Spatiotemporal dynamics of a reaction-diffusion nutrient-algae model

Abstract

Nutrient Algae models are important for policymakers, scientists, and environmental managers. The growth of algae in aquatic environments and how it is impacted by nutrient availability and other factors is studied using nutrient algae models. The development of community resilience to environmental changes, the safe use of water by humans and other species, as well as the appropriate management of aquatic resources and ecosystem health, are all facilitated by nutrient algae models. In this manuscript, a qualitative and computational analysis of the nutrient algae model is presented where partial differential equations explain how nutrients and algae coexist in the water column that provides the system of equations. We used a model of algae production and used it to explore the best ways to optimise algae farms to meet the best objectives. Two finite difference methods are used for the nutrient algae model. The computational results for the diffusion model are obtained to describe the growth of algae depending on nutrients that are poorly mixed in the water column. For algal growth, an asymmetric resource supply mechanism is formed by nutrients from the water bottom and inorganic carbon from the water surface. From this analysis, two schemes are derived, which are both von Neumann stable and consistent with the underlying model. The positive solutions are studied with the initial conditions that are non-negative in the qualitative analysis for the model. The M-matrix theory and the induction technique are used to show the positivity of the implicit scheme. The effectiveness of the implicit and explicit schemes for the test problem is demonstrated through simulations drawn for various parameter values. Equilibrium points are shown graphically for various values of the parameters. Furthermore, the numerical solutions for the system of equations were obtained to validate the theoretical results.