Ecoacoustics is increasingly mathematical: short-time Fourier transforms produce spectrograms; information-theoretic indices quantify diversity; optimal transport (Wasserstein distance) compares whole soundscapes; and stochastic-process models capture the temporal dynamics of biophony. This module collects the mathematical tools you will need to read modern ecoacoustic literature, with emphasis on those that come from analysis (Sobolev spaces, optimal transport) and from probability (point processes, entropy).

Module 11

Mathematical Foundations

Ecoacoustics is fundamentally a quantitative science. This module consolidates the core mathematical framework.

Soundscape–Ecosystem Mutual Information \[ I(\mathcal{S}; \mathcal{E}) = \int\!\int p(\mathbf{s}, \mathbf{e})\,\log\frac{p(\mathbf{s},\mathbf{e})}{p(\mathbf{s})\,p(\mathbf{e})}\,d\mathbf{s}\,d\mathbf{e} \] Maximising $I$ over acoustic features is the core optimization problem of ecoacoustic index design.

Generalized Master Equation for Acoustic State \[ \frac{\partial \rho_{\mathcal{S}}(t)}{\partial t} = -i\mathcal{L}_0\,\rho_{\mathcal{S}}(t) - \int_0^t \mathcal{K}(t-\tau)\,\rho_{\mathcal{S}}(\tau)\,d\tau + \mathcal{F}(t) \] $\mathcal{K}(t-\tau)$ is the memory kernel encoding non-Markovian dawn chorus entrainment and chorus synchronization.

Acoustic Restoration as Optimal Transport \[ W_2(\mu_0,\mu_*)^2 = \inf_{\gamma \in \Gamma(\mu_0,\mu_*)} \int_{\mathcal{F}\times\mathcal{F}} \|f - f'\|^2\,d\gamma(f,f') \] $\mu_0$ degraded vs $\mu_*$ reference acoustic spectral distribution. $W_2$ measures the minimum 'work' to restore the soundscape.

Acoustic Landscape Connectivity (Fiedler) \[ \lambda_2(\mathbf{L}) = \min_{\mathbf{x} \perp \mathbf{1},\, \|\mathbf{x}\|=1} \mathbf{x}^\top \mathbf{L}\, \mathbf{x} = \min_{\mathbf{x} \perp \mathbf{1}} \frac{\sum_{(i,j)\in E} W_{ij}(x_i - x_j)^2}{\sum_i x_i^2} \] Large $\lambda_2$ = efficient acoustic information propagation. Habitat fragmentation reduces $\lambda_2$, isolating acoustic communities.

Research Frontier

The convergence of Villani's hypocoercivity framework, Nakajima-Zwanzig non-Markovian kinetics, and ecoacoustic monitoring opens thermodynamic ecoacoustics — treating soundscape evolution as entropy production in a driven, dissipative ecological system.

Why optimal transport here

Optimal-transport metrics (Wasserstein W₂, Sinkhorn-regularised divergences) provide a principled way to compare two soundscapes that share no exact spectral coincidences. A degraded forest can be compared to a reference forest by computing the minimum-cost remapping of one's spectral mass onto the other's. This connects directly to the Villani hypocoercivity machinery used in the Mycorrhizal Networks course — both relax dynamical systems to equilibrium distributions, and the same gradient-flow language underlies the analysis.

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