Dynamical analysis in a hybrid bioeconomic system with multiple time delays and strong Allee effect

Abstract

In this paper, a multiple delayed differential–algebraic prey–predator system is established, where commercial harvesting on predator and strong Allee effect in prey growth are considered. Three time delays are introduced to represent maturation delay for prey (τ1), reaction delay to changes of prey surviving environment due to strong Allee effect (τ2) and gestation delay for predator (τ3), respectively. Positivity of solutions and uniform persistence of system are discussed. In absence of time delay, existence of singularity induced bifurcation and local stability analysis are investigated due to variation of economic interest of commercial harvesting. In presence of multiple time delays, existence of Hopf bifurcation and local stability analysis are discussed by analyzing associated characteristic equation. By using new normal form of multiple delayed differential–algebraic system and center manifold theorem, properties of Hopf bifurcation are studied. Furthermore, existence of global continuation of periodic solutions bifurcating from interior equilibrium is discussed by using a global Hopf bifurcation theorem.