## Tokyo Journal of Mathematics

### Partial Survival and Extinction of Species in Discrete Nonautonomous Lotka-Volterra Systems

Yoshiaki MUROYA

#### Abstract

In this paper, we consider the partial survival and extinction of species in model governed by the following discrete model of nonautonomous Lotka-Volterra type: $\left\{ \begin{array}{@{}ll} N_i(p+1) = N_i(p) \exp\{ c_i(p)-\displaystyle \sum_{j=1}^n \displaystyle \sum_{l=0}^m a_{ij}^l(p) N_j(p-k_l) \}\,, \quad p \geq 0\,, \quad 1 \leq i \leq n\,, \\ N_i(p) = N_{ip} \geq 0\,, \quad p \leq 0\,, \quad \mbox{and} \quad N_{i0}>0\,, \quad 1 \leq i \leq n\,, \\ \end{array} \right.$ where each $c_i(p)$ and $a_{ij}^l(p)$ are bounded for $p \geq 0$ and $\displaystyle \sum_{l=0}^m (\displaystyle \inf_{p \geq 0} a_{ii}^l(p))>0\,,\quad \ a_{ij}^l(p) \geq 0\,, \quad i \leq j \leq n\,, \quad 1 \leq i \leq n\,, \quad k_l \geq 0\,, \quad 0 \leq l \leq m\,.$ To the above discrete system, we extend results on the {\it principle of competitive exclusion} in nonautonomous Lotka-Volterra differential systems which has been established by Shair Ahmad (1999, {\it Proceedings of the American Mathematical Society} {\bf 127}, 2905--2910), that is, if the coefficients satisfy certain inequalities, then any solution with positive components at some point will have all of its last $n-1$ components tend to zero, while the first one will stabilize at a certain solution of a discrete logistic equation.

#### Article information

Source
Tokyo J. Math., Volume 28, Number 1 (2005), 189-200.

Dates
First available in Project Euclid: 5 June 2009

https://projecteuclid.org/euclid.tjm/1244208288

Digital Object Identifier
doi:10.3836/tjm/1244208288

Mathematical Reviews number (MathSciNet)
MR2149632

Zentralblatt MATH identifier
1081.39014

Subjects
Primary: 34K20: Stability theory
Secondary: 92D25: Population dynamics (general)

#### Citation

MUROYA, Yoshiaki. Partial Survival and Extinction of Species in Discrete Nonautonomous Lotka-Volterra Systems. Tokyo J. Math. 28 (2005), no. 1, 189--200. doi:10.3836/tjm/1244208288. https://projecteuclid.org/euclid.tjm/1244208288

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