Open Access

Diffusive induced global dynamics and bifurcation in a predator-prey system

Advances in Difference Equations20172017:323

https://doi.org/10.1186/s13662-017-1318-8

Received: 14 March 2017

Accepted: 10 August 2017

Published: 11 October 2017

Abstract

In this paper, a diffusive Leslie-type predator-prey model is investigated. The existence of a global positive solution, persistence, stability of the equilibria and Hopf bifurcation are studied respectively. By calculating the normal form on the center manifold, the formulas determining the direction and the stability of Hopf bifurcations are explicitly derived. Finally, our theoretical results are illustrated by a model with homogeneous kernels and one-dimensional spatial domain.

Keywords

existence of a global positive solution global stability permanence Lyapunov function Hopf bifurcation

1 Introduction

Bifurcation phenomena are observed and studied in a variety of fields, such as engineering [1], ecological [2], chemical [3], biological [46], control [7, 8] and neural networks [913]. A classical predator-prey model is investigated as follows:
$$ \textstyle\begin{cases} \dot{x}(t)= x(1- x )-\frac{x^{2}y}{ax^{2}+bx+1},\\ \dot{y}(t)=y(\delta- \frac{\beta y}{x}). \end{cases} $$
(1.1)
The study shows that model (1.1) has complex bifurcation phenomena and dynamic behavior, such as the existence of both Hopf bifurcation and Bogdanov-Takens bifurcation. For more detailed results on model (1.1), one can refer to [14].
As we all know, the spatial heterogeneity of predator and prey distributions is more or less obvious. The predator or prey can flow from higher density regions to lower density ones, which is called normal diffusion. And specifically, to the best of our awareness, few papers which discuss the dynamic behavior of model (1.1) with diffusion have appeared in the literature. This paper attempts to fill this gap in the literature. To achieve this goal, we look at the following model:
$$ \textstyle\begin{cases} \frac{\partial u}{\partial t}=d_{1}\triangle u+ u(1-u)-\frac {u^{2}v}{au^{2}+bu+1}, & (x,t)\in(0, l\pi)\times[0,\infty), t>0,\\ \frac{\partial v}{\partial t}=d_{2} \triangle v+ v( \delta- \frac{\beta v}{u}),& (x,t)\in(0, l\pi)\times[0,\infty), t>0,\\ u_{x}(t,0)=v_{x}(t,0)=0,\\ u_{x}(t,l\pi)=v_{x}(t,l\pi)=0, & t>0,\\ u(0,x)=u_{0}(x)\geq0, \qquad v(0,x)=v_{0}(x)\geq0, & x\in[0, l\pi], \end{cases} $$
(1.2)
where \(u_{i}=u_{i}(x,t),i=1,2, \triangle =\frac{\partial^{2}}{\partial x^{2}}\) denotes the Laplacian operator, \(d_{1}, d_{2}\) are diffusion coefficients of prey and predator, respectively. Ω is a bounded subset with sufficiently smooth boundary Ω in \(R^{n}\), where \(R^{n}\) is an arbitrary positive-integer N-dimensional space. is the Laplacian operator in the region Ω. The no-flux boundary condition means that the statical environment Ω is isolated.

In the biological field, the study of the stability and bifurcations of the reaction-diffusion models describing a variety of biological phenomena is one of the fundamental problems, which can be conducive to understand the dynamic behavior. Recently, the stability of the steady state [15, 16] and that of the non-constant steady state solution [1720] are investigated. The main contributions of this article are as follows: (a) the existence of the global positive solution and the persistence of system (1.2) are discussed in detail; (b) the Hopf bifurcation analysis of system (1.2) is provided; (c) the Lyapunov function is constructed to complete the proof of global stability of the equilibria.

The rest of this article is organized as follows. In Section 2, the existence of a global positive solution of (1.2) as well as the boundedness are considered. In Section 3, the persistence of the system is investigated. In Section 4, the stability of the constant equilibria, the existence of Hopf bifurcations and the stability of a bifurcating periodic solution around the interior constant equilibrium are investigated. In Section 5, the global stability of the constant equilibria is proved by the Lyapunov function method under certain conditions investigated. Finally, simulations are presented to verify our theoretical analysis.

