KCC-theory and its applications to coral reef modelling

Abstract

The systems of second order differential equations (SODE) have played a very important role in the study of physics and biological models, in particular, the Volterra-Hamilton system is one of the most useful SODE in ecological problems. We develop the required background of Finsler geometry and classical equation of ecological models to study some aspects of the tra- jectories of a Volterra-Hamilton system and other subject called semispray. Some geometrical invariants, called KCC-invariants, there are five, are computed to study aspects of the solution trajectories of a semispray. We use Volterra-Hamilton systems theory and their associated cost functional to study the population dynamics and productive processes of coral reefs together with their symbiont algae in recovery from bleaching and show that the cost of production remains the same after the process. The KCC-theory geometrical invariants are determined for the model proposed to describe the renewed symbiotic interaction between coral and algae.