Systems and control in mathematical biology

We seek connections, such as the phenomenon of feedback, between control theory and biological systems.

Computer graphic showing dna strand and microbes
We seek to use tools from mathematical control theory to describe biological systems.

Quoting The Way Life Works by Hoagland & Dodson (1995) "Feedback is a central feature of life. The process of feedback governs how we grow, respond to stress and challenge, and regulate factors such as body temperature, blood pressure, and cholesterol level. The mechanisms operate at every level, from the interaction of proteins in cells to the interaction of organisms in complex ecologies." We seek to use tools from mathematical control theory, including feedback, to describe, and often with a view to control, biological systems. Motivating examples include the structure and composition of ecosystems, the potential for biological invasion, and the spread of disease.

Mathematical approaches

We typically work with deterministic dynamical systems for models of biological processes, which include difference and integro-difference equations in discrete-time, or ordinary or partial differential equations in continuous-time. Many biological models are inherently uncertain, and so we also consider difference and differential inclusions. We make both analytic and computational assertions. Robustness is a key feature of many of the control schemes we propose, as they may be operating in noisy, uncertain, or variable, environments.

Applications

A recent application has involved using tools from systems and control theory, namely the concept of common Lyapunov functions, to better predict how populations may use dispersal to survive in heterogeneous, yet seemingly adverse, environments. Roughly, sufficient dispersal may cause instability, in much the same sense as that originally proposed by Turing in the context of pattern formation. Other applications include identifying readily computable proxies for predicting when a biological invasion shall be successful. These proxies do not rely on eigenvalues of a linearisation of the assumed resident-invader dynamics.

Staff working in this area

Dr Chris Guiver
Dr Jane White
Dr Ben Adams