GLOBAL QUALITATIVE-ANALYSIS OF A RATIO-DEPENDENT PREDATOR-PREY SYSTEM

Ratio-dependent predator-prey models are favored by many animal ecolog ists recently as more suitable ones for predator-prey interactions whe re predation involves searching process. However, such models are not well studied in the sense that most results are local stability relate d. In this paper, we consider the global behaviors of solutions of a r atio-dependent predator-prey systems. Specifically, we shall show that ratio dependent predator-prey models are rich in boundary dynamics, a nd most importantly, we shall show that if the positive steady state o f the so-called Michaelis-Menten ratio-dependent predator-prey system is locally asymptotically stable, then the system has no nontrivial po sitive periodic solutions. We also give sufficient conditions for each of the possible three steady states to be globally asymptotically sta ble. We note that for ratio-dependent systems, in general, local asymp totic stability of the positive steady state does not even guarantee t he so-called persistence of the system, and therefore does not imply g lobal asymptotic stability. To show that the system has no nontrivial positive periodic solutions. we employ the so-called divergency criter ion for the stability of limit cycles in planar systems and some criti cal transformations.