An extended Fibonacci sequence associated with the discrete hungry Lotka-Volterra system

The integrable hungry Lotka-Volterra (hLV) system stands for a prey-predator model in mathematical biology. The discrete-time hLV (dhLV) system is derived from a time discretization of the hLV system. The solution to the dhLV system is known to be represented by using the Casorati determinant. In this paper, we show that if the entries of the Casorati determinant become an extended Fibonacci sequence at the initial discrete time, then those are also an extended Fibonacci sequence at any discrete time. In other words, the extended Fibonacci sequence always appears in the entries of the Casorati determinant under the time evolution of the dhLV system with suitable initial setting. We also show that one of the dhLV variables converges to the ratio of two successive extended Fibonacci numbers as the discrete time goes to infinity.