Analysis of stability and Hopf bifurcation in a fractional Gausstype predator–prey model with Allee effect and Holling typeIII functional response
 Kanokrat Baisad^{1} and
 Sompop Moonchai^{2}Email authorView ORCID ID profile
https://doi.org/10.1186/s1366201815359
© The Author(s) 2018
Received: 14 November 2017
Accepted: 22 February 2018
Published: 5 March 2018
Abstract
The Kolmogorov model has been applied to many biological and environmental problems. We are particularly interested in one of its variants, that is, a Gausstype predator–prey model that includes the Allee effect and Holling typeIII functional response. Instead of using classic first order differential equations to formulate the model, fractional order differential equations are utilized. The existence and uniqueness of a nonnegative solution as well as the conditions for the existence of equilibrium points are provided. We then investigate the local stability of the three types of equilibrium points by using the linearization method. The conditions for the existence of a Hopf bifurcation at the positive equilibrium are also presented. To further affirm the theoretical results, numerical simulations for the coexistence equilibrium point are carried out.
Keywords
1 Introduction
Fractional calculus is an extension of classical calculus that generalizes the order of derivatives and integrals to a noninteger order. Fractional calculus was first discussed more than 300 years ago by Leibniz and L’Hôpital [1], who considered derivatives of order \(\alpha =\frac{1}{2}\). In the last few decades, fractional calculus has been widely used in many fields such as engineering [2–4], mechanics [5, 6], physics [7, 8], chemistry [9, 10], and biology [11–13]. For some dynamic phenomena, models with fractional order derivatives provide a better and more efficient description than those with classical derivatives [14]. Models with fractional order derivatives can take different forms depending on the system understudy and purpose of the model. Among them, fractional differential equations (FDEs) and fractional partial differential equations (FPDEs) are most often applied to represent continuous and deterministic systems. FDEs and FPDEs have been mathematically studied from several different perspectives, yielding methods for solving FDEs [15–19] and FPDEs [20–26] and characterization of the asymptotic behavior of solutions [27–30].
Recently, many researchers have used fractional differential equations to develop models for predator–prey interactions [31–34], atmospheric dispersion [35], pharmacokinetics [36], HIV infection [37], and bioreactors [38]. The predator–prey model has been applied to describe the relationship between two species in biological systems in which one a predator feeds on the other its prey [39]. Fractional order predator–prey models have been used by many researchers, who also discussed the stability and numerical solutions of the models [32, 33, 40, 41]. Moreover, some studies showed the conditions for existence of a Hopf bifurcation [42–44] and the appearance of a limit cycle [45].
The Kolmogorov model is a general continuous time predator–prey model, consisting of a system of first order differential equations [46]. There are many types of Kolmogorov models such as the Lotka–Volterra model [47], Gausstype models [48], Hsu model [49], Kuang and Freedman model [50], and Huang and Merrill model [51]. Among them, the Gausstype predator–prey models have been widely used to formulate population models [52–55]. A crucial phenomenon in biology is the decrease in per capita growth rate at low population densities [56], which was first described by Warder Clyde Allee in the 1930s [57]. Several researchers are interested in the Allee effect for multispecies systems and have included the Allee effect term in Gausstype predator–prey models [58, 59]. In addition, many population models [60–62] are often associated with a functional response, which refers to the predation rate per predator as a function of prey density [63, 64] and predator density [65–67]. There are a variety of functional responses such as the Holling typeI–III functions [68], Holling typeIV function [69], the simplified Holling typeIV function [70], the Beddington–DeAngelis function [66], the Crowley–Martin function [71].

the prey population is affected by the Allee effect,

the functional response is Holling typeIII, and

the predator is a generalist species; see details in Ref. [73].

r is the growth rate of the prey.

K is the environmental capacity of the prey.

m is the minimum viable population.

s is the maximum per capita consumption rate.

a is the amount of prey at which predation rate is maximal.

p is the conversion efficiency of reduction rate of the predator.

c is the natural per capita death rate of the predator.
In model (1), the Allee effect is defined by the term \(r ( 1\frac{x}{K} ) ( xm ) x\) and the Holling typeIII functional response is represented by the term \(\frac{sx^{2}}{x^{2}+a ^{2}}\). This functional response describes a behavior in which the number of prey consumed per predator initially increases quickly as the density of prey grows and levels off with further increase in prey density [74].
