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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]-c(t)xy m(t)y+x,y ' =y[-d(t)+f(t)x m(t)y+x],

where, instead of the traditional prey-dependent functional response x m+x, the functional response is 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.

34D05Asymptotic stability of ODE
92D25Population dynamics (general)
34C25Periodic solutions of ODE
34C29Averaging method