Pattern formation in a coupled population-economy system with asymmetric density-dependent diffusion
摘要
This paper constructs a coupled population-economy reaction-diffusion model with asymmetric density-dependent diffusion, aiming to theoretically reveal how the mechanism of “congestion inhibiting diffusion and agglomeration enhancing diffusion” drives the self-organization of urban and regional spatial patterns. The critical conditions for Turing instability and the characteristic wavelength are rigorously derived, clarifying the regulatory role of density-dependent parameters on the instability threshold. Furthermore, the amplitude equations are derived using weakly nonlinear analysis, revealing the selection and stability mechanisms of spatial patterns near the bifurcation point. The interaction between Hopf bifurcation and Turing instability is also explored, and by establishing coupled amplitude equations, the resulting rich spatiotemporal dynamics are characterized. Numerical simulations are in excellent agreement with theoretical predictions and further reveal that, compared to linear diffusion, the density-dependent diffusion significantly accelerates the formation of spatial heterogeneous structures by introducing dynamic feedback mechanisms.