Global dynamics of a predator-prey model with time delay and stage structure for the prey. (English) Zbl 1223.34115

Summary: We consider the following delay differential system
\[ \begin{aligned} & \dot x_1(t)=ax_2(t)-r_1x_1(t)-bx_1(t),\\ & \dot x_2(t)=bx_1(t)-r_2x_2(t)-b_1x^2_2(t)-\frac{a_1x_2(t)y(t)}{1+mx_2(t)}\,,\\ & \dot y(t)=\frac{a_2x_2(t-\tau)y(t-\tau)}{1+mx_2(t-\tau)}-ry(t).\end{aligned} \]
By analyzing the characteristic equations, the local stability of each equilibrium of the system is discussed, and the existence of a Hopf bifurcation at the coexistence equilibrium is established. By means of the persistence theory, it is proven that the system is permanent if the coexistence equilibrium exists. By using Lyapunov functionals and LaSalle’s invariant principle, it is shown that the trivial equilibrium is globally stable when both the predator-extinction equilibrium and the coexistence equilibrium are not feasible, and that the predator-extinction equilibrium is globally asymptotically stable if the coexistence equilibrium does not exist, and sufficient conditions are derived for the global stability of the coexistence equilibrium. Numerical simulations are carried out to illustrate the main theoretical results.


34K60 Qualitative investigation and simulation of models involving functional-differential equations
34K25 Asymptotic theory of functional-differential equations
34K18 Bifurcation theory of functional-differential equations
34K20 Stability theory of functional-differential equations
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