# Existence and uniqueness of monotone wavefronts in a nonlocal resource-limited model

## TL;DR

This paper establishes the existence and uniqueness of monotone wavefronts in a nonlocal resource-limited model, broadening the understanding of wave solutions and revealing new non-oscillating fronts through numerical simulations.

## Contribution

It provides a broad and explicit existence theorem for monotone wavefronts in a nonlocal model, including the first numerical observation of non-oscillating and non-monotone fronts.

## Key findings

- Existence of monotone wavefronts for a broad parameter range
- Numerical evidence of non-oscillating and non-monotone fronts
- Uniqueness of monotone fronts up to translation

## Abstract

We are revisiting the topic of travelling fronts for the food-limited (FL) model with spatio-temporal nonlocal reaction. These solutions are crucial for understanding the whole model dynamics. Firstly, we prove the existence of monotone wavefronts. In difference with all previous results formulated in terms of `sufficiently small parameters', our existence theorem indicates a reasonably broad and explicit range of the model key parameters allowing the existence of monotone waves. Secondly, numerical simulations realized on the base of our analysis show appearance of non-oscillating and non-monotone travelling fronts in the FL model. These waves were never observed before. Finally, invoking a new approach developed recently by Solar $et\ al$, we prove the uniqueness (for a fixed propagation speed, up to translation) of each monotone front.