Dispersal diversity buffers species vulnerability to local extinction

The gist This framework presents a general framework, grounded in the statistical physics of complex systems, that integrates key processes governing species survival into a single measurable quantity: the competitive balance.

Theoretical Framework and Metrics

The study introduces an analytical framework to quantify each species’ proximity to its competitive exclusion boundary in metacommunities This approach employs a parsimonious ecological model explicitly incorporating basic processes taking place in a heterogeneous spatial environment Leveraging analytical methods from the physics of disordered systems, it derives ecologically interpretable metrics that quantify how species exploit locally available space and how they are affected by community-mediated interactions These metrics define species vulnerability as their proximity to the competitive-exclusion threshold, explicitly accounting for the ecological context in which they are embedded

The key metrics derived from the model include:

1. Space-use fraction (pαi): This is defined as Fraction of the total effective space available in site (or subplot) i occupied by species α

2. Total fraction of occupied space: This is defined as Average space-use fraction, or total fraction of effective available space occupied by species α (across the entire sampled area)

3. Vacancy-Adjusted Abundance (VAA) (xαi): This metric is defined as Space-use of species α in subplot i, normalized by the locally available (unoccupied) space

4. Dispersal heterogeneity (δ−1α): This is defined as the inverse shape parameter of the VAA distribution, δ−1α, which is equal to the squared coefficient of variation of the VAA distribution

5. Effective dispersal strength (β¯−1α): This is defined as Scale (or inverse rate) parameter of the VAA distribution

6. Competitive balance (Bα): This is defined as Ratio between community-wide factors that increase and decrease abundance

7. Species vulnerability (Wα): This is defined as Rescaled distance to each species’ persistence threshold

Model Dynamics and Inference

The model describes the temporal dynamics of species abundances in a habitat composed of N colonizable sites, incorporating basic processes such as death, reproduction, dispersal, colonization, and competition for space The dynamics are governed by a set of differential equations that capture local logistic growth with rate Kαii (eq. 2), linear death with rate eαi (eq. 3), local effective competition term proportional to K′αβ,ii (eq. 4), and colonization depending on local space availability and dispersal from other sites with rate Kα,ij (eq. 5)

The kernel matrix Kα is not measured directly but its elements are assumed to be drawn from a statistical distribution Q(α)N characterized by a small number of species-specific parameters The vacancy-adjusted abundance (VAA), xα = ηα/eα, is derived from the stationary state of the model and follows a Gamma distribution Fitting empirical VAA distributions to the theoretical prediction enables inference of this kernel, which encodes species-specific spatial strategies

Validation and Key Findings

The framework was validated using tropical and temperate forest data, showing that greater heterogeneity in dispersal strategies reduces vulnerability to competitive exclusion for a given abundance The study found that species with positive vulnerabilities have a stationary abundance that approaches zero, consistent with the theory Furthermore, the analysis revealed a clear and significant negative correlation between σpα and Wα in all three datasets, indicating that even if two species were to have the same average abundances but different spatial variability, the one with more heterogeneous dispersal would tend to be a superior competitor

Ecological Insights and Trade-offs

The competitive balance is not simply additive but is coupled non-linearly through a shared term that makes community-level coexistence emerge as a collective property The vulnerability of species α is quantified by the distance from the metacommunity’s position in the competitive balance space to its own persistence boundary, defined as Wα < 0 indicating persistence The results show that abundance and vulnerability are strongly correlated only for species with vulnerability close to zero, i.e., close to the extinction threshold A key insight is that equal competitive balance does not imply that species have identical traits but reflects a global trade-off at the community level

The study concludes by showing that "at a given competitive balance species with a high abundance tend to have less spatially variable dispersal, allowing them to colonize a greater number of patches effectively, while species with more heterogeneous dispersal patterns (broader kernel distributions) will be more subject to crowding effects" This provides a mechanistic link between dispersal heterogeneity, abundance, and survival

Limitations and Future Directions

A limitation is that the vulnerability metric does not provide information on the timescales over which different species approach their steady-state, but only predicts the final outcome Furthermore, distinct mechanisms of local competition and facilitation cannot be disentangled in our framework Future work should incorporate stochastic effects and accounting for temporal fluctuations to move beyond deterministic proximity to exclusion