The most famous definitions of fractional order derivatives are the Grünwald–Letnikov [75], Riemann–Liouville [76], and Caputo [77] definitions. The Caputo definition is very useful because in this case the derivative of a constant is zero and the initial conditions for the fractional order differential equations can be provided in the same manner as for the classical integer case, which has a clear physical meaning [78]. This paper aims to construct a fractional Gausstype predator–prey model with Allee effect and Holling typeIII functional response in the Caputo sense by modifying model (1). We analyze the local stability of the equilibria for the model using the linearization method. Moreover, given a derivative order, the conditions for the existence of a Hopf bifurcation of the positive equilibrium are obtained.
This article is organized as follows: In Sect. 2, the definition of a fractional order derivative in the Caputo sense and some important theorems for fractional order systems are introduced. The model is then developed along with the existence and uniqueness of a nonnegative solution of the model in Sect. 3. In Sect. 4, we analyze the local stability of the equilibrium points and examine the conditions for the existence of a Hopf bifurcation at the positive equilibrium. Some numerical examples to support the theoretical results are shown in Sect. 5. Finally, the conclusions of this study are presented in Sect. 6.
2 Preliminaries
In this section, the definition of a fractional derivative in the Caputo sense is given. Furthermore, definitions of an equilibrium point and some important theorems of the local stability of a fractional order system are presented.
Definition 2.1
([77])
Theorem 2.2
([80])
Theorem 2.3
([77])
An equilibrium point of system (2) and the stability of the equilibrium point are described by the following definition and theorems.
Definition 2.4
([81])
A point \(X^{*}\) is called an equilibrium point of system (2) if and only if \(f(t,X^{*})=0\).
Theorem 2.5
([33])
Let \(J(X^{*})\) denote the Jacobian matrix of system (2) evaluated at equilibrium point \(X^{*}\). The eigenvalues of \(J(X^{*})\) are \(\lambda_{i}\), where \(i=1,\ldots,n\). Then equilibrium point \(X^{*}\) is a saddle point if some eigenvalues \(\lambda_{i}\) satisfy \(\vert \arg (\lambda_{i}) \vert >\frac{\alpha \pi }{2}\) and some others satisfy \(\vert \arg (\lambda_{i}) \vert < \frac{\alpha \pi }{2}\).
Theorem 2.6
 (i)
equilibrium point \(X^{*}\) is locally asymptotically stable if and only if all eigenvalues \(\lambda_{i}\), \(i=1,\ldots,n\) of \(J(X^{*})\) satisfy \(\vert \arg (\lambda_{i}) \vert >\frac{\alpha \pi }{2}\),
 (ii)
equilibrium point \(X^{*}\) is stable if and only if all eigenvalues \(\lambda_{i}\), \(i=1,\ldots,n\) of \(J(X^{*})\) satisfy \(\vert \arg (\lambda_{i}) \vert \geq \frac{\alpha \pi }{2}\) and eigenvalues with \(\vert \arg (\lambda_{i}) \vert =\frac{\alpha \pi }{2}\) have the same geometric multiplicity and algebraic multiplicity, and
 (iii)
equilibrium point \(X^{*}\) is unstable if and only if there exist eigenvalues \(\lambda_{i}\) for some \(i=1,\ldots,n\) of \(J(X^{*})\) which satisfy \(\vert \arg (\lambda_{i}) \vert < \frac{\alpha \pi }{2}\).
3 Fractional order model
In this section, we construct the fractional Gausstype predator–prey model with Allee effect and Holling typeIII functional response by modifying model (1). We subsequently show the existence and uniqueness of a nonnegative solution.
3.1 Existence and uniqueness of a nonnegative solution
Let \(\mathbb{R}^{2}_{+}=\{X=(x,y)^{T}\in \mathbb{R}^{2} \mid x \geq 0, y\geq 0\}\) be the nonnegative quadrant of the xyplane.
From model (3), we can verify that \(f_{i}\), \(\frac{ \partial f_{i}}{\partial x}\), and \(\frac{\partial f_{i}}{\partial y} \) for \(i=1,2 \) are continuous in \(\mathbb{R}^{2}_{+}\). According to a lemma in reference [83], we find that \(f(X(t))\) satisfies a local Lipschitz condition with respect to X in \(\mathbb{R}^{2}_{+}\). Therefore, by Remark 3.8 in Ref. [82], fractional order model (3) has a unique solution in \(\mathbb{R}^{2}_{+}\).
