Dynamics of coral reef models in the presence of parrotfish

Abstract

The decline of coral reefs characterized by macroalgae increase has been a global threat. We consider a slightly modified version of an ordinary differential equation (ODE) model proposed in Blackwood, Hastings, and Mumby [Theor. Ecol. 5 (2012), pp. 105–114] that explicitly considers the role of parrotfish grazing on coral reef dynamics. We perform complete stability, bifurcation, and persistence analysis for this model. If the fishing effort (f) is in between two critical values f 0 and f 1, then the system has a unique interior equilibrium, which is stable if f 0 < f < f 1 and unstable if f 1 < f < f 0. If f is less (more) than these critical values, then the system has up to two (zero) interior equilibria. Also, we develop a more realistic delay differential equation (DDE) model to incorporate the time delay and treating it as the bifurcation parameter, and we prove that Hopf bifurcation about the interior equilibria could occur at critical time delays, which illustrate the potential importance of the inherent time delay in a coral reef ecosystem. Recommendations for Resource Managers One serious threat to coral reefs is overfishing of grazing species, including high level of algal abundance. Fishing alters the entire dynamics of a reef (Hughes, Baird, & Bellwood, 2003), for which the coral cover was predicted to decline rapidly (Mumby, 2006). One major issue is to reverse and develop appropriate management to increase or maintain coral resilience. We have provided a detailed local and global analysis of model (Blackwood, Hastings, & Mumby, 2012) and obtained an ecologically meaningful attracting region, for which there is a chance of stable coexistence of coral–algal–fish state. The healthy reefs switch to unhealthy state, and the macroalgae–parrotfish state becomes stable as the fishing effort increases through some critical values. Also, for some critical time delays, a switch between healthy and unhealthy reef states occurs through a Hopf bifurcation, which can only appear in the delay differential equation (DDE) model. Eventually, for large enough time delay, oscillations appear and an unhealthy state occurs.

Zigzag persistence for coral reef resilience using a stochastic spatial model

“…The MHE model describes the temporal evolution of the fraction of a reef covered by either coral (C), macroalgae (M) or algal turf (T), where T = 1 − C − M (assuming the seabed area is fully covered). Previous work has extended the MHE model to account for the effect of ocean acidification [41], natural disasters [42] and more complex fishing dynamics [43,44].…”

“…In [19], Mumby, Edwards and Hastings proposed the following model (MHE model) to describe such interactions in a simple three-component ODE system:C˙=rCT−dC−aCMandM˙=aCM−gMM+T+γMT.The MHE model describes the temporal evolution of the fraction of a reef covered by either coral ( C ), macroalgae ( M ) or algal turf ( T ), where T = 1 − C − M (assuming the seabed area is fully covered). Previous work has extended the MHE model to account for the effect of ocean acidification [41], natural disasters [42] and more complex fishing dynamics [43,44]. Other models consider multiple competing coral and macroalgae types [45] with complex intra-species dynamics [46].…”