Modeling Community Dynamics Through Environmental Effects, Species Interactions and Movement

Abstract

Understanding how communities respond to environmental change is frustrated by the fact that both species interactions and movement affect biodiversity in unseen ways. To evaluate the contributions of species interactions on community growth, dynamic models that can capture nonlinear responses to the environment and the redistribution of species across a spatial range are required. We develop a time-series framework that models the effects of environment–species interactions as well as species–species interactions on population growth within a community. Novel aspects of our model include allowing for species redistribution across a spatial region, and addressing the issue of zero inflation. We adopt a hierarchical Bayesian approach, enabling probabilistic uncertainty quantification in the model parameters. To evaluate the impacts of interactions and movement on population growth, we apply our model using data from eBird, a global citizen science database. To do so, we also present a novel method of aggregating the spatially biased eBird data collected at point-level. Using an illustrative region in North Carolina, we model communities of six bird species. The results provide evidence of nonlinear responses to interactions with the environment and other species and demonstrate a pattern of strong intraspecific competition coupled with many weak interspecific species interactions. Supplementary materials accompanying this paper appear online.

A Appendix

1.1 A.1 Posterior Update for Growth Coefficients

Sampling the ESI coefficient matrix ${\textbf {P}}$ can be achieved in a similar fashion, so here we provide details about sampling ${\textbf {A}}$. Let $\Sigma = \textrm{diag}(\sigma _{\gamma , j}^2)$. The full conditional posterior distribution for $vec({\textbf {A}})$ is UJ multivariate Normal with mean ${\textbf {m}}$ and covariance ${\textbf {M}}$, where ${\textbf {M}}= (\sum _{t=1}^{T-1} C_t)^{-1}$, ${\textbf {m}}={\textbf {M}}\sum _{t=1}^{T-1} c_t$, and

$$\begin{aligned} \begin{aligned}&C_t =S^{-1} \otimes ({\textbf {U}}_t' {\textbf {U}}_t )\\&c_t = C_t \otimes vec(({\textbf {U}}_t' {\textbf {U}}_t )^{-1} {\textbf {U}}_t' ({\textbf {W}}_{t}^{*} - ({\tilde{{\textbf {W}}}}_{t} + {\textbf {V}}' {\textbf {P}}))) \end{aligned} \end{aligned}$$

(5)

Recall that ${\textbf {A}}$ is a sparse matrix, structured such that most elements are fixed at 0. Thus, we only require updating a subset of the elements of ${\textbf {A}}$, or equivalently $vec({\textbf {A}})$. Define ${\textbf {A}}_{u}$ to be a vector of this subset of elements of ${\textbf {A}}$ to be updated, and vector ${\textbf {A}}_{u'}$ the elements of ${\textbf {A}}$ fixed at 0. Here, u holds indices of nonzero elements in $vec({\textbf {A}})$, and $u'$ the indices of the zero elements in $vec({\textbf {A}})$. Rather than sampling all of $vec({\textbf {A}})$, we reduce dimensionality by conditioning on ${\textbf {A}}_{u'} = {\textbf {0}}$. We reorganize $vec({\textbf {A}})$, ${\textbf {m}}$, and ${\textbf {M}}$ such that the elements u are ordered before $u'$:

$$\begin{aligned} vec({\textbf {A}}) = \begin{bmatrix} {\textbf {A}}_u \\ {\textbf {A}}_{u'}\end{bmatrix} \qquad {\textbf {m}}= \begin{bmatrix} {\textbf {m}}_u \\ {\textbf {m}}_{u'}\end{bmatrix} \qquad {\textbf {M}}= \begin{bmatrix} {\textbf {M}}_{uu} &{} {\textbf {M}}_{uu'} \\ {\textbf {M}}_{u'u} &{} {\textbf {M}}_{u'u'} \end{bmatrix} \end{aligned}$$

Multivariate normal theory yields the conditional distribution from which we sample:

$$\begin{aligned} {\textbf {A}}_{u} | {\textbf {A}}_{u'}, {\textbf {P}}, \varvec{\Sigma } \sim MVN_{|u|J}({\textbf {m}}_{u} - {\textbf {M}}_{u u'} {\textbf {M}}_{u'u'}^{-1} {\textbf {m}}_{u'}, {\textbf {M}}_{uu} - {\textbf {M}}_{u u'} {\textbf {M}}_{u'u'}^{-1} {\textbf {M}}_{u'u})\end{aligned}$$

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1.2 A.2 Redistribution Matrix

We first define local redistribution from BLOB $B_i$ to $B_k$. Using a kernel $f(;\theta )$ and uniform distribution over $B_i$, the probability $r^{*}_{ik}$ of moving from i to k is:

$$\begin{aligned} r^{*}_{ik} = \frac{1}{B_i} \int _{B_k} \int _{B_i} f(x-y; \theta ) dx dy \end{aligned}$$

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for $x \in B_i$, $y \in B_k$. Then, we take the probability $r_{ik}$ of local dispersal from $B_{i}$ to $B_{k}$ as

$$\begin{aligned} r_{ik} = \frac{r^{*}_{ik}}{\sum _{k'}r^{*}_{ik'}} \end{aligned}$$

(8)

We also allow for long distance redistribution from $B_i$. That is, it is possible for birds located in BLOB i at time t to redistribute to any other BLOB on the map at time $t+1$. The probability of long distance redistribution to $B_k$ from $B_i$, $s_{ik}$, is proportional to its area:

$$\begin{aligned} s_{ik} = \frac{|B_{k}|}{\sum _{k' \ne i} |B_{k'}| } \end{aligned}$$

(9)

Letting l denote the proportion of dispersal that could occur due to long-distance, redistribution from $B_i$ to any receiving $B_k$ takes the form ${\textbf {H}}_{[k,i]} = (1-l)r_{ik} + ls_{ik}$, which is column-normalized to preserve abundances.