Theoretical foundations, methodological advances, and applications of joint species distribution models with a focus on the HMSC framework in ecology

doi:10.17520/biods.2025364

Abstract

Abstract:

Background: Understanding how environmental filtering, biotic interactions, and neutral processes jointly shape species distributions and community structure is a central question in modern community ecology. However, traditional diversity indices, ordination analyses, and single-species distribution models (SDMs) cannot simultaneously integrate species associations, environmental gradients, functional traits, and phylogenetic relationships, thereby limiting their ability to disentangle community assembly mechanisms.

Framework: Joint species distribution models (JSDMs), particularly the hierarchical modelling of species communities (HMSC) framework, offer a unified and flexible Bayesian tool for community-level mechanistic inference. This study provides a systematic review of the statistical structure, mathematical foundations, and inferential mechanisms of HMSC, and establishes a complete analytical workflow encompassing data organization, model specification, Markov Chain Monte Carlo (MCMC) estimation, model evaluation, ecological interpretation, and predictive applications. A step-by-step tutorial, Joint Species Distribution Modelling with HMSC, accompanies the review and illustrates the practical implementation of HMSC through bryophyte community data and fully reproducible R code.

Theory: In the theoretical component, we clarify how HMSC integrates environmental gradients, species traits, phylogenetic relationships, and spatial structure within a unified hierarchical Bayesian framework to distinguish statistical signals of environmental filtering, biotic filtering, and dispersal limitation. Methodologically, we dissect the mathematical structure of latent variable models, elucidate the boundaries of ecological interpretation for residual correlations, and provide a conceptual basis for differentiating species co-occurrence signals from environmental effects and unobserved factors. We further compare HMSC with other mainstream JSDMs implementations and traditional community analytical methods, highlighting their relative advantages and ecological applicability.

Applications: On the applied side, we synthesize the rapidly expanding use of JSDMs across forest, wetland, grassland, marine, urban, and microbial ecosystems, demonstrating their value in conservation planning, invasive species risk assessment, co-occurrence network analysis, and scenario-based forecasting. With the advancement of GPU-accelerated computation, migration learning, and high-dimensional modelling frameworks, HMSC greatly improves ecological niche estimation and distribution prediction for rare species and enables community modelling for tens of thousands of taxa. Overall, JSDMs and HMSC in particular represent a methodological shift from single-species prediction toward integrative, multi-species and multi-dimensional ecological modelling. They provide an efficient, scalable, and uncertainty-aware platform that strengthens ecological theory testing, enhances understanding of community assembly mechanisms, and supports biodiversity conservation and management decisions.

Key words: community ecology, joint species distribution models, hierarchical modelling of species communities, environmental filtering, neutral processes, Bayesian models

Jiqi Gu, Jiangshan Lai, Ying Wang, Haoran Wu, Xue Zhang, Xiaotong Song, Xiaoming Shao, Anru Lou. Theoretical foundations, methodological advances, and applications of joint species distribution models with a focus on the HMSC framework in ecology[J]. Biodiv Sci, 2026, 34(1): 25364.

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URL: https://www.biodiversity-science.net/EN/10.17520/biods.2025364

https://www.biodiversity-science.net/EN/Y2026/V34/I1/25364

Figures/Tables 6

Fig. 1 Schematic diagram of the multi-scale framework of community assembly processes and corresponding types of observational data. The figure illustrates the different temporal and spatial scales at which community assembly processes occur, including global, regional, and local scales. Community assembly processes can be divided into speciation, neutral processes (dispersal, ecological drift, etc.), biotic filtering (species interactions), and environmental filtering (selection by abiotic factors). Blue arrows indicate the direction of mechanisms, and green arrows denote the pathways of different community assembly processes.

Table 1 Indices and their ranges in the HMSC framwork

索引及其范围 Index and range	含义说明 Description
i = 1,..., n	样方(采样单元) Sampling plots (Sampling units)
j = 1,..., n_s	物种 Species
k = 1,..., n_c	环境协变量 Environmental covariates
l = 1,..., n_t	物种性状 Species traits
h = 1,..., n_f	潜在因子 Latent factors
u = 1,..., nᵤ	层级单元 Hierarchical units
q = 1,..., d	空间坐标维度 Spatial coordinate dimensions
r = 1,..., nᵣ	随机效应 Random effects

