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Volume speciale delle Memorie dell'Accademia di Scienze Fisiche, Matematiche e Naturali della Società Nazionale di Scienze, Lettere ed Arti in Napoli.

Proc. IX Int. Conf. "Waves and Stability in Continuous Media" (Bari, October 6-11,1997)

edited by L.Dekker,G.Savastano, G.C. Vansteenkiste

In this paper a numerical and symbolic procedure is developed to determine periodic solutions of planar autonomous systems. The proposed method supplies the Fourier series of the periodic solutions as well as their period in the presence of isochronous centers and unstable limit cycles.

The influence of toxicants on n competitive populations in an unsimplified chemostat model is investigated, where the nutrient input and washout rate are allowed to be time variable. The threshold between persistence and extinction of the populations has been obtained for sufficiently small initial densities.

The problem of the equilibrium stability for the chemostat equation is considered in the special case of a biotic species feeding on one limiting nutrient and predated by another biotic species. Both the biotic species through the decomposition process can return with delay to the chemostat fraction of dead biomass as new nutrient. The delay kernels of nutrient recycling are assumed to be general L2(0, +∞) non-negative functions which admit up to second order finite moments. Two approaches are adopted: the first one applies when both the biotic species have a self-regulating term in their evolution equations and can be worked out without linearizing the equations. This approach does not require any further constraint on delay kernels, whereas it introduces constraints on the remaining parameters of the model. The second approach applies to the linearized equations when the predator self-regulating term is set equal to zero. In this case the stability condition requires a constraint on the average time delays in the recycling processes. Both the approaches are performed by constructing suitable Krasovskii-Lyapunov functionals for the related functional retarded differential equations.

A mathematical model is proposed to describe the dynamics of users of one product in two different patches. Advertisement force, contact rate between users and non-users, population dispersal rate and returning rate from user class to non-user class are chosen as key parameters. For the product with a long life-span, it is assumed that the returning rate is proportional to the number of users. It is shown that this model has a unique positive equilibrium which is globally stable. For the product with a short life-span, time delays are introduced to represent the duration of the product in two different markets and the stability of a positive equilibrium is analyzed in one reasonable case. Periodic advertisements are also incorporated and the existence and uniqueness of positive periodic solutions are investigated.

In this paper the Comparison Method involving vector Lyapunov functions is applied to study the partial stability properties (i.e. with respect to only a part of the state variables) for large scale systems through the decomposition aggregation approach. Some general theorems are stated, which are based on classic results of the partial stability theory. An application of the results is given for a problem of partial controllability.

This paper studies periodic solutions of two types of population models with time delays and periodic coefficients. The first represents one species growth with delayed recruitments and the second models a single species despersal among the n patches. By developing the theory of monotone and concave operators to functional differential equations, we establish conditions under which each model admits a positive periodic solution which attracts all positive solutions.

In this paper a theorem concerning the asymptotic partial stability for large-scale systems is given. By describing high-order systems as collections of lower order interconnected subsystems, the theorem takes the form of conditions on the interconnection structure so that asymptotic partial stability property of isolated subsystems infer the same property of the over-all system. An application is given for a problem of partial controllability in the large.