Pattern transitions in a nonlocal logistic map for populations

TL;DR

This paper investigates pattern solutions in a doubly nonlocal logistic map, revealing different stationary states, their stability, and transitions, with applications to nonlinear optics phenomena.

Contribution

It introduces a comprehensive analysis of pattern solutions in a nonlocal logistic map, including stability criteria and connections to optical phenomena.

Findings

Identified three types of stationary solutions: uniform, wavelike, and Gaussian.

Derived analytical expressions for nonlinear pattern behavior.

Established criteria for pattern stability and transitions.

Abstract

In this work, we study the pattern solutions of doubly nonlocal logistic map that include spatial kernels in both growth and competition terms. We show that this map includes as a particular case the nonlocal Fisher-Kolmogorov equation, and we demonstrate the existence of three kinds of stationary nonlinear solutions: one uniform, one cosine type that we refer to as wavelike solution, and another in the form of Gaussian. We also obtain analytical expressions that describe the nonlinear pattern behavior in the system, and we establish the stability criterion. We define thermodynamics grandeurs such as entropy and the order parameter. 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 nonlinear optics.