On Direction Of Hopf Bifurcation In Four-Dimensional Competitive Lotka-Volterra System

In this paper,an explicit formula of the first order focus value of a class of high dimensional competitive Lotka-Volterra(LV)systems is given.Using the formula,we study the direction of Hopf bifurcation in four dimensional competitive LV system,and construct examples of four dimensional competitive LV system which can be easily verified to have at least one limit cycle by manual calculation.Moreover,we show a close relationship between four dimensional competitive LV system with equal intrinsic growth rates and three dimensional LV system,for example,their close orbits are corresponding one by one,the positive equilibria have the same stability,and sign of the first order focus values of the equilibrium do not change.By using available 3-D competitive LV systems with 2-4limit cycles,we obtain 4-D competitive LV systems with 2-4 limit cycles.