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Stability and Hopf bifurcation analysis in a fractionalorder delayed paddy ecosystem
Advances in Difference Equations volumeÂ 2018, ArticleÂ number:Â 315 (2018)
Abstract
By introducing a delayed fractionalorder differential equation model, we deal with the dynamics of the stability and Hopf bifurcation of a paddy ecosystem with three main components: rice, weeds, and inorganic fertilizer. In the system, there exists an equilibrium for rice and weeds extinction and an equilibrium for rice extinction or weeds extinction. We obtain sufficient conditions for the stability and Hopf bifurcation by analyzing their characteristic equation. Some numerical simulations validate our theoretical results.
1 Introduction
Rice is one of the major grain crops in the world. China is the largest rice producer and consumer country in the world, where over 60% of the population is staple food for rice. Throughout the world rice producing countries, it is a major research topic to improve rice yield and quality. Obviously, there are a lot of factors affecting the production of rice, such as weed, insect, microorganism, inorganic fertilizer, light intensity, moisture, and so on. These factors interact and transform each other to form a complex nonlinear relationship. It is a common research method to analyze the interaction of all factors in a population system by using mathematical models [1â€“11]. As far as we know, there are only a few mathematical models that have been established for paddy ecosystems [12â€“14].
A differential equation model of a paddy ecosystem in fallow season was proposed by Xiang et al. [14]. They revealed the interaction between weeds and inorganic fertilizer and found that in the system, there exists a stable node, an unstable saddle point, or a saddlenode point. By considering the effects of herbivores on the paddy ecosystem in fallow season, Xiang, Wu, and Zhou found that the content of inorganic fertilizer is improved by putting some herbivores into the paddy ecosystem in fallow season. They also found that the system can exhibit Hopf bifurcation phenomenon and gave the critical value of Hopf bifurcation by taking a system parameter as the bifurcation parameter [13]. Wang et al. [12] further studied the interaction of rice, weeds, and inorganic fertilizer in a paddy ecosystem. They discussed the existence and stability of equilibria in a paddy ecosystem. They also found that there exist Hopf bifurcations in such a system.
The three models mentioned have been restricted to integerorder (delay) differential equations [12â€“14]. In recent more than 20 years, the research of fractionalorder differential equations has been the concern of many scholars. According to the study and numerical experiments in different fields such as physical, mechanical, and engineering problems, many phenomena can be described more successfully by using factionalorder differential equation models. In view of this, some scholars have used fractional differential equations to study the interaction relationship of biological populations [15â€“20]. Recently, some researchers have also concerned about the existence of Hopf bifurcation of fractionalorder models [21â€“27]. Abdelouahab et al. [21] obtained the Hopf bifurcation conditions of a threedimensional fractionalorder system without time delay, and Li et al. [23] obtained the Hopf bifurcation conditions of a fourdimensional fractionalorder system without time delay. For general delayed fractionalorder systems, the Hopf bifurcation conditions were proposed by Xiao et al. [26] in 2017.
It is of practical significance to study whether there exists a Hopf bifurcation in a paddy ecosystem. If a Hopf bifurcation exists in a paddy ecosystem, the stability of the system will be destroyed. An unstable paddy ecosystem brings difficulties and uncertainties to management of rice production. Therefore, we want to delay or eliminate the Hopf bifurcation by using the existence conditions of Hopf bifurcation. On the other hand, in case the paddy ecosystem has come up with a Hopf bifurcation, we should try to harvest rice at the peak of its biomass to increase rice yield.
In this paper, we establish a factionalorder differential equation model with delay for the interaction among the main components of a paddy ecosystem. We give a detailed stability analysis of the system equilibria and study the existence of Hopf bifurcation by using the Hopf bifurcation conditions proposed by Xiao et al. [26].
2 Preliminaries
Considering a general delayed fractionalorder system
the time delay \(\tau> 0\), where \(Y(t)=(y_{1}(t),y_{2}(t),\ldots ,y_{n}(t))^{T} \in\mathbb{R}^{n}\), and \(D^{\alpha}\) is the Caputo fractional derivative defined as
where \(\Gamma(q)=\int_{0}^{\infty}e^{t}t^{q1}\,dt\) is the gamma function, \(m \in\mathbb{N}\), and \(m1 < \alpha< m\). When \(\alpha=m\), \(D^{\alpha}f(t)=f^{(m)}(t)\). In this paper, we suppose \(0 < \alpha\leq1\).