2 Existence of a global positive solution of (1.2)

For convenience, we introduce some notations. For \(x, y \in R\), we denote \(x \wedge y:\min\{x,y\}\), \(x\vee y:=\max\{x,y\}\), \(x_{+}:x\vee 0\), \(x_{-}:(-x)\vee0\) and extend these notations to real-valued functions. If \((\vee,\geq)\) is a partially ordered vector space, we denote its positive cone by \(\vee_{+}:=\{v\in\vee:v\geq0\}\). For \(p\in [1, {+}\infty),p>N/2\), let \(X=\{L^{p}(\Omega)\}^{2}\). Assume \(W=(u,v)\in X\), and its norm is defined as \(\Vert W\Vert =\Vert u\Vert _{p}+\Vert v\Vert _{p}\). Now, model (1.2) can be rewritten as an abstract differential equation
$$ \textstyle\begin{cases} \frac{dz}{dt}=Az+H(z), & t\in(0, \infty), \\ z(0)=z_{0}, \end{cases} $$
(2.1)
where \(z=(u(x,t), v(x,t))^{T}, z_{0}=(u_{0},v_{0})^{T}\),
$$ A= \begin{pmatrix} \triangle-I&0\\ 0&\triangle-I \end{pmatrix} , $$
(2.2)
with
$$D(A)= \biggl\{ (u,v)\in\bigl\{ C^{2}(\bar{\Omega})\bigr\} ^{2}, \frac{\partial u}{\partial \nu}\biggl\vert _{\partial\Omega}=\frac{\partial v}{\partial \nu }\biggr\vert _{\partial\Omega}=0 \biggr\} $$
and
$$ H(z)= \begin{pmatrix}u(2-u-\frac{uv}{au^{2}+bu+1}) \\ v(\delta+1-\frac{\beta v}{u}) \end{pmatrix}. $$
(2.3)

Next, we prove the existence of the local solution for (2.1). First, we give out the following lemma.

Lemma 2.1

For every \(z_{0}\in X _{+}\), the Cauchy problem (2.1) has a unique maximal local solution
$$ z\in\bigl([0,T_{\max});X \bigr)\cap C^{1} \bigl((0,T_{\max});X\bigr)\cap C^{1}\bigl((0,T_{\max});D(A) \bigr), $$
(2.4)
which satisfies the following Duhamel formula for \(t\in[ 0,T_{\max})\):
$$ z(t)=e^{tA_{p}}z_{0}+ \int^{t}_{0}e^{(t-s)A_{p}}H\bigl[z(s)\bigr]\,ds, $$
(2.5)
where \(A_{p}\) is an operator and the closure of A in X. Moreover, if \(T_{\max}<\infty\), then \(\lim \sup_{ T^{-}_{\max}} \Vert z(t)\Vert =\infty\).

Proof

Since the operator \(A_{p}\) is the closure of A in X, it generates an analytic, condensed, strong continuous operator semi-group \(e^{tA_{p}}\). Furthermore, for \(t>0\), we have
$$ \bigl\Vert e^{tA_{p}} w\bigr\Vert \leq e^{-t} \Vert w\Vert ,\quad \forall w\in X. $$
(2.6)
Moreover, we observe that \(H:D(A)\rightarrow X\) is locally Lipschitz on a bounded set. Via Theorem 7.2.1 in [21], we complete his proof. □

Lemma 2.2

For the initial-boundary problem of model (1.2), the components of its solution u(x,t), v(x,t) are nonnegative.

Proof

To prove \(u(x,t), v(x,t)\geq0\),
$$ \textstyle\begin{cases} \frac{\partial u^{\prime}}{\partial t}=d_{1}\triangle u^{\prime}+u^{\prime}_{+}(1-u^{\prime}_{+}-\frac {u^{\prime}_{+}v^{\prime}_{+}}{a{u^{\prime}_{+}}^{2}+bu^{\prime}_{+}+1}),\\ \frac{\partial v^{\prime}}{\partial t}=d_{1}\triangle v^{\prime} +cv^{\prime}_{+}(\delta -\frac{\beta v^{\prime}_{+}}{u^{\prime}_{+}}), \end{cases} $$
(2.7)
with the boundary and initial conditions
$$\textstyle\begin{cases} \frac{\partial u^{\prime}}{\partial\nu}\vert _{\Omega }=\frac{\partial v^{\prime}}{\partial\nu}\vert _{\Omega}=0,\\ u^{\prime}(x,0)=u_{0},\qquad v^{\prime}(x,0)=v_{0}. \end{cases} $$
After multiplying (2.7) by \(u^{\prime}_{-}\) and \(v^{\prime}_{-}\), respectively, and integrating by parts on the domain Ω, we obtain
$$\textstyle\begin{cases} -\frac{1}{2} \frac{d}{dt} \int_{\Omega} \vert \partial u^{\prime}_{-}\vert ^{2} _{\Omega}\,dx- d_{1}\int_{\Omega} \vert \nabla\partial u^{\prime}_{-}\vert ^{2}\,dx=0,\\ -\frac{1}{2} \frac{d}{dt}\int_{\Omega} \vert \partial v^{\prime}_{-}\vert ^{2} _{\Omega }\,dx- d_{2}\int_{\Omega} \vert \nabla\partial v^{\prime}_{-}\vert ^{2}\,dx=0. \end{cases} $$
Hence, for \(t\in(0,T_{\max})\),
$$\bigl\Vert u^{\prime}_{-}(\cdot,t)\bigr\Vert ^{2}_{2}+\bigl\Vert v^{\prime}_{-}( \cdot,t)\bigr\Vert ^{2}_{2}\leq\bigl\Vert u^{\prime}_{-}(\cdot,0)\bigr\Vert ^{2}_{2}+ \bigl\Vert v^{\prime}_{-}(\cdot,0)\bigr\Vert ^{2}_{2}=0. $$
Consequently, \(u^{\prime}(x,t)\geq0, v^{\prime}(x,t)\geq0\).

