This document resumes the equation used in our model and details the different processes. Our coexistence model is developed following this equation (Beverton-Holt equation):
$$ N_{t+1, i, x} = \frac{R_{i, x} \times N_{t, i, x}}{1 + A \times \alpha_i} $$
with Nt + 1, i, x the abundance of species i in patch x at time t + 1, Ri, x the growth of species i in patch x, A the constant for inter-specific competition and αi the competition coefficient for species i. We can detail both Ri, x and αi:
$$ \begin{aligned} \alpha_i = \sum_{j = 1, j \neq i}^{S} N_{t, j, x} \times \sigma_{ij}\\ R_{i, x} = k \times \exp\left(- \frac{(t_i - t_{opt(x)})^2}{2\times l^2}\right) \end{aligned} $$
with σij the functional similarity between species i and j; we see that αi gets bigger as species i and j are more similar and as species j is more abundant. And with ti the trait of species i, topt(x) the environmental value of patch x and l2 the width of the environmental filter across all patches.
If we replace αi and Ri, x in the first equation it gives:
$$ N_{t+1, i, x} = \frac{ k \times \exp\left(- \frac{(t_i - t_{opt(x)})^2}{2\times l^2}\right) \times N_{t, i, x}}{ 1 + A \times \displaystyle\sum_{j = 1, j \neq i}^{S} N_{t, j, x} \times \sigma_{ij} } $$
δij is the functional dissimilarity coefficient between species i and species j. It reflects the fact that the more two species are dissimilar the less competition they experience from each other.
First we can compute δij as the euclidean distance between species using multiple traits:
$$ \delta_{ij} = \sqrt{\sum_{r = 1}^{T} (t_{i, r} - t_{j, r})^2}, $$
with T the number of traits, ti, r trait r of species i and tj, r trait r of species j.
Different traits r can contribute to limiting similarity in different proportions ar (we have $\sum_{r = 1}^{T} a_r = 1$). If want to take the contribution of traits to competition we have to modify the dissimilarity computation by weighting them by a factor $\sqrt{a_r}$, as follow:
$$ \begin{aligned} \delta_{ij} = \sqrt{\sum_{r = 1}^{T} \left[ \sqrt{a_r} (t_{i, r} - t_{j, r})\right]^2}\\ \delta_{ij} = \sqrt{\sum_{r = 1}^{T} a_r \times (t_{i, r} - t_{j, r})^2}, \end{aligned} $$
with ar the contribution of trait r in limiting similarity and we have $\sum_{r = 1}^{T} a_r = 1$.
In order to consider that the effect of limiting similarity is non-linear we use an exponent on distances s: $$ \begin{aligned} {\delta_{ij}}' &= {\delta_{ij}}^s\\ &= \left(\sqrt{\sum_{r = 1}^{T} a_r \times (t_{i, r} - t_{j, r})^2}\right)^s \end{aligned} $$
We use max (δij′) − δij′ in above-mentioned equations to consider limiting similarity (closer species experience higher inter-specific competition) thus we can use a similarity term:
$$ \begin{aligned} \sigma_{ij} = \max({\delta_{ij}}') - {\delta_{ij}}',\quad \sigma_{ii} = 0 ~\forall~i \in [1, S] \\ \end{aligned} $$
The equation above only considers inter-specific competition when j ≠ i in the sum. We can however add intra-specific competition when j = i. Each site has a species-specific carrying capacity K as the number of individuals approaches this carrying capacity the intra-specific competition increases:
αii = B × Nt, i, x
Thus the equation becomes:
$$ N_{t+1, i, x} = \frac{ k \times \exp\left(- \displaystyle\frac{(t_i - e_x)^2}{2\times l^2}\right) \times N_{t, i, x}}{ 1 + A \displaystyle\sum_{j = 1,~j \neq i}^{S} N_{t, j, x} \times \sigma_{ij} + B \times N_{t,i,x} } $$
with B the coefficient scaling intra-specific competition.
Because several traits contribute to the growth term depending on their contribution we can rewrite the growth term as:
$$ R_{i, x} = \sum_{r = 1}^{T} b_r \times k \times \exp\left(- \frac{(t_{i, r} - t_{opt, x})^2}{2\times l^2} \right) $$
with r the trait number, 0 ≤ br ≤ 1 the contribution of this trait to growth (and $\sum_{r = 1}^{T} b_r = 1$), ti, r the trait number r of species i.
