arXiv subjects

Davide Cusseddu

Publications and source records attributed to Davide Cusseddu.

5 recordsLinked to original sources

arXiv2026

On a phenotype-structured Shigesada--Kawasaki--Teramoto model: Turing instability and pattern selection under fast phenotype switching

The Shigesada-Kawasaki-Teramoto (SKT) model has become a classical modelling framework for studying spatial segregation and cross-diffusion-driven pattern formation in competing populations. This model assumes phenotypic homogeneity, but phenotypic variability persists within any population and can strongly influence both ecological and evolutionary dynamics. In this paper, we present a generalised phenotype-structured formulation of the SKT model that accounts for phenotypic variability. In this formulation, the competing populations are continuously structured across some phenotype state spaces. Population members move and compete in phenotype-dependent ways, and can also switch between different phenotype states. First we show how a form of the classical SKT model, wherein parameters are written in terms of continuous weighted averages of the phenotype-dependent functions of the generalised structured model, with weights given by the phenotype distributions of the two populations, can be obtained in the quasi-invariant regime of fast phenotype switching. Then, still assuming fast phenotype switching and extending classical Turing-like linear and weakly nonlinear analyses, we explore the conditions for the emergence of spatial patterns, identify a Turing-type bifurcation threshold leading to pattern formation, and investigate the nature of such a bifurcation (super- or sub-critical) as well as the stability of the patterned state. The results obtained make it possible to draw connections between phenotype-dependent model functions and the emergence of population-scale aggregate spatial dynamics, showing in particular how phenotype distributions can act as effective control parameters for Turing instability and pattern selection. These findings are complemented by numerical simulations, which validate the formal asymptotics and confirm the predictions of the pattern formation analyses.

q-bio.PE↗

arXiv2025

Reaction-diffusion systems from kinetic models for bacterial communities on a leaf surface

Many mathematical models for biological phenomena, such as the spread of diseases, are based on reaction-diffusion equations for densities of interacting cell populations. We present a consistent derivation of reaction-diffusion equations from systems of suitably rescaled kinetic Boltzmann equations for distribution functions of cell populations interacting in a host medium. We show at first that the classical diffusive limit of kinetic equations leads to linear diffusion terms only. Then, we show possible strategies in order to obtain, from the kinetic level, macroscopic systems with nonlinear diffusion and also with cross-diffusion effects. The derivation from a kinetic description has the advantage of relating reaction and diffusion coefficients to the microscopic parameters of the interactions. We present an application of our approach to the study of the evolution of different bacterial populations on a leaf surface. Turing instability properties of the relevant macroscopic systems are investigated by analytical methods and numerical tools, with particular emphasis on pattern formation for varying parameters in two-dimensional space domains.

math.AP↗

arXiv2024

Pattern formation of bulk-surface reaction-diffusion systems in a ball

Weakly nonlinear amplitude equations are derived for the onset of spatially extended patterns on a general class of n-component bulk-surface reaction-diffusion systems in a ball, under the assumption of linear kinetics in the bulk and coupling Robin-type boundary conditions. Linear analysis shows conditions under which various pattern modes can become unstable to either generalised pitchfork or transcritical bifurcations depending on the parity of the spatial wavenumber. Weakly nonlinear analysis is used to derive general expressions for the multi-component amplitude equations of different patterned states. These reduced-order systems are found to agree with prior normal forms for pattern formation bifurcations with O(3) symmetry and they provide information on the stability of bifurcating patterns of different symmetry types. The analysis is complemented with numerical results using a dedicated bulk-surface finite-element method. The theory is illustrated in two examples: a bulk-surface version of the Brusselator and a four-component bulk-surface cell-polarity model.

nlin.PS↗

arXiv2022

Numerical investigations of the bulk-surface wave pinning model

The bulk-surface wave pinning model is a reaction-diffusion system for studying cell polarisation. It is constituted by a surface reaction-diffusion equation, coupled to a bulk diffusion equation with a non-linear boundary condition. Cell polarisation arises as the surface component develops specific patterns. Since proteins diffuse much faster in the cell interior than on the membrane, in the literature, the bulk component is often assumed to be spatially homogeneous. {\re Therefore, the model can be reduced to a single surface equation}. However, {\re in real applications} a spatially non-uniform bulk component might be an important player to take into account. In this paper, we study, through numerical computations, the role of the bulk component and, more specifically, how different bulk diffusion rates might affect the polarisation response. We find that the bulk component is indeed a key factor in {\re determining the surface polarisation response}. Moreover, for certain geometries, it is the spatial heterogeneity of the bulk component that triggers the polarisation response, which might not be possible in a reduced model. Understanding how polarisation depends on bulk diffusivity might be crucial when studying models of migrating cells, which are naturally subject to domain deformation.

physics.bio-ph↗

arXiv2018

A coupled bulk-surface model for cell polarisation

Several cellular activities, such as directed cell migration, are coordinated by an intricate network of biochemical reactions which lead to a polarised state of the cell, in which cellular symmetry is broken, causing the cell to have a well defined front and back. Recent work on balancing biological complexity with mathematical tractability resulted in the proposal and formulation of a famous minimal model for cell polarisation, known as the wave pinning model. In this study, we present a three-dimensional generalisation of this mathematical framework through the maturing theory of coupled bulk-surface semilinear partial differential equations in which protein compartmentalisation becomes natural. We show how a local perturbation over the surface can trigger propagating reactions, eventually stopped in a stable profile by the interplay with the bulk component. We describe the behavior of the model through asymptotic and local perturbation analysis, in which the role of the geometry is investigated. The bulk-surface finite element method is used to generate numerical simulations over simple and complex geometries, which confirm our analysis, showing pattern formation due to propagation and pinning dynamics. The generality of our mathematical and computational framework allows to study more complex biochemical reactions and biomechanical properties associated with cell polarisation in multi-dimensions.

q-bio.CB↗