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.
