Computational Algorithm for Some Problems with Variable Geometrical Structure

International audienceThe work is devoted to the computational algorithm for a problem of plant growth. The plant is represented as a system of connected intervals corresponding to branches. We compute the concentration distributions inside the branches. The originality of the problem is that the geometry of the plant is not a priori given. It evolves in time depending on the concentrations of plant hormones found as a solution of the problem. New branches appear in the process of plant growth. The algorithm is adapted to an arbitrary plant structure and an arbitrary number of branches

Reaction-diffusion waves with nonlinear boundary conditions

International audienceA reaction-di usion equation with nonlinear boundary condition is considered in a two-dimensional in nite strip. Existence of waves in the bistable case is proved by the Leray-Schauder method

Mathematical Modeling Reveals That the Administration of EGF Can Promote the Elimination of Lymph Node Metastases by PD-1/PD-L1 Blockade

In the advanced stages of cancers like melanoma, some of the malignant cells leave the primary tumor and infiltrate the neighboring lymph nodes (LNs). The interaction between secondary cancer and the immune response in the lymph node represents a complex process that needs to be fully understood in order to develop more effective immunotherapeutic strategies. In this process, antigen-presenting cells (APCs) approach the tumor and initiate the adaptive immune response for the corresponding antigen. They stimulate the naive CD4+ and CD8+ T lymphocytes which subsequently generate a population of helper and effector cells. On one hand, immune cells can eliminate tumor cells using cell-cell contact and by secreting apoptosis inducing cytokines. They are also able to induce their dormancy. On the other hand, the tumor cells are able to escape the immune surveillance using their immunosuppressive abilities. To study the interplay between tumor progression and the immune response, we develop two new models describing the interaction between cancer and immune cells in the lymph node. The first model consists of partial differential equations (PDEs) describing the populations of the different types of cells. The second one is a hybrid discrete-continuous model integrating the mechanical and biochemical mechanisms that define the tumor-immune interplay in the lymph node. We use the continuous model to determine the conditions of the regimes of tumor-immune interaction in the lymph node. While we use the hybrid model to elucidate the mechanisms that contribute to the development of each regime at the cellular and tissue levels. We study the dynamics of tumor growth in the absence of immune cells. Then, we consider the immune response and we quantify the effects of immunosuppression and local EGF concentration on the fate of the tumor. Numerical simulations of the two models show the existence of three possible outcomes of the tumor-immune interactions in the lymph node that coincide with the main phases of the immunoediting process: tumor elimination, equilibrium, and tumor evasion. Both models predict that the administration of EGF can promote the elimination of the secondary tumor by PD-1/PD-L1 blockade

Properness and Topological Degree for Nonlocal Reaction-Diffusion Operators

International audienceThe paper is devoted to integro-differential operators, which correspond to non-local reaction-diffusion equations considered on the whole axis. Their Fredholm property and properness will be proved. This will allow one to define the topological degree

Dynamical system model of decision making and propagation

International audienceIndividual decision making is described as a bistable dynamical system. It can be influenced by the environment represented by other individuals, public opinion, all kinds of visual, oral and other information. We will study how the interaction of the individual decision making with the environment results in various patterns of decision making in the society

Solvability Conditions for Some non Fredholm Operators

We obtain solvability conditions for some elliptic equations involving non Fredholm operators with the methods of spectral theory and scattering theory for Schrodinger type operators. Though the Fredholm property is not satisfied, the solvability conditions are formulated in terms of orthogonality of the right-hand side to solutions of the homogeneous adjoint equation

How morphology of artificial organisms influences their evolution

International audienceThe principle of natural selection implies that variations are transmitted from parents to offsprings. The individuals with advantageous variations have better fitness. Consequently, such variations spread in the population and influence its evolution. This schematic description is conventionally accepted but it jumps over an important step: how variations are related to fitness. In the other words, how the phenotype is related to the reproduction and mortality rates. It is important to note that this relation will not be imposed by the assumptions of the model but it should follow from the morphology of the artificial organisms. In order to study this question, we will introduce in this work virtual populations of artificial organisms and will observe their behavior. The main idea of this study is that we prescribe individual characteristics of the organisms (size, form) but not their behavior in the search for resources. The model presented below will allow us to study on a simple example the interaction between morphology and natural selection, or, in a more general formulation, the evolution of the phenotype. 1.1. Artificial life models Artificial life models are largely used to study behavior of biological organisms at the individual level, their collective behavior and evolution. We will consider a complete life cycle model which includes the genotype of the organisms in its relation to the phenotype, the mechanism of motion and food search determined by the morphology of the organisms, and reproduction (Fig. 1). A B S T R A C T The purpose of this work is to study virtual populations of artificial organisms with their genotype, morphology, mechanism of motion, search and competition for food, reproduction, mutations. The genotype determines the phenotype (morphology), while morphology determines efficiency of motion and success in the search for food in the competition with other individuals; sufficient amount of food allows reproduction. Ensemble of these elements constitutes the minimal model to study natural selection of artificial organisms. Considering only some of them, as it is often the case in artificial life models, can be used for the optimization of some properties (for example, robot's gait or embryo's form) but not to study natural selection in the evolutionary context. Artificial organisms are considered in this work in the form of polygons (triangles) on the plane. Their genotype is given by three positive numbers associated to the vertices and their morphology is determined by the lengths of the sides equal the sum of the numbers in the adjacent vertices. Behavior of the individuals and their success in the search for food depend on their morphology. More efficient individuals will reproduce more than the others and will transmit their advantageous variations to their offsprings. Hence we can observe how natural selection chooses more efficient morphology and how it evolves due to random mutations. We develop an individual based model where the individuals recognize food and move to it with the speed determined by their morphology (and not prescribed in the algorithm). If they have enough food, they survive and reproduce. Therefore morphology and evolution are tightly interconnected and should be studied together. Dynamics of such populations appears to be different from the dynamics described by conventional models of competition and evolution of species. In particular, a new phenotype can emerge due to a different strategy of foraging (related to a different morphology) and not only due to a difference in consumed resources with the existing phenotype. We also observe that realization of Cope's rule (increase of body size in the process of evolution) can depend on parameters of the model.