\sum_i p_i^{1/2} \right)^2} \end{equation}

\noindent and synchrony of growth rates is

\begin{equation} \phi_N = \frac{\sum_i p_i^{-1}}{\left(\sum_i p_i^{-1/2} \right)^2} \end{equation}

\noindent where $p_i$ is the average frequency of species i, $p_i = N_i/N_T$. When all species have identical abundances and $p_i = 1/S$, where S is species richness, the both synchrony values equal 1/S [@Loreau2008a].

References

atredennick/community_synchrony documentation built on May 10, 2019, 2:10 p.m.