A requirement for the biological significance of the model, that the solution is nonnegative for \(t>t_{0} \) if the initial solution of model (3) starts in \(\mathbb{R}^{2}_{+}\), will be considered in the next subsection.
3.2 Nonnegative solution
Theorem 3.1
If \(x(t_{0})\geq 0\) and \(y(t_{0})\geq 0\), then the solution \(X ( t ) \) of model (3) remains in \(\mathbb{R} ^{2}_{+}\).
Proof
This theorem is proved by contradiction.
Let \(X_{0}=\bigl( {\scriptsize\begin{matrix}{} x_{0} \cr y_{0} \end{matrix}} \bigr) \in \mathbb{R}^{2}_{+}\), and \(X(t)\) for \(t\geq t_{0}\) be the solution of model (3).
Suppose there exists a solution \(X(t)\) that moves away from \(\mathbb{R}^{2}_{+}\). Then there are two possibilities: it passes through either the xaxis or yaxis. The proof will be divided into these two cases.
Considering both Case 1 and Case 2, we conclude that the solution of model (3) remains in \(\mathbb{R}^{2}_{+}\) if the initial solution starts in this region. □
4 Main results
4.1 Equilibrium points and stability analysis
 (i)
the extinction equilibrium point: \(E_{0}=(0,0)\),
 (ii)
the predatorfree equilibrium points: \(E_{1}=(m,0)\) and \(E_{2}=(K,0)\),
 (iii)the coexistence equilibrium point: \(E_{3}=(x^{*},y^{*})\), where$$\begin{aligned} x^{*}=\sqrt{\frac{ca^{2}}{pc}} \quad \textrm{and}\quad y^{*}=\frac{r}{sx^{*}} \biggl( 1\frac{x^{*}}{K} \biggr) \bigl(x^{*}m\bigr) \bigl(x^{*^{2}}+a ^{2}\bigr). \end{aligned}$$(11)
Since the populations of prey and predator are nonnegative, equilibrium point \(E_{3}=(x^{*},y^{*})\) exists whenever \({p>c}\).
By using Theorems 2.5 and 2.6, we obtain the stabilities of the four equilibrium points, which are discussed in the following theorems.
Theorem 4.1
Equilibrium point \(E_{0}=(0,0)\) of model (3) is locally asymptotically stable.
Proof
Therefore, by Theorem 2.6, we conclude that equilibrium point \(E_{0}=(0,0)\) is locally asymptotically stable. □
Theorem 4.2
Equilibrium point \(E_{1}=(m,0)\) of model (3) is unstable and is a saddle point if \(\frac{(pc)m^{2}}{ca^{2}}<1\).
Proof
If \(\frac{(pc)m^{2}}{ca^{2}}<1\), then \(\lambda_{2}<0\). Hence, \(\arg (\lambda_{2})=\pi \), which results in \(\vert \arg (\lambda_{2}) \vert >\frac{ \alpha \pi }{2}\).
According to Theorem 2.5, equilibrium point \(E_{1}=(m,0)\) is a saddle point if \(\frac{(pc)m^{2}}{ca^{2}}<1\). □
Theorem 4.3
Equilibrium point \(E_{2}=(K,0)\) of model (3) is a saddle point if \(\frac{(pc)K^{2}}{ca^{2}}>1\) and is locally asymptotically stable if \(\frac{(pc)K^{2}}{ca^{2}}<1\).
Proof
Therefore, by Theorem 2.5 and 2.6, equilibrium point \(E_{2}=(K,0)\) is a saddle point if \(\frac{(pc)K^{2}}{ca^{2}}>1\) and is locally asymptotically stable if \(\frac{(pc)K^{2}}{ca^{2}}<1\). □
Theorem 4.4
 (i)
\(\operatorname {tr}( J ( x^{*},y^{*} ) ) \leq 0\).
