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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

x ˙ 1 =(r 1 (t)-a 11 (t)x 1 )x 1 -f(t,x 1 )x 2 ,x ˙ 2 =r 2 (t)-a 21 (t)x 2 x 1 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 )=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, m0, n0 and r i (t), a ij (t), i,j=1,2, are continuous, positive and ω-periodic functions.

First, by using a continuation theorem of coincidence degree theory, the authors discuss the existence of positive ω-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:
34C60Qualitative investigation and simulation of models (ODE)
34C25Periodic solutions of ODE
92D25Population dynamics (general)
47N20Applications of operator theory to differential and integral equations
34D20Stability of ODE
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