Periodic orbits

Numerical simulation of standard biological systems like the MacArthur-Rosenzweig model (Holling type II predator response) as well as analytic results on highly symmetric systems show that limit cycles of the system (6.5) undergo some systematic changes if quiescent phases are introduced. From the local stability analysis at a stationary point it is evident that introducing a quiescent phase works against Hopf bifurcations. Suppose we have a system depending on some parameter a which undergoes a Hopf bifurcation. A stationary state is stable for a < 0 and unstable for a > 0 in such a way that a pair of eigenvalues crosses the imaginary axis at a = 0. The stability Theorem 3.1 suggests that by introducing a quiescent phase the Hopf bifurcation is shifted to some parameter value a > 0. This is what indeed happens in concrete examples.

Example 6.4 (Bilinsky and Hadeler, 2008) The MacArthur-Rosenzweig model with quiescence reads u uv u = au(l ——J — b--p\u + q\w

It is known that the two-dimensional system without quiescence has either a stable coexistence point or a unique (stable) limit cycle. In the latter case the system with quiescence either has no limit cycle at all or again a limit cycle, this time in dimension four, whereby the "size" of the projection in the active phase gets smaller. If the coexistence state in the problem without quiescence is stable then it is strongly stable. For every choice of P and Q there is a gain in stability. Let S and t be the determinant and the trace at the coexistence state of the system without quiescence. In the t, S-plane the stability domain is given by S > 0, t < 0. For given P, Q the stability domain extends into the range t > 0. The boundary of the stability domain can be found explicitly as a curve t = 4>{S) with ^(0) =0 and 0(S) > 0 for S > 0 (Bilinsky and Hadeler, 2008).

In numerical experiments, the four-dimensional limit cycle of (6.17) can be visualized by presenting the total population sizes u + w and v + z for prey and predator. In this projection the effect of a quiescent phase is not easily recognized because the position of the (projection of) the stationary point is shifted. It is easier to project to the u, v-plane and also to the w, z-plane. Then one sees that the "size" of the projected limit cycle in the u, v-plane is smaller than the limit cycle in the system without quiescence and gets ever smaller if the rates are increased. Eventually the limit cycle may contract to the stationary point.

Here "size" is used as a phenomenological description. For the typical egg-shaped limit cycles of predator prey models area and circumference and diameter all shrink (see Figure 6.1). It is interesting to observe that at the same time the projection onto the w, z-plane gets larger. For symmetric systems the shrinking of the periodic orbit can be rigorously proven, see Hadeler and Hillen (2006).

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