 (ii)
\(\operatorname {tr}( J ( x^{*},y^{*} ) ) >0\), \(\operatorname {tr}^{2} ( J ( x^{*},y^{*} ) ) 4\det ( J ( x^{*},y^{*} ) ) <0\), and \(\vert \operatorname {tr}^{2} ( J ( x^{*},y^{*} ) ) 4 \det ( J ( x^{*},y^{*} ) ) \vert ^{1/2}>\operatorname {tr}( J ( x^{*},y^{*} ) ) \tan ( \frac{\alpha \pi }{2} ) \).
Proof
(i) Assume that \(\operatorname {tr}( J ( x^{*},y^{*} ) ) \leq 0\). The proof will be divided into three cases.
Case 1: If \(\operatorname {tr}( J ( x^{*},y^{*} ) ) =0\), then, by Eq. (15), we obtain a pair of complex conjugate eigenvalues \(\lambda_{1}\) and \(\lambda_{2}=\bar{\lambda_{1}}\). Since \(\operatorname {Re}(\lambda_{1})=\operatorname {Re}(\lambda_{2})=\operatorname {tr}( J ( x^{*},y^{*} ) ) =0\), we have \(\arg (\lambda_{1})=\frac{ \pi }{2}\) and \(\arg (\lambda_{2})=\frac{\pi }{2}\) leading to \(\vert \arg (\lambda_{1}) \vert >\frac{\alpha \pi }{2}\) and \(\vert \arg (\lambda_{2}) \vert >\frac{ \alpha \pi }{2}\).
By Theorem 2.6, equilibrium point \(E_{3}=(x^{*},y^{*})\) is locally asymptotically stable.
Case 2: If \(\operatorname {tr}( J ( x^{*},y^{*} ) ) <0\) and \(\operatorname {tr}^{2} ( J ( x^{*},y^{*} ) ) 4\det ( J ( x^{*},y^{*} ) ) \geq 0\), then according to Eq. (15), the eigenvalues of \(J(x^{*},y^{*})\) are \(\lambda_{1}<0\) and \(\lambda_{2}<0\). Consequently, we obtain the condition \(\vert \arg ( \lambda_{1}) \vert >\frac{\alpha \pi }{2}\) and \(\vert \arg (\lambda_{2}) \vert >\frac{ \alpha \pi }{2}\).
Hence, by Theorem 2.6, the equilibrium point \(E_{3}=(x^{*},y ^{*})\) is locally asymptotically stable.
Case 3: If \(\operatorname {tr}( J ( x^{*},y^{*} ) ) <0\) and \(\operatorname {tr}^{2} ( J ( x^{*},y^{*} ) ) 4\det ( J ( x^{*},y^{*} ) ) <0\), then, by Eq. (15), we obtain a pair of complex conjugate eigenvalues \(\lambda_{1}\) and \(\lambda_{2}=\bar{\lambda_{1}}\). Since \(\operatorname {tr}( J ( x^{*},y ^{*} ) ) <0\), we have \(\operatorname {Re}(\lambda_{1})= \operatorname {Re}(\lambda_{2})=\operatorname {tr}( J ( x^{*},y^{*} ) ) <0\). Thus, \(\vert \arg (\lambda_{1}) \vert >\frac{\alpha \pi }{2}\) and \(\vert \arg (\lambda _{2}) \vert >\frac{\alpha \pi }{2}\).
Therefore, by Theorem 2.6, equilibrium point \(E_{3}=(x^{*},y ^{*})\) is locally asymptotically stable.
By considering Case 1, Case 2, and Case 3, we conclude that equilibrium point \(E_{3}=(x^{*},y^{*})\) is locally asymptotically stable if \(\operatorname {tr}( J ( x^{*},y^{*} ) ) \leq 0\).