The equilibrium \(Y^{*}\) of system (1) is the solution to equation \(F(Y,Y)=0\).
The corresponding linearized system of (1) at an equilibrium \(Y^{*}\) is of the form
The characteristic equation of system (2) is
If \(\tau=0\), then system (2) is simplified as
where the coefficient matrix \(M=A+B\).
Based on the characteristic equation \(\Delta(\lambda)=0\) and the coefficient matrix M, we have the following stability result on the delayed fractionalorder system (2) [28].
Lemma 1
If \(\alpha\in(0, 1)\), then all the eigenvalues Î» of M satisfy \(\arg(\lambda) >\pi/2\), and the characteristic equation \(\Delta(\lambda) = 0\) has no purely imaginary roots for any \(\tau> 0\), then the zero solution of system (2) is Lyapunov globally asymptotically stable.
The Hopf bifurcation conditions were proposed in [26] for the general delayed fractionalorder system (1). If the following conditions hold, then system (1) undergoes a Hopf bifurcation at the equilibrium \(Y^{*}\) when \(\tau= \tau_{0}\).

(1)
All the eigenvalues of the coefficient matrix M of the linearized system of (1) satisfy \(\arg(\lambda) > \alpha\pi/2\).

(2)
The characteristic equation \(\Delta(\lambda)=0\) of the linearized system of (1) has a pair of purely imaginary roots \(\pm i\omega _{0}\) when \(\tau= \tau_{0}\).

(3)
\(\frac{d\operatorname {Re}(\lambda(\tau))}{d\tau} \vert _{\tau=\tau_{0}} > 0\), where \(\operatorname {Re}(\cdot)\) denotes the real part of a complex number.
3 Fractionalorder model of a paddy ecosystem
Wang et al. [12] have considered the following paddy ecosystem with three main components, rice, weeds, and inorganic fertilizer:
where \(r(t)\) and \(p(t)\) denote the rice and weeds biomasses per unit area at timeÂ t, respectively, and \(u(t)\) denotes the inorganic fertilizer content per unit area at time t. The system can reflect the interactions among rice, weeds, and inorganic fertilizer. The first two equations in system (4) indicate that the growth of rice \(r(t)\) and weeds \(p(t)\) are affected by soil fertility \(u(t)\), light and other factors \(s_{i}\), and there is natural death \(d_{1}r(t)\) and \(d_{2}p(t)\) for the rice and weeds. The coefficients \(c_{1}\) and \(c_{2}\) represent rice and weeds utilization rate of inorganic fertilizer, light energy, and other factors, respectively. The third equation in system (4) shows that the inorganic fertilizer in soil partly comes from fertilization b and partly comes from organic fertilizer such as decaying leaves of rice and weeds, \(d_{1}r(t\tau)\) and \(d_{2}p(t\tau)\), which can be transformed to inorganic fertilizer after some time Ï„ by microbial. Natural loss \(d_{3}u(t)\) also reduces the content of inorganic fertilizers in soil.
Using the Caputo fractionalorder derivative of order \(\alpha\in (0,1)\), a fractionalorder delayed paddy ecosystem is established as follows:
where \(c_{3}\) and \(c_{4}\) are the conversion rates from organic fertilizer \(d_{1}r(t\tau)\) and \(d_{2}p(t\tau)\) to inorganic fertilizer \(u(t)\), respectively. The meaning of other symbols in system (5) are consistent with system (4). Similarly, the parameters in system (5) are nonnegative and satisfy the following conditions: \(0< c_{i} < 1\), \(b \geq0\), \(\tau\geq0\), \(s_{i}>0\), and \(d_{i}>0\). We also introduce the following notation [12]:
where \(\theta_{1}\) is called the relative mortality of rice, and \(\theta _{2}\) is called the relative mortality of weeds.
