title: Alternative Stable States, Coral Reefs, and Smooth Dynamics with a Kick.
abstract: We consider a computer simulation, which was found to be faithful to time series data for Caribbean coral reefs, and an analytical model to help understand the dynamics of the simulation. The analytical model is a system of ordinary differential equations (ODE), and the authors claim this model demonstrates the existence of alternative stable states. The existence of an alternative stable state should consider a sudden shift in coral and macroalgae populations, while the grazing rate remains constant. The results of such shifts, however, are often confounded by changes in grazing rate. Although the ODE suggest alternative stable states, the ODE need modification to explicitly account for shifts or discrete events such as hurricanes. The goal of this paper will be to study the simulation dynamics through a simplified analytical representation. We proceed by modifying the original analytical model through incorporating discrete changes into the ODE. We then analyze the resulting dynamics and their bifurcations with respect to changes in grazing rate and hurricane frequency. In particular, a "kick" enabling the ODE to consider impulse events is added. Beyond adding a "kick" we employ the grazing function that is suggested by the simulation. The extended model was fit to the simulation data to support its use and predicts the existence cycles depending nonlinearly on grazing rates and hurricane frequency. These cycles may bring new insights into consideration for reef health, restoration and dynamics.
authors.affiliation: Department of Mathematical Sciences, Florida Atlantic University, 777 Glades Rd, Boca Raton, FL, 33431, USA. steve.ippolito@gmail.com.
authors.forename: Stephen
authors.initials: S
authors.lastname: Ippolito
authors.affiliation: Department of Mathematical Sciences, Florida Atlantic University, 777 Glades Rd, Boca Raton, FL, 33431, USA.
authors.forename: Vincent
authors.initials: V
authors.lastname: Naudot
authors.affiliation: Department of Biological Sciences, Florida Atlantic University, 3200 College Ave., Davie, FL, 33314, USA.
authors.forename: Erik G
authors.initials: EG
authors.lastname: Noonburg
mesh_terms: D000818:Animals; D037421:Anthozoa / Q000254:growth & development; D044822:Biodiversity; D017691:Caribbean Region; D003198:Computer Simulation; D057894:Coral Reefs; D017753:Ecosystem; D055641:Mathematical Concepts; D008954:Models, Biological; D011157:Population Dynamics; D012624:Seaweed / Q000254:growth & development
keywords: Alternative stable states; Competitive dynamics; Coral reefs; Dirac delta distribution; Dynamical systems