Diffusive Resource–Consumer Dynamics With the Simplest Learning Mechanism and Nonlocal Memory Usage

ABSTRACT To describe cognitive consumers' movement, we study a diffusive resource–consumer model with nonlocal memory usage described by a system of parabolic equations, which is coupled with spatial memory dynamics described by a linear learning equation. The analysis in this paper is mathematically challenging and originally resolves Open Problem 21 in [1] with the simplest learning mechanism. Under certain conditions, the system possesses a unique globally Hölder continuous weak solution. We investigate the existence of steady‐state bifurcation, as well as the bifurcation direction. Finally, a numerical simulation illustrates and verifies our key theoretical result.