Stochastic partial differential equation model for environmental DNA dynamics in river environments

Abstract

Environmental DNA (eDNA) has emerged as a novel tool for quantifying the seasonal abundance of aquatic species in water bodies; however, its mathematical modeling is still at a germinating stage because of its mechanistic uncertainties. We propose a first-step mathematical and computational framework for the eDNA dynamics of migratory fish based on a novel stochastic partial differential equation model with a delayed source input. The model governs spatiotemporal eDNA concentration in rivers where the source comes from a stochastic differential equation for the migration dynamics of the fish. The affine nature of the model facilitates its theoretical analysis, including the guarantee of well-posedness and the closed-form derivation of the Laplace functional despite the proportionality coefficient of the multiplicative noise term being non-Lipschitz. We also propose a discretization scheme for the model that theoretically generates nonnegative numerical solutions. We finally apply the proposed model to eDNA concentration data sampled from midstream reaches of a river system and perform sensitivity analysis.