Competition models

For the $\delta$ model and Avery's model ...

Model Specification for Avery's Model

Data

• $Nobs$: number of focal tree observations
• $S$: number of species
• $N_{neighbors}$: total number of neighbor records

For each focal tree $i = 1,\dots,Nobs$:

• $y_i$: observed ring width
• $tree_{sp_i}$: species of focal tree
• $tree_{N_{neighbs_i}}$: number of neighbors of tree $i$
• neighbor records for tree $i$ are stored from index $a_i$ to $b_i$, where:

$$ a_i = tree_{startidxs_i} $$

$$ b_i = tree_{endidxs_i} $$

For each neighbor $j = 1,\dots,N_{neighbors}$:

• $neighbor_{BA_j}$: basal area of neighbor
• $neighbor_{sp_j}$: species identity of neighbor

Other inputs:

• $BA0$: base BA for neighbors
• $\Sigma$: species trait covariance matrix

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Transformed Data

Relative neighbor basal area:

$$ neighbor_{relBA_j} = \frac{neighbor_{BA_j}}{BA0} $$

Cholesky factor:

$$ L = chol(\Sigma) $$

so that:

$$ \Sigma = LL^T $$

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Parameters

For each species $s = 1,\dots,S$:

• $\mu_s$: base growth mean
• $basevar_s$: lognormal scale parameter
• $traits_s$: latent species trait

Global parameters:

• $\alpha > 0$: effect of neighbor basal area
• $\beta_1 > 0$: overall competition coefficient
• $\beta_2$: trait-distance decay parameter
• $\kappa$: maximum phylogenetic competition strength

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Priors

$$ \alpha \sim Normal(0,\ 0.1) $$

$$ \beta_1 \sim Normal(1,\ 1) $$

$$ \beta_2 \sim Normal(1,\ 1) $$

$$ \kappa \sim Normal(0.1,\ 0.1) $$

For each species $s$:

$$ \mu_s \sim Normal(0.7,\ 0.3) $$

$$ basevar_s \sim Normal(0.5,\ 0.3), \quad basevar_s > 0 $$

Latent traits:

$$ traits \sim MVN(0,\Sigma) $$

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Competition Model

For focal tree $i$, let neighbor set be:

$$ j = a_i,\dots,b_i $$

Basal Area Competition

For each neighbor $j$:

$$ BAcomp_{ij} = neighbor_{relBA_j}^{-\alpha} $$

Combined basal area effect:

$$ BAprod_i = \prod_{j=a_i}^{b_i} BAcomp_{ij} $$

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Phylogenetic Competition

Trait distance:

$$ d_{ij} = |traits_{neighbor_{sp_j}} - traits_{tree_{sp_i}}| $$

Competition term:

$$ phylocomp_{ij} = 1 - \kappa e^{-\beta_2 d_{ij}} $$

Combined phylogenetic effect:

$$ Phyloprod_i = \prod_{j=a_i}^{b_i} phylocomp_{ij} $$

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Total Competition Multiplier

$$ totalcomp_i = \beta_1 \cdot BAprod_i \cdot Phyloprod_i $$

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Likelihood

Ring width is modeled as lognormal:

$$ y_i \sim LogNormal \left( \mu_{tree_{sp_i}} + \log(totalcomp_i),\ basevar_{tree_{sp_i}} \right) $$

Equivalent form:

$$ \log y_i \sim Normal \left( \mu_{tree_{sp_i}} + \log(totalcomp_i),\ basevar_{tree_{sp_i}} \right) $$

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Model Specification of $\delta$ model

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Data

• $N_f$: number of focal trees
• $N_total$: total number of neighbouring species

For each focal tree $i= 1, \dots, Nf$:

• $N_i$: number of neighbouring species around focal tree $i$
• $bf_i$: basal area of focal tree
• $y_i$: observed annual tree growth (mm)
• neighbor records for tree $i$ are stored from index $a_i$ to $b_i$, where:

$$ a_i = tree_{startidxs_i} $$

$$ b_i = tree_{endidxs_i} $$

For each neighboring species $j=i,\dots,N_{neighbors}$:

• $b_j$: basel area of neighbouring species
• $corr_j$: phylogenetic correlation between focal tree and neighbour

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Model Constants

• $bf_0$: reference basal area constant (2)
• $BA_{compet0}$: baseline competition constant (16)

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Parameter Definitions

• $y_0$: baseline growth rate |
• $\gamma$: scaling factor controlling effect of focal tree basal area
• $k$: exponent controlling how strongly phylogenetic similarity affects competition
• $\beta$: strength of competition effect
• $\sigma$: standard deviation of log growth

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Priors

The model uses the following prior distributions:

$$ \sigma \sim \text{Normal}\left(0, \frac{0.095}{2.57}\right) $$

$$ y_0 \sim \text{Normal}\left(0, \frac{10}{2.57}\right) $$

$$ k \sim \text{Beta}(2,2) $$

$$ r \sim \text{Beta}(4,2) $$

$$ \beta \sim \text{Normal}\left(0,\frac{\log(1.1)}{2.57}\right) $$

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Resource Availability

The available resources for focal tree $i$ are modeled as

$$ A_i = \gamma \cdot (bf_i - bf_0) $$

where $bf_i$ is the basal area of the focal tree.

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Competition Term

Competition from neighbouring species is calculated as

$$ C_i = \beta (B_i - BA_{compet0}) $$

where the phylogenetically weighted basal area of neighbours is

$$ B_i = \sum_{j=a_i}^{b_i} b_{j} \cdot (corr_{j})^k $$

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Likelihood

For each focal tree $i$:

$$ \log(y_i) \sim \text{Normal}(\mu_i, \sigma) $$

where the expected log-growth is

$$ \mu_i = \log(y_0) + A_i - C_i $$

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Next steps ....

Towards the end of the summer (17 August 2026) Lizzie and Devina met and agreed on next steps:

1. Write up where we are at, including with the MORA data results (including comparing 1 year growth to averaged growth)
2. Devina asked about ecological predictions and we discussed different effects for different species, smaller trees experiencing more competition and lower competition across elevations. She will start working on these (for elevation, we will just subset the data by elevation and run the model).
3. Next would be work on another dataset.

... plus things we discussed to do that I am forgetting.

On 31 August 2026:

1. This is basically done, Lizzie and Victor just needs to review it.
2. This is in progress, we should review the results by species names (by end of September, to present to Janneke). And also ideally do the elevation for TSHE and THPL (check which species are common across elevations).
3. We are en route on HF and BCI!