Persistence and global stability for discrete models of nonautonomous Lotka-Volterra type.
The paper is concerned with the following discrete model of Lotka-Volterra type \[ \begin{gathered} N_{i}(p+1)=N_{i}(p)\exp\left\{ c_{i}(p)-a_{i}(p)N_{i}(p)-\sum_{j=1}^{n} \sum_{l=0}^{m}a^{l}_{ij}(p)N_{i}(p-k_{l})\right\}, \;p\geq 0,\\ N_{i}(p)=N_{ip}\geq0\;\text{ for} \;p\leq0,\;\text{and} \;N_{i0}>0, \;\;\;\;1\leq i\leq n, \end{gathered} \] where each \(c_{i}(p)\), \(a_{i}(p)\), \(a^{l}_{ij}(p)\) are bounded for \(p\geq0\); \(\inf_{p\geq0}a_{i}(p)>0\), \(a^{0}_{ii}(p)\equiv0\) for \(1\leq i\leq n\); \(a^{l}_{ij}(p)\geq0\) for \(1\leq i\leq j\leq n\), \(0\leq l\leq m\); \(k_{0}=0\), integers \(k_{l}\geq0\) for \(1\leq l\leq m\). By using the technique developed recently by \textit{S. Ahmad} and \textit{A. C. Lazer} [Nonlinear Anal., Theory Methods Appl. 40 A, 37--49 (2000; Zbl 0955.34041)] for competitive Lotka-Volterra differential system the author establishes similar averaged conditions for the persistence and global asymptotic stability of the above discrete system. These conditions depend on the infimum and supremum bounds of the coefficients \(a_{i}(p)\), \(a^{l}_{ij}(p)\), and the lower and upper averages \(m[c_{i}]=\liminf_{q\to\infty}\{A(c_{i},p_{1},p_{2}): p_{2}-p_{1}\geq q\}\), \(M[c_{i}]=\limsup_{q\to\infty}\{A(c_{i},p_{1},p_{2}): p_{2}-p_{1}\geq q\}\), where \(A(c_{i},p_{1},p_{2})=(p_{2}-p_{1})^{-1} \sum_{p=p_{1}}^{p_{2}-1}c_{i}(p)\).
discrete nonautonomous Lotka-Volterra model
scientific article; zbMATH DE number 1866580
scientific article