Marcus’s formulation of stochastic algae population dynamics subject to power-type abrasion

Bloom of nuisance filamentous green algae on riverbed in regulated river reaches has been an aquatic environmental problem. Algae population dynamics contain growth and abrasion, the latter being due to mechanical disturbances by sediment-laden flows during flood events. A recent experimental study revealed that the abrasion of algae population follows power-type decay which is slower than exponential one that has been conventionally assumed. We propose a stochastic differential equation model to describe the algae population dynamics driven by both continuous sigmoidal growth and flood-induced abrasion. The power-type decay is modeled using a Marcus’s formulation of jumps. Our model is nonlinear whose coefficients are not Lipschitz continuous. We prove the unique existence, boundedness, and closed-form representation of the solution based on a nonlinear transformation method and a contradiction argument. Moments bounds of the solution are obtained as well, with which we conclude that the solution, the algae population, never goes extinct by the abrasion with probability 1. We also conduct Monte Carlo simulation concerning a sediment replenishment project to suppress the algae bloom.