Limit cycles for competitor–competitor–mutualist Lotka–Volterra systems

It is known that a limit cycle (or periodic coexistence) can occur in a competitor–competitor–mutualist Lotka–Volterra system { x ̇ 1 = x 1 ( r 1 − a 11 x 1 − a 12 x 2 + a 13 x 3 ) , x ̇ 2 = x 2 ( r 2 − a 21 x 1 − a 22 x 2 + a 23 x 3 ) , x 3 ̇ = x 3 ( r 3 + a 31 x 1 + a 32 x 2 − a 33 x 3 ) , where r i , a i j are positive real constants [X. Liang, J. Jiang, The dynamical behavior of type- K competitive Kolmogorov systems and its applications to 3-dimensional type- K competitive Lotka–Volterra systems, Nonlinearity 16 (2003) 785–801]. In this paper, we shall construct an example with at least two limit cycles, and furthermore, we will show that the number of periodic orbits (and hence a fortiori of limit cycles) is finite. It is also shown that, contrary to three-dimensional competitive Lotka–Volterra systems, nontrivial periodic coexistence does occur even if none of the three species can resist invasion from either of the others. In this case, new amenable conditions are given on the coefficients under which the system has no nontrivial periodic orbits. These conditions imply that the positive equilibrium, if it exists, is globally asymptotically stable.