Generalized models – A new tool for the investigation of ecological systems

Abstract

Ecological systems are commonly studied by very concrete conventional models or very abstract random matrix models. Here we review and extend the approach of generalized structural kinetic modeling, that offers an intermediate way between these extremes. Generalized models describe systems with a specific structure, but do not restrict the processes in the model to specific functional forms. The approach is based on the construction of a locally linear model in every point of parameter space, in such a way that each element of the model is directly accessible to measurement and has a well defined ecological interpretation. Here we show that generalized models can be used to study the local asymptotic stability of steady states and reveal certain features of the global dynamics. Among other examples we present results on spatial predator-prey system and a complex food web.

Dynamical analysis of evolution equations in generalized models

“…Only subsequently it was recognized as a general approach [22] and applied to a wide verity of applications. A comprehensive list of publications is [21,22,23,4,54,59,53,56,19,18,55,17,34,51,50,58,47,3,13,61]. In these applications generalized models for instance revealed essential factors for the stability of food webs [23]; resolved a discrepancy between different modeling approaches in ecology [54]; implicated a Hopf bifurcation as a cause of Paget's disease in humans [61]; and provided insights in the stability of mitochondrial metabolism [51].…”

“…Generalized modelling for partial differential equations has first been developed for reactiondiffusion equations in the context of spatial predator-prey models [17,4]. A more general class of partial differential equations (PDEs) that can be analyzed by dynamical systems techniques is given by (semilinear) parabolic equations.…”

Nonlocal generalized models of predator-prey systems

“…Because the parameter α can always be removed by scaling the variable Y and re-labelling the functions we will always assume α = 1 from now on. Generalized models [20,33] directly work with the formulation (1) without specifying functional forms for S, G and M. Previous works on generalized models [14,15,12,52] focused on analyzing the dynamics close to stationary states. Beyond the structure of the equations (1) this analysis requires only the assumption that steady states exist in the class of models under consideration.…”