Fisher-KPP Equations And Chemotaxis Models on Metric Graphs

Abstract

This dissertation is devoted to the study of population dynamics on network-like structures both in the absence and presence of chemical signals that guide population movement. Mathematically, population movement in response to certain chemical signals is often modeled by so-called Keller-Segel chemotaxis systems. In the absence of chemotaxis sensitivity and assuming logistic population growth, such systems reduce to the so-called Fisher-KPP equation. In this dissertation, we investigate the global existence and dynamical aspects of these models on both bounded and unbounded metric graphs.

The first part of our study is focused on bounded or compact graphs. We start by proving the existence of local classical solutions for parabolic-parabolic and parabolic-elliptic Keller-Segel chemotaxis models with general non-linear coefficients following polynomial growth assumptions. Next, we prove global uniform boundedness of solutions for Keller-Segel models with logistic source without any condition on the chemotaxis sensitivity. Moreover, we study dynamical aspects of the two chemotaxis systems with a logistic source term on compact metric graphs. It can be easily verified that the positive constant solution of these systems on compact graphs is globally stable in the absence of chemotaxis sensitivity, i.e, χ = 0, in which case the systems reduce to the Fisher-KPP equation. In this dissertation, we determine a threshold value χ*> 0 of the chemotaxis sensitivity parameter that separates the regimes of local asymptotic stability and instability, and, in addition, determine the parameter intervals that facilitate global asymptotic convergence of solutions with positive initial data to constant steady states. In addition, we provide a sequence of bifurcation points for the chemotaxis sensitivity parameter that yields non-constant steady state solutions. In particular, we show that the first bifurcation point coincides with threshold value χ* for a generic compact metric graph. Finally, we supply numerical computation of bifurcation points for several graphs.

Next, we focus on population dynamics on unbounded graphs. Here, we study the global existence, spreading speed, and traveling wave solutions of the Fisher-KPP equation on unbounded graphs with finitely many edges. Note that there are several studies on front propagation phenomena in bistable equations on unbounded metric graphs with finitely many edges. From these works, it is known that in such equations the network structure of the underlying environment may block the propagation of the fronts. We prove that, unlike the case in bistable equations, the network structure of the environments doesn’t block the propagation of fronts in Fisher-KPP equations. In particular, we show that the Fisher-KPP equation exhibits the same spreading speed c* as on the real line. Moreover, it admits a generalized traveling wave solution connecting the stable positive constant solution and the trivial solution, with an averaged speed c for any c > c*. We concluded by mentioning current and future works in the study of global existence and long-time dynamics of chemotaxis equations on unbounded graphs.