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Global analysis in some delayed ratio-dependent predator-prey systems. (English) Zbl 0946.34061
This paper contains the boundedness of solutions, permanence, and the global stability of the boundary equilibrium \((k,0)\) for the following delayed systems: \[ \begin{cases} x'= ax(1-{x\over k})- cxy/(my+ x),\\ y'= y(-d+ fx(t- \tau)/(my(t- \tau)+ x(t- \tau)));\end{cases}\tag{1} \] \[ \begin{cases} x'= ax(1-{x\over k})- cv(x) y,\\ y'=dy(1- fy(t- \tau)/x(t- \tau)).\end{cases}\tag{2} \] Sufficient conditions are included for the positive equilibrium (when it exists) of system (1) to be globally asymptotically stable. The authors obtain an explicitly expressed region of attraction for the positive equilibrium of system (2) under suitable assumptions. The paper ends with a brief discussion which includes local stability results for the positive equilibrium of system (1) and (2).
Reviewer: R.S.Dahiya (Ames)

MSC:
34K12 Growth, boundedness, comparison of solutions to functional-differential equations
92D25 Population dynamics (general)
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