Mathematical insights into the influence of delay and external recruitment on coral-macroalgae system

Abstract

As anthropogenic pressures on coral reef ecosystems continue to increase due to global warming and mass coral bleaching events, there is growing interest in developing conservation strategies to restore degraded coral reefs. One such approach is the transplantation of coral larvae onto degraded reefs. In this paper, we use a delayed coral-macroalgae model to explore the effects of external recruitment of coral. By analyzing the local and global stability of macroalgae-free equilibrium and coexistence equilibrium in the system, we find that sustained external recruitment of coral will favor coral competition with macroalgae. In addition, we choose delay as a bifurcation parameter and demonstrate that Hopf bifurcation may occur at a critical delay near the coexistence equilibrium. Interestingly, the delay may cause the globally asymptotically stable equilibrium in the ODE system to become unstable, resulting in the appearance of periodic solutions. Furthermore, we analyze population dynamics using optimal control theory and determine the effect of minimum external recruitment on the population dynamics. In the numerical simulation section, parameters of coral-macroalgae dynamics are estimated by using a 12-year (2005–2017) benthic cover dataset of coral reefs in the Gulf of Mannar, southeastern India. The theoretical results are validated and supported by numerical simulations.