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Zbl 1032.34044
Fan, Meng; Wang, Qian; Zou, Xingfu
Dynamics of a non-autonomous ratio-dependent predator-prey system.
(English)
[J] Proc. R. Soc. Edinb., Sect. A, Math. 133, No.1, 97-118 (2003). ISSN 0308-2105; ISSN 1473-7124/e

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.
[Pei-xuan Weng (Guangzhou)]
MSC 2000:
*34D05 Asymptotic stability of ODE
92D25 Population dynamics
34C25 Periodic solutions of ODE
34C29 Averaging method

Keywords: predator-prey system; ratio-dependent functional response; stability; periodic solution; almost-periodic solution

Cited in: Zbl 1215.34104

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