A linear stochastic model for phytoplankton production in a marine ecosystem

A linear stochastic process is formulated as an approximation to a nonlinear ecosystem model. The formulation traces a chemical nutrient as it undergoes random exchanges between the phytoplankton, zooplankton, and the euphotic zone of an aquatic ecosystem. The formulation of a linear process allows the derivation of a partial-differential equation for the cumulant-generating function of the process. The use of this equation leads to a system of linear, deterministic differential equations for the cumulants of the process. Closed-form solutions are obtained for the means of the ecosystem components. These means display the characteristic oscillations and phase-plane spirals of predator-prey models. A variance which increases almost linearly with time suggests a non-stable probability distribution. Predation dynamics are reflected in the correlation structure of the solutions.