Tag - Mathematical biology
We consider a non-local tumour growth model of phase-field type, describing the evolution of tumour cells through proliferation in the presence of a nutrient. The model consists of a coupled system, incorporating a non-local Cahn-Hilliard equation for the tumour phase variable and a reaction-diffusion equation for the nutrient. The optimal control problem aims to identify a suitable therapy, capable of guiding the evolution of the tumour towards a predefined target. We first establish novel regularity results for the PDE system by applying maximal regularity theory in weighted Lp spaces. Such a technique enables us to prove the local existence and uniqueness of a regular solution in a quite general framework, which also includes chemotaxis effects to some extent, but restricts us to regular double-well potentials. Then, by leveraging the time-regularisation properties of the weighted spaces and some global boundedness estimates, we further extend the solution to a global one. In a second version of the model, we add a viscous regularisation term which allows us to prove the existence and uniqueness of a global regular solution under more general hypotheses. Indeed, we can also include singular double-well potentials and cross-diffusive chemotactic effects, at the expense of some additional hypotheses on the controls. These results provide the foundation for addressing the optimal control problem in both cases. Specifically, we prove the existence of an optimal therapy and then, by studying the Fréchet-differentiability of the control-to-state operator and introducing the adjoint system, we derive first-order necessary optimality conditions. We finally discuss some open questions and future research directions.
Two key questions in biodiversity conservation relate to how we manage land/sea-scapes for wildlife and at what point in time we can expect to observe the consequences of our actions. Mathematically, these questions relate to (multiple) steady states and transient dynamics. I will consider two theoretical and inferential examples under each of these questions. I will first examine how the use of suburban and agroforestry ecosystems can be optimised for biodiversity gains. I will then explore frameworks for the analysis of delayed impacts in large-scale bird community data and very fine scale grazer rewilding experiments. The convergence of ideas from dynamical systems and modern statistical inference is a very exciting development in quantitative ecology. Thorny rewilding applications are one of the top areas where such theoretical advances can gain real traction and produce tangible impact.
Rewilding aims to create ecosystems and landscapes whose dynamics are driven by natural processes. This idea has been defined verbally, but its theoretical aspects remain at the early stages of development. I will discuss aspects of rewilding that could be explored by theorists, also drawing on the related topic of ecological restoration. 1) Complex systems. Rewilding has been described as an attempt to recreate complex systems, whose key features are complex trophic structures, stochastic disturbances and heightened dispersal. 2) Resilience and critical thresholds. Concepts from complex systems science that are linked to non-linearity, such as regime shifts, ecological resilience and ecological feedbacks, could be employed to help explain variation in rewilding outcomes. 3) Dispersal. Rewilding is focused on ‘natural colonization’, meaning that outcomes depend critically on dispersal abilities and the permeability of landscapes. Ultimately, development of theory on these and related concepts may help us understand the various trajectories that rewilding may take.
A key component of most rewilding projects is the translocation of plant and animal species to restore ecosystem function, however translocations are a complex process both biologically and socially. Here I will explore the reintroduction process and suggest the areas where mathematical and statistical methods could aid decision making and the understanding of complex biological systems.
There are trade-offs in practice with rewilding particularly around food and nutrition security but also around other societal dimensions as globally represented by the UN Sustainable Development Goals (SDGs). We need solutions that are synergistic.
Building on previous collaborations, we propose transdisciplinary, methodological approaches with a focus on qualitative and quantitative scenario planning to develop both aspirational and plausible solutions based on science and society.
In a recent work, R. Carmona, Q. Cormier and M. Soner proposed a mean field game based on the classical Kuramoto model, originally motivated by systems of chemical and biological oscillators. Such MFG model exhibits several stationary equilibria, and the question of their ability to capture long time limits of dynamic equilibria is largely open. I will discuss in the talk how to show that, up to translations, there are two possible stationary equilibria only - the incoherent and the synchronised one - provided that the interaction parameter is large enough. Finally, I will present some local stability properties of the synchronised equilibrium.
The first seminar included an overview of rewilding, highlighting some of the areas in which we expect that the mathematical sciences could have an impact. This was followed by a response/discussion led by mathematical scientists. There were then opportunities for informal discussion as well. The session closed with a wrap-up in which we captured the most promising lines to follow up at the next seminar.
Climate change is having profound effects on the incidence of vector borne disease, such as dengue, chikungunya and West Nile virus. However, developing effective measures of disease risk on a global scale are challenged by the complex ways in which environmental variation acts in vector-host-pathogen systems. One way in which insect vectors, such as mosquitos, respond to environmental variation is to change their traits. For example, if food was scarce for juvenile mosquitos then when they become adults they are smaller, and lay fewer eggs to ensure there is less competition for food in the next generation. So the environment of the juvenile determines the trait the individual has as an adult. In this way the individuals adapt to the environment. Current models over-simplify the interaction between vector traits and environmental variation and so risk misestimating disease risk. Here, we derive a mathematical framework for capturing the interaction of vector traits and population dynamics. I show how this new mathematical framework leads to both interesting mathematical questions and can be used to help explain the location, magnitude and timing of historical dengue outbreaks.
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