Three-Dimensional Competitive Systems

If mifipl) 1, then mif p e)-e) 1 for sufficiently small e. Since the hypotheses imply that A2 ai/(w i/(/ 2)-l), it follows that A2 — Using the first comparison forxj, the equations 

We interrupt the analysis of the inhibitor model in order to present some theorems on competitive systems which are needed in the analysis. Consider the system 

Such systems are said to be competitive (as noted in Chapter 1). When (6.1) represents a population growth equation, (6.2) indicates that an increase in the size of one component inhibits the growth of the others. Such a system is not necessarily order-preserving, so the theory of monotone systems does not apply. However, if solutions exist for all time and if one runs time backwards (more correctly, if one makes the change of variables / = —t and regards t as time ), then the corresponding dynamical system is monotone. More formally, the system 

Remark 6.1. If two points are ordered at time t, then the trajectories through them have been ordered for all previous time. That is, a implies that 7t(x, t) T (y, t) for all t 0. 

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A major difference between competitive and cooperative systems is that cycles may occur as attractors in competitive systems. However, three-dimensional systems behave like two-dimensional general autonomous equations in that the possible omega limit sets are similarly restricted. Two important results are given next. These allow the Poincare-Bendix-son conclusions to be used in determining asymptotic behavior of three-dimensional competitive systems in the same manner used previously for two-dimensional autonomous systems. The following theorem of Hirsch is our Theorem C.7 (see Appendix C, where it is stated for cooperative systems). 

Theorem 7.1 guarantees the coexistence of both the Xi and X2 populations when Ec exists. However, it does not give the global asymptotic behavior. The further analysis of the system is complicated by the possibility of multiple limit cycles. Since this is a common difficulty in general two-dimensional systems, it is not surprising that such difficulties occur in the analysis of three-dimensional competitive systems. 

SWl] H. Smith and P. Waltman (1987), A classification theorem for three dimensional competitive systems, Journal of Differential Equations 70 325-32. 

Hi4] M. Hirsch (1990), Systems of differential equations that are competitive or cooperative, IV Structural stability in three dimensional systems, SIAM Journal on Mathematical Analysis 21 1225-34. 

The observed black-eye patterns are not understood. They may arise from a competition among the increasing number of modes that become unstable as the system is driven far beyond the onset of patterns. For example, the second harmonic modes may develop a resonant interaction with the fundamental modes, leading to the black eyes. Alternatively, three-dimensional effects could lead to the black-eye patterns. 

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