Modelling the spread and growth of Caulerpa taxifolia in closed waterways in southern Australia using cellular automata

Abstract

This study presents the mathematical development of a cellular automata model for the species Caulerpa taxifolia for closed or intermittently closed waterways along the Australian coast. The model is used to assess the spatial coverage of C. taxifolia by describing changes in growth, spread and total biomass for the species. Building upon a foundation model developed by the authors, this study was designed to enhance the predictive capabilities of a model based upon a discrete version of Laplace's equation. The improvements relate to several components integrated into the Laplacian coefficients; a periodic function which represents the seasonal variations in growth, the incorporation of the prevailing wind to represent the most likely direction of spread, and growth restrictions based on lake depth. The additional complexity improved the predictive capability of the model. Cellular Automata (CA) have been used to model changing plant distributions over the last 20 years, providing efficient models for complex environmental systems, particularly of exotic species. In this project the discrete CA algorithm is designed to determine the state or biomass B = {−1, 0, 1, 2} of the current cell using the primary rule of the discrete Laplacian system. Biomass of −1 refers to land, and the other values represent the relative quantity of the weed in the cell; none, sparse or dense respectively. Cell interactions are governed by the coefficients of the Laplacian system which is discussed.The foundation model incorporated simple rules, not unlike those of John Conway's Game of Life. The biomass of the surrounding cells at time t determines the state of the central cell at time t + 1. The boundary conditions were catered for by allocating a biomass of negative one to the land cells adjacent to the water.