Periodic Competitive Planar Systems

Further analysis of the system (3.2) requires understanding the dynamics generated by the Poincare map in the interior of Q. For general periodic systems, such an understanding is beyond current knowledge. Fortunately, system (3.2) - in addition to being two-dimensional - has the property that it is competitive. A beautiful theory for such systems has recently been constructed. The next section is devoted to the principal result of this theory. 

The study of mathematical models of competition has led to the discovery of some very beautiful mathematics. This mathematics, often referred to as monotone dynamical systems theory, was largely developed by M. W. Hirsch [Hil Hi3], although others have made substantial contributions as well. In this section we describe a result that was first obtained in a now classical paper of DeMottoni and Schiaffino [DS] for the special case of periodic Lotka-Volterra systems. Later, it was recognized by Hale and Somolinos [HaS] and Smith [S4 S5] that the arguments in [DS] hold for general competitive and cooperative planar periodic systems. The result says that every bounded solution of such a system converges to a periodic solution that has the same period as the differential equation. 

We proceed to state and prove this result. Consider the system 

The requirement (4.2) means that (4.1) is a competitive system - an increase in X2 has a negative effect on the growth rate of x and vice versa. The system is said to be a cooperative system if the reverse inequalities hold in (4.2). Our interest here will be in the competitive case since (3.2) satisfies (4.2) in 2, as is easily checked. However, the cooperative case is 

ye then we write x y whenever x, yj holds for / = 1,2. We write X Ky whenever X yi and X2 y2- If one imagines the Xi axis pointing east and the X2 axis pointing north, then x y means that y lies to the northeast of x and x - k y means that y lies to the southeast of x. 

---

In Remark 6.1 we have used < rather than because K is the usual positive cone. A consequence of these remarks is that planar cooperative or competitive systems do not have periodic orbits. For example, consider a planar cooperative system and let P be an arbitrary point on a periodic orbit. Impose the standard two-dimensional coordinate system at P. The orbit cannot be tangent to both the x and the y axes at P and so must have points in both quadrants II and IV (the sets unordered with respect to P), since points in quadrant I or III would be ordered. The orbit cannot pass through P again and cannot have points in quadrants I... 

---