Chaos via Torus destruction in models of dengue fever and predator-prey systems, implications for data analysis

Seminário de Análise

Orador: Nico Stollenwerk, Biomathematics and Statistics group, CMAF-CIO

Data: 8/3/2017

Hora: 13h

Local: Sala 1.5, ed. VII

Abstract:

In the analysis of multi-strain models describing dengue fever epidemiology we found complex bifurcation structures, and especially the  appearance of deterministic chaos after torus bifurcations. Since the fluctuations of severe dengue fever cases e.g. in Thailand and its provinces  can be well described by such models, the next step is to investigate algorithms for time series analysis even under deterministic chaos. Based on  dynamic noise one can obtain estimations of likelihood functions and apply the whole toolbox of parameter estimation and model evaluation  in principle, however still under technical difficulties of long computer runs. To understand the transition into chaos after torus bifurcations  better we therefore searched for simpler population models than the already quite high dimensional dengue models. One of the best candidates  is the in itself two-dimensional seasonal Rosenzweig-MacArthur model which was described to undergo torus bifurcations where also regions of  deterministic chaos were found by increasing parameters crossing torus bifurcations. The chaotic regions could only be speculated and  exemplified by individual simulations but their parameter regions could only be guessed fuzzily. We investigated these models again and now not  only with AUTO to detect bifurcation structures, but also with Lyapunov spectra in which the same bifurcation lines could be detected by  looking at subdominant Lyapunov exponents reaching zero and also chaotic parameter regions could be detected. Surprisingly, the previously  only fuzzy guessed chaotic regions turned out to be part of Arnol’d tongues on the tori after the torus bifucations. For the analysis of empirical  data, of course stochastic versions of the models have to be investigated, in the case of the Rosenzweig-MacArthur model time scale separable  stoichiometric versions.