Table 2 Data matrices and their dimensions in the core model of HMSC framework

数据矩阵 Data matrix	数据维度 Data dimension	含义说明 Description
𝐘, 元素 $γ i j$ 𝐘, elements $γ i j$	$n × n s$	群落数据 Community data
𝐗, 元素 $x i k$ 𝐗, elements $x i k$	$n × n c$	环境数据 Environmental data
𝐓, 元素 $t j l$ 𝐓, elements $t j l$	$n s × n t$	物种性状数据 Species trait data
𝐂, 元素 $c j 1 j 2$ 𝐂, elements $c j 1 j 2$	$n s × n s$	系统发育数据 Phylogenetic data
𝚷, 元素 $π i u$ 𝚷, elements $π i u$	$n × n u$	研究设计 Study design
𝐒, 元素 $s u q$ 𝐒, elements $s u q$	$n u × d$	空间坐标 Spatial coordinates

Table 2 Data matrices and their dimensions in the core model of HMSC framework

数据矩阵 Data matrix	数据维度 Data dimension	含义说明 Description
𝐘, 元素 $γ i j$ 𝐘, elements $γ i j$	$n × n s$	群落数据 Community data
𝐗, 元素 $x i k$ 𝐗, elements $x i k$	$n × n c$	环境数据 Environmental data
𝐓, 元素 $t j l$ 𝐓, elements $t j l$	$n s × n t$	物种性状数据 Species trait data
𝐂, 元素 $c j 1 j 2$ 𝐂, elements $c j 1 j 2$	$n s × n s$	系统发育数据 Phylogenetic data
𝚷, 元素 $π i u$ 𝚷, elements $π i u$	$n × n u$	研究设计 Study design
𝐒, 元素 $s u q$ 𝐒, elements $s u q$	$n u × d$	空间坐标 Spatial coordinates

Table 3 Parameters in the core model of the HMSC framework and their interpretations

类别 Category	参数 Parameter	类型 Type	含义 Description
固定效应 Fixed effect	L^F, 元素 $L i j F$ L^F, elements $L i j F$	$n × n s$ 矩阵 $n × n s$ matrix	固定效应的线性预测量 Linear predictor of fixed effects
固定效应 Fixed effect	B, 元素 $β k j$ B, elements $β k j$	$n c × n s$ 矩阵 $n c × n s$ matrix	物种生态位 Species ecological niches
固定效应 Fixed effect	M, 元素 $μ k j$ M, elements $μ k j$	$n c × n s$ 矩阵 $n c × n s$ matrix	基于性状的物种生态位期望值 Trait‐based expected values of species niches
固定效应 Fixed effect	ρ	标量 Scalar	物种生态位的系统发育信号 Phylogenetic signal in species niches
固定效应 Fixed effect	Γ, 元素 $γ k j$ Γ, elements $γ k j$	$n c × n t$ 矩阵 $n c × n t$ matrix	性状对生态位的影响 Effects of traits on species niches
固定效应 Fixed effect	V, 元素 $V k 1 k 2$ V, elements $V k 1 k 2$	$n c × n c$ 矩阵 $n c × n c$ matrix	物种生态位的残差协方差 Residual covariance of species niches
随机效应 Random effect	Lᴿ, 元素 $L i j R$ Lᴿ, elements $L i j R$	$n × n s$ 矩阵 $n × n s$ matrix	随机效应的线性预测量 Linear predictor of random effects
随机效应 Random effect	H, 元素 $η u h$ H, elements $η u h$	$n u × n f$ 矩阵 $n u × n f$ matrix	样地载荷 Site loadings
随机效应 Random effect	α, 元素 $α h$ α, elements $α h$	长度为 $n f$ 的向量 Vector of length $n f$	样地载荷的空间尺度 Spatial scale of site loadings
随机效应 Random effect	Λ, 元素 $λ h j$ Λ, elements $λ h j$	$n f × n s$ 矩阵 $n f × n s$ matrix	物种载荷 Species loadings
随机效应 Random effect	Ω, 元素 $Ω j 1 j 2$ Ω, elements $Ω j 1 j 2$	$n s × n s$ 矩阵 $n s × n s$ matrix	物种间的关联关系 Interspecific association matrix
随机效应 Random effect	Φ, 元素 $φ h j$ Φ, elements $φ h j$	$n f × n s$ 矩阵 $n f × n s$ matrix	物种载荷的局部收缩项 Local shrinkage parameters of species loadings
随机效应 Random effect	δ, 元素 $δ l$ δ, elements $δ l$	长度为 $n f$ 的向量 vector of length $n f$	物种载荷的全局收缩项 Global shrinkage parameters of species loadings
数据模型 Data model	L, 元素 $L i j$ L, elements $L i j$	$n × n s$ 矩阵 $n × n s$ matrix	线性预测量 Linear predictor
数据模型 Data model	Σ, 元素 $σ j 2$ Σ, elements $σ j 2$	$n s × n s$ 对角矩阵 $n s × n s$ diagonal matrix	残差方差 Residual variances

