Abstract
The integrable Lotka-Volterra (LV) system stands for a prey-predator model in mathematical biology. The discrete LV (dLV) system is derived from a time-variable discretization of the LV system. The solution to the dLV system is known to be represented by using the Hankel determinants. In this paper, we show that, if the entries of the Hankel determinants become m-step Fibonacci sequences at the initial discrete-time, then those are also so at any discrete time. In other words, the m-step Fibonacci sequences always arrange in the entries of the Hankel determinants under the time-variable evolution of the dLV system with suitable initial setting. Here the 2-step and the 3-step Fibonacci sequences are the famous Fibonacci and Tribonacci sequences, respectively. We also prove that one of the dLV variables converges to the ratio of two successive and sufficiently large m-step Fibonacci numbers, for example, the golden ratio in the case where m=2, as the discrete-time goes to infinity. Some examples are numerically given.
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