Periodic Reporting for period 1 - MesoCroMo (A Mesoscopic approach to Cross-diffusion Modelling in population dynamics)
Berichtszeitraum: 2024-01-01 bis 2025-12-31
The description of natural phenomena with mathematical models is a fundamental step to understand the key physical mechanisms that dictate their evolution. By varying the scale of observation, these mechanisms can be formulated using microscopic first principles and conservation laws, and their effect is then made visible in the macroscopic equations that govern the evolution of the relevant observable quantities characterizing the phenomenon under study. A typical example is provided by gas dynamics, where the collisions between molecules determine the behavior of the physical quantities (density, temperature, pressure, etc.) describing the whole gaseous mixture. It is then crucial to determine the rigorous mathematical connections between these alternative formulations, to ensure that they are in fact coherent. The challenging task of the MesoCroMo project has been to construct a new modeling approach able to characterize, across multiple scales of observation, the evolution of multi-species systems in the context of population dynamics. In particular, the proposed approach had to be general and flexible enough to be adapted in a wide variety of settings, most notably for ecological and epidemiological applications. The ultimate goal of the project was the justification of macroscopic cross-diffusion models, as suitable asymptotic limits of Boltzmann-like equations. These models find numerous real-world applications due to their ability to capture relevant collective behaviors, emerging from the complex spatial movement of a species when its diffusion is influenced by the presence of another. As such, the project will also stand as a powerful mean to bridge the mathematical interest with other disciplines like physics, chemistry and biology.
The specific objectives of the MesoCroMo project have been to
1) use the classical kinetic theory of reactive gaseous mixtures as a benchmark to set up an appropriate microscopic framework for inelastic interactions in multi-species populations, and to introduce new biologically meaningful Boltzmann-like operators modeling such interactions at a mesoscopic level;
2) determine simplified Fokker-Planck-type equations with kinetic nonconservative reaction operators describing birth/death processes and cooperative/competitive dynamics, and derive the corresponding macroscopic models;
3) extend the Fokker-Planck equations to include spatially inhomogeneous dynamics, and investigate the coupled diffusion/fast-reaction regimes of its parameters to derive macroscopic cross-diffusion-like systems;
4) characterize the equilibria of these simplified kinetic equations, study the stability of the equilibria, and analyze the trends to equilibrium of the kinetic solutions;
5) develop and implement an asymptotic-preserving numerical scheme able to simulate the evolution of the newly introduced Fokker-Planck equations and to capture their limit macroscopic models.
The specific objectives of the MesoCroMo project have been to
1) use the classical kinetic theory of reactive gaseous mixtures as a benchmark to set up an appropriate microscopic framework for inelastic interactions in multi-species populations, and to introduce new biologically meaningful Boltzmann-like operators modeling such interactions at a mesoscopic level;
2) determine simplified Fokker-Planck-type equations with kinetic nonconservative reaction operators describing birth/death processes and cooperative/competitive dynamics, and derive the corresponding macroscopic models;
3) extend the Fokker-Planck equations to include spatially inhomogeneous dynamics, and investigate the coupled diffusion/fast-reaction regimes of its parameters to derive macroscopic cross-diffusion-like systems;
4) characterize the equilibria of these simplified kinetic equations, study the stability of the equilibria, and analyze the trends to equilibrium of the kinetic solutions;
5) develop and implement an asymptotic-preserving numerical scheme able to simulate the evolution of the newly introduced Fokker-Planck equations and to capture their limit macroscopic models.
The main achievement of the MesoCroMo project is the development of a general multiscale approach to model nonconservative interactions in population dynamics. At its core is the design of biologically meaningful kinetic equations that bridge microscopic processes with macroscopic observables. The approach is flexible in that it can be applied in various ecological and epidemiological contexts, to derive relevant generalizations of well-known models like the SIR and Lotka-Volterra equations. By adapting the method to include spatial dynamics in the kinetic equations and by suitably merging diffusion and fast-reaction asymptotics, it has then been possible to justify the derivation of more complex macroscopic systems that involve cross-diffusion effects.
In more details, the research conducted during the MesoCroMo project has contributed to accomplish the following results.
1) Design of microscopic interactions and mesoscopic operators for multi-species systems. The novel idea has been to define the relevant microscopic trait as the size of the interacting groups from each species, and to characterize the exchange between groups through a combination of deterministic nonconservative interaction functions and random fluctuations. The balance of these microscopic exchanges has been accounted for by weak forms of Boltzmann-like operators, which have been used to construct the integro-differential mesoscopic system governing the evolution of the size distributions of the species.
