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We study a new nonlocal approach to the mathematical modelling of the Chemotaxis problem, which describes the random motion of a certain population due a substance concentration. Considering the initial-boundary value problem for the…
The main objective of this article is to study the dynamic transition and pattern formation for chemotactic systems modeled by the Keller-Segel equations. We study chemotactic systems with either rich or moderated stimulant supplies. For…
Motivated by an ongoing collaboration with clinical oncologists and pathologists, we develop a hybrid partial differential equation--ordinary differential equation (PDE--ODE) framework that captures (i) competition between susceptible and…
It is known that in two dimensions the classical Keller-Segel model can lead to cell aggregation. This behavior can be controlled by adding a logistic growth term with quadratic decay. Researchers have tried to find weaker damping…
Mathematical models that describe the tumor growth process have been formulated by several authors in order to understand how cancer develops and to develop new treatment approaches. In this study, it is aimed to investigate the long-time…
This paper consists of a detailed and novel stochastic optimal control analysis of a coupled non-linear dynamical system. The state equations are modeled as additional food provided prey-predator system with Holling Type-III functional…
The paper arXiv:1804.11290 contains well-posedness and regularity results for a system of evolutionary operator equations having the structure of a Cahn-Hilliard system. The operators appearing in the system equations were fractional…
Three-dimensional cultures of cells are gaining popularity as an in vitro improvement over 2D Petri dishes. In many such experiments, cells have been found to organize in aggregates. We present new results of three-dimensional in vitro…
The interactions between tumor cells and the immune system play a crucial role in cancer evolution. In this study, we explore how these interactions influence cancer progression by modeling the relationships among naive T cells, effector T…
We study the stationary Keller--Segel chemotaxis models with logistic cellular growth over a one-dimensional region subject to the Neumann boundary condition. We show that nonconstant solutions emerge in the sense of Turing's instability as…
Cancer is a very complex phenomenon that involves many different scales and situations. In this paper we consider a free boundary problem describing the evolution of a tumor colony and we derive a new asymptotic model for tumor growth. We…
We consider a state-constrained optimal control problem of a system of two non-local partial-differential equations, which is an extension of the one introduced in a previous work in mathematical oncology. The aim is to minimize the tumor…
We introduce mesoscopic and macroscopic model equations of chemotaxis with anomalous subdiffusion for modelling chemically directed transport of biological organisms in changing chemical environments with diffusion hindered by traps or…
This article presents a partial differential equation (PDE) of Keller-Segel (KS) type that reproduces patterns commonly observed during the growth of brain microvasculature. We provide mathematical insights into the mechanisms underlying…
A Cahn-Hilliard equation with stochastic multiplicative noise and a random convection term is considered. The model describes isothermal phase-separation occurring in a moving fluid, and accounts for the randomness appearing at the…
A mathematical model for time development of metastases and their distribution in size and carrying capacity is presented. The model is used to theoretically investigate anti-cancer therapies such as surgery and chemical treatments…
In this paper, we consider a two-dimensional diffuse interface model for the phase separation of an incompressible and isothermal binary fluid mixture with matched densities. This model consists of the Navier--Stokes equations, nonlinearly…
The aim of this notes is to give a concise introduction to control theory for systems governed by stochastic partial differential equations. We shall mainly focus on controllability and optimal control problems for these systems. For the…
We propose a diffuse interface model to describe tumor as a multicomponent deformable porous medium. We include mechanical effects in the model by coupling the mass balance equations for the tumor species and the nutrient dynamics to a…
We consider a system of two coupled integro-differential equations modelling populations of healthy and cancer cells under therapy. Both populations are structured by a phenotypic variable, representing their level of resistance to the…