Uniqueness and asymptotic stability properties of the critical solution of the prey-predator retarded equation model

Abstract

The challenge of modeling time-varying phenomenon using retarded equations is of great interest in mathematical sciences. This is because the non-instantaneous reaction of the state parameters is addressed. In this research, the Volterra prey/predator model system is modified by introducing time-lag functions f (t - h) into the state parameters to account for the non-instantaneous reaction of the state parameters to changes. The compactness, contraction and continuity properties of the functional on the Banach space are utilized to establish the uniqueness of the integral solution of the critical point. The asymptotic stability properties of the critical solution are investigated using the quadratic matrix equation and symmetric linear matrix inequality test. Results obtained shows that the system is asymptotically stable, if the recruitment rate of the prey is kept higher than the recruitment rate of the predator at all time in the system. This is a stronger condition for stability and sustainability of the system, compared to the stability result of the equivalent ordinary differential equation model of the system.

Keywords: Retarded system, prey/predator model, uniqueness of solution, asymptotic stability, and linear matrix inequality test

Journal of the Nigerian Association of Mathematical Physics, Volume 19 (November, 2011), pp 85 – 90