2) Model reduction to Fokker-Planck equations. By considering a suitable quasi-invariant regime of the parameters that appear in the microscopic interactions, the next step has been to reduce the Boltzmann models to simpler kinetic Fokker-Planck equations. The structure of the latter is systematically given by a degenerate drift-diffusion operator prescribing the relaxation toward local equilibrium and a nonconservative reaction-type operator encoding the evolution of the underlying macroscopic dynamics. As an application of the method, it has been possible to derive kinetic generalizations of the SIR and Lotka-Volterra equations.
3) Extension to cross-diffusion modeling. With the introduction of transport operators to describe the change in spatial orientation of the species, we have considered a model extension to spatially inhomogeneous Fokker-Planck equations. By considering a parabolic rescaling of the space-time variables, we have then studied a suitable asymptotic regime of fast nonconservative reactions to derive macroscopic models that include cross-diffusion effects. In particular, the method is currently in development to prove that the SKT equations can be justified in this modeling context.
4) Analytical study. We have been able to compute explicitly the mesoscopic equilibria of these degenerate Fokker-Planck equations, which are Gamma-type distributions whose coefficients depend on the macroscopic densities of the species and on the microscopic parameters of the problem. We have carried out a stability analysis to ensure that the relevant moments of the distributions converge to the expected macroscopic equilibria. Moreover, by means of the entropy method we have performed a rigorous study of convergence to equilibrium, proving that the solutions to the Fokker-Planck equations converge toward the correct (local and global) equilibrium Gamma distributions.
5) Numerical study. We have designed and implemented a semi-implicit structure-preserving numerical solver able to approximate the evolution of the Fokker-Planck equations under study. The scheme is robust and it preserves the conservation properties of the equations, the positivity of the solutions and their trends to equilibrium. Moreover, it is asymptotic-preserving, in the sense that it is able to capture the limit macroscopic dynamics in suitable singular regimes of the parameters, without resolving the time step.
In more details, the research conducted during the MesoCroMo project has contributed to accomplish the following results.
1) Design of microscopic interactions and mesoscopic operators for multi-species systems. The novel idea has been to define the relevant microscopic trait as the size of the interacting groups from each species, and to characterize the exchange between groups through a combination of deterministic nonconservative interaction functions and random fluctuations. The balance of these microscopic exchanges has been accounted for by weak forms of Boltzmann-like operators, which have been used to construct the integro-differential mesoscopic system governing the evolution of the size distributions of the species.
2) Model reduction to Fokker-Planck equations. By considering a suitable quasi-invariant regime of the parameters that appear in the microscopic interactions, the next step has been to reduce the Boltzmann models to simpler kinetic Fokker-Planck equations. The structure of the latter is systematically given by a degenerate drift-diffusion operator prescribing the relaxation toward local equilibrium and a nonconservative reaction-type operator encoding the evolution of the underlying macroscopic dynamics. As an application of the method, it has been possible to derive kinetic generalizations of the SIR and Lotka-Volterra equations.
3) Extension to cross-diffusion modeling. With the introduction of transport operators to describe the change in spatial orientation of the species, we have considered a model extension to spatially inhomogeneous Fokker-Planck equations. By considering a parabolic rescaling of the space-time variables, we have then studied a suitable asymptotic regime of fast nonconservative reactions to derive macroscopic models that include cross-diffusion effects. In particular, the method is currently in development to prove that the SKT equations can be justified in this modeling context.
4) Analytical study. We have been able to compute explicitly the mesoscopic equilibria of these degenerate Fokker-Planck equations, which are Gamma-type distributions whose coefficients depend on the macroscopic densities of the species and on the microscopic parameters of the problem. We have carried out a stability analysis to ensure that the relevant moments of the distributions converge to the expected macroscopic equilibria. Moreover, by means of the entropy method we have performed a rigorous study of convergence to equilibrium, proving that the solutions to the Fokker-Planck equations converge toward the correct (local and global) equilibrium Gamma distributions.
5) Numerical study. We have designed and implemented a semi-implicit structure-preserving numerical solver able to approximate the evolution of the Fokker-Planck equations under study. The scheme is robust and it preserves the conservation properties of the equations, the positivity of the solutions and their trends to equilibrium. Moreover, it is asymptotic-preserving, in the sense that it is able to capture the limit macroscopic dynamics in suitable singular regimes of the parameters, without resolving the time step.