So the equation becomes:
$$ N_{t+1, i, x} = \frac{ \sum_{r = 1}^{T} b_r \times k \times \exp\left(- \frac{(t_{i, r} - t_{opt, x})^2}{2\times l^2} \right) \times N_{t, i, x} }{ 1 + A \displaystyle\sum_{j = 1,~j \neq i}^{S} N_{t, j, x} \times \sigma_{ij} + B \times N_{t,i,x} } $$
We consider hierarchical competition by considering its influence on growth rate. To do so, we break down the growth rate term Rix into two components:
Rix = Ri, x, env + Ri, x, hierarch with Rix the total growth rate as numerator of the global equation; Ri, x, env the growth rate based on the environmental fitting of the species as developed above; and Ri, x, hierarch the effect of hierarchical competition on growth rate that we develop beneath.
We want hierarchical competition to depends on the species present in the community and the distance of species such that smallest species get their growth reduced by tallest species in proportion of their distances and abundances. So the hierarchical competition term Ri, x, hierarch of a species i in patch x at time t can be:
$$ R_{t, i, x, hierarch} = H \sum_{ \substack{j = 1\\ t_j \ge t_i}}^S (t_j - t_i) N_{t,j,x} $$ with H the scale for hierarchical competition tj the trait of species j, ti the trait of species i and Nt, j, x the abundance of species j at time t in patch x. To account for the fact that hierarchical competition has a negative influence on growth we add a minus sign in front of the equation which becomes:
$$ R_{t, i, x, hierarch} = - H \sum_{ \substack{j = 1\\ t_j \ge t_i}}^S (t_j - t_i) N_{t,j,x} $$
If we want to consider multiple traits with different contributions we can generalize the equation using the same principle as for limiting similarity:
$$ R_{t, i, x, hierarch} = - H \sum_{r = 1}^T c_r \sum_{ \substack{j = 1\\ t_{j, r} \ge t_{i, r}}}^S (t_{j, r} - t_{i, r}) N_{t,j,x} $$ with cr the hierarchical weight of trait r among the T traits, with ∑cr = 1.
Finally we can consider that hierarchical competition is not linear with the trait difference but instead that taller species have a saturating influence on smaller species (a tree of 10m or 50m will have almost similar influence for light interception for grasses 0.5m high). To account for that we can scale the distances using a hierarchical competition exponent h:
$$ R_{t, i, x, hierarch} = - H \sum_{r = 1}^T c_r \sum_{ \substack{j = 1\\ t_{j, r} \ge t_{i, r}}}^S N_{t,j,x} (t_{j, r} - t_{i, r})^h $$
The general equation with hierarchical competition added becomes:
$$ N_{t+1, i, x} = \frac{ \left[ k \displaystyle\sum_{r = 1}^{T} b_r \times \exp\left(- \frac{(t_{i, r} - t_{opt, x})^2}{2\times l^2} \right) - H \sum_{r = 1}^T c_r \sum_{ \substack{j = 1\\ t_{j, r} \ge t_{i, r}}}^S N_{t,j,x} (t_{j, r} - t_{i, r})^h \right] \times N_{t, i, x} }{ 1 + A \displaystyle\sum_{j = 1,~j \neq i}^{S} N_{t, j, x} \times \sigma_{ij} + B \times N_{t,i,x} } $$
In our model we consider dispersal between patches as if all patches were in a meta-community. At each time step a proportion d of inviduals for each species move across patches. d percent of individuals leave each patch and move evenly across patches.
The dispersion is thus: $$ \begin{aligned} N_{t, i, x, \text{leaving}} &= d \times N_{t, i, x} \\ N_{t, i, x, \text{sedentaries}} &= N_{t, i, x} - N_{t, i, x, \text{leaving}} = N_{t, i, x} - d \times N_{t, i, x} \\ N_{t, i, x, \text{sedentaries}} &= (1 - d) N_{t, i, x}\\ \end{aligned} $$
For then for each species all immigrants are split evenly across patches:
$$ N_{t, i, \text{migrants}} = \sum_{y = 1}^{P} d \times N_{t, i, y} $$ with P the total number of patches.
