A superprocess model for eDNA transport in random river environment

Abstract

Environmental DNA (eDNA) is increasingly used to monitor aquatic organisms in rivers, yet predicting its concentration and assessing the reliability of such predictions remain challenging because river flow fluctuates with long memory and models of eDNA dynamics are inevitably misspecified. We propose a stochastic modeling framework for eDNA concentration in rivers that addresses both issues. The eDNA concentration is described by a stochastic partial differential equation with transport, decay, source, and square-root multiplicative noise, and the flow velocity is modeled by a superposition of Ornstein-Uhlenbeck-type processes with long memory. We show that the long memory of the flow velocity induces superdiffusive transport of eDNA, with the variance of the transported distance growing superlinearly in the ensemble sense. The proposed model admits a closed-form Laplace functional, based on which we prove the existence and uniqueness in law of the solution on an unbounded domain. We further evaluate the effects of model uncertainty through a robust control formulation based on the relative entropy, in which the lower and upper bounds of the expected eDNA concentration, together with the worst-case distortions, are obtained in closed form. The framework is applied to eDNA and flow velocity data collected for the upstream migration of Plecoglossus altivelis altivelis in a river in Japan. The results demonstrate that the closed-form bounds provide a computationally inexpensive tool for quantifying the range of eDNA concentrations that can be expected under model uncertainty.