Nonautonomous Lotka–Volterra Systems, II

✍ Scribed by Ray Redheffer

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 526 KB
• Volume: 132
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.1996.0168

✦ Synopsis

Being a continuation of Part I, this paper needs only a brief introduction. For i, j=1, ..., n the letters a i and b ij denote continuous real-valued functions of t and u i satisfies

In Sections 1 6 we assume

where \ is a positive constant. These inequalities collectively will be called the A L inequalities. They are die to Ahmad and Lazer , except that in their work the hypothesis b ii \ is replaced by the more subtle condition

This is used in Section 7, where we also keep the multipliers d i that were present in Ahmad and Lazer's original formulation. In the counterexample

The A L problem is here defined as follows: If u, v are two positive solutions of E, does it follow necessarily that u(t)&v(t) Ä 0 as tÄ ? Although this problem has been in circulation for several years, the answer is still unknown for n=2, 3, 4, 5. For obvious reasons, it would be desirable to base a affirmative solution on the weaker condition b ii #, article no. 0168