(ii) Assume that \(\operatorname {tr}( J ( x^{*},y^{*} ) ) >0\), \(\operatorname {tr}^{2} ( J ( x^{*},y^{*} ) ) 4\det ( J ( x^{*},y^{*} ) ) <0\) and \(\vert \operatorname {tr}^{2} ( J ( x ^{*},y^{*} ) ) 4\det ( J ( x^{*},y^{*} ) ) \vert ^{1/2}> \operatorname {tr}( J ( x^{*},y^{*} ) ) \tan ( \frac{ \alpha \pi }{2} ) \). By Eq. (15), we have a pair of complex conjugate eigenvalues \(\lambda_{1}\) and \(\lambda_{2}= \bar{\lambda_{1}}\) such that \(\operatorname {Im}(\lambda_{1})= \operatorname {Im}(\lambda_{2})={ [ 4\det ( J(x^{*},y^{*}) ) \operatorname {tr}^{2} ( J(x^{*},y^{*}) ) ] }^{1/2}>0\) and \(\operatorname {Re}(\lambda_{1})=\operatorname {Re}(\lambda_{2})= \operatorname {tr}( J(x^{*},y^{*}) ) >0\). From the assumptions, we obtain \(\operatorname {Im}(\lambda_{1})>\operatorname {Re}(\lambda _{1})\tan ( \frac{\alpha \pi }{2} ) \) and \(\operatorname {Im}( \lambda_{2})>\operatorname {Re}(\lambda_{2})\tan ( \frac{\alpha \pi }{2} ) \). These imply that \(\frac{\alpha \pi }{2}<\arg ( \lambda_{1})<\frac{\pi }{2}\) and \(\frac{\pi }{2}<\arg (\lambda_{2})<\frac{ \alpha \pi }{2}\), which satisfy the conditions \(\vert \arg (\lambda_{1}) \vert >\frac{ \alpha \pi }{2}\) and \(\vert \arg (\lambda_{2}) \vert >\frac{\alpha \pi }{2}\), respectively.
According to Theorem 2.6, equilibrium point \(E_{3}=(x^{*},y ^{*})\) is locally asymptotically stable. □
Theorem 4.5
 (i)
\(\operatorname {tr}^{2} ( J ( x^{*},y^{*} ) ) 4 \det ( J ( x^{*},y^{*} ) ) \geq 0\) and \(\operatorname {tr}( J ( x^{*},y^{*} ) ) >0\).
 (ii)
\(\operatorname {tr}^{2} ( J ( x^{*},y^{*} ) ) 4 \det ( J ( x^{*},y^{*} ) ) <0\), \(\operatorname {tr}( J ( x^{*},y^{*} ) ) >0\), and \(\vert \operatorname {tr}^{2} ( J ( x^{*},y^{*} ) ) 4 \det ( J ( x^{*},y^{*} ) ) \vert ^{1/2}<\operatorname {tr}( J ( x^{*},y^{*} ) ) \tan ( \frac{\alpha \pi }{2} ) \).
Proof
(i) Assume that \(\operatorname {tr}^{2} ( J ( x^{*},y^{*} ) ) 4 \det ( J ( x^{*},y^{*} ) ) \geq 0\) and \(\operatorname {tr}( J ( x^{*},y^{*} ) ) >0\). From Eq. (13) in Theorem 4.4, we have \(\det (J(x^{*},y^{*}))>0\). Then, by Eq. (15), we obtain \(\lambda_{1}>0\) and \(\lambda _{2}>0\), which lead to \(\vert \arg (\lambda_{1}) \vert <\frac{\alpha \pi }{2}\) and \(\vert \arg (\lambda_{2}) \vert <\frac{\alpha \pi }{2}\).
Hence, by Theorem 2.6, the equilibrium point \(E_{3}=(x^{*},y ^{*})\) is unstable.
(ii) Assume that \(\operatorname {tr}^{2} ( J ( x^{*},y^{*} ) ) 4 \det ( J ( x^{*},y^{*} ) ) <0\), \(\operatorname {tr}( J ( x^{*},y^{*} ) ) >0\), and \(\vert \operatorname {tr}^{2} ( J ( x^{*},y^{*} ) ) 4\det ( J ( x^{*},y^{*} ) ) \vert ^{1/2}<\operatorname {tr}( J ( x ^{*},y^{*} ) ) \tan ( \frac{\alpha \pi }{2} ) \). From Eq. (15), we obtain a pair of complex conjugate eigenvalues \(\lambda_{1}\) and \(\lambda_{2}=\bar{\lambda_{1}}\) such that \(\operatorname {Im}(\lambda_{1})=\operatorname {Im}(\lambda_{2})= [ 4 \det ( J(x^{*},y^{*}) ) \operatorname {tr}^{2} ( J(x^{*}, y ^{*}) ) ] ^{1/2}>0\) and \(\operatorname {Re}(\lambda_{1})= \operatorname {Re}(\lambda_{2})=\operatorname {tr}( J(x^{*},y^{*}) ) >0\). By the assumptions, we obtain \(\operatorname {Im}(\lambda_{1})<\operatorname {Re}(\lambda_{1})\tan ( \frac{ \alpha \pi }{2} ) \) and \(\operatorname {Im}(\lambda_{2})< \operatorname {Re}(\lambda_{2})\tan ( \frac{\alpha \pi }{2} ) \). These imply that \(0<\arg (\lambda_{1})<\frac{\alpha \pi }{2}\) and \(\frac{\alpha \pi }{2}<\arg (\lambda_{2})<0\), then \(\vert \arg (\lambda _{1}) \vert <\frac{\alpha \pi }{2}\) and \(\vert \arg (\lambda_{2}) \vert <\frac{\alpha \pi }{2}\).
