Patterns of boundedness of a rational system in the plane

Abstract

We investigate the boundedness character of non-negative solutions of a rational system in the plane. The system contains 10 parameters with non-negative real values and consists of 343 special cases, each with positive parameters. In 342 out of the 343 special cases, we establish easily verifiable necessary and sufficient conditions, explicitly stated in terms of 10 parameters, which determine the boundedness character of solutions of the system. In the remaining special case, we conjecture the boundedness character of solutions. It is interesting to note that this special case can be transformed to the well-known May's Host-Parasitoid model.

On a second-order rational difference equation and a rational system

“…with positive parameters and nonnegative initial conditions. There is a conjecture in [3], originating in [10], which claims that for each solution of the system #(6, 25) the sequence {x n } ∞ n=0 arising from the x component of the solution is bounded. The system #(6, 25) is now the only rational system in the plane for which the boundedness character has not been established yet.…”