Now, we know that \(( u^{\prime}(x,t), v^{\prime}(x,t))\) is a solution of (1.2), and according to Lemma 2.1, we finally get that \(u(x,t)\geq0, v(x,t)\geq0, t\in(0,T_{\max})\). Next, we prove the global existence of the positive solution for (1.2). Via Lemmas 2.1 and 2.2, we only need to show that the solution of model (1.2) is uniformly bounded, i.e., dissipation. For this, we first introduce the following lemma. □

Lemma 2.3

([22])

Suppose that \(z(x,t)\) satisfies the following equation:
$$\textstyle\begin{cases} \frac{\partial z}{\partial t} =d \triangle z+z(1-z),& x\in\Omega, t>0, \\ \frac{\partial z}{\partial n}=0, & x\in\Omega, t>0,\\ z(x,0)=z_{0}(x)\geq0, \not\equiv0, & x\in\Omega. \end{cases} $$
Then \(\lim_{t\rightarrow\infty}z(x,t)=1\) for any \(x\in\Omega\).

Theorem 2.4

The solution \((u(x,t), v(x,t))\) of (1.2) satisfies the following inequalities:
$$\limsup_{t\rightarrow\infty}\max_{x\in\Omega}u(x,t)\leq1,\qquad \limsup_{t\rightarrow\infty}\max_{x\in\Omega}v(x,t)\leq \frac{\delta}{\beta}. $$

Proof

$$\textstyle\begin{cases} \frac{\partial u}{\partial t} \leq d_{1} \triangle u+u(1-u), & x\in\Omega, t>0, \\ \frac{\partial u}{\partial n}=0, & x\in\Omega, t>0,\\ u(x,0)=u_{0}(x), & x\in\Omega. \end{cases} $$
By the comparison principle and Lemma 2.3, we obtain easily the first inequality. Therefore, there exist \(T_{1}>0\) and sufficiently small \(0<\epsilon\leq1\) such that \(u(x,t)\leq1+\epsilon\) for any \(x\in\Omega\) and \(t>T_{1}\). For the second equation, we have the following inequality:
$$\textstyle\begin{cases} \frac{\partial v}{\partial t} \leq d_{2} \triangle v+v(\delta-\frac{\beta v}{1+\epsilon}), & x\in\Omega, t>0, \\ \frac{\partial v}{\partial n}=0, & x\in\Omega, t>0,\\ v(x,0)=v_{0}(x), & x\in\Omega. \end{cases} $$
By the comparison principle, for the above \(0<\epsilon\leq1\), there exists \(T_{2}(>T_{1})\) such that, for any \(t>T_{2}\),
$$v(x,t)\leq\frac{\delta(1+\epsilon)}{\beta}+\epsilon. $$
For the arbitrariness of \(\epsilon>0\), we obtain
$$\limsup_{t\rightarrow\infty}\max_{x\in\Omega}v(x,t)\leq \frac{\delta }{\beta},\quad \forall x\in\Omega. $$
As a result, we complete this proof. By Lemmas 2.1, 2.2 and Theorem 2.4, we can get the following theorem. □

Theorem 2.5

System (1.2) has a unique, nonnegative and bounded solution \(Z(x,t)=(u(x,t), v(x,t))\) such that
$$z(x,t)\in C\bigl([0,\infty),X \bigr)\cap C^{1}\bigl((0, \infty);X\bigr) \cap C \bigl((x,\infty),D(A)\bigr). $$

3 Permanence

In this section, we show that any nonnegative solution \((u(x,t), v(x,t))\) of (1.2) lies in a certain bounded region as \(t\rightarrow \infty\) for all \(x\in\Omega\).

We make the following hypothesis: \((\mathrm{H}_{0})\) \(\frac{\delta}{b\beta}<1\).

Theorem 3.1

If \((\mathrm{H}_{0}) \) holds, the solution (u,v) of (1.2) satisfies the following inequalities:
$$\liminf_{t\rightarrow\infty}\min_{x\in\Omega}u(x,t)\geq1- \frac{\delta }{b\beta}, \qquad\liminf_{t\rightarrow\infty}\min_{x\in\Omega}v(x,t) \geq \frac{\delta(1-\frac{\delta}{b\beta})}{\beta}. $$

Proof

From the first equation of (1.2) and Theorem 3.1, we have
$$\frac{\partial u}{\partial t} \geq d_{1} \triangle u+u\biggl(1-u- \frac{\frac {\delta(1+\epsilon)}{\beta}+\epsilon}{b }\biggr). $$
By the comparison principle and Lemma 2.3, we obtain easily
$$\liminf_{t\rightarrow\infty}\min_{x\in\Omega}u(x,t)\geq1- \frac{\delta }{b\beta}. $$
From the second equation of (1.2), we have
$$\frac{\partial v}{\partial t} \geq d_{2} \triangle v+v\biggl(\delta- \frac{\beta v}{1-\frac{\delta}{b\beta}}\biggr). $$
This implies
$$\liminf_{t\rightarrow\infty}\min_{x\in\Omega}v(x,t)\geq \frac{\delta (1-\frac{\delta}{a\beta})}{\beta}. $$
From Theorems 2.4 and 3.1, we can easily obtain the following theorem. □

Theorem 3.2

If \((\mathrm{H}_{0}) \) holds, model (1.2) is permanent.

4 The stability of equilibria and the existence of Hopf bifurcations

4.1 The existence of equilibrium of model (1.2)

The system has always constant equilibria \(E_{0}(1,0)\). By the direct calculation, there exists a unique constant positive equilibrium denoted by \(E_{\ast}(u_{0}, v_{0})\) if and only if the following cubic equation
$$ au^{3}+\biggl(\frac{\delta}{\beta}+b-a \biggr)u^{2}+(1-b)u-1=0 $$
(4.1)
holds in the interval \((0, 1)\). Note that the third-order algebraic equation (4.1) can have one, two, or three positive roots in the interval \((0,1)\) which can be evaluated by using the root formula of the third-order algebraic equation. Correspondingly, system (1.2) can have one, two, or three positive equilibria. Regarding the number of positive equilibria in the interval \((0,1)\) of (1.2), [14] gives the result in detail as follows.

Lemma 4.1

([14])

Let \(\Theta=(\frac{\delta}{\beta }+b-a)^{2}+3a(b-1)\) and \(\Delta=-4\Theta^{3}+(27a^{2}+9a(1-b)(\frac{\delta}{\beta}+b-a)-2(\frac {\delta}{\beta}+b-a)^{3})^{2}\).
  1. (H1)

    If \(\Delta>0\), then model (1.2) has a unique positive equilibrium \(E_{\ast}\), which is an elementary and anti-saddle equilibrium.

     
  2. (H2)

    If \(\Delta=0\) and \(\Theta>0\), then the model has two different positive equilibria: an elementary anti-saddle equilibrium \(E_{\ast}\) and a degenerate equilibrium \(E_{1}\).

     
  3. (H3)

    If \(\Delta=0\) and \(\Theta=0\), the model has a unique positive equilibrium \(E_{2}(\frac{3}{1-b}, \frac{3\delta}{\beta(1-b)})\), which is a degenerate equilibrium.

     
  4. (H4)

    If \(\Delta<0, \frac{\delta}{\beta}=a-b-\sqrt{3a(1-b)}\) and \(-2\sqrt{a}< b<1\), then model (1.2) has three different positive equilibria \(E^{\ast}_{1}\), \(E^{\ast}_{2}\) and \(E^{\ast}_{3}\), which are all elementary equilibria and \(E^{\ast}_{3}\) is a saddle.

     

Remark

We further consider the following condition (H1). We suppose (H1) is satisfied throughout the rest of this paper.

