Investigating Coexistence and Extinction in a Four-Species Trophic System Using Random Matrix Theory

Random matrix theory has played a pivotal role in understanding the complexity of biological systems, especially in elucidating the stability and coexistence of species in ecological communities. In this study, we apply random matrix theory at a trophic level consisting of four species with different initial populations. We incorporate varying interaction probabilities, denoted as p, to explore how the structure and strength of interspecific interactions affect species persistence and extinction rates. By modeling the system’s dynamics through a suite of mathematically derived equations and generating adjacency matrices under different values of p, we produce multiple scenarios that highlight the interplay between cooperation and competition. Our numerical simulations yield a series of graphs illustrating the likelihood of coexistence, extinction trajectories, and the effect of interaction intensity. The results underscore the delicate balance between competition and mutual benefit, shedding light on conditions in which biodiversity is maintained or lost. In our ensuing discussion, we reflect on theoretical implications, potential applications in conservation biology, limitations of our approach, and directions for future research.