## Periodicity in a ratio-dependent predator-prey system with stage structure for predator.(English)Zbl 1103.34060

The author studies the following delayed ratio-dependent predator-prey model with periodic coefficients and stage structure of the predator \begin{aligned} \frac{dx(t)}{dt} &=x(t)(r(t)-a(t)x(t))-\frac {b(t)x(t)y_2(t)}{my_2(t)+x(t)}, \\ \frac{dy_1(t)}{dt} & = \frac{c(t)x(t-\tau(t))y_2(t-\tau(t))}{my_2(t-\tau(t))+x(t-\tau(t))}-(D(t)+\nu_1(t))y_1(t), \tag{*} \\ \frac{dy_2(t)}{dt} &= D(t)y_1(t)-\nu_2(t)y_2(t), \end{aligned} where $$x(t)$$ is the density of the prey, $$y_1(t)$$ is the density of the immature predator, and $$y_2(t)$$ is the density of the mature predator. $$r(t)$$, $$a(t)$$, $$b(t)$$, $$c(t)$$, $$D(t)$$, $$\nu_1(t)$$, and $$\nu_2(t)$$ are all continuously positive periodic functions with period $$\omega$$. The initial conditions for system (*) take the form \begin{aligned} & x(\theta)=\phi(\theta)\geq 0,\qquad y_1(\theta)=\psi_1(\theta)\geq 0,\qquad y_2(\theta)=\psi_2(\theta)\geq 0, \\ & \theta\in [-\tau,0], \quad \phi(0)>0,\quad \psi_1(0)>0,\quad \psi_2(0)>0, \end{aligned}\tag{**} where $$\tau=\max_{0\leq t\leq \omega}\{\tau(t)\}$$, $$\Phi=(\phi(\theta),\psi_1(\theta),\psi_2(\theta))\in C([-\tau,0], \mathbb{R}^3_{+0})$$ with $$\mathbb{R}^3_{+0}=\{(x_1,x_2,x_3)$$ $$:x_i\geq 0, i=1,2,3\}$$. It is easy to see that the solution of (*) with initial conditions (**) exists and remains positive for all future time. First, based on the coincidence degree theory, some sufficient conditions are established for the global existence of positive periodic solutions of (*) with initial conditions (**). Then, under the assumption that $$\tau(t)\equiv 0$$, the global attractivity of the positive periodic solutions is studied. By constructing a suitable Lyapunov function, the author also provides sufficient conditions guranteeing the existence of a unique globally attractive positive periodic solution of system (*).

### MSC:

 34K13 Periodic solutions to functional-differential equations 34K20 Stability theory of functional-differential equations 92D25 Population dynamics (general)
Full Text: