# Introduction to Delay Models and Their Wave Solutions

## TL;DR

This paper reviews delay population models in ecology, focusing on wave solutions, stability, and recent advances, highlighting their ability to capture complex ecological phenomena like invasions and species interactions.

## Contribution

It provides a comprehensive overview of recent developments in traveling and stationary wave solutions of delay models in ecological contexts.

## Key findings

- Existence of stationary and traveling wave solutions
- Stability analysis of wave solutions
- Formation of wavefronts in delay models

## Abstract

In this paper, a brief review of delay population models and their applications in ecology is provided. The inclusion of diffusion and nonlocality terms in delay models has given more capabilities to these models enabling them to capture several ecological phenomena such as the Allee effect, waves of invasive species and spatio-temporal competitions of interacting species. Moreover, recent advances in the studies of traveling and stationary wave solutions of delay models are outlined. In particular, the existence of stationary and traveling wave solutions of delay models, stability of wave solutions, formation of wavefronts in the special domain, and possible outcomes of delay models are discussed.