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.

1. 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. 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]. Land use/land cover change (LUCC) is as a key link connecting the four circles of the earth system, being an ideal entry point for the study of natural and human processes, and has rapidly become the focus of global change research [3, 4]. LUCC dynamic change is to describe, evaluate, interpret, and predict the quantity, quality, spatial distribution, classification, rate, and process of land use/land cover change by accumulating spatiotemporal continuous land use/land cover data and using the mathematical model [5]. The driving forces of land use change are mainly divided into natural and social systems. Notably, 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]. Generally speaking, it is the interwoven factors in these two systems that cause land use changes [9], but compared with the natural factors (such as climate, soil, and topography), human factors (such as regional policy, economic development, and population growth) play a dominant role [10].

Since the key of LUCC research is to understand the driving force and driving mechanism [11], a variety of system analysis and mathematical statistic methods have been well applied in the research of driving force. For example, the internal and human driving force of land use change is studied through the analysis of the multiple linear regression model [12]. The system dynamic model was used to quantitatively diagnose the contribution of each driving factor to land use change [13]. The CLUE comprehensive model was used to establish the interaction between land-use change and its influencing factors, and land change and its spatial distribution were simulated [14–16]. Therefore, the comprehensive mathematical model will be a development trend of LUCC research at present and in the future.

In 1997, Dobson et al. considered the land model with four coupled differential equations [17]:where stands for an original area of pristine forest habitat, denotes the agricultural land, unused land is , and is the population density with time , for more detail, one can read [17] and the references cited therein. In [18], Chen et al. studied the dynamical analysis of the land model with Holling-II type land reclamation rate, and they found the positive equilibrium point, the basic reproductive rate , and, furthermore, some sufficient conditions for the global stability of the positive equilibrium and one of the boundary equilibrium.

On the one hand, it is well known that time delays are unavoidable for population modelling. It is therefore important to consider the dynamics for the population models with time delays, and dynamic complex analysis is obviously one of the most important problems. For example, delay differential equations have been applied in the stability of population dynamic systems [19–25], impulsive effects, and control systems [26–28].

On the other hand, there are two types of differential equations, namely, integer-order and fractional-order differential equations. Traditionally, the fractional-order differential equations have enjoyed a preference over integer-order differential equations because of the mathematical tractability of fractional systems. Over the past 30 decades, fractional-order differential equations have been applied in many fields, such as design and control of various ecological systems [29–32], secure communication [33–35], and system control [36–42].

Recently, the existence of Hopf bifurcation of differential equations has been studied as an important qualitative behavior of integer-order differential equations [43]. Moreover, fractional calculus is merged into complicated, dynamical systems which extremely renovate the theory of the design and control performance for complex systems. It has been discovered that physical phenomena in nature can be depicted more accurately by fractional-order models in comparison with classical integer-order ones [44, 45]. Some scholars introduced fractional calculus into population models and constructed fractional population or epidemic models [46–50], fractional dynamic systems, and fractional neural networks [51–53]. However, 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:where is the fractional order, represent the land of wood and grass, survival land, and unused land, respectively, and is the density of the predator population at time ; (land area of survival) of a period of time becomes unused land (area ), which in turn recovers through natural succession or ecological restoration to become a forest after a time interval . The unused land may also be restored to survival viable land after a time interval . Average clearing ability is described by constant . is the growth rate into the population. is the time delay required for the gestation of the mature population.

The main contributions can be summed up in three aspects:(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)The influences of the order on the Hopf bifurcation are obtained

Throughout this paper, we address the following assumptions.

Assumption 1. .

Assumption 2. .
Suppose Assumptions 1 and 2 hold, then system (2) has an unique positive equilibrium point , which is described byThe remainder of the current paper is organized as follows: In Section 2, some definitions and lemmas of fractional calculus are recalled. In Section 3, the fractional land model with Holling-II type land reclamation rate and time delay is investigated by using time delay as the bifurcation parameter, and the conditions of Hopf bifurcation are presented. One numerical example is given to illustrate the effectiveness of our main results in Section 4. Finally, conclusions are drawn in the last section.

2. Basic Tools of Fractional Calculus

In this section, we recall some definitions and lemmas of fractional calculus, which can be used in the proofs of the main results in Section 3.

There are several definitions of fractional derivatives. The Grnwald–Letnikov definition, the Riemann–Liouville definition, and the Caputo definition are usually used to deal with fractional-order systems. Since the Caputo derivative only requires the initial conditions which are based on the integer-order derivative and represents well-understood features of physical situation, it is more applicable to real-world problems. Hence, the Caputo fractional-order derivative is adopted in this paper.

Definition 1. (see [44]). The fractional integral of order for a function is defined aswhere , , is the gamma function, and .

Definition 2. (see [44]). The Caputo fractional derivative of order for a function is defined bywhere and is a positive integer such that .
Moreover, when ,

Lemma 1. (see [41]). The following autonomous systemwhere , , and , is asymptotically stable if and only if . In this case, each component of the states decays towards 0 like . Also, this system is stable if and only if , and those critical eigenvalues that satisfy have geometric multiplicity one.

3. Main Results

In this section, by applying the previous analytic technique, we shall investigate the stability and bifurcation of system (2) with time delay by taking time delay as the bifurcation parameter. The conditions of delay-induced bifurcation will be derived.

From reference [18], the dynamics of system (2) crucially depend on the basic reproduction number , which is given by

Let , and then we have

Taking advantage of the Taylor expansion formula, the linearized system of system (9) at the zero equilibrium iswhere .

The associated characteristic equation of system (10) can be obtained aswhich leads towhere

Let and , and from equation (12), we have

Assume that is a root of equation (12), . Substituting into equation (12) and separating the real and imaginary parts, one can have

Applying equation (15), direct calculation yieldswhere

From (16), we obtain

Definewhere is defined by equation (18).

To derive the condition of the occurrence for Hopf bifurcation, we further give the following assumption.

Assumption 3. where

Lemma 2. Let be the root of equation (18) near satisfying , , then the following transversality condition holds

Proof. By using implicit function theorem and differentiating (12) with respect to , we havewhere are the derivatives of .
Thus, we obtainwhereLet and be the real and imaginary parts of , respectively. We further suppose that and are the real and imaginary parts of , respectively, and , are the real and imaginary parts of , respectively.
Hence,Therefore, based on Assumption 3, the transversality condition is satisfied. This completes the proof of Lemma 2.
Denotewhere