4 The stability of equilibria and Hopf bifurcation
Similarly to [12], system (5) always has an equilibrium for rice and weeds extinction
If \(b/d_{3} > \theta_{1}\), then system (5) has an equilibrium for weeds extinction
If \(b/d_{3} > \theta_{2}\), then system (5) still has an equilibrium for rice extinction
For an equilibrium \((r^{*},p^{*},u^{*})\) of system (5), we make a coordinate transformation \(x=rr^{*}\), \(y=pp^{*}\), \(z=uu^{*}\); then system (5) can be converted to
Obviously, the linearized system of (6) is
Its characteristic equation is
When the time delay \(\tau= 0\), the coefficient matrix of system (7) is
Next, we consider the stability of the three equilibria of system (5).
Case (I) for the equilibrium \((r_{1}^{*},p_{1}^{*},u_{1}^{*})\). At this case, we have the following conclusion of the stability of the equilibrium.
Theorem 1
If \(b/d_{3}<\min\{\theta_{1}, \theta_{2}\}\), then the equilibrium for rice and weeds extinction of system (5) \((r_{1}^{*}, p_{1}^{*}, u_{1}^{*})\) is locally asymptotically stable. Otherwise, if \(b/d_{3}>\min\{\theta_{1},\theta_{2}\}\), then the equilibrium \((r_{1}^{*}, p_{1}^{*}, u_{1}^{*})\) is unstable.
Proof
At the equilibrium \((r_{1}^{*},p_{1}^{*},u_{1}^{*})\), by (8) the characteristic equation of the linearized system is
So the eigenvalues satisfy \(\lambda_{1}^{\alpha}=d_{3}\), \(\lambda _{2}^{\alpha}=c_{1}s_{1}(b/d_{3}\theta_{1}) \), and \(\lambda_{3}^{\alpha }=c_{2}s_{2}(b/d_{3}\theta_{2}) \). Therefore the characteristic equation \(\Delta(\lambda) = 0 \) has no purely imaginary roots for any \(\tau> 0\).
Similarly, the eigenvalues of the coefficient matrix M are \(\lambda _{1} = d_{3}<0\), \(\lambda_{2} = c_{1}s_{1}(b/d_{3}\theta_{1})\), and \(\lambda_{3}=c_{2}s_{2}(b/d_{3}\theta_{2})\).
If \(b/d_{3}<\min\{\theta_{1}, \theta_{2}\}\), then the eigenvalues of the matrix M \(\lambda_{2} < 0 \) and \(\lambda_{3} < 0\). By Lemma 1 the equilibrium \((0,0,0)\) of system (7) is Lyapunov globally asymptotically stable. Therefore, the equilibrium \((r_{1}^{*}, p_{1}^{*}, u_{1}^{*})\) of system (5) is locally asymptotically stable.
If \(b/d_{3}>\min\{\theta_{1}, \theta_{2}\}\), then at least one of the eigenvalues \(\lambda_{2}\) and \(\lambda_{3 }\) of the matrix M is positive. Therefore the equilibrium \((r_{1}^{*}, p_{1}^{*}, u_{1}^{*})\) is unstable under this condition.â€ƒâ–¡
Case (II) for the equilibrium \((r_{2}^{*},p_{2}^{*},u_{2}^{*})\).
To discuss the stability of the other two equilibria, we introduce the polynomial of degree 4 with real coefficients \(a=(1,a_{1},a_{2},a_{3},a_{4})\)
and the cubic polynomial equation
If \(a_{1} > 0\), \(a_{2}<0\), \(a_{3}>0\), and \(a_{4}>0\), then \(4a_{2}a_{4}a_{1}^{2}a_{4}a_{3}^{2}<0\). Hence equation (11) must have a positive real root, denoted by \(\nu_{a}\). It is noted that equation (11) can also be expressed in the form
Therefore from equation (12) it follows that the positive real root \(\nu_{a}\) must satisfy \(\frac{\nu^{2}_{a}}{4}a_{4} \geq0\). Let
and
where \(\mathrm{sgn}(\cdot)\) is the sign function.