4.2 The existence of Hopf bifurcation

In this section we assume that (H1) holds. We first assume that \(E_{*}(u_{0}, v_{0})\) is any point of system (1.2). Now we define the real-valued Sobolev space
$$X:=\bigl\{ (u,v)\in\bigl[H^{2}(0, l\pi)\bigr]^{2}:(u_{x}, v_{x})\vert _{x=0,l\pi}=0\bigr\} , $$
and the complexification of X
$$X_{C}:=X\oplus iX=\{x_{1}+ix_{2}:x_{1},x_{2} \in X\}. $$
The linearized system of (1.2) at any point \(E(u_{0},v_{0})\) has the form
$$ L= \begin{pmatrix} A &-B\\ \frac{\delta^{2}}{\beta}&-\delta \end{pmatrix} $$
(4.2)
with the domain \(D_{L(s)}=X_{C}\), where
$$A=1-2u_{0}-\frac{u_{0}v_{0}(bu_{0}+2)}{(au^{2}_{0}+bu_{0}+1)^{2}},\qquad B=\frac {u^{2}_{0}}{au^{2}_{0}+bu_{0}+1}. $$
Obviously, \(B>0\). From Wu [23], we obtain that the characteristic equation of (4.2) is
$$ \lambda y-d\triangle y-L\bigl(e^{\lambda}y\bigr)=0,\quad y\in \operatorname{dom}(d\triangle), y\neq 0. $$
(4.3)
It is well known that the eigenvalue problem
$$-\varphi^{\prime\prime}=\mu\varphi,\quad x \in(0, l\pi): \varphi^{\prime}(0)= \varphi^{\prime}(l\pi)=0 $$
has eigenvalues \(\mu_{n}=\frac{n^{2}}{l^{2}}\) (\(n=0,1,2, \ldots\)) with corresponding eigenfunctions
$$\varphi(x)_{n}=\cos\frac{n\pi}{l}, \quad n=0,1,\ldots. $$
Substituting
$$y=\sum^{\infty}_{k=0} \begin{pmatrix} y_{1n} \\ y_{2n} \end{pmatrix} \cos\frac{n\pi}{l} $$
into the characteristic Eq. (4.3), we have
$$ \begin{pmatrix}A-\frac{d_{1} n^{2}}{l^{2}}&-B \\ \frac{\delta^{2}}{\beta} &-\delta -\frac{d_{1} n^{2}}{l^{2} } \end{pmatrix} \begin{pmatrix}y_{1n}\\ y_{2n} \end{pmatrix} =\lambda \begin{pmatrix} y_{1n}\\ y_{2n} \end{pmatrix} , \quad n=0,1,2,\ldots. $$
Therefore the characteristic Eq. (4.3) is equivalent to
$$ \operatorname{det}(\lambda I -M_{n}-L)=0, $$
(4.4)
where I is the \(2\times2\) identity matrix and \(M_{n}=\frac{n^{2}}{l^{2}} \operatorname{diag}\{d_{1},d_{2}\}, n=0,1,2, \ldots\) . It follows from (4.4) that the characteristic equations for the equilibrium \((u_{0},v_{0})\) are the following sequence of quadratic transcendental equations:
$$ \Delta_{n}(\lambda, \tau)=\lambda^{2}-T_{n} \lambda+D_{n} =0, $$
(4.5)
where
$$ \textstyle\begin{cases} T_{n}=-(d_{1}+d_{2})\frac{ n^{2}}{l^{2}}-\delta+A, \\ D_{n}= \frac{d_{1}d_{2} n^{4}}{l^{4}}+(d_{1}\delta-d_{2}A)\frac{n^{2}}{l^{2}}+\delta (\frac{B\delta}{\beta}- A). \end{cases} $$
(4.6)
The eigenvalues are given by
$$ \lambda_{1,2}=\frac{T_{n}\pm\sqrt{T^{2}_{n}-4 D_{n}}}{2},\quad n=0,1,2,\ldots. $$
(4.7)
We make the following hypothesis with fixing δ:
$$(\mathrm{H}_{5}) \quad A< \rho\delta \quad\mbox{and}\quad ( \mathrm{H}_{6})\quad \frac{ B\delta}{\beta}< A < \frac{d_{1}}{d_{2}}\delta, $$
where \(\rho=\min\{1, \frac{d_{1}}{d_{2}},\frac{B}{\beta}\}\).

Lemma 4.2

  1. (i)

    Suppose \((\mathrm{H}_{1})\) and \((\mathrm{H}_{5})\) are satisfied. Then all the roots of Eq. (4.5) have a negative real part.

     
  2. (ii)

    Suppose \((\mathrm{H}_{1})\) and \((\mathrm{H}_{6})\) hold, then Eq. (4.5) has at least one positive real part.

     

Proof

  1. (i)

    Obviously, by \((\mathrm{H}_{5})\) \(T_{n}<0\) for \(n=0,1,2, \ldots\) , we know all the roots of Eq. (4.5) have negative real parts.

     
  2. (ii)

    If \((\mathrm{H}_{6})\) holds, we know that \(D_{n}\) is monotone increasing with regard to n, and \(\lim_{n\rightarrow\infty} D_{n} = \infty\) and \(D_{0}<0\). There exists an integer \(n^{\ast}_{0}\geq1\) such that \(D_{n}<0\) when \(n=0,1,2,\ldots, n^{\ast}_{0} \). Hence, there are \(D_{n}<0\), \(T^{2}_{n}-4D_{n}>T^{2}_{n}\), \(\lambda_{1} >0\) and \(\lambda_{2} <0 \) for \(n=0,1,2,\ldots, n^{\ast}_{0}\) and \(D_{n}>0\), \(T^{2}_{n}-4D_{n}< T^{2}_{n}\), \(\lambda_{1} >0\) and \(\lambda_{2} >0\) for \(n \geq n^{\ast}_{0}\). These complete the proof of (i) and (ii).

     

From the discussion, we have the following theorem. □

Theorem 4.3

  1. (i)

    Suppose \((\mathrm{H}_{1})\) and \((\mathrm{H}_{5})\) hold, the equilibrium \(E_{*}\) is stable.

