Asymptotic Stability of a Mathematical Model of Coffee Trees Rust Control

Coffee leaf rust, caused by the fungus Hemileia vastatrix, is one of the main diseases of coffee plantations worldwide. Snails of the species Bradybaena similaris feed on the rust and have been used as biological agents to control it, but, being invasive, they must themselves be kept in check. We study a model of this control coupling an SIS epidemiological component with a Lotka–Volterra predator–prey component, in which rust-infected trees are the prey and snails are the predators. The model includes a finite carrying capacity of coffee trees, two discrete delays—the rust latent period and the delay in snail maturation and gestation—and stochastic perturbations proportional to the deviation of the state from an equilibrium, which yields a system of stochastic delay differential equations. We obtain four equilibria—extinction, an endemic snail-free state and two coexistence states—with their conditions of existence, and establish their stability in probability by the method of Lyapunov functionals. The extinction state and the lower-density coexistence state are unstable, while the endemic snail-free state and the higher-density coexistence state are stable in probability under explicit conditions on the parameters, delays and noise intensity. At the stable coexistence equilibrium the rust persists without eradication and the density of healthy trees stays above half the carrying capacity.