Lemma 2
Suppose that \(a_{i} > 0\) (\(i=1,3,4\)). If \(a_{2}<0\) and \(\Delta_{a} \geq0\), then there are only two positive real roots of the equation \(f_{a}(\xi)=\xi^{4}+a_{1}\xi^{3}+a_{2}\xi ^{2}+a_{3}\xi+a_{4} = 0\). If \(a_{2}\geq0\), or \(a_{2}<0\) and \(\Delta_{a} < 0\), then the equation \(f_{a}(\xi) = 0\) has no positive real root.
Proof
The polynomial \(f_{a}(\xi)\) can be decomposed into the product of two quadratic polynomials
and
The discriminants of the polynomials \(f_{1}(\xi)\) and \(f_{2}(\xi)\) are \(\Delta_{a}\) and
respectively. If \(\Delta_{a} \geq0\), then the equation \(f_{1}(\xi)=0\) has two real roots \(\xi_{a1}\) and \(\xi_{a2}\). From
we haven \(\xi_{a1}>0\) and \(\xi_{a2}>0\).
If \(\Delta_{1} \geq0\), then the equation \(f_{2}(\xi)=0\) has two real roots \(\xi_{a3}\) and \(\xi_{a4}\). From
we have \(\xi_{a3}<0\) and \(\xi_{a4}<0\).
Therefore equation \(f_{a}(\xi)= 0\) has only two positive real roots.
Otherwise, if \(a_{i}>0\) (\(i=1,3,4\)) and \(a_{2}\geq0\), it is obvious that the equation \(f_{a}(\xi)=0\) has no positive real root. If \(a_{i}>0\) (\(i=1,3,4\)), \(a_{2} < 0\), and \(\Delta_{a} < 0\), then the equation \(f_{1}(\xi)= 0\) has no real root, and so the equation \(f_{a}(\xi)=0\) has no positive real root.â€ƒâ–¡
Let the coefficients of polynomial (10) be as follows:
Since \(0< \alpha< 1\), we have that \(a_{1} > 0\), \(a_{3}>0\), and \(a_{4}>0\).
The conclusion of the stability of the equilibrium \((r_{2}^{*}, p_{2}^{*}, u_{2}^{*})\) is as follows.
Theorem 2
Suppose that \(b/d_{3} > \theta_{1}\).

(I)
If \(\theta_{1} > \theta_{2}\), then the equilibrium \((r_{2}^{*}, p_{2}^{*}, u_{2}^{*})\) is unstable.

(II)
If \(\theta_{1} < \theta_{2}\) and \(a_{2} \geq0\), or \(\theta_{1} < \theta_{2}\), \(a_{2}<0\), and \(\Delta_{a} < 0\), then the equilibrium \((r_{2}^{*}, p_{2}^{*}, u_{2}^{*})\) is locally asymptotically stable for \(\tau\geq0\).

(III)
If \(\theta_{1} < \theta_{2}\), \(a_{2}<0\), and \(\Delta_{a} > 0\), then there exists a positive number \(\tau_{a}\) such that when \(\tau\in [0, \tau_{a})\), the equilibrium \((r_{2}^{*}, p_{2}^{*}, u_{2}^{*})\) is locally asymptotically stable; when \(\tau> \tau_{a}\), the equilibrium \((r_{2}^{*}, p_{2}^{*}, u_{2}^{*})\) is unstable; and a Hopf bifurcation emerges at \(\tau=\tau_{a}\).
Proof
By (8) the characteristic equation of the linearized system is
It has one real eigenvalue satisfying \(\lambda_{1}^{\alpha }=c_{2}s_{2}(\theta_{1}\theta_{2}) \), the real part of which cannot be zero. Its other eigenvalues are the roots of the equation
By (9) the characteristic equation of the coefficient matrix M is
It has one real eigenvalue \(\lambda_{1}=c_{2}s_{2}(\theta_{1}\theta _{2})\). Its other eigenvalues are
Obviously, the real parts of \(\lambda_{2,3}\) are less than zero.
(I) If \(\theta_{1} > \theta_{2}\), then the eigenvalue of the coefficient matrix M \(\lambda_{1}> 0\). It indicates that the equilibrium \((r_{2}^{*}, p_{2}^{*}, u_{2}^{*})\) is unstable.