Finally the number of inviduals in the patch after dispersal becomes:
$$ \begin{aligned} N_{t, i, x} = N_{t, i, x, \text{sedentaries}} + \frac{N_{t, i, \text{migrants}}}{P} \\ N_{t, i, x} = (1 - d) N_{t, i, x} + d \frac{\displaystyle\sum_{y = 1}^{P} N_{t, i, y}}{P}\\ N_{t, i, x} = (1 - \frac{P - 1}{P}d) N_{t, i, x} + d \frac{\displaystyle\sum_{y = 1, y \neq x}^{P} N_{t, i, y}}{P} \\ N_{t, i, x} = N_{t, i, x} \underbrace{- d N_{t,i,x} + d \frac{\displaystyle\sum_{y = 1}^{P} N_{t, i, k}}{P}}_{D_{t,i,x}} \\ D_{t, i, x} = - d N_{t,i,x} + d \frac{\displaystyle\sum_{y = 1}^{P} N_{t, i, y}}{P} \end{aligned} $$
If a single trait contributes to everything than the equation simplifies as there is no need to to account for the difference in contribution to the different processes the final equation (and as such ar = br = cr = 1)
$$ N_{t+1, i, x} = \frac{ \left[ k \exp\left(- \frac{(t_i - t_{opt, x})^2}{2\times l^2} \right) - H \displaystyle\sum_{ \substack{j = 1\\ t_j \ge t_i}}^S N_{t,j,x} (t_j - t_i)^h \right] \times N_{t, i, x} }{ 1 + A \displaystyle\sum_{j = 1,~j \neq i}^{S} N_{t, j, x} \times \sigma_{ij} + B \times N_{t,i,x} } + D_{t, i, x} $$ If we fully develop the limiting similarity term it becomes:
$$ N_{t+1, i, x} = \frac{ \left[ k \exp\left(- \frac{(t_{i, r} - t_{opt, x})^2}{2\times l^2} \right) - H \displaystyle\sum_{ \substack{j = 1\\ t_{j, r} \ge t_{i, r}}}^S N_{t,j,x} (t_{j, r} - t_{i, r})^h \right] \times N_{t, i, x} }{ 1 + A \displaystyle\sum_{j = 1,~j \neq i}^{S} N_{t, j, x} \times \left[ \max_{\forall~i,j \in [1, S]^2}[(t_{i} - t_{j})^s] - \left(t_{i} - t_{j}\right)^s \right] + B \times N_{t,i,x} } + D_{t, i, x} $$
To account for the various contributions of multiple processes to the different processes the final equation is:
$$ N_{t+1, i, x} = \frac{ \left[ k \displaystyle\sum_{r = 1}^{T} b_r \times \exp\left(- \frac{(t_{i, r} - t_{opt, x})^2}{2\times l^2} \right) - H \sum_{r = 1}^T c_r \displaystyle\sum_{ \substack{j = 1\\ t_{j, r} \ge t_{i, r}}}^S N_{t,j,x} (t_{j, r} - t_{i, r})^h \right] \times N_{t, i, x} }{ 1 + A \displaystyle\sum_{j = 1,~j \neq i}^{S} N_{t, j, x} \times \sigma_{ij} + B \times N_{t,i,x} } + D_{t, i, x} $$
with br and cr, the contribution respectively of trait to growth and hierarchical competition with T the number of traits.
If we develop fully the similarity term σij it gives:
$$ N_{t+1, i, x} = \frac{ \left[ k \displaystyle\sum_{r = 1}^{T} b_r \times \exp\left(- \frac{(t_{i, r} - t_{opt, x})^2}{2\times l^2} \right) - H \sum_{r = 1}^T c_r \displaystyle\sum_{ \substack{j = 1\\ t_{j, r} \ge t_{i, r}}}^S N_{t,j,x} (t_{j, r} - t_{i, r})^h \right] \times N_{t, i, x} }{ 1 + A \displaystyle\sum_{j = 1,~j \neq i}^{S} N_{t, j, x} \times \left[\max_{\forall~i,j \in [1, S]^2} \left[ \left(\sqrt{\sum_{r = 1}^{T} a_r \times (t_{i, r} - t_{j, r})^2}\right)^s \right] - \left(\sqrt{\sum_{r = 1}^{T} a_r \times (t_{i, r} - t_{j, r})^2}\right)^s \right] + B \times N_{t,i,x} } + D_{t, i, x} $$