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Coexistence for systems governed by difference equations of Lotka- Volterra type. (English) Zbl 0638.92019
The authors consider the question of coexistence and survival of species in models governed by a system of Lotka-Volterra difference equations \[ x_ i'=x_ i \exp \{r_ i-\sum^{n}_{j=1}a_{ij}x_ j\}\quad (i=1,...,n). \] Here \(x=(x_ 1,...,x_ n)\) T is the population vector at one generation and x’ is the corresponding vector at the next generation. The system is said to be permanent if there is a compact set M in the interior \(int({\mathbb{R}}\) \(n_+)\) of \({\mathbb{R}}\) \(n_+\) such that for every initial value in \(int({\mathbb{R}}\) \(n_+)\) the orbit enters and remains in M. The authors assume that the system is dissipative in the sense that there is a bounded global attractor. Define \(f_ i(x)=r_ i- \sum^{n}_{j=1}a_{ij}x_ j\). Then the authors prove the following sufficient condition:
Theorem. If the system is dissipative and if there are positive real numbers \(p_ i\) such that \(\sum^{n}_{i=1}p_ if_ i(x\) \(*)>0\) for each equilibrium point x * in \(\partial {\mathbb{R}}\) \(n_+\), then the system is permanent.
Necessary conditions for dissipativity and for permanence are also discussed. Furthermore, concepts of weak persistence, robust dissipative, etc., are introduced and analyzed. Some special results are found for the case \(n=3\).
Reviewer: K.Cooke

MSC:
92D25 Population dynamics (general)
39A10 Additive difference equations
39A12 Discrete version of topics in analysis
92D40 Ecology
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