Complex Behavior Analysis of a Fractional-Order Land Dynamical Model with Holling-II Type Land Reclamation Rate on Time Delay

Zhouhong Li,Li Wu,Binggeng Xie,Yuan Zhang

doi:10.1155/2020/1053283

Abstract

In this paper, a fractional-order land model with Holling-II type transformation rate and time delay is investigated. First of all, the variable-order fractional derivative is defined in the Caputo type. Second, by applying time delay as the bifurcation parameter, some criteria to determine the stability and Hopf bifurcation of the model are presented. It turns out that the time delay can drive the model to be oscillatory, even when its steady state is stable. Finally, one numerical example is proposed to justify the validity of theoretical analysis. These results may provide insights to the development of a reasonable strategy to control land-use change.

Highlights

Since the 20th century, the global environment has changed at an unprecedented speed, and a series of major global environmental problems pose a serious threat to the survival and development of mankind
From the perspective of earth system science, the generation of global environmental problems can be recognized as the result of the interaction between the earth’s atmosphere, hydrosphere, biosphere, lithosphere, and human activities [1, 2]
Hydrology, topography, and geomorphology [6, 7] are the main driving forces in the natural system, while economic development, population growth, and policies are the main driving forces in the social and economic systems [8]. It is the interwoven factors in these two systems that cause land use changes [9], but compared with the natural factors, human factors play a dominant role [10]

Summary

Introduction

Since the 20th century, the global environment has changed at an unprecedented speed, and a series of major global environmental problems pose a serious threat to the survival and development of mankind. To the best of our knowledge, there are few studies to investigate the existence of Hopf bifurcation to the fractional-order land model with time delay. Motived by the above ideas, in this paper, we will consider the following fractional-order land model with Holling-II type response and time delay:. (1) A novel fractional-order land model with Holling-II type land reclamation rate and time delay is formulated (2) Two primary dynamical properties—stability and oscillation—of the delayed fractional-order land model are investigated (3) e influences of the order on the Hopf bifurcation are obtained roughout this paper, we address the following assumptions.

Basic Tools of Fractional Calculus

Main Results

Numerical Example and Application