Generalized models as a universal approach to the analysis of nonlinear dynamical systems

Abstract

We present a universal approach to the investigation of the dynamics in generalized models. In these models the processes that are taken into account are not restricted to specific functional forms. Therefore a single generalized models can describe a class of systems which share a similar structure. Despite this generality, the proposed approach allows us to study the dynamical properties of generalized models efficiently in the framework of local bifurcation theory. The approach is based on a normalization procedure that is used to identify natural parameters of the system. The Jacobian in a steady state is then derived as a function of these parameters. The analytical computation of local bifurcations using computer algebra reveals conditions for the local asymptotic stability of steady states and provides certain insights on the global dynamics of the system. The proposed approach yields a close connection between modelling and nonlinear dynamics. We illustrate the investigation of generalized models by considering examples from three different disciplines of science: a socioeconomic model of dynastic cycles in china, a model for a coupled laser system and a general ecological food web.

Collapse of an ecological network in Ancient Egypt

“…Because changes in productivity can alter population-level responses to species interactions, we performed a sensitivity analysis to determine whether and to what extent changes in primary productivity influence estimates of PSW. We address changing habitat productivity by incorporating the following assumptions: (i) when productivity is high, the per-capita contribution of herbivores to population growth increases, such that the impact of herbivore density on growth is elevated; (ii) because prey are plentiful, the growth of predator populations is not limited by prey density (46). Conversely, when primary productivity decreases (as is assumed to have occurred over the Holocene), herbivore population growth becomes nutrient limited, such that changes in herbivore density have a smaller impact on population growth, whereas predator population growth becomes limited by herbivore density.…”

The dynamics of hybrid metabolic-genetic oscillators

“…13. Consider a system of ordinary differential equations dX dt ¼ IðX; YÞ À OðX; YÞ; dY dt ¼ FðX; YÞ À GðX; YÞ:…”

“…17 One of our primary approaches to studying the metabolator is based on a non-dimensionalization technique for studying dynamical systems commonly referred to as generalized modeling (GM). 10,12,13,25,26 In GM, a change of variables is applied to a dynamical system so that many of the otherwise unconstrained parameters in the system (e.g., rate constants) are replaced by "elasticity" parameters with well-defined ranges. Then, the dynamics of the system can be studied around an arbitrary and potentially unknown steady-state.…”

Spatio-temporal organization in a morphochemical electrodeposition model: Hopf and Turing instabilities and their interplay

“…[2,3]. In [3] the authors use the concept of generalized models developed in [31] to show that in a modified diffusive predator-prey Rosenzweig-McArthur model, spatially irregular self-sustained non-stationary patterns or even spatio-temporal chaos can appear for parameter values in the neighbourhood of the TH bifurcation point. Instead, in [2], where a two species reaction-diffusion predator-prey system with a ratio-dependent functional response is considered, spatio-temporal chaos is not found in the close vicinity of the TH bifurcation, but rather for parameter values far from the bifurcation point.…”