Delay-induced Hopf bifurcation in a coral-macroalgae system under carbon-driven acidification stress

Abstract

Ocean acidification can reduce coral growth, alter coral-macroalgae interactions, and weaken reef resilience. This study develops a delayed coral-macroalgae-acidification-pressure model describing live coral cover, fleshy macroalgal cover, and a dimensionless effective acidification-pressure index. The model incorporates coral-macroalgae overgrowth, external stress input, natural relaxation, macroalgal-associated attenuation, coral-associated feedback, and a delayed coral response to acidification pressure. We establish positivity and boundedness of solutions, determine biologically feasible equilibria, derive local stability conditions, and analyze the effect of delay on the coexistence state. The analysis identifies a critical delay at which the stable coexistence equilibrium loses stability through a Hopf bifurcation. For the baseline parameter set, the bifurcation is supercritical and generates a local branch of orbitally asymptotically stable periodic orbits. Numerical simulations, sensitivity analysis, and robustness tests support the analytical findings and show how coral stress sensitivity, external forcing, and reef-scale feedbacks influence stability thresholds and coral persistence. These results provide a mechanistic framework for examining how delayed ecological responses to acidification may promote oscillatory reef dynamics.