Mathematical modeling of biological interactions: formalisms and contemporary applications

Abstract

This paper introduces an advanced mathematical framework for describing biological interactions through spectral graph theory. We examine differential equations coupled with various Laplacians (classical, signless, and normalized) to address competition, predation, and evolutionary cooperation. We cover stability analyses, bifurcations, and computational validation using empirical data. Network topological properties, including connectivity and modularity, significantly influence ecosystem robustness and the spread of epidemic phenomena. Our results illustrate the effectiveness of combining graph theory and differential equations for modeling large ecological and genomic networks at scale. Moreover, future perspectives are discussed, highlighting the integration of spectral preconditioners and deep learning. This approach provides a solid foundation for research in mathematical biology and the management of natural resources.