As to nutrient enrichment, for a low preference value (i.e., predator preference on consumer), NSW exhibits complex behaviour for a considerably wide range of K, whereas SW is stable, presenting

limit cycles only for high values within the examined K range.

In some cases, there appear to be

limit cycles; the pattern for inequality resembles a series of "Kuznets curves," i.e., inequality increases and then decreases over time, and this pattern appears to repeat itself.

Thus, the problem arises to find in the space of coefficients {[a.sub.ij]} the manifold on which the corresponding maps f have a center at the origin and to investigate the

limit cycles bifurcations of such maps.

Langford's fundamental work in the field, the topics here include flow invariant subspaces for lattice dynamical systems, low- to high-dimensional behavior in waves in extended systems, mixed mode oscillations due to the generalized Canard phenomenon, bioremediation of waste in a porous medium, bifurcation of gyroscopic systems near a O:1 resonance, high dimensional data clustering from a dynamical systems point of view, and the computation of

limit cycles as the second part of Hilbert's tenth problem.

We have already seen that some specifications of this matrix can lead to

limit cycles, but now we focus on cases in which there are no

limit cycles.

In particular, [1] describes topics about

limit cycles and some theorems of usefulness in typical specific nonlinear problems such as Vanderpol's equation.

The parameters we chose to demonstrate

limit cycles in a three-species food chain ([x.sub.[c.sub.i]]= 0.15, [x.sub.p] 0.08, [y.sub.[c.sub.i]], = 1.5, [y.sub.p] = 1.5, [R.sub.[O.sub.i]]= 0.2, [C.sub.[O.sub.i]] = 0.5) were consistent with the same set of biologically plausible food webs used in our chaos example.

In this case, however, various dynamic features are observed, including the

limit cycles and the foci, which are not observed in the adiabatic systems.

(1) Can adaptative change of the capture rate parameter [C.sup.*] stabilize a predator-prey system that would exhibit

limit cycles in the absence of adaptive change?

This is of some importance for the analysis because with its help, the conditions giving rise to a Hopf Bifurcation leading to stable

limit cycles can be characterized.(3) From the conditions mentioned above, det J = [(K/2).sup.2] + [r.sup.2](K/2), det J [greater than] [(K/2).sup.2], it becomes clear that K [greater than] 0 is a necessary prerequisite for persistent oscillations of the variables.

However, they do not consider further, the same as the phenomenon of

limit cycles bifurcation in continuous models, the fact that the discrete models can exhibit the interesting dynamic behavior of two invariant closed curves bifurcating from an equilibrium point.

Existence of

limit cycles in the Solow model with delayed-logistic population growth.