Therefore, by Theorem 2.6, equilibrium point \(E_{3}=(x^{*},y ^{*})\) is unstable. □
4.2 Existence of a Hopf bifurcation
A Hopf bifurcation occurs when a system has a pair of complex conjugate eigenvalues of the Jacobian matrix at an equilibrium point and when the stability of the equilibrium point changes from stable to unstable as a bifurcation parameter crosses a critical value [84, 85]. From the results of Sect. 4.1, we observe that the order of derivatives has an effect on the stability of model (3). Thus, we can choose the order to be the bifurcation parameter. The conditions for existence of a Hopf bifurcation in a fractional order system are different from integer order systems. There are some studies that considered the existence of Hopf bifurcations in fractional order systems [86, 87]. In this study, we use the conditions for the existence of a Hopf bifurcation which were introduced by Xiang Li and Ranchao Wu [87]. The conditions for the existence are modified for our system as given below.
Theorem 4.6
([87]; Existence of Hopf bifurcation)
 (a)
the Jacobian matrix of the system (3) at the equilibrium point has a pair of complex conjugate eigenvalues \(\lambda_{1,2}=\theta \pm i\omega \), where \(\theta >0\);
 (b)
\(m(\alpha^{*})=0 \), where \(m(\alpha )= \frac{\alpha \pi }{2}\min_{1\leq i\leq 2} \vert \arg (\lambda_{i}) \vert \);
 (c)
\(\frac{dm(\alpha )}{d\alpha } _{\alpha =\alpha^{*}}\neq 0 \).
Remark 1
Condition (b) indicates that \(\alpha^{*}\) is the switching point of stability and condition (c) guarantees that \(m(\alpha )\) can change sign when the bifurcation parameter α passes through the critical value \(\alpha^{*}\) [87].
We then establish the conditions for the existence of a Hopf bifurcation at the positive equilibrium point as the order of derivatives crosses a critical value. The result is presented as the following theorem.
Theorem 4.7
Model (3) undergoes a Hopf bifurcation at equilibrium point \(E_{3}=(x^{*},y^{*})\) when bifurcation parameter α passes through the critical value \(\alpha^{*}=\frac{2}{\pi }\arctan [ ( \vert \operatorname {tr}^{2} ( J(x^{*}, y^{*}) ) 4\det ( J(x^{*},y^{*}) ) \vert ^{1/2})/( \operatorname {tr}( J(x^{*},y^{*}) ) ) ] \) if \(\operatorname {tr}^{2} ( J ( x^{*},y^{*} ) ) 4\det ( J ( x^{*},y^{*} ) ) <0 \) and \(\operatorname {tr}( J ( x^{*}, y^{*} ) ) >0\).
Proof
Assume that \(\operatorname {tr}^{2} ( J ( x^{*},y^{*} ) ) 4 \det ( J ( x^{*},y^{*} ) ) <0\), \(\operatorname {tr}( J ( x^{*},y^{*} ) ) >0\), and the critical value \(\alpha^{*}=\frac{2}{\pi }\arctan [ \frac{ \vert \operatorname {tr}^{2} ( J(x^{*},y^{*}) ) 4\det ( J(x^{*},y^{*}) ) \vert ^{1/2}}{ \operatorname {tr}( J(x^{*},y^{*}) ) } ] \).
