Mathematical modeling and simulation of the spatial dynamics of voles populations in eastern France

• Main Author: Nguyen, Thi Nhu Thao
• Other Authors: Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté COMUE (UBFC)-Université Bourgogne Franche-Comté COMUE (UBFC), Université Bourgogne Franche-Comté, Ulrich Jerry Razafison, Carlotta Donadello, Giuseppe Maria Coclite
• Format: Doctoral or Postdoctoral Thesis
• Language: English
• Published: HAL CCSD 2020
• Subjects: Finite volumes method; Parabolic–hyperbolic equation; Compensated compactness; Nonlocal Boundary value problem; Prey-Predator systems; Transport equations; Méthode des volumes finis; Équation parabolique – hyperbolique; Compacité par compensation; Problème aux limites non locales; Systèmes proie-Prédateur; Équations de transport; [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]; Microtus arvalis
• Online Access: https://theses.hal.science/tel-03148955; https://theses.hal.science/tel-03148955/document; https://theses.hal.science/tel-03148955/file/these_A_NGUYEN_ThiNhuThao_2020.pdf

The main objective of the thesis is to propose and analyze mathematical models based on partial differential equations (PDE) to describe the spatial dynamics of two species of voles (Microtus arvalis and Arvicola terrestris), which are particularly monitored in Eastern France. The models that we have proposed are based on PDE which describe the evolution of the density of the population of voles as a function of time, age and position in space. We have two complementary approaches to represent the dynamics. In the first approach, we propose a first model that consists of a scalar PDE depending on time, age, and space supplemented with a non-local boundary condition. The flux is linear with constant coefficient in the direction of age but contains a non-local term in the directions of space. Moreover, the equation contains a second order term in the spatial variables only. We have demonstrated the existence and stability of weak entropy solutions for the model by using, respectively, the Panov's theorem of the multidimensional compensated and a doubling of the variables type argument. In the second approach we were inspired by a Multi Agent model proposed by Marilleau-Lang-Giraudoux, where the spatial dynamics of juveniles is decoupled from local evolution in each plot. To apply this model, we have introduced a directed graph whose nodes are the plots. In each node, the evolution of the colony is described by a transport equation with two variables, time and age, and the movements of dispersion, in space, are represented by the passages from one node to the other. We have proposed a discretization of the model, by finite volume methods, and noticed that this approach manages to reproduce the qualitative characteristics of the spatial dynamics observed in nature. We also proposed to consider a predator-prey system consisting of a hyperbolic equation for predators and a parabolic-hyperbolic equation for preys, where the prey's equation is analogous to the first model of the vole populations. The drift term in the ...