Hidden States of the Lotka-Volterra Equations

Abstract

Classically, population dynamics are described by the deterministic Lotka-Volterra equations. These equations do not accurately reflect reality, where stochastic influences have a big impact on populations of species, due to unpredictable environmental and biological factors. Using the Wiener process to include these influences allows for more realistic results, but means that solutions can deviate to unrealistic population numbers. This thesis provides a self-contained proof of the existence, uniqueness and boundedness of solutions of such systems.
In many cases, population data for all species is unavailable. To make accurate estimates for the unknown or hidden data, the extended Kalman filter can be applied. Which, through a combination of the data and the mathematical model, creates an estimate for the population. An exponential bound for the error of this estimation is derived in expectation