Nonlinear feedback loops and robustness: Integrating density dependence, uncertainty and environmental stress in modelling biological invasions

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

The key beneficiaries of this work will be: the academic community who will gain novel developments in systems and control motivated by ecological questions; applied ecologists (including ecotoxicologists, pest managers, conservation biologists, farmers and horticulturalists) who will gain novel perspectives from a control systems approach to populations under environmental pressures; the general public who will benefit from potential applications of the proposed research to the prediction and control of specific biological invasions affecting their local environment; and individuals who will benefit from training and education. The project will adopt a novel control systems approach to understanding the dynamics of coupled, resident-invader dynamics. This approach will provide new insights because it is a change from the dominant modeling and analysis approaches in invasion biology. We will engage with the scientific community via standard activities (conferences, publishing our research in high-profile journals, and presenting scientific seminars). To enhance these standard routes to impact, we will draw on a number of networks at Bath and Exeter. At Exeter we will be able to draw on high profile, science strategy themes in Climate Change & Sustainable Futures and Systems Biology. At Bath we will be able to link with research activities in Complexity Science, Control Theory and Mathematical Biology. We will conduct the following additional activities. 1. An International Workshop. Two-thirds of the way through the project, we will host an international workshop on Control Systems Tools for Ecology and Evolution. This workshop will include 8 international, invited speakers from across control theory, systems engineering, theoretical and applied ecology. The workshop will be timed directly before the British Ecological Society (BES) Annual Conference in 2013. The workshop will allow us to showcase our results, and allow to respond to direct feedback from applied and th(Beoretical ecologists. 2. Engagement with Applied Ecology. In addition to the natural links we have to the applied ecology community through one of the Co-Is (Hodgson), we will also interact directly with applied ecologists through the international workshop described above. We will apply through Hodgson (a member of the BES Council) to host a Special Symposium at the annual BES Conference on 'Predicting Biological Invasions'. 3. Public Engagement and Communication. Public engagement will be a key focus for impact. Systems & control, as a mathematical discipline, is typically quite theoretical in focus. However, in this project, we will be developing parts of systems & control theory for applications to important biological processes. We therefore have an opportunity not only to address significant applications oriented aspects, but also to convey, in a convincing manner, the importance of some of the key issues of control theory. At the core of our project are infinite-dimensional Integral Projection Models. These models can account for continuous-size in describing the state of a population. But they can also be used to describe spatially distributed populations. We will pick up on this feature in our public engagement activity. Alongside the novel science, mathematics and systems engineering, we aim to create a `toy' model of two spatially distributed populations. Modelling these spatial invasions in open-access code will provide an access point to our work in a number of activities: (i) Website. We will create a graphically-rich educational website including a GUI based around the toy model; (ii) School visits programme. We will take the toy model on the road, visiting schools in the South West. We will run small workshops in the classroom based around the GUI. (iii) Publicity. We will draw on the successes of the Centre for Ecology and Conservation in promoting its work to publicise the role of systems & control in biocience