Table 3 Parameters in the core model of the HMSC framework and their interpretations

类别 Category	参数 Parameter	类型 Type	含义 Description
固定效应 Fixed effect	L^F, 元素 $L i j F$ L^F, elements $L i j F$	$n × n s$ 矩阵 $n × n s$ matrix	固定效应的线性预测量 Linear predictor of fixed effects
固定效应 Fixed effect	B, 元素 $β k j$ B, elements $β k j$	$n c × n s$ 矩阵 $n c × n s$ matrix	物种生态位 Species ecological niches
固定效应 Fixed effect	M, 元素 $μ k j$ M, elements $μ k j$	$n c × n s$ 矩阵 $n c × n s$ matrix	基于性状的物种生态位期望值 Trait‐based expected values of species niches
固定效应 Fixed effect	ρ	标量 Scalar	物种生态位的系统发育信号 Phylogenetic signal in species niches
固定效应 Fixed effect	Γ, 元素 $γ k j$ Γ, elements $γ k j$	$n c × n t$ 矩阵 $n c × n t$ matrix	性状对生态位的影响 Effects of traits on species niches
固定效应 Fixed effect	V, 元素 $V k 1 k 2$ V, elements $V k 1 k 2$	$n c × n c$ 矩阵 $n c × n c$ matrix	物种生态位的残差协方差 Residual covariance of species niches
随机效应 Random effect	Lᴿ, 元素 $L i j R$ Lᴿ, elements $L i j R$	$n × n s$ 矩阵 $n × n s$ matrix	随机效应的线性预测量 Linear predictor of random effects
随机效应 Random effect	H, 元素 $η u h$ H, elements $η u h$	$n u × n f$ 矩阵 $n u × n f$ matrix	样地载荷 Site loadings
随机效应 Random effect	α, 元素 $α h$ α, elements $α h$	长度为 $n f$ 的向量 Vector of length $n f$	样地载荷的空间尺度 Spatial scale of site loadings
随机效应 Random effect	Λ, 元素 $λ h j$ Λ, elements $λ h j$	$n f × n s$ 矩阵 $n f × n s$ matrix	物种载荷 Species loadings
随机效应 Random effect	Ω, 元素 $Ω j 1 j 2$ Ω, elements $Ω j 1 j 2$	$n s × n s$ 矩阵 $n s × n s$ matrix	物种间的关联关系 Interspecific association matrix
随机效应 Random effect	Φ, 元素 $φ h j$ Φ, elements $φ h j$	$n f × n s$ 矩阵 $n f × n s$ matrix	物种载荷的局部收缩项 Local shrinkage parameters of species loadings
随机效应 Random effect	δ, 元素 $δ l$ δ, elements $δ l$	长度为 $n f$ 的向量 vector of length $n f$	物种载荷的全局收缩项 Global shrinkage parameters of species loadings
数据模型 Data model	L, 元素 $L i j$ L, elements $L i j$	$n × n s$ 矩阵 $n × n s$ matrix	线性预测量 Linear predictor
数据模型 Data model	Σ, 元素 $σ j 2$ Σ, elements $σ j 2$	$n s × n s$ 对角矩阵 $n s × n s$ diagonal matrix	残差方差 Residual variances

Fig. 2 Schematic overview of the complete analytical workflow and outputs of hierarchical modelling of species communities (HMSC) framework. The figure illustrates five key steps of HMSC-based modelling and inference: (1) model construction and data integration, in which species distribution data are jointly modelled with environmental variables, functional traits, phylogenetic relationships, and spatial random effects within a unified hierarchical Bayesian framework; (2) MCMC convergence diagnostics, where trace plots, posterior distributions, effective sample size (ESS), and potential scale reduction factors (PSRF) are used to assess convergence and fitness parameter mixing; (3) model fit evaluation and comparison, in which predictive performance is quantified using root mean square error (RMSE), area under curve (AUC), and R²; (4) parameter estimation and ecological interpretation, including environmental response parameters, trait and phylogenetic effects, residual correlation structures, and variance partitioning; and (5) model prediction and application, illustrating species responses to key environmental gradients with associated uncertainty intervals for community-level prediction and scenario-based analyses.

Fig. 3 Changes in the number of JSDMs publications from 2011 to 2024

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