Encontros IST-IME 

05/09/2016, 11:20 — 12:00 — Anfiteatro Pa2, Pavilhão de Matemática
Teresa Faria, Universidade de Lisboa

Asymptotic behaviour for some classes of non-autonomous delay differential equations

We study the global asymptotic behaviour of solutions for some families of $n$-dimensional non-autonomous delay differential equations (DDEs), which encompass a large number of structured population models.

Some classes of monotone DDEs (with possible infinite delay) are first analysed: by using comparative results from the theory of cooperative systems [4], some criteria for persistence and permanence are given [2]. We then consider a family of non-autonomous DDEs obtained by adding a non-monotone delayed perturbation to a linear homogeneous cooperative system of ODEs. By exploiting the stability and the monotone character of the linear ODE, and by using comparison techniques with auxiliary monotone systems, we are able to establish sufficient conditions for both the extinction of all the populations and the permanence of the system [3].

In the case of DDEs with autonomous coefficients, sharper results are obtained, even in the case of reducible community matrices, improving or extending criteria in recent literature (see e.g. [1]).

References

1. J. Arino, L. Wang, G. S. K. Wolkowicz, An alternative formulation for a delayed logistic equation, J. Theor. Biol. 241 (2006), 109-119.
2. T. Faria, Persistence and permanence for a class of functional differential equations with infinite delay, J. Dyn. Diff. Equ.(2016). DOI 10.1007/s10884-015-9462-x.
3. T. Faria, R. Obaya, A. M. Sanz, Asymptotic behaviour for non-monotone delayed perturbations of monotone non-autonomous linear ODEs, submitted (2016). http://arxiv.org/abs/1607.05033.
4. H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, Amer. Math. Soc., Providence, RI, 1995.