Define \(\theta =\frac{1}{2}\operatorname {tr}(J(x^{*},y^{*}))\) and \(\omega =\frac{1}{2} \vert \operatorname {tr}^{2} ( J(x^{*},y^{*}) ) 4 \det ( J(x^{*},y^{*}) ) \vert ^{1/2}\). By the second assumption, we have \(\theta >0\). According to the first assumption and Eq. (15), we have a pair of complex conjugate eigenvalues \(\lambda_{1,2}=\theta \pm i\omega \), with \(\theta >0\). Hence, condition (a) in Theorem 4.6 holds.
Therefore, from Theorem 4.6, model (3) undergoes a Hopf bifurcation at equilibrium point \(E_{3}=(x^{*},y^{*})\) when bifurcation parameter α passes through the critical value \(\alpha^{*}\). □
5 Numerical simulations
In this section, we present numerical examples and simulate model (3) to illustrate the results of Sect. 4. However, we only show the numerical examples for the coexistence equilibrium point. There are many numerical methods for solving nonlinear fractional differential equations such as the predictorcorrector method [15, 88], Adomian decomposition method [16], variational iteration method [17], and Adams method [89]. However, the Adams method is often used for solving nonlinear fractional differential equations [90–92] and is useful for studying the dynamic behavior (especially long time behavior) of the solutions [45]. Thus, in this study, the Adams method is used to solve model (3) by the Matlab software.
The goals of the numerical simulations are to identify the effects of different parameter values and of variations of the derivative order α on the dynamic behavior of the model (3). The discussion of these effects is divided into the two following subsections.
5.1 The effect of different parameter values
We assign three different sets of parameters. In each set, we calculate the coexistence equilibrium point \(E_{3}=(x^{*},y^{*})\) of the model by using Eq. (11). With these parameters, the determinant and trace of the Jacobian matrix \(J(x^{*},y^{*})\) are calculated by using Eqs. (13) and (14).
The simulations presented in the two figures above indicate that the stability of the equilibrium points does not depend on the order \(\alpha \in (0,1)\). This result is consistent with the theorems in the previous section.
The simulation results in Fig. 3 indicate that the order α of model (3) has an effect on the stability of equilibrium point \(E_{3}\). Next, we demonstrate how the order α can cause a Hopf bifurcation at \(E_{3}\) for model (3).
5.2 The effect of varying the order α
In Figs. 4(a) and 4(b), numerical simulations are shown for the orders \(\alpha =0.79\) and \(\alpha =0.80\), respectively, in which both values satisfy \(\alpha <\alpha^{*}\). By using these orders and the parameters which are specified above, condition (ii) in Theorem 4.4 is satisfied. Consequently, equilibrium point \(E_{3}=(30,42)\) is locally asymptotically stable. Figures 4(a) and 4(b) show that the trajectory converges to equilibrium point \(E_{3}=(30,42)\).
However, if we use \(\alpha =0.83\) and \(\alpha =0.95\) in which both values satisfy \(\alpha >\alpha^{*}\), then, according to Theorem 4.5, equilibrium point \(E_{3}=(30,42)\) is unstable. The simulation results indicate that the trajectory diverges from equilibrium point \(E_{3}\), which is shown in Figs. 4(c) and 4(d) for the orders \(\alpha =0.83\) and \(\alpha =0.95\), respectively.
The above results indicate that equilibrium point \(E_{3}=(30,42)\) loses its stability when the order α is increased to pass through the critical value \(\alpha^{*}\), which implies that a Hopf bifurcation occurs. This result is consistent with Theorem 4.7.
6 Conclusions
In this paper, we have studied the dynamic behavior of a fractional Gausstype predator–prey model with Allee effect and Holling typeIII functional response. The model was constructed by starting with the first order model as shown in Eq. (1). We showed the existence and uniqueness of a nonnegative solution of the model. We used the linearization method to classify the local stability of the three types of equilibrium points. In addition, we obtained the conditions and critical value \(\alpha^{*}\) for occurrence of a Hopf bifurcation at the positive equilibrium point. Finally, numerical simulations were used to show the dynamic behavior of interaction between prey and predator and to verify the validity of the theoretical results. The simulation results illustrated that the order α is a factor affecting the dynamic behavior and is responsible for a Hopf bifurcation. Moreover, the numerical results showed the appearance of an attracting limit cycle of model (3). However, this limit cycle cannot be an exact periodic solution of the model.
Declarations
Acknowledgements
This research was partially supported by Department of Mathematics, Faculty of Science, Chiang Mai University.
Authors’ contributions
Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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