Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system

Abstract

Abstract. The recent broad interest on ratio-dependent based predator functional response calls for detailed qualitative study on ratio-dependent predator-prey differential systems. A first such attempt is documented in the recent work of Kuang and Beretta(1998), where Michaelis-Menten-type ratio-dependent model is studied systematically. Their paper, while contains many new and significant results, is far from complete in answering the many subtle mathematical questions on the global qualitative behavior of solutions of the model. Indeed, many of such important open questions are mentioned in the discussion section of their paper.Through a simple change of variable, we transform the Michaelis-Menten-type ratiodependent model to a better studied Gause-type predator-prey system. As a result, we can obtain a complete classification of the asymptotic behavior of the solutions of the MichaelisMenten-type ratio-dependent model. In some cases we can determine how the outcomes depend on the initial conditions. In particular, open questions on the global stability of all equilibria in various cases and the uniqueness of limit cycles are resolved. Biological implications of our results are also presented.

Population models with singular equilibrium

“…Thus we have proved the existence of an elliptic sector in model (14), which was overlooked in the original analysis of Hwang and Kuang [29]. This dynamical regime corresponds to the initial growth of the total population size, and the eventual population extinction can be preceded by a relatively normal population evolution.…”

“…We note that if δ > 1 then, from (14), d dt (x + y) < 0 follows. Thus, we shall consider only the case when 0 < δ < 1.…”

“…Solutions of (14) are considered in R 2 + . Using the transformation v = x/y, Hwang and Kuang reduced model (14) to a Gause-type system which was completely studied.…”

Rich dynamics of a ratio-dependent one-prey two-predators model

“…It turns out that, in many aspects, the ratio-dependent models actually provide the richest dynamics, while the prey-dependent ones provide the least in dynamical behavior , Jost et al(1999), Hsu et al(2001) and Berezovskaya et al(2001)). Since the ratio-dependent form use one less parameter than the Beddington-DeAngelis one, it is thus more appealing for both analytical and experimental applications (Jost and Arditi(2000)).…”

“…The following Michaelis-Menten type ratio-dependent predator-prey system was studied extensively , Jost et al(1999), Hsu et al(2001), Berezovskaya et al(2001), and the references cited))…”