(II) If \(\theta_{1}<\theta_{2}\), then the real eigenvalue of the coefficient matrix M \(\lambda_{1}<0\). So the eigenvalues \(\lambda _{j}\) of M satisfy \(\arg(\lambda_{j})>\frac{\pi}{2}>\frac{\alpha\pi }{2}\) (\(j=1,2,3\)).
Assume that equation (17) has a purely imaginary root \(\lambda=i\xi=\xi(\cos\frac{\pi}{2}+i\sin\frac{\pi}{2})\) (\(\xi>0\)). Substituting it into (17) gives
Separating its real and imaginary parts yields
and
So we have
Since \(\sin^{2}\tau\xi+ \cos^{2}\tau\xi= 1\), we have
that is,
where \(a_{i}\) (\(i=1,2,3,4\)) are given in (15) and (16); note that \(a_{1},a_{3},a_{4}>0\). If \(a_{2}\geq0\), or \(a_{2}<0\) and \(\Delta_{a} < 0\), then the equation \(f_{a}(\xi)=0\) has no positive real root by Lemma 2. This leads to that equation (20) has no any positive real number Î¾. Therefore the real parts of any roots of (17) must be negative for any \(\tau> 0 \). This shows that the equilibrium \((r_{2}^{*}, p_{2}^{*}, u_{2}^{*}) \) is locally asymptotically stable for any \(\tau\geq0\).
(III) If \(\theta_{1} < \theta_{2}\), \(a_{2}<0\), and \(\Delta_{a} > 0\), then the equation \(f_{a}(\xi)=0\) has two unequal positive real roots by Lemma 2; denote the larger root by \(\xi_{+}\). So equation (20) has a positive real root \(\xi_{a}\) satisfying \(\xi_{a}^{\alpha }=\xi_{+}\).
Notice that \(\sin\tau_{a}\xi_{a} <0\) from (19). If \(\cos\tau _{a}\xi_{a} >0\), then from (19) we have
If \(\cos\tau_{a}\xi_{a}<0\), then we have
Next, we verify the transversal condition. Taking the derivative of Î» with respect to Ï„ in (17), we have
So we obtain
Thus, its real part is
Substituting (18) and (19) into this expression, we obtain
Since \(\xi_{+}\) is the larger single root of the equation \(f_{a}(\xi )=0\) and the highest order power coefficient of the polynomial is positive, we have \(f'(\xi_{+})>0\). Therefore the transversal condition is satisfied, and thus a Hopf bifurcation occurs at \(\tau=\tau_{a}\).â€ƒâ–¡
Remark 1
From (15) we know that if \((s_{1}r_{2}^{*}+d_{3})^{2} \geq 2s_{1}r_{2}^{*}d_{1}\), then \(a_{2} \geq0\). When \((s_{1}r_{2}^{*}+d_{3})^{2} < 2s_{1}r_{2}^{*}d_{1}\), we let
Obviously, if \(0< \alpha< \alpha_{a}\), then \(a_{2}>0\). Otherwise, if \(\alpha_{a} < \alpha< 1\), then \(a_{2}<0\).
Case (III) for the equilibrium \((r_{3}^{*},p_{3}^{*},u_{3}^{*})\).
To discuss the stability of the equilibrium \((r_{3}^{*}, p_{3}^{*}, u_{3}^{*})\), similarly to case (II), we introduce a polynomial of degree 4 with real coefficients \(q=(1,q_{1},q_{2},q_{3},q_{4})\) as follows:
where
Since \(0< \alpha< 1\), it is obvious that \(q_{1} > 0\), \(q_{3}>0\), and \(q_{4}>0\).
Similarly to case (II), we have the following stability conclusion of the equilibrium \((r_{3}^{*}, p_{3}^{*}, u_{3}^{*})\).
Theorem 3
Suppose that \(b/d_{3} > \theta_{2}\).

(I)
If \(\theta_{1}< \theta_{2}\), then the equilibrium \((r_{3}^{*}, p_{3}^{*}, u_{3}^{*})\) is unstable.

