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Kong, Fanze

University of British Columbia

This thesis is devoted to the analysis of several reaction-diffusion equations involving advection, which serve as paradigms to describe biological, ecological and financial phenomena since they admit rich dynamics and generate various self-organized patterns. In Chapter 2, we focus on population models which are proposed to quantitatively explore the effect of large biased movement on strong Allee effect. Concerning the population dynamics of single species, we intensively study the existence and local stability of strong localized patterns. Moreover, we extend our analysis into the multi-populations counterpart and explore the coexistence of two competing species. Chapter 3 and Chapter 4 focus on the qualitative properties of localized patterns to Keller-Segel models with logistic growth, in which two distinct regimes are considered: the strong chemotactic movement and the small chemical diffusion rate. More specifically, in Chapter 3, we are concerned with the former regime: in the one-dimensional setting, we construct the single boundary spike and show its local stability; in the two-dimensional case, we prove the existence of multi-spots. Chapter 4 is devoted to the latter regime: first of all, the asymptotic forms of N-spike quasi-equilibria and equilibria are derived formally; next, the Differential-Algebraic Equations (DAEs) governing the slow motions of collective locations of the N-spike quasi-pattern are formulated. Finally, the instability thresholds of N-spike pattern associated with large and small eigenvalues are computed explicitly. In Chapter 5, we consider chemotaxis-fluid systems with the self-consistent coupling in 2-D and prove the global existence of classical solutions with the subcritical cellular mass, and then construct the boundary spots rigorously. In Chapter 6, we are concerned with the stationary mean-field games (MFG) system in some unbounded domain that involve a decreasing cost. Assume the local coupling is mass critical and imposing some technical assumptions on potential functions, we first establish the Gagliardo-Nirenberg type inequality corresponding to the potential-free MFG system. Next, the existence of ground states are classified in terms of the mass of population density. Finally, we investigate the asymptotic profiles of minimizers as the mass increases to the critical one.

Text

2024-07-17

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/88660

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/