Prostate cancer progression and therapeutic resistance present significant clinical challenges, particularly in the transition to castration-resistant disease. Although androgen deprivation therapy and second-generation drugs have improved patient outcomes, resistance frequently develops, reflecting tumour heterogeneity and the influence of its microenvironment. This talk presents two interdisciplinary studies that address these issues through data-driven mathematical approaches. We show how integrating experimental data with mathematical and statistical modelling can improve our understanding of prostate cancer dynamics and inform more effective, context-specific therapeutic strategies. The first study examines drug resistance and tumour evolution under treatment. We develop a multi-scale hybrid modelling framework to capture processes occurring across different temporal scales. Partial differential equations describe the behaviour of drugs and other chemicals in the tumour microenvironment (over the ‘fast’ timescale), while a cellular automaton captures the dynamics of tumour cells (over the ‘slow’ timescale). Through computational analysis of the model solutions, we examine the spatial dynamics of tumour cells, assess the efficacy of different drug therapies in inhibiting prostate cancer growth, and identify effective drug combinations and treatment schedules to limit tumour progression and prevent metastasis. The second study focuses on the role of host–microbiome interactions in obesity-associated prostate cancer. Using data from experiments with the TRAMP mouse model, we apply statistical and machine learning methods, including generalised linear models, Granger causality, and support vector regression, to characterise microbial dynamics and their responses to treatment. These findings are incorporated into a dynamical systems framework that captures microbiome–tumour co-evolution under therapeutic and dietary perturbations, providing insight into how dietary fat and combination therapies involving enzalutamide and phytocannabinoids influence tumour progression and gut microbiota composition.

Reaction systems are continuos-time dynamical systems with polynomial right-hand side, and are very common in biochemistry, cell signaling, population dynamics, and many other biological applications. We discuss global stability (i.e., the existence of a globally attracting point) and persistence (i.e., robust absence of extinction) for large classes of reaction systems. In particular, we describe recent progress on the proof of the Global Attractor Conjecture (which says that vertex-balanced reaction systems are globally stable) and the Persistence Conjecture (which says that weakly-reversible reaction systems are persistent), and how these results can be extended outside their classical setting using the notion of “disguised reaction systems". We will also discuss analogous results for the case where reaction systems are replaced by generalized Lotka-Volterra systems of arbitrary degree. 

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