Asymptotic analysis for an extended discrete Lotka–Volterra system related to matrix eigenvalues

Abstract

The integrable discrete hungry Lotka–Volterra (dhLV) system is easily transformed to the qd-type dhLV system, which resembles the recursion formula of the qd algorithm for computing matrix eigenvalues. Some of the qd-type dhLV variables play a role for assisting the time evolution of the others. This property does not appear in the original dhLV system. In this article, we first show the existence of a centre manifold for the qd-type dhLV system. With the help of the centre manifold theory, we next investigate the local convergence of the qd-type dhLV system, and then clarify the monotonicity related to the qd-type dhLV variables at the final phase of the convergence.

Keywords:

• the discrete hungry Lotka–Volterra system
• centre manifold theory
• asymptotic behaviour

AMS Subject Classifications::

• 37L10
• 37K10
• 37J25

Acknowledgements

The authors would like to thank the reviewer for his/her careful reading and beneficial suggestions. This was partially supported by Grants-in-Aid for Young Scientist (B) No. 20740064 and Scientific Research (C) No. 23540163 of Japan Society for the Promotion of Science (JSPS).