Abstract
In this paper, we investigate a delayed differential algebraic prey–predator system, where commercial harvesting on predator and additive Allee effect on prey are considered. A discrete time delay is utilized to represent gestation delay of the predator population. Positivity of solutions and uniform persistence of system are discussed. In the absence of time delay, by taking economic interest as a bifurcation parameter, some sufficient conditions associated with additive Allee effect and economic interest are derived to show that the proposed system undergoes singularity-induced bifurcation around the interior equilibrium. In the presence of time delay, combined dynamic effects of time delay and additive Allee effect on population dynamics are discussed in the case of positive economic interest of commercial harvesting. Existence of Hopf bifurcation and local stability switch around the interior equilibrium are studied as gestation delay crosses the critical value. Furthermore, properties of Hopf bifurcation are investigated based on the center manifold theorem and the norm form of a delayed singular system. Existence of global continuation of periodic solutions bifurcating from interior equilibrium is discussed by using a global Hopf bifurcation theorem. Numerical simulations are provided to show consistency with theoretical analysis.
Highlights
Compared with the system established in [12], an algebraic equation is introduced into system (3), which concentrates on dynamic effect of economic interest of commercial harvesting on population dynamics and provides a straightforward way to investigate complex dynamics due to variation of economic interest
By using the global Hopf bifurcation theorem for general functional delayed differential equations introduced in [28], the existence of global continuation of periodic solutions bifurcating from interior equilibrium Sw∗ will be discussed in the following part
Further computations show that γ2 = 1.0329 > 0, ι2 = –0.4127 < 0, and T = 1.4193 > 0, it follows from Theorem 4.1 that the Hopf bifurcation is supercritical, the direction of the Hopf bifurcation is τ > τ1∗d and these bifurcating periodic solutions from the interior equilibrium Sw∗ at τ1∗d are stable
Summary
It is necessary to investigate combined dynamics of time delay and additive Allee effect on population dynamics of a harvested prey–predator system with commercial harvesting. In the absence of time delay, the existence of singularity-induced bifurcation is investigated under the case of additive Allee effect on prey. Compared with the system established in [12], an algebraic equation is introduced into system (3), which concentrates on dynamic effect of economic interest of commercial harvesting on population dynamics and provides a straightforward way to investigate complex dynamics due to variation of economic interest. Compared with the work done in [12], we can investigate combined dynamic effects of time delay and additive Allee effect on population dynamics by analyzing the local stability and bifurcation phenomenon of system (3) in this paper. Which derives that system (3) with initial conditions (4) is uniformly persistent
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