Stability and approximate analytic solutions of the fractional Lotka-Volterra equations for three competitors

This paper studies the fractional Lotka-Volterra equations for three competitors, since the fractional derivatives possess the properties of good memory and have great biological significance. First of all, the equilibrium points and asymptotic stability for the equations are studied by the stability analysis method. As expected, the fractional-order differential equations are, at least, as stable as their integer-order counterpart. Second, some approximate analytic solutions for this systems are obtained by the stability analysis method and the homotopy perturbation method, which are expressed in the form of the Mittag-Leffler function. The results show that it takes less time for the population to get close to the equilibrium point as the time derivatives increase. Comparing with the classical ones, the fractional Lotka-Volterra equations, no matter whether the fractional derivatives have big or small order, if both take more time, even multiplied time, for the system to reach the equilibrium point, then that may explain the memory properties. Furthermore, some numerical analyses are carried out and verify the theoretical analysis.

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