Zhanyuan Hou

Fixed point global attractors and repellors in competitive Lotka–Volterra systems

Zeeman and Zeeman [E.C. Zeeman and M.L. Zeeman, From local to global behavior in competitive Lotka–Volterra systems, Trans. Amer. Math. Soc. 355 (2003), pp. 713–734] show that if a strongly competitive Lotka–Volterra system (i) has a unique interior fixed point p and (ii) the carrying simplex Σ lies below (above) the strongly balanced tangent plane to Σ at p then the system has no periodic orbits and p is a global attractor (repellor) relative to Σ. Condition (ii) is then translated into the definiteness of a certain quadratic function on the tangent plane, which is equivalent to the definiteness of an (N − 1) × (N − 1) real symmetric matrix that can be computed. Here we adapt these methods to show that the above conclusions are still true without the assumption (i). Hence, our results apply to globally attracting or repelling fixed points on the boundary, as well as in the interior, of . Moreover, the algebraic condition for global attraction also implies global asymptotic stability of the fixed point. We also show that the global attraction holds not just relative to Σ, but also relative to the interior of the first quadrant.

Permanence criteria for Kolmogorov systems with delays

In this paper, a class of Kolmogorov systems with delays are studied. Sufficient conditions are provided for a system to have a compact uniform attractor. Then Jansens result (J. Math. Biol. Vol. 25 (1987) 411-422) for autonomous replicator and Lotka-Volterra systems has been extended to delayed nonautonomous Kolmogorov systems with periodic or autonomous Lotka-Volterra subsystems. Thus, simple algebraic conditions are obtained for partial permanence and permanence. An outstanding feature of all these results is that the conditions are irrelevant of the size and distribution of the delays.

Asymptotic behaviour and bifurcation in competitive Lotka–Volterra Systems

Abstract A conjecture about global attraction in autonomous competitive Lotka–Volterra systems is clarified by investigating a special system with a circular matrix. Under suitable assumptions, this system meets the condition of the conjecture but Hopf bifurcation occurs in a particular instance. This shows that the conjecture is not true in general and the condition of the conjecture is too weak to guarantee global attraction of an equilibrium. Sufficient conditions for global attraction are also obtained for this system.