Hirsch (1988) proved that if the origin is a repeller, then there exists a carrying simplex which attracts all nontrivial orbits for (3.32) and it is homeomorphic to probability simplex by radial projection. Note that dissipation implies that the infinity is also a repeller.
Smith (1986) investigated C2 diffeomorphisms T on the nonnegative orthant K which possesses the properties (see the hypotheses in Smith (1986)) of the Poincare map induced by C2 strong competition system
= XiFi(t; xi, x2, • • •, xn), Xi >0, ¿ = 1,2,---, n, (3.33)
dt where Fi is 2n-periodic in t, Fi(t; 0) > 0, and (3.33) has a globally attracting 2n-periodic solution on each positive coordinate axis. This implies that the origin is a repeller for T and it has a global attractor r. He proved that the boundaries of the repulsion domain of the origin and the global attractor relative to the nonnegative orthant are a compact unordered invariant set homeomorphic to the probability simplex by radical projection. He conjectured both boundaries coincide, serving as a unique carrying simplex. Introducing a mild additional restriction on T, which is generically satisfied by the Poincare map of the competitive Kolmogorov system (3.33), Wang and Jiang (2002) proved this conjecture and that the unstable manifold of m—periodic point of T is contained in this carrying simplex. Diekmann, Wang and Yan (2008) have showed the same result holds by dropping one of the hypotheses in Smith's original conjecture so that the result is easier to use in the setting of competitive mappings. Hirsch (2008) introduces a new condition—strict sublinear-ity in a neighborhood of the global attractor, to give a new existence criterion for the unique carrying simplex. The uniqueness of the carrying simplex is important in classifying the dynamics of lower dimensional competitive systems, for example the 3-dimensional Lokta-Volterra competition system (Zeeman (1993)). The classification of many three dimensional competitive mappings (see Davydova, Diekmann and van Gils (2005a, 2005b), Hirsch (2008) and references therein) are still open, and the uniqueness of the carrying simplex is one of the reasons.
Note that if one reverses the time t to —t in the n-dimensional competition system (3.32), then the system becomes a monotone system with both the origin and the infinity stable (under the assumption that the origin and the infinity are repellers). However this new system is not strongly monotone as required in Jiang et.al. (2004) and Lazzo and Schmidt (2005). Thus the existence of the carrying simplex cannot follow from Jiang et.al. (2004) and Lazzo and Schmidt (2005) except in the case of n = 2. Indeed this is one of the main difficulties in Hirsch (1988), Wang and Jiang (2002), and Diekmann, Wang and Yan (2008).
We conclude our discussion of threshold manifolds with a model of biochemical feedback control circuits. More details on the modeling can be found in, for example, Murray (2003) or Smith (1995). A segment of DNA is assumed to be translated to mRNA which in turn is translated to produce an enzyme and it in turn is translated to another enzyme and so on until an end product molecule is produced. This end product acts on a nearby segment of DNA to produce a feedback loop, controlling the translation of DNA to mRNA. Let xi be the cellular concentration of mRNA, let x2 be the concentration of the first enzyme, and so on, finally let xn be the concentration of their substrate. Then this biochemical control circuit is described by the system of equations xi' = g(xn) - aixi, xi = xi-i - aixi, 2 < i < n, (3.34)
where a4 > 0 and the feedback function g(u) is a bounded continuously differen-tiable function satisfying
Hence it models a positive feedback. For the Griffith model (Griffith (1968)) we have xp g(xn) = —^T (3.36)
1 +xn where p is a positive integer (the Hill coefficient). For the Tyson-Othmer model (Tyson and Othmer (1978)) we have
where p is a positive integer and K > 1. The solution flow for (3.34) is strongly monotone (see Smith (1995) for detail). The steady states for (3.34) are in one-to-one correspondence with solutions of g(u) = au (3.38)
where a = n a4. Suppose that the line v = au intersects the curve v = g(u) (u > 0) transversally. Then every non-negative steady state for (3.34) is hyperbolic, which implies that the number of steady states for (3.34) is odd for either the Griffith or Tyson-Othmer model. For most of biological parameters in the Griffith or Tyson-Othmermodel, there are exactly three steady states (Selgrade (1979,1980,1982) and Jiang (1992,1994)). In this case, the least steady state and the greatest steady state are asymptotically stable and intermediate one is a saddle point through which there is an invariant threshold manifold whose norm is positive. In the multistable case, there n1
invariant threshold manifolds which separate the attracting domains for
stable steady states (see Jiang et.al. (2004)). From a general result of Mallet-Paret and Smith (1990), we know that on each invariant threshold manifold every orbit either converges to the saddle point or is asymptotic to a nontrivial unstable periodic orbit. For n < 3, all orbits tend to the corresponding saddle point on threshold manifolds, which was proved by using topological arguments in Selgrade (1979,1980), the Du-lac criterion for 3-dimensional cooperative system in Hirsch (1989) and a Lyapunov function in Jiang (1992); for n > 5, in the bistable case for the Griffith or Tyson-Othmer model, there may exist Hopf bifurcation on the unique threshold manifold (see Selgrade (1982)). But for n = 4, whether there is a nontrivial periodic orbit or not on threshold manifold is an open problem. In Jiang (1994), it was proved that for 4-dimensional Griffith or Tyson-Othmer model all orbits are convergent to a steady state via Lyapunov method for parameters with biological significance.
Hetzer and Shen (2005) added a third equation to the classical Lotka-Volterra equations for two competing species, which describes explicitly the evolution of toxin, called an inhibitor. The equations in rescaled form are u = u(1 — u — d1v — d2 w),
W = v — (g1u + g2)w, where dud2,p,f,g1,g2 > 0. Note that 0(0,0,0), Ex(1,0,0), and Ey(0,1,g—1) are non-negative steady states of (3.39). Observing that O is a saddle, not a repeller, Hetzer and Shen (2005) studied the long-time behavior for (3.39) and the existence of threshold manifold in the bistable case, where they called a "thin separatrix" following Hsu, Smith and Waltman (1996), Smith and Thieme (2001). Jiang and Tang (2008) gave a complete classification for dynamical behavior for (3.39) and proved that the bistability occurs if and only if a* > 0, b* < 0, c* > 0, A* = (b*)2 — 4a*c* > 0, 2a* + b* > 0, a* + b* + c* > 0,
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