(II)
If \(\theta_{1} > \theta_{2}\) and \(q_{2} \geq0\), or \(\theta_{1}> \theta_{2}\), \(q_{2}<0\), and \(\Delta_{q} < 0\), then the equilibrium \((r_{3}^{*}, p_{3}^{*}, u_{3}^{*})\) is locally asymptotically stable for \(\tau\geq0\).

(III)
If \(\theta_{1}> \theta_{2}\), \(q_{2}<0\), and \(\Delta_{q} > 0\), then there exists a positive number \(\tau_{q}\) such that the equilibrium \((r_{3}^{*}, p_{3}^{*}, u_{3}^{*})\) is locally asymptotically stable for \(\tau\in[0, \tau_{q})\) and unstable when \(\tau> \tau_{q}\). A Hopf bifurcation emerges at the equilibrium \((r_{3}^{*}, p_{3}^{*}, u_{3}^{*})\) when \(\tau= \tau_{q}\).
The proof of Theorem 3 is similar to Theorem 2 and is omitted here.
Remark 2
From (25) we know that if \((s_{2}p_{3}^{*}+d_{3})^{2} \geq 2s_{2}p_{3}^{*}d_{2}\), then \(q_{2} \geq0\). When \((s_{2}p_{3}^{*}+d_{3})^{2} < 2s_{2}p_{3}^{*}d_{2}\), we let
Obviously, if \(0< \alpha< \alpha_{q}\), then \(q_{2}>0\). Otherwise, if \(\alpha_{q} < \alpha< 1\), then \(a_{q}<0\).
5 Examples
In this section, we give two examples to confirm our theoretical results obtained in Sect.Â 4 and use the predictorâ€“corrector scheme to calculate their numerical solutions [29]. In system (5), we let \(\alpha= 0.98\), \(c_{1}=0.8\), \(c_{2}=0.1\), \(c_{3}=0.9\), \(c_{4}=0.6\), \(s_{1}=0.6\), \(s_{2}=0.1\), \(d_{1}=0.9\), \(d_{2}=0.1\), and \(d_{3}=0.1\).
First, we take \(b=1.1\). System (5) has three equilibria: the equilibrium for rice and weeds extinction \((0, 0, 11)\), the equilibrium for weeds extinction \((2.8968, 0, 1.875)\), and the equilibrium for rice extinction \((0, 0.1064, 10)\). By computing we have \(b/d_{3}= 11\), \(\theta_{1} = 1.875\), \(\theta_{2} = 10\), and \(a_{2} \approx0.2562 > 0\). So the inequality \(b/d_{3} > \theta _{2}> \theta_{1}\) holds. By Theorem 2 the equilibrium \((2.8968, 0, 1.875) \) is asymptotically stable for any \(\tau\geq0\) as illustrated in Fig.Â 1 (where \(\tau= 13.5\)). By Theorems 1 andÂ 3 the equilibria \((0, 0, 11)\) and \((0, 0.1064, 10)\) are unstable.
In succession, we take \(b=0.3\) again. System (5) has two equilibria: the equilibrium for rice and weeds extinction \((0, 0, 3)\) and the equilibrium for weeds extinction \((0.3571, 0, 1.875)\). Because \(b/d_{3} = 3\), \(\theta_{1} = 1.875\), and \(\theta_{2} = 10\), the inequality \(\theta_{2} > b/d_{3} > \theta_{1}\) holds. By Theorem 1 the equilibrium \((0, 0, 3)\) is unstable. From (15) and (16) we have \(a_{1} \approx0.0197\), \(a_{2} \approx0.2862\), \(a_{3} \approx0.0038\), \(a_{4} \approx0.0179\). Equation (11) has a positive real root \(\nu_{a} \approx0.2677\). From (13) we obtain \(M_{a} \approx0.7443\) and \(N_{a} \approx7.8279\). Because the discriminant \(\Delta_{a} \approx0.000866 > 0\) and \(a_{2}<0\), the equation \(f_{a}(\xi)=0\) has two unequal positive real roots by Lemma 2, and the larger root \(\xi_{+} \approx0.3819 \). So equation (20) has a positive real root \(\xi_{a}\) satisfying \(\xi_{a}^{\alpha}=\xi_{+}\). Substituting \(\xi_{a}\) into (18), we get \(\cos\tau\xi_{a} \approx0.3677>0\). So we obtain the Hopf bifurcation critical value \(\tau_{a} \approx 13.5888\) by using (21). Therefore, by Theorem 2 the equilibrium \((0.3571, 0, 1.875)\) is asymptotically stable when \(\tau \in[0, 13.5888)\) as illustrated in Fig.Â 2 (where \(\tau =13.5\)); Otherwise, the equilibrium \((0.3571, 0, 1.875) \) is unstable, and a Hopf bifurcation emerges at \(\tau\approx13.5888\) (see Fig.Â 3, where \(\tau=13.59\)).