The paper provides a tractable approach for studying metacommunity dynamics by bridging ecological theory and statistical physics, offering a novel perspective on the fundamental principles governing biodiversity patterns and their resilience to perturbations

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The model dynamics are governed by the following set of differential equations: dpαi(t) dt = −eαpαi(t) + (1 − X S β=1 pβi(t)) X N j=1 Kα,ij pαj (t). (1)

The kernel matrix Kα is an effective species-specific interaction–colonization kernel. It quantifies both how species interact and how they disperse between patches The kernel is not measured directly from field data, nor do we attempt to reconstruct each individual kernel entry Kα,ij separately

The model dynamics are governed by the following set of differential equations: dpαi(t) dt = Kα,iipαi(t) (1 − pαi(t)) - eαipαi(t) - X S β=1 pβi(t) K′ αβ,ii pαi(t) + (1 − X S β=1 pβi(t)) X N j=1 Kα,ijpαj (t). (5)

The kernel matrix Kα is an effective species-specific interaction–colonization kernel. It quantifies both how species interact and how they disperse between patches

Dynamical Mean Field Theory

In real ecosystems, estimating the colonization kernel directly is challenging One simplification is to assume that in large, heterogeneous environments lacking macroscopic barriers, the kernel elements are randomly distributed, mirroring the underlying landscape heterogeneity We adopt the assumption of “homogeneous inhomogeneity,” where all patches are statistically equivalent

Under this framework, we make the following scaling assumptions for the characteristic function of the kernel distribution in the limit of a large number of sites N → ∞: Z dKe−iKzQ (α)N (K) = e F(α)(z)/N. (6)

With this assumption, the full system reduces to a set of S effective stochastic equations, one for each species [42]: dpα(t) dt = −eαpα(t) + (1 − X S β=1 pβ(t)) ηα(t), (7) where ηα(t) is a stochastic process, the properties of which summarize the dynamics of the coupled system in one representative patch

Approximate Stationary Solution Defines Vacancy-Adjusted Abundance

We focus on the long-term behavior of Eq. 7, and ask which species will persist The stationary state involves solving Eq. 9 for xα = ηα/eα, which captures the ratio of abundance to the amount of free space available

Assuming that the cumulant generating function of the kernel distribution, F(α)(z), is analytic, we derive an explicit solution for xα = ηα/eα as x α,ij ∼ Lα(x) ≡ β¯δα α Γ(δa) x −1+δa e −β¯a xΘ(x), where β¯a = βαeα/⟨pα⟩, and ⟨pα⟩ denotes the abundance of species α averaged across sites

Coexistence Transition Boundaries

The persistence condition Bα > 1 holds for a species in isolation When other species are present, the critical transition to persistence depends on the competitive balance of all species, forming a S − 1 dimensional manifold described by parametric equations

Perturbative Expansion Leads to Vulnerability Definition

To circumvent the problem of the combinatorial explosion of transition boundaries, we use a perturbative expansion in powers of 1/S to define vulnerability Wα <ref:

1. Bold header: Adaptive vulnerability assessment based on competitive balance

This improved system can quantify species vulnerability as a distance in competitive balance space by computing "Wα = 1/∆α [Bα]α − Bα - ∆α < 0, explicitly incorporating the entanglement among dispersal strategies, spatial distribution patterns, and total population size."

2. Bold header: Inferring species-specific interaction-dispersal kernels from spatial abundance data

The system can infer species-specific dispersal strategies without requiring direct measurements of dispersal or interaction parameters by fitting the analytical solution of the model to empirical VAA distributions to obtain the values of δα and β¯α.

3. Bold header: Context-dependent persistence prediction for species-rich communities

The AI can predict persistence using a metric that depends on the ecological context, allowing it to determine if a species persists by checking if its vulnerability is below the threshold, as shown in Fig. 4D, rather than relying solely on abundance.

4. Bold header: Predicting optimal spatial scales for parameter estimation

The system can determine the optimal M (total carrying capacity of each subplot) by minimizing the total CDF mean-squared error across all selected species when fitting empirical VAA distributions to their theoretical Gamma distribution.

5. Bold header: Distinguishing abundance from vulnerability in species-rich systems

The system can identify that abundance and vulnerability are strongly correlated only for species with vulnerability close to zero, i.e., close to the extinction threshold, providing a mechanism for distinguishing risk from low population size.