A bifurcation path to chaos in a time-delay fisheries predator–prey model with prey consumption by immature and mature predators

Stage-structure models have been extensively applied in predator–prey systems. In this paper, we consider an application to fisheries. We assume that there is a single prey population and a predator population that can be separated by reproduction ability into an immature and a mature stage, with a time delay for the immature to mature transition. Our model includes the new assumption that both predators are able to hunt the same prey, although at different rates. It is proved that the system has three nonnegative equilibrium points, namely, a trivial point with all populations zero, a predator-free equilibrium point, and a coexistence equilibrium point with all three populations non-zero. It is proved that the trivial equilibrium point is always unstable, that the predator-free equilibrium point is stable if and only if the coexistence equilibrium point does not exist, and that the coexistence point can either be stable for all time delays or become unstable if a Hopf bifurcation exists at a critical time delay. Numerical simulations show that the behavior of the system can become extremely complicated as the time delay is increased, with the long-time behavior changing from a stable coexistence equilibrium, to a limit cycle with one local maximum and minimum per cycle (Hopf bifurcation), to limit cycles with an increasing number of local maxima and minima per cycle, and finally to chaotic-type solutions.