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F.J. Thomsen
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Subject of this thesis is the study of differential equations describing complex ecosystems. More broadly, the theme is the study of the dynamics of interacting nonnegative components. The physical entities represented by these components depend on the specific context of the system. Aside from the populations of animal or microbial species in an ecosystem, examples of interest to us here range from concentrations of chemical species, or expressions of genes in regulatory networks, to epidemiological compartments. The phase space for such systems is naturally constrained to (subsets of) the nonnegative cone in Euclidean space. This thesis studies three distinct types of differential equations describing nonnegative components in three distinct areas of application.
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Subject of this thesis is the study of differential equations describing complex ecosystems. More broadly, the theme is the study of the dynamics of interacting nonnegative components. The physical entities represented by these components depend on the specific context of the system. Aside from the populations of animal or microbial species in an ecosystem, examples of interest to us here range from concentrations of chemical species, or expressions of genes in regulatory networks, to epidemiological compartments. The phase space for such systems is naturally constrained to (subsets of) the nonnegative cone in Euclidean space. This thesis studies three distinct types of differential equations describing nonnegative components in three distinct areas of application.
Adaptive therapy is a recent paradigm in cancer treatment aiming at indefinite, safe containment of the disease when cure is judged unattainable. In modeling this approach, inherent limitations arise due to the structure of the vector fields and the bounds imposed by toxic side-effects of the drug. In this work we analyze these limitations in a minimal class of models describing a cancer population with a slowly co-evolving drug resistance trait. Chemotherapeutic treatment is introduced as any bounded time-varying input, forcing the cells to adapt to a changing environment. We leverage the affine structure and low dimension of the system to explicitly construct controllable subsets of the state space enclosing sets of equilibria. We show that these controllable sets entirely determine the asymptotic behavior of all trajectories that cannot lead to a cure.
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Adaptive therapy is a recent paradigm in cancer treatment aiming at indefinite, safe containment of the disease when cure is judged unattainable. In modeling this approach, inherent limitations arise due to the structure of the vector fields and the bounds imposed by toxic side-effects of the drug. In this work we analyze these limitations in a minimal class of models describing a cancer population with a slowly co-evolving drug resistance trait. Chemotherapeutic treatment is introduced as any bounded time-varying input, forcing the cells to adapt to a changing environment. We leverage the affine structure and low dimension of the system to explicitly construct controllable subsets of the state space enclosing sets of equilibria. We show that these controllable sets entirely determine the asymptotic behavior of all trajectories that cannot lead to a cure.
Journal article
(2025)
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Christo Morison, Małgorzata Fic, Thomas Marcou, Javier Redondo Antón, Alexander Stein, Frank Bastian, Hana Krakovská, Mohammadreza Satouri, Frederik J. Thomsen, More authors...
Cooperation arises in nature at every scale, from within cells to entire ecosystems. Public goods games (PGGs) are used to represent scenarios characterised by the conflict/dilemma between choosing cooperation as a socially optimal strategy and defection as an individually optimal strategy. Evolutionary game theory is often used to analyse the dynamics of behaviour emergence in this context. Here, we focus on PGGs arising in the disease modelling of cancer evolution and the spread of infectious diseases. We use these two systems as case studies for the development of the theory and applications of PGGs, which we succinctly review. We also posit that applications of evolutionary game theory to decision-making in cancer, such as interactions between a clinician and a tumour, can learn from the PGGs studied in epidemiology, where cooperative behaviours such as quarantine and vaccination compliance have been more thoroughly investigated. Furthermore, instances of cellular-level cooperation observed in cancers point to a corresponding area of potential interest for modellers of other diseases, be they viral, bacterial or otherwise. We aim to demonstrate the breadth of applicability of PGGs in disease modelling while providing a starting point for those interested in quantifying cooperation arising in healthcare.
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Cooperation arises in nature at every scale, from within cells to entire ecosystems. Public goods games (PGGs) are used to represent scenarios characterised by the conflict/dilemma between choosing cooperation as a socially optimal strategy and defection as an individually optimal strategy. Evolutionary game theory is often used to analyse the dynamics of behaviour emergence in this context. Here, we focus on PGGs arising in the disease modelling of cancer evolution and the spread of infectious diseases. We use these two systems as case studies for the development of the theory and applications of PGGs, which we succinctly review. We also posit that applications of evolutionary game theory to decision-making in cancer, such as interactions between a clinician and a tumour, can learn from the PGGs studied in epidemiology, where cooperative behaviours such as quarantine and vaccination compliance have been more thoroughly investigated. Furthermore, instances of cellular-level cooperation observed in cancers point to a corresponding area of potential interest for modellers of other diseases, be they viral, bacterial or otherwise. We aim to demonstrate the breadth of applicability of PGGs in disease modelling while providing a starting point for those interested in quantifying cooperation arising in healthcare.
Journal article
(2021)
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F.J. Thomsen, Lisa Rapp, Fabian Bergmann, Walter Zimmermann
In this work we identify and investigate a novel bifurcation in conserved systems. This secondary bifurcation stops active phase separation in its nonlinear regime. It is then either replaced by an extended, system-filling, spatially periodic pattern or, in a complementary parameter region, by a novel hybrid state with spatially alternating homogeneous and periodic states. The transition from phase separation to extended spatially periodic patterns is hysteretic. We show that the resulting patterns are multistable, as they show stability beyond the bifurcation for different wavenumbers belonging to a wavenumber band. The transition from active phase separation to the hybrid states is continuous. Both transition scenarios are systems-spanning phenomena in particle conserving systems. They are predicted with a generic dissipative model introduced in this work. Candidates for specific systems, in which these generic secondary transitions are likely to occur, are, for example, generalized models for motility-induced phase separation in active Brownian particles, models for cell division or chemotactic systems with conserved particle dynamics.
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In this work we identify and investigate a novel bifurcation in conserved systems. This secondary bifurcation stops active phase separation in its nonlinear regime. It is then either replaced by an extended, system-filling, spatially periodic pattern or, in a complementary parameter region, by a novel hybrid state with spatially alternating homogeneous and periodic states. The transition from phase separation to extended spatially periodic patterns is hysteretic. We show that the resulting patterns are multistable, as they show stability beyond the bifurcation for different wavenumbers belonging to a wavenumber band. The transition from active phase separation to the hybrid states is continuous. Both transition scenarios are systems-spanning phenomena in particle conserving systems. They are predicted with a generic dissipative model introduced in this work. Candidates for specific systems, in which these generic secondary transitions are likely to occur, are, for example, generalized models for motility-induced phase separation in active Brownian particles, models for cell division or chemotactic systems with conserved particle dynamics.