Persistence in predator-prey systems with ratio-dependent predator influence

✍ Scribed by H.I. Freedman; R.M. Mathsen

• Publisher: Springer
• Year: 1993
• Tongue: English
• Weight: 567 KB
• Volume: 55
• Category: Article
• ISSN: 1522-9602
• DOI: 10.1016/s0092-8240(05)80190-9

✦ Synopsis

Predator-prey models where one or more terms involve ratios of the predator and prey populations may not be valid mathematically unless it can be shown that solutions with positive initial conditions never get arbitrarily close to the axis in question, i.e. that persistence holds. By means of a transformation of variables, criteria for persistence are derived for two classes of such models, thereby leading to their validity. Although local extinction certainly is a common occurrence in nature, it cannot be modeled by systems which are ratio-dependent near the axes.

1. Introduction. Continuous models, usually in the form of differential equations, have formed a large part of the traditional mathematical ecology literature. In order to carry out the required mathematical analyses, they are assumed to be sufficiently smooth over their entire domain of definition so that solutions to initial value problems exist uniquely and are continuable for all positive time. In particular, in the case of predator-prey systems the domain must include the prey and predator axes, since one must know the dynamics of each population in the absence of the other (see, e.g. Freedman, 1980; Freedman and Waltman, 1984 and refs therein). Siinilar statements can be made with respect to competition models, mutualism models or models which incorporate combinations of these (Freedman, 1980;Freedman and Waltman, 1985).

Recently, however, several manuscripts have been written, and more are likely to appear, which incorporate terms involving the ratio of prey-topredator or predator-to-prey. These terms have appeared in place of predator functional response terms or in the predator growth terms (Arditi and