Nonlinear solution for a doubly nonlocal population dynamics: pattern transitions

In this work, we investigate the pattern solutions of doubly non-local Fisher population equation that include spatial kernels in both growth and competition terms. We show the existence of two types of stationary nonlinear solutions: one cosine, which we refer to as a wavelike solution and another in the form of Gaussians. We obtain analytical expressions that describe the nonlinear pattern behavior in the system and establish the stability criterion. Based on this, the pattern-no-pattern and pattern-pattern transitions are properly analyzed. We show that these pattern solutions may be related to the recently observed peak-adding phenomenon in non-linear optics.