Nonautonomous Lotka–Volterra Systems, I

✍ Scribed by Ray Redheffer

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 700 KB
• Volume: 127
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.1996.0081

✦ Synopsis

This paper is concerned with the Lotka Volterra system E: u* i =u i \ a i (t)& : n j=1 b ij (t) u j + , t>t 0 ; u i (t 0 )>0 for i=1, ..., n, where a i and b ij are continuous real-valued functions of the real variable t. Each u i in E satisfies an equation of the form u* i =u i , i , where , i (t) is continuous. Hence u i >0 holds trivially on the interval of existence of u. If u i admits a bound on (t 0 , T) independent of T, then u exists on (t 0 , ). The equation is interpreted for t # R by setting t 0 =& and replacing the condition u(t 0 )>0 by u(t 1 )>0 for some value t 1 . Unless the hypothesis t # R is mentioned explicitly, t>t 0 and t 0 # R.

We use the notation a=vector(a i )=a(t), B=matrix(b ij )=B(t), u=vector(u i )=u(t).

The letters c, d, d denote positive vectors of R n partially ordered as follows:

Inequalities between vector-valued functions are interpreted accordingly; in particular, inf u>0 means inf u i (t)>0 for i=1, ..., n. For any real-valued function , we define

, & (t)=min(,(t), 0), , + (t)=max(,(t), 0).

(1)

If a condition involving i, j or t is stated without further explanation, it holds for i, j=1, ..., n and for t>t 0 . To simplify the statement, it is assumed in Theorem 1 that b ii >0, i=1, 2, ..., n.