Influences of the Order of Derivative on the Dynamical Behavior of Fractional-Order Antisymmetric Lotka–Volterra Systems

Abstract

This paper studies the dynamic behavior of a class of fractional-order antisymmetric Lotka–Volterra systems. The influences of the order of derivative on the boundedness and stability are characterized by analyzing the first-order and 0<α<1-order antisymmetric Lotka–Volterra systems separately. We show that the order does not affect the boundedness but affects the stability. All solutions of the first-order system are periodic, while the 0<α<1-order system has no non-trivial periodic solution. Furthermore, the 0<α<1-order system can be reduced on a two-dimensional space and the reduced system is asymptotically stable, regardless of how close to zero the order of the derivative used is. Some numerical simulations are presented to better verify the theoretical analysis.