     
  2. (ii)

    Suppose \((\mathrm{H}_{1})\) and \((\mathrm{H}_{6})\) hold, the equilibrium \(E_{*}\) is unstable.

     
Next, we consider the occurrence of Hopf bifurcation of \(E_{*}\) by regarding A as a bifurcation parameter with fixing δ. In the light of (4.7), it is known that (4.5) has purely imaginary roots if and only if
$$ A =A_{n}= (d_{1}+d_{2}) \frac{ n^{2}}{l^{2}} +\delta,\quad n=0,1,\ldots, $$
(4.8)
and \(D_{n}>0 \) holds. From (4.8) we know that \(A_{n}\) is monotone increasing with regard to n, and
$$\lim_{n\rightarrow\infty} A_{n} = \infty,\quad \textit{and}\quad A_{0}=\delta>0. $$
Hence, there is \(A_{n}>0\) for \(n=0,1,2,\ldots\) . Bringing \(A_{n}\) into \(D_{n}\), we get
$$ D_{n}= - \frac{d_{2} n^{4}}{l^{4}}-2\frac{d_{2} n^{2} }{l^{2}}\delta+ \biggl(\frac{B}{\beta }-1\biggr)\delta^{2}. $$
(4.9)
If \(D_{0}=\delta^{2} (\frac{B}{\beta}- 1) >0\) holds, then there exists an integer \(n^{\ast}_{1}\geq1\) such that \(D_{n}>0\) when \(n=0,1,2,\ldots, n^{\ast}_{1} \) and \(D_{n}\leq0\) when \(n \geq n^{\ast}_{1}\).

Let \(n^{\ast}= [n^{\ast}_{1}]\) and \(\Lambda= \{A_{n^{\ast }-1}>A_{n^{\ast}-2}>A_{n^{\ast}-3}>\cdots>A_{0} \}\).

Lemma 4.4

If \(\frac{B}{\beta}-1>0\) is satisfied, then Eq. (4.5) has purely imaginary roots if and only if \(A \in\Lambda\).

Proof

Let \(\lambda_{n}(A)=\alpha_{n}(A)+i\omega_{n}(A), n=0,1,\ldots,n^{\ast}\) be the roots of Eq. (4.8) satisfying \(\alpha_{n}(A)=0, \omega_{n}(A)=\sqrt{D_{n}(A)}\). If \(\frac{B}{\beta}-1>0\) holds, we know that \(D_{n}\) is monotone decreasing with regard to n, and \(\lim_{n\rightarrow\infty} D_{n}= -\infty\) and \(D_{0}>0\). There exists an integer \(n^{\ast}_{1}\geq1\) such that \(D_{n}>0\) when \(n=0,1,2,\ldots, n^{\ast}\). These complete the proof of Lemma 4.4. From expression (4.7) we have the following conclusion. □

Lemma 4.5

Suppose \(\frac{B}{\beta}-1>0\) is satisfied, then all the roots of Eq. (4.5) with \(A=A_{n}\) (\(n\rightarrow\infty\)) have at least one root with a positive real part except the imaginary roots \(\pm i\sqrt{D_{n}(A_{n})}, n \in(0,1, \ldots, n^{\ast} )\).

Proof

If \(\frac{B}{\beta}-1>0\) holds, we know that \(D_{n}\) is monotone decreasing with regard to n, and \(\lim_{n\rightarrow\infty} D_{n} = -\infty\). Hence, there are \(D_{n}<0\), \(T^{2}_{n}-4D_{n}>T^{2}_{n}\), \(\lambda_{1} >0\) and \(\lambda_{2} <0 \) for \(n>n^{\ast}\). In combination with Lemma 4.4, we complete the proof of Lemma 4.5. □

Based on the above analysis, we have the following theorem.

Theorem 4.6

Suppose \(\frac{B}{\beta}-1>0\) is satisfied. Then model (1.2) undergoes a Hopf bifurcation at the origin when \(A=A_{n}\) for \(0\leq n\leq n^{\ast} \). Moreover:
  1. (i)

    The bifurcation periodic solution from \(A=A_{0}\) is spatially homogeneous, which coincides with the periodic solution of the corresponding ODE system.

     
  2. (ii)

    The bifurcation periodic solution from \(A=A_{n}\) is spatially non-homogeneous for \(0< n\leq n^{\ast}\).

     

4.3 Direction and stability of Hopf bifurcation

From the previous section, we know that model (1.2) undergoes a Hopf bifurcation at the origin when \(A=A_{n}\), \(n=0,1,\ldots, n^{\ast} \), where \(n^{\ast}\) and \(A_{n}\) are defined as in (4.8).

