Nonlinear Feedback Loops and Robustness in Modelling Biological Invasions

Project: Research council

Project Details


Invasion is a key biological process. It plays a crucial role in structuring ecological systems. Invasive species threaten native biodiversity worldwide. The dynamics of biological invasions are influenced by environmental stresses, exploitation, habitat fragmentation and pollution. The ``difficulty of testing for patterns from real case studies has led to a potentially fruitful increase in mathematical ... models of invasion'', [Lodge, 1993]. But such ecological models are simplifications of reality. They suffer problems with data quality and poor parameterization, resulting in models of invasion processes that are highly uncertain. Analysis of these models is also often oversimplified. Transient dynamics are ignored as a minor irritation. Most research focuses on long-term dynamics so that modelling techniques in ecology remain dominated by asymptotic analysis of eigenvalues obtained by linearising around steady states. But ecological systems rarely reach stable steady states: instead they are prone to transients and are regularly and intensely disturbed. We need to embrace robustness concepts, built around feedback systems and control theory techniques, and tailor them to problems specific to modelling of biological invasions. This feedback control approach will have a significant impact on improving our understanding of nonlinear dynamics in invasion biology and then evolutionary ecology. The interplay between ecology and systems and control is captured in three ecological challenges and the control theorist's response:The Ecologist's Challenge #1. Models of population dynamics are uncertain and density dependent. They are prone to transients and are regularly and intensely disturbed. So what can control theory offer?The Control Theorist's Response #1: Ecological processes can be decomposed into linear and nonlinear elements, with feedbacks between biological signals and external influences. Absolute stability theory plays a fundamental role in the analysis of feedback control systems. It links to passivity theory, storage functions, naturally defined energy and the key concept of input-to-state stability. Absolute stability theory is most applicable where nonlinear feedbacks are sector bounded.The Ecologist's Challenge #2. Ecologists have a well developed technique of invasion exponent analysis based on infinitesimal disturbances to population structure and modal analysis of asymptotic stability around carrying capacity attractor dynamics. However, this modal analysis fails when the linearised population dynamics are non-normal. So what can robustness approaches offer to overcome these perceived shortcomings?The Control Theorist's Response #2: Pseudospectrum analysis is a powerful tool for teasing out the interactions between non-normal modes and transient vs. asymptotic dynamics. Simulations of invading populations suggest that pseudospectrum analysis outperforms invasion exponent analysis in predicting the onset of invasion as ``biological fitness'' increases.The Ecologist's Challenge #3. In resident-invader systems, traditional analysis would predict that a resident was not invadable if the invasion exponent is negative. But this simple approach of computing invasion exponents does not accommodate unmodelled effects and external disturbances. Highly negative invasion exponents do not correspond to highly non-invadable resident populations. The ecology community needs quantitative and predictive tools to assess the stability and robustness of a resident-invader system.The Control Theorist's Response #3. We know from the robustness paradigm that uncertainty in models and external disturbance can lead to fragile predictions. We know that systems which optimise certain performance (fitness) measures are destabilized by small uncertainties or disturbances. Resident-invader dynamics epitomise the systems much studied in feedback control where two uncertain systems are coupled in feedback
Effective start/end date1/10/1128/02/15


  • Engineering and Physical Sciences Research Council