The key results of the MesoCroMo project have been the introduction of new kinetic operators modeling nonconservative reactive-like interactions in multi-species systems, the justification of relevant macroscopic models for population dynamics, the analytical study of the trends to equilibrium for solutions to degenerate Fokker-Planck equations with time-dependent coefficients, the design of robust asymptotic/structure-preserving numerical schemes to simulate these drift-diffusion processes, and a mesoscopic formulation of reaction-cross-diffusion systems that offers a new insightful multiscale perspective on the problem.
With a combination of modeling, theoretical and numerical approaches, the MesoCroMo project has provided several novelties with respect to the state of the art. An original use of models and methods coming from the kinetic theory of reactive gaseous mixtures has allowed to develop an innovative connection between macroscopic fast-reaction limits and mesoscopic relaxation to thermodynamical equilibrium, leading to a deeper understanding of the derivation of cross-diffusion operators from reaction-diffusion dynamics. As reaction-cross-diffusion systems appear in many different areas of physics, chemistry and biology, these mesoscopic equations and newly developed tools could be systematically applied to investigate a large class of problems, in connection with the modeling of multi-species systems. Every outcome of the project has established new interesting connections between these different fields and has brought together the corresponding scientific communities in the numerous dissemination activities performed during the action. Moreover, the analytical results have been numerically validated thanks to the design of a robust asymptotic-preserving scheme, which will be made available online encouraging the reuse of these models by other researchers to simulate real-world data. In particular, the methods and outcomes of MesoCroMo will have the potential to find application in many biological contexts (multi-species chemotaxis, spread of epidemics, angiogenesis), thus assessing the interdisciplinary value of the action.
The project will give an impulse to existing and new lines of research, since it naturally opens the way to next fundamental questions about the rigorous derivation of cross-diffusion equations from kinetic population models, in line with analogous studies performed in the context of multicomponent fluid dynamics. This would involve several critical points regarding the existence and uniqueness of solutions for degenerate kinetics equations in perturbative regimes around non-equilibrium states. Moreover, the models and methods that have been developed through MesoCroMo could find future application in relevant real-world scenarios like the conservation/restoration of ecosystems, the modeling of epidemics spread or the understanding of socio-economic behavioral interactions of individuals and groups, thus being of interest for a large class of scientists, from mathematicians, to ecologists, epidemiologists and economists. All these aspects ensure a long-term scientific impact for the project.
With a combination of modeling, theoretical and numerical approaches, the MesoCroMo project has provided several novelties with respect to the state of the art. An original use of models and methods coming from the kinetic theory of reactive gaseous mixtures has allowed to develop an innovative connection between macroscopic fast-reaction limits and mesoscopic relaxation to thermodynamical equilibrium, leading to a deeper understanding of the derivation of cross-diffusion operators from reaction-diffusion dynamics. As reaction-cross-diffusion systems appear in many different areas of physics, chemistry and biology, these mesoscopic equations and newly developed tools could be systematically applied to investigate a large class of problems, in connection with the modeling of multi-species systems. Every outcome of the project has established new interesting connections between these different fields and has brought together the corresponding scientific communities in the numerous dissemination activities performed during the action. Moreover, the analytical results have been numerically validated thanks to the design of a robust asymptotic-preserving scheme, which will be made available online encouraging the reuse of these models by other researchers to simulate real-world data. In particular, the methods and outcomes of MesoCroMo will have the potential to find application in many biological contexts (multi-species chemotaxis, spread of epidemics, angiogenesis), thus assessing the interdisciplinary value of the action.
The project will give an impulse to existing and new lines of research, since it naturally opens the way to next fundamental questions about the rigorous derivation of cross-diffusion equations from kinetic population models, in line with analogous studies performed in the context of multicomponent fluid dynamics. This would involve several critical points regarding the existence and uniqueness of solutions for degenerate kinetics equations in perturbative regimes around non-equilibrium states. Moreover, the models and methods that have been developed through MesoCroMo could find future application in relevant real-world scenarios like the conservation/restoration of ecosystems, the modeling of epidemics spread or the understanding of socio-economic behavioral interactions of individuals and groups, thus being of interest for a large class of scientists, from mathematicians, to ecologists, epidemiologists and economists. All these aspects ensure a long-term scientific impact for the project.