Global qualitative analysis of a ratio-dependent predator-prey system

Abstract

Ratio-dependent predator-prey models are favored by many animal ecologists recently as more suitable ones for predator-prey interactions where predation involves searching process. However, such models are not well studied in the sense that most results are local stability related. In this paper, we consider the global behaviors of solutions of a ratio-dependent predator-prey systems. Specifically, we shall show that ratio dependent predator-prey models are rich in boundary dynamics, and most importantly, we shall show that if the positive steady state of the so-called Michaelis-Menten ratio-dependent predator-prey system is locally asymptotically stable, then the system has no nontrivial positive periodic solutions. We also give sufficient conditions for each of the possible three steady states to be globally asymptotically stable. We note that for ratio-dependent systems, in general, local asymptotic stability of the positive steady state does not even guarantee the so-called persistence of the system, and therefore does not imply global asymptotic stability. To show that the system has no nontrivial positive periodic solutions. we employ the so-called divergency criterion for the stability of limit cycles in planar systems and some critical transformations.

Continuous Traveling Waves for Prey-Taxis

“…h(v, n) = μv n+dv , has attracted the attention of researchers for its rich behaviors at (v, n) = (0, 0) (Kuang and Beretta, 1998;Jost et al, 1999). This singular function causes a singularity for a prey and predator local dynamics, which prevents a linear stability analysis at the leading edge of a wave.…”

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

“…Specifically, it will not produce the paradox of biological control and the so-called paradox of enrichment. It also allows mutual extinction as a possible outcome of a given predator-prey interaction (Kuang and Beretta (1998), Jost et al (1999)). …”

“…In this paper we study following ratio-dependent predator-prey system which was discussed in Kuang and Beretta (1998) (see also Jost et al (1999)):…”

“…Clearly, with this assumption, both F and G are continuous on the closure of R 2 + where R 2 + = {(u, y)| u > 0, y > 0}. Kuang and Beretta (1998) presented some global qualitative analysis of solutions of system (1.2). The authors showed that ratio-dependent predator-prey models are rich in boundary dynamics.…”

“…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.…”