In this section, we study the direction of Hopf bifurcation and the stability of the bifurcating periodic solution by applying the center manifold theorem and the normal form theorem of partial differential equations. We first transform \(E_{\ast} \) of (1.2) to the origin via the translation: \(\hat{u}=u-u_{0}, \hat{v}-v_{0}\) and drop the hats for simplicity of notation, then (1.2) is transformed into
$$ \textstyle\begin{cases} \frac{\partial u}{\partial t}=d_{1}\nabla u+f(\delta,u,v), \\ \frac{\partial v}{\partial t}=d_{2} \nabla v+g(\delta,u,v), \end{cases} $$
(4.10)
where
$$\begin{aligned}& f(A,u,v)= u+u_{0}-(u+u_{0})^{2}- \frac{(u+u_{0})^{2}(v+v_{0})}{a(u+u_{0})^{2}+b(u+u_{0})+1}, \\& g(A,u,v) =(v+v_{0}) \biggl( \delta- \frac{\beta(v+v_{0})}{u+u_{0}}\biggr). \end{aligned}$$
So we have
$$q:= \begin{pmatrix} a_{0}\\b_{0} \end{pmatrix} = \begin{pmatrix} 1 \\ \frac{\delta^{2}}{\beta(\delta+i\omega)} \end{pmatrix} ,\qquad q^{\ast}:= \begin{pmatrix} a^{\ast}_{0}\\b^{\ast}_{0} \end{pmatrix} = \begin{pmatrix} \frac{1}{2l\pi} \\ \frac{\beta}{2l\pi(-i\omega+\delta )\delta^{2}} \end{pmatrix}. $$
The straightforward computation yields
$$\textstyle\begin{cases} f_{uu}=-2-\frac{2v_{0}(bu_{0}+1)}{(au^{2}_{0}+bu_{0}+1)^{2}}+\frac {u_{0}v_{0}(bu_{0}+2)(4au_{0}+2b)}{(au^{2}_{0}+bu_{0}+1)^{3}},\\ f_{uv}=-\frac{bu^{2}_{0}+2u_{0}}{(au^{2}_{0}+bu_{0}+1)^{2}},\\ f_{uuv}=-\frac{2bu_{0}+2}{(a^{2}u_{0}+bu_{0}+1)^{2}}+\frac {u_{0}(bu_{0}+2)(4au_{0}+2b)}{(au^{2}_{0}+bu_{0}+1)^{3}},\\ f_{uuu}=\frac {2v_{0}(12abu^{2}_{0}+2b^{2}u_{0}+8au_{0}+2b-b)}{(au^{2}_{0}+bu_{0}+1)^{2}}+\frac {4v_{0}(4abu^{3}_{0}+b^{2}u^{2}_{0}+4au^{2}_{0}+2bu_{0})}{(au^{2}_{0}+bu_{0}+1)^{3}}+\frac {4v_{0}(2abu^{2}_{0}+b^{2}u_{0}+2au_{0}+b)}{(au^{2}_{0}+bu_{0}+1)^{4}},\\ f_{vv}=0,\\ g_{uu}=-2\frac{\delta^{2}}{\beta u_{0}},\\ g_{uv}=2\frac{\delta}{u_{0}},\\ g_{uuv}=-4\frac{\delta}{u^{2}_{0}},\\ g_{uvv}=\frac{2\beta}{u^{2}_{0}},\\ g_{vv}=-\frac{2\beta}{u_{0}}. \end{cases} $$
Moreover,
$$\textstyle\begin{cases} Q_{qq}= \bigl({\scriptsize\begin{matrix}{} c_{n}\cr d_{n} \end{matrix}}\bigr) \cos^{2}\frac{nx}{l}= \bigl({\scriptsize\begin{matrix}{} f_{uu}a^{2}_{n}+2f_{uv}a_{n}b_{n}+f_{vv}b^{2}_{n} ,\cr g_{uu}a^{2}_{n}+2g_{uv}a_{n}b_{n}+g_{vv}b^{2}_{n} \end{matrix}}\bigr) , \\ Q_{q\bar{q}}= \bigl({\scriptsize\begin{matrix}{} e_{n} \cr f_{n} \end{matrix}}\bigr) \cos^{2}\frac{nx}{l}= \bigl({\scriptsize\begin{matrix}{} f_{uu}\vert a_{n}\vert ^{2}+2f_{uv}(a_{n}\overline{b}_{n}+ \overline{a}_{n} b_{n})+f_{vv}\vert b_{n}\vert ^{2} \cr g_{uu}\vert a_{n}\vert ^{2}+2g_{uv}(a_{n} \overline{b}_{n}+\overline{a}_{n} b_{n})+g_{vv}\vert