| Abstract: |
| Competition between macroalgae and corals for occupying the available space in sea bed is an important ecological process underlying coral-reef dynamics. While herbivorous reef-fish play a beneficial role in decreasing the growth of macroalgae, macroalgal toxicity and overfishing of herbivores leads to the proliferation of macroalgae in coral reef ecosystem, which eventually changes the community structure towards macroalgae-dominated reef ecosystem. We analyze a mathematical model of interactions between coral, macroalgae and herbivores to investigate coral-macroalgal phase shifts by assuming that the growth of herbivorous Parrotfish is limited by coral cover. It is observed that in presence of macroalgal toxicity and overfishing of Parrotfish the system exhibits hysteresis through saddle-node bifurcation and transcritical bifurcation. We examine the effects of macroalgal toxicity and herbivore-harvesting in the resilience of coral-macroalgae coexistence steady state. Further, we study the non-autonomous version of the model by incorporating synchronous or asynchronous seasonal variations in different parameters. By using of Mawhin`s continuous theorem of coincidence degree theory, a sufficient condition is obtained for the existence of a positive periodic solution. Computer simulations have been carried out to illustrate different analytical results. |
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