Positive Steady-State Solutions of a Competing Reaction-Diffusion System

✍ Scribed by W.H. Ruan; C.V. Pao

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 639 KB
• Volume: 117
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.1995.1059

✦ Synopsis

This paper is concerned with positive steady-state solutions of a coupled reaction-diffusion system which models the coexistence problem of two competing species in ecology. The main purpose of the paper is to determine the set 1 of natural growth rate (\left(r_{1}, r_{2}\right)) of the two competing species so that the coupled system possesses positive solutions. It is shown that (A) is a connected unbounded region in (\mathbf{R}{+}^{2}) whose boundary consists of two monotone nondecreasing curves (r{1}=H_{2}\left(r_{2}\right)) and (r_{2}=H_{1}\left(r_{1}\right)). For every (\left(r_{1}, r_{2}\right)) inside (A) the coupled system has positive solutions and for (\left(r_{1}, r_{2}\right)) outside (A) there exists no positive solution. The functions (H_{1}\left(r_{1}\right)) and (H_{2}\left(r_{2}\right)) are constructed in terms of the limit of the corresponding timedependent solution with a specific initial function. I 1995 Academic Press, Jse