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Periodic solutions for a semi-ratio-dependent predator-prey system with Holling IV functional response. (English) Zbl 1201.34073
The authors consider nonautonomous semi-ratio-dependent predator-prey systems $$\dot x_1 = (r_1 (t)-a_{11}(t)x_1)x_1 - f(t,x_1)x_2, \quad \dot x_2= \left ( r_2 (t)-a_{21} (t)\frac{x_2}{x_1} \right ) x_2$$ with initial conditions $x_i (0) >0$, $i=1,2$ and where the nonmonotonic functional response $f(t,x_1)$ is given by $f(t,x_1) = \frac{a_{12}(t)x_1}{m^2 + nx_1 + x_1^2}$. Here, $x_1$ and $x_2$ denote the density of the prey and the predator, respectively, $m \neq 0$, $n \geq 0$ and $r_i (t)$, $a_{ij} (t)$, $i,j= 1,2,$ are continuous, positive and $\omega$-periodic functions. First, by using a continuation theorem of coincidence degree theory, the authors discuss the existence of positive $\omega$-periodic solutions of the previous system. Then, by constructing a Lyapunov function, they establish a sufficient condition for the uniqueness and global asymptotic stability of such positive periodic solution.

##### MSC:
 34C60 Qualitative investigation and simulation of models (ODE) 34C25 Periodic solutions of ODE 92D25 Population dynamics (general) 47N20 Applications of operator theory to differential and integral equations 34D20 Stability of ODE
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##### References:
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