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Dynamics of a non-autonomous ratio-dependent predator-prey system. (English) Zbl 1032.34044
The authors consider the Lotka-Volterra-type predator-prey model with Holling type-II functional response $x'=x[a(t)-b(t)x]-\frac{c(t)xy}{m(t)y+x},\qquad y'=y[-d(t)+\frac{f(t)x}{m(t)y+x}],$ where, instead of the traditional prey-dependent functional response $$\frac{x}{m+x}$$, the functional response is $$\frac{x/y}{m+x/y}$$ is given, which is a ratio-dependent response. Assume that $$a,b,c,d,f,m$$ are bounded continuous functions. Some properties such as positive invariance, permanence, nonpersistence and globally asymptotic stability for the given system are discussed. If $$a,b,c,d,f,m$$ are periodic or almost-periodic, the existence, uniqueness and stability of a positive periodic solution or a positive almost-periodic solution are also investigated. The methods used in this paper are comparison method, coincidence degree theory and Lyapunov function.

##### MSC:
 34D05 Asymptotic properties of solutions to ordinary differential equations 92D25 Population dynamics (general) 34C25 Periodic solutions to ordinary differential equations 34C29 Averaging method for ordinary differential equations
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