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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

${x}^{\text{'}}=x\left[a\left(t\right)-b\left(t\right)x\right]-\frac{c\left(t\right)xy}{m\left(t\right)y+x},\phantom{\rule{2.em}{0ex}}{y}^{\text{'}}=y\left[-d\left(t\right)+\frac{f\left(t\right)x}{m\left(t\right)y+x}\right],$

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 stability of ODE 92D25 Population dynamics (general) 34C25 Periodic solutions of ODE 34C29 Averaging method