Stationary states and stability

From a biological point of view we want to know how the dynamics of the system (6.5) is changed by introducing quiescent phases. This problem is also interesting from a mathematical point of view. Some aspects concerning global existence of solutions and of compact global attractors are presented in Hadeler and Hillen (2006).

v = —u v — pu + qx —u — pv + qy pu — qx pv — qy.

General results on global attractors are surprisingly difficult. On the other hand we have some detailed results on stationary points and their stability and some preliminary results for periodic orbits.

At first glance introducing quiescent phases seems similar to introducing delays. For delay equations we know that combining a negative feedback with sufficiently large delays leads to oscillations and then periodic orbits. Quite on the contrary, quiescent phases stabilize against oscillations.

Suppose u is a stationary point of the system (6.5), i.e., f (u) = 0. Then

is a stationary point of (6.6). Let A = f '(u) be the Jacobian matrix of (6.5) at the stationary point. Then the Jacobian matrix of (6.6) is given by

The eigenvalue problem of the matrix B is equivalent to that of the matrix pencil

Equal rates: In the case of equal rates we have P = pI, Q = qI, the matrices P,Q, A commute and we can apply the spectral mapping theorem to the pencil (6.15). To each eigenvalue n of the matrix A there are two eigenvalues Ai and A2, ordered by KA2 < KA1, which can be obtained from the equation

This is a very simple quadratic equation. In principle the two solutions can be represented by an explicit formula. The problem is that n is a complex number. The following can be shown. Always KA2 < 0. Hence A2 does not affect stability. Stability is governed by the eigenvalue A1.

Now there are three quite distinct cases: If n = 0 then A1 = 0. If n is real then A1 is located between n and 0. Hence, with respect to real eigenvalues, quiescence does not change stability. If n is complex (with nonvanishing imaginary part) then, generally speaking, for eigenvalues with positive real parts the real parts are decreased by introducing quiescence and may eventually become negative. This effect is most prominent for eigenvalues with large imaginary parts, i.e., high frequency oscillations are damped. Detailed information is given by the following theorem.

Theorem 6.1 (Hadeler, 2008a) Let n = a + i[3 bean eigenvalue of the linearization of (6.5) at a steady state u. Then the linearization of (6.6) at (u,pu/q) has two corresponding eigenvalues A1 ,A2 with < KA1. The eigenvalues n and A1,A2 are related as follows:

(a) Let n = a e IR. Then A1, A2 are real. (a.i) If a < 0 then A2 < a < A1 < 0. (a.ii) If a = 0 then A2 = -(p + q) < 0 = A1.

(a.iii) If a > 0 then X2 < 0 < X1 < a. (b) Let n = a ± ifj, ¡3 > 0. Then KA2 < 0. (b.i) If a < 0 then KAi < 0. (b.ii) If a > 0 then KA1 < a. (b.iii) If a < 0 and

32 + (p + q + a)2 + 4ap > 0 and ¡2(q + a)+a(p + q + a)2 > 0, then KA1 < a.

32 > 4aq — (p + q — a)2 and3(p — a) > a(p + q — a)2, then KA1 < 0.

Unequal rates: If the matrices P and Q are not multiples of the identity and the various types of particles go quiescent and return with pairwise distinct rates, then the situation is quite different and the stability problem has about the same complexity as the Turing stability problem. Indeed, here as in the Turing problem we have a given stable matrix and a matrix pencil depending on positive diagonal matrices. So far only the case n =2 of two types has been dealt with (Hadeler, 2008a). Recall that a 2 x 2 matrix A = (aj) is stable if tr A = a11 + a22 < 0 and det A = a11a22 — a12a21 > 0, and strongly stable (in the sense of Turing) if, in addition, a11 < 0, a21 < 0. (A is excitable if A is stable, but not strongly stable.) Suppose that A is stable. Then the matrix B is stable for all choices of P and Q if and only if A is strongly stable. Thus, if A is excitable in the sense of Turing, the system may become destabilized by introducing quiescent phases with suitably chosen distinct rates. The problem for n > 2 is open.

However, there are classes of problems for which additional mathematical tools are available (Hadeler and Thieme, 2008). For example, if the system (6.5) is cooperative then the system (6.6) is cooperative as well; or if the system (6.5) is competitive then the system for v and —w is competitive as well.

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