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Soft interfaces formed by lipid membranes are fundamental to living cells, synthetic cells, and membrane-based soft materials. However, a quantitative framework linking molecular organization with nonlinear interfacial mechanics remains…
The elapsed-time model describes the behavior of interconnected neurons through the time since their last spike. It is an age-structured non-linear equation in which age corresponds to the elapsed time since the last discharge, and models…
Age-structured models capture the dynamic behavior of populations over time and result in nonlinear integro-partial differential equations (IPDEs). These processes arise in various fields such as biotechnology, economics, or demography.…
We provide a review of recent advancements in nonlocal continuous models for migration, mainly from the perspective of its involvement in embryonal development and cancer invasion. Particular emphasis is placed on spatial nonlocality…
Cell-cell adhesion plays a vital role in the development and maintenance of multicellular organisms. One of its functions is regulation of cell migration, such as occurs, e.g. during embryogenesis or in cancer. In this work, we develop a…
We derive amplitude equations for interface dynamics in pattern forming systems with long-range interactions. The basic condition for the applicability of the method developed here is that the bulk equations are linear and solvable by…
We study dynamics emergent from a two-dimensional reaction--diffusion process modelled via a finite lattice dynamical system, as well as an analogous PDE system, involving spatially nonlocal interactions. These models govern the evolution…
We propose a variational formulation for the nonequilibrium thermodynamics of discrete open systems, i.e., discrete systems which can exchange mass and heat with the exterior. Our approach is based on a general variational formulation for…
A nonlinear inequality is formulated in the paper. An estimate of the rate of growth/decay of solutions to this inequality is obtained. This inequality is of interest in a study of dynamical systems and nonlinear evolution equations. It can…
In this paper, we studied phase-space analysis of a certain mathematical model of tumor growth with an immune responses and chemotherapy therapy. Mathematical modelling of this process is viewed as a potentially powerful tool in the…
We discuss pore dynamics in osmotically stressed vesicles. A set of equations which govern the liposomal size, internal solute concentration, and pore diameter is solved numerically. We find that dependent on the internal solute…
We present a theoretical agent-based model of cell evolution under the action of cytotoxic treatments, such as radioteraphy or chemoteraphy. The major features of cell cycle and proliferation, cell damage and repair, and chemical diffusion…
Apoptosis is an important area of research because of its role in keeping a mature multicellular organism's number of cells constant hence, ensuring that the organism does not have cell accumulation that may transform into cancer with…
We numerically address the stability analysis of linear age-structured population models with nonlocal diffusion, which arise naturally in describing dynamics of infectious diseases. Compared to Laplace diffusion, models with nonlocal…
We introduce and analyse a continuum model for an interacting particle system of Vicsek type. The model is given by a non-linear kinetic partial differential equation (PDE) describing the time-evolution of the density $f_t$, in the single…
A connection is established between discrete stochastic model describing microscopic motion of fluctuating cells, and macroscopic equations describing dynamics of cellular density. Cells move towards chemical gradient (process called…
Continuum mathematical models for collective cell motion normally involve reaction-diffusion equations, such as the Fisher-KPP equation, with a linear diffusion term to describe cell motility and a logistic term to describe cell…
An overdamped system with a linear restoring force and two multiplicative colored noises is considered. Noise amplitudes depend on the system state $x$ as $x$ and $|x|^{\alpha}$. An exactly soluble model of a system is constructed due to…
A class of coupled cell-bulk ODE-PDE models is formulated and analyzed in a two-dimensional domain, which is relevant to studying quorum sensing behavior on thin substrates. In this model, spatially segregated dynamically active signaling…
We describe a percolation-type approach to modeling of the processes of aging and certain other properties of tissues analyzed as systems consisting of interacting cells. Tissues are considered as structures made of regular healthy,…