Eur. Phys. J. Plus (2021) 136: 256
https://doi.org/10.1140/epjp/s13360-021-01251-z

Regular Article

Spatial patterns in a vegetation model with internal competition and feedback regulation

¹ Department of Mathematics and Informatics, Center of Belhadj Bouchaib Ain Temouchent, BP 284 RP, 46000, Ain Temouchent, Algeria
² Laboratoire d’Analyse Non Linéaire et Mathématiques Appliquées. Département de Mathématiques, Université Abou Bakr Belkaïd, 13000, Tlemcen, Algeria
³ Faculty of Exact sciences and informatics, Mathematic Department, Hassiba benbouali university, Chlef, Algeria
⁴ Department of Engineering Science, Kermanshah University of Technology, Kermanshah, Iran
⁵ Department of Mathematics, Faculty of Engineering and Natural Sciences, Bahçeşehir University, 34349, Istanbul, Turkey
⁶ Department of Mathematics, National Institute of Technology, 831014, Jamshedpur, Jharkhand, India

^d skumar.math@nitjsr.ac.in

Received: 15 December 2020
Accepted: 18 February 2021
Published online: 24 February 2021

Abstract

The vegetation patterns are a characteristic particularity of semiarid zones, which can be the future of modern ecology for its importance. This paper aims to study a diffusive vegetation model for the subject of studying the complex patterns generated by the presence of Turing-Hopf bifurcation. The main focus is on analyzing the effect of the locative internal rivalry between plants and feedback regulation on the pattern formations. The cross-diffusion produced by the positive feedback regulation generates surprising dynamics such as Hopf bifurcation, Turing bifurcation, Turing-Hopf bifurcation, which confirms the imbalance of the distribution of the vegetation in the semi-desert regions. For analyzing the spatiotemporal behavior near the Turing-Hopf bifurcation point, the Amplitude equation restricted at this point is used. Further, by using numerical simulations, the complex dynamics induced by the positive feedback redistribution and inner competition are explored.