Dynamics of Coral-Macroalgae Interactions under Crowding

Abstract: We study a planar ODE model for the benthic competition between coral, macroalgae, and algal turf on a reef, extending the classical model of Mumby, Hastings, and Edwards by a nonlinear, density-dependent coral mortality that accounts for crowding. The strength of crowding is set by an exponent $δ>0$ that reshapes the coral nullcline and enriches the bifurcation structure of the system. We establish positive invariance of the biologically relevant region and the absence of periodic orbits, classify the three boundary equilibria together with their local stability, and reduce the coexistence problem to a single scalar equation whose shape, in particular its concavity, controls the number and local stability of the interior equilibria. The grazing intensity $g$ organizes the dynamics through two thresholds $g_0<g_1$ determining the stability of the coral- and macroalgae-dominated states, and a further threshold $g^\star$ at which two interior equilibria collide. We prove that the system undergoes transcritical bifurcations at the boundary equilibria and a saddle-node bifurcation of interior equilibria, and we discuss the implications for coral reef resilience and hysteresis. We complement these results with numerical simulations that illustrate the bifurcation sequence across the grazing regimes.