Effect of Delay on Coral Reef Ecosystem with Disease Presence in Coral Species

Taking into account the effect of incubation time delay, a coral reef system with disease present in coral species is studied in this paper. We present the qualitative analysis of the model including existence and positivity of its solutions. Local asymptotic stability of the biologically feasible equilibria in case of both delayed and non-delayed system is discussed. The Hopf-bifurcation analysis for the system around the interior equilibrium is also performed by treating time delay as a bifurcation parameter. The duration of time delay parameter is estimated by means of the Laplace transformation when all three species of the system preserve the stability. We derive the direction of Hopf-bifurcation, the stability and period of bifurcating periodic solutions by applying the normal form theory and central manifold theorem. We perform numerical investigation and explore their biological implication to support our theoretical results. Our findings indicate that incubation delay is responsible for destabilizing the system and can generate a cycle. In addition, conversion rate of infected colonies to healthy coral can prevent the oscillation within the system, and controls disease transmission among coral species.