Complexity Forum: Andrey Morozov (Leicester)
Speaker: Andrey Morozov (Leicester)
Title: Detecting structural sensitivity in biological models: developing a new framework
Abstract: When we construct mathematical models to represent a given real-world system, there is always a degree of uncertainty with regards to the model specification, whether with respect to the choice of parameters or to the choice of formulation of model functions. This can become a real problem in some cases, where choosing two different functions with close shapes in a model can result in substantially different model predictions. This phenomenon is known as structural sensitivity, and is a significant obstacle to improving the predictive power of models, particularly in fields where it is not possible to derive the functions suitable for representing system processes from theory or physical laws, such as the biological sciences. In this talk, I shall revisit the notion of structural sensitivity and its relation to the property of structural (in)stability, and propose a general approach to reveal structural sensitivity which is a far more powerful technique than the conventional approach consisting of fixing a particular functional form and varying its parameters, since we consider the infinite-dimensional neighbourhood of a given model's unknown functions. In particular, a rigorous method to unearth sensitivity with respect to the local stability of a system's equilibrium points will be discussed. Then, I implement the method to explore sensitivity in several well-known multicomponent ecological models, demonstrate the existence of structural sensitivity in these models and show that conventional methods based on variation of parameters alone will often miss such sensitivity. I shall discuss the consequences that structural sensitivity and the resulting model uncertainty may have for the modelling of biological systems. In particular, I shall consider that structural sensitivity may allow models to represent far more complex dynamics than the dimension of the state-space may suggest, and that in a structurally sensitive model, the concept of a "concrete" bifurcation structure may no longer be relevant. In this case, we can only describe bifurcations with a certain probability. I suggest several ways to define the probability of a bifurcation taking place when there is uncertainty in the parameterization in model functions. I call bifurcations in such systems the fuzzy bifurcations and I provide an example of construction of a fuzzy bifurcation for a class of predator-prey biological models.