Hysteresis in coral reefs under macroalgal toxicity and overfishing

Macroalgae and corals compete for the available space in coral reef ecosystems.While herbivorous reef fish play a beneficial role in decreasing the growth of macroalgae, macroalgal toxicity and overfishing of herbivores leads to proliferation of macroalgae. The abundance of macroalgae changes the community structure towards a macroalgae-dominated reef ecosystem. We investigate coral-macroalgal phase shifts by means of a continuous time model in a food chain. Conditions for local asymptotic stability of steady states are derived. It is observed that in the presence of macroalgal toxicity and overfishing, the system exhibits hysteresis through saddle-node bifurcation and transcritical bifurcation. We examine the effects of time lags in the liberation of toxins by macroalgae and the recovery of algal turf in response to grazing of herbivores on macroalgae by performing equilibrium and stability analyses of delay-differential forms of the ODE model. Computer simulations have been carried out to illustrate the different analytical results.