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Dynamics of a class of nonautonomous semi-ratio-dependent predator–prey systems with functional responses. (English) Zbl 1029.34042

The authors give an in-depth study of various properties of solutions to the system

${x}^{\text{'}}=x\left[a\left(t\right)-b\left(t\right)x\right]-c\left(t,x\right)y,\phantom{\rule{1.em}{0ex}}{y}^{\text{'}}=y\left[d\left(t\right)-e\left(t\right)y/x\right],$

which contains a number of special cases arising in applications. State of the art is shown by an extensive bibliography. Under several hypotheses on the coefficients of the system, the ultimate boundedness of solutions and the permanence of the system (as defined in the paper) are proved.

By adding the assumption of periodicity (almost-periodicity) of the coefficients, the existence and the uniqueness of positive periodic (almost-periodic) solutions are proved. In addition, the considered solutions are globally asymptotically stable (as defined in the paper) in all the three cases.

##### MSC:
 34D40 Ultimate boundedness (MSC2000) 34D05 Asymptotic stability of ODE