6 Conclusions
We have proposed a delayed fractionalorder differential equation model that reflects the interaction among rice, weeds, and inorganic fertilizer in a paddy ecosystem. If \(\alpha=1\) and \(c_{3}=c_{4}=1\), then system (5) degenerates into system (4), which was studied in [12]. The equilibria and their existence conditions of system (5) are the same as those of system (4), where those conditions are related to the relative mortality of rice and weeds, \(\theta_{1}\) and \(\theta_{2}\), and to the ratio of fertilizer supply and loss \(b/d_{3}\), but not to other parameters. Under the condition \(b/d_{3} < \min\{\theta_{1}, \theta_{2}\}\), there is aÂ unique stable equilibrium \((0, 0, u_{1}^{*})\) in each of the two systems. If \(b/d_{3} > \max\{\theta_{1}, \theta_{2}\}\), then each of the two systems has three equilibria: the equilibrium for rice and weeds extinction \((0, 0, u_{1}^{*})\), the equilibrium for weeds extinction \((r_{2}^{*}, 0, u_{2}^{*})\), and the equilibrium for rice extinction \((0, p_{3}^{*}, u_{3}^{*})\), where the equilibrium \((0, 0, u_{1}^{*})\) is unstable, and \((r_{2}^{*}, 0, u_{2}^{*})\) is also unstable when \(\theta _{1}>\theta_{2}\), or \((0, p_{3}^{*}, u_{3}^{*})\) is unstable when \({\theta_{1}<\theta_{2}}\). Under the condition \(\theta_{1} < b/d_{3} < \theta_{2}\), there exist two equilibria \((0, 0, u_{1}^{*})\) and \((r_{2}^{*}, 0, u_{2}^{*})\). Under the condition \(\theta_{2} < b/d_{3} < \theta_{1}\), there exist two equilibria \((0, 0, u_{1}^{*})\) and \((0, p_{3}^{*}, u_{3}^{*})\).
We also generalize the conditions of stabilities of equilibria and Hopf bifurcation obtained by Wang et al. [12]. If we take \(\alpha =1\) and \(c_{3}=c_{4}=1\), then we have \(a_{1}=a_{3}=0\), \(a_{2}=(s_{1}r_{2}^{*}+d_{3})^{2}2s_{1}r_{2}^{*}d_{1}\), and \(a_{4}=s_{1}^{2}r_{2}^{*2}d_{1}^{2}(1c_{1}^{2})\) from (15) and (16). Equation (11) has a positive root \(\nu_{a}=2\sqrt {a_{4}}\). So we obtain \(\Delta_{a}= a_{2}2\sqrt{a_{4}}\) from (14). If \(\Delta_{a}<0\), then we have
It is condition (5) of Theorem 2 in [12]. Similarly, from \(\Delta_{a}>0\) we can obtain condition (6) in [12]. Moreover, substituting \(\alpha=1\) and \(c_{3}=1\) into the Hopf bifurcation critical value formulas (21) and (22), we can obtain formulas (12) and (13) in [12], respectively.
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This work was partly supported by Hunan province science and technology project (grants 2015JC3101), and the scientific research fund of Hunan provincial education department (grants 14B090).
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Zhou, X., Wu, Z., Wang, Z. et al. Stability and Hopf bifurcation analysis in a fractionalorder delayed paddy ecosystem. Adv Differ Equ 2018, 315 (2018). https://doi.org/10.1186/s1366201817193
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DOI: https://doi.org/10.1186/s1366201817193