Dynamics of coupled nonlinear diffusion–reaction systems

Abstract

We study two coupled nonlinear Fisher–Kolmogorov equations with anomalous diffusion. Because analytical solutions are difficult to obtain, we use a reduced collective-coordinate method based on two time-dependent power-law assumptions. This approach reduces the original partial differential equations to a simpler system of coupled ordinary differential equations that we can analyze semi-analytically. The simplified dynamics show two different populations. One of them follows logistic-like growth, described by an autonomous Bernoulli equation, and settles into a steady state. The other one grows quickly and reaches a point where the reduced model no longer works. By numerical methods, we solve the coupled nonlinear Fisher–Kolmogorov equations directly to compare both results. Our findings offer a simplified perspective on interacting nonlinear diffusion–reaction systems with unusual transport and nonlinear coupling. The comparison of the reduced system with the numerical results yields the interval of time where both solutions coincide. This approach provides a tractable route to study emergent dynamical structures in coupled nonlinear diffusion–reaction systems.