b_{n}\vert ^{2} \end{matrix}}\bigr) , \\ Q_{qq\bar{q}}= \bigl({\scriptsize\begin{matrix}{} g_{n}\cr h_{n} \end{matrix}}\bigr) \cos^{2}\frac{nx}{l}= \bigl({\scriptsize\begin{matrix}{} f_{uuu}\vert a_{n}\vert ^{2}b_{n}+f_{uuv}(2\vert a_{n}\vert ^{2}b_{n}+a^{2}_{n} \overline{b}_{n})+f_{uvv}\vert b_{n}\vert ^{2}a_{n}+b_{n}^{2}\overline{a}_{n}+f_{vvv}\vert b_{n}\vert ^{2}b_{n}\cr g_{uuu}\vert a_{n}\vert ^{2}b_{n}+g_{uuv}(2\vert a_{n}\vert ^{2}b_{n}+a^{2}_{n} \overline{b}_{n})+g_{uvv}\vert b_{n}\vert ^{2}a_{n}+b_{n}^{2} \overline{a}_{n}+g_{vvv}\vert b_{n}\vert ^{2}b_{n} \end{matrix}}\bigr). \end{cases} $$
So
$$ \textstyle\begin{cases} c_{0}=f_{uu} +2f_{uv} b_{0}+f_{vv}b^{2}_{0}= [-2-\frac {2v_{0}(bu_{0}+1)}{(au^{2}_{0}+bu_{0}+1)^{2}}+\frac {u_{0}v_{0}(bu_{0}+2)(4au_{0}+2b)}{(au^{2}_{0}+bu_{0}+1)^{3}}]\\ \phantom{c_{0}=} {}+2[-\frac {bu^{2}_{0}+2u_{0}}{(au^{2}_{0}+bu_{0}+1)^{2}}][-\frac{\delta^{2}}{\beta(\delta+i\omega )}],\\ d_{0}=g_{uu} +2g_{uv} b_{0}+g_{vv}b^{2}_{0}=[-2\frac{\delta^{2}}{\beta u_{0}}]+2[2\frac{\delta}{u_{0}}][-\frac{\delta^{2}}{\beta(\delta+i\omega )}]^{2}, \\ e_{0}=f_{uu}+2f_{uv}( \overline{b}_{0}+b_{0})+f_{vv}\vert b_{0}\vert ^{2} \\ \quad=[-2-\frac{2v_{0}(bu_{0}+1)}{(au^{2}_{0}+bu_{0}+1)^{2}}+\frac {u_{0}v_{0}(bu_{0}+2)(4au_{0}+2b)}{(au^{2}_{0}+bu_{0}+1)^{3}}]\\ \phantom{e_{0}=}{}+2[-\frac {bu^{2}_{0}+2u_{0}}{(au^{2}_{0}+bu_{0}+1)^{2}}][-\frac{\delta^{2}}{\beta(\delta+i\omega )}-\frac{\delta^{2}}{\beta(\delta-i\omega)}],\\ f_{0}=g_{uu}+2g_{uv}( \overline{b}_{0}+b_{0})+g_{vv}\vert b_{0}\vert ^{2} \\ \phantom{f_{0}}=[-2\frac{\delta^{2}}{\beta u_{0}}]+2[2\frac{\delta}{u_{0}}][-\frac{\delta^{2}}{\beta(\delta +i\omega)}-\frac{\delta^{2}}{\beta(\delta-i\omega)}]+[-\frac{2\beta }{u_{0}}][ -\frac{\delta^{2}}{\beta(\delta+i\omega)} ]^{2},\\ g_{0}=f_{uuu} +f_{uuv}(2 b_{0}+ \overline{b}_{0})+f_{uvv}(2\vert b_{0}\vert ^{2} +b_{0}^{2})+f_{vvv}\vert b_{0}\vert ^{2}b_{0}\\ \phantom{g_{0}} =[-4\frac{\delta}{u^{2}_{0}}] +[-\frac{2bu_{0}+2}{(a^{2}u_{0}+bu_{0}+1)^{2}}+\frac {u_{0}(bu_{0}+2)(4au_{0}+2b)}{(au^{2}_{0}+bu_{0}+1)^{3}}][-2\frac{\delta^{2}}{\beta (\delta+i\omega)}-\frac{\delta^{2}}{\beta(\delta-i\omega)}], \\ h_{0}= g_{uuu} +g_{uuv}(2 b_{0}+ \overline{b}_{0})+g_{uvv}(2\vert b_{0}\vert ^{2} +b_{0}^{2} )+g_{vvv}\vert b_{0}\vert ^{2}b_{0}\\ \phantom{h_{0}}=[6\frac{\delta^{2}}{\beta u^{2}_{0}}] +[-4\frac{\delta}{u^{2}_{0}}][ -2\frac {\delta^{2}}{\beta(\delta+i\omega)}-\frac{\delta^{2}}{\beta(\delta-i\omega )}]\\ \phantom{h_{0}=}{}+[\frac{2\beta}{u^{2}_{0}}][2\vert -\frac{\delta^{2}}{\beta(\delta+i\omega )}\vert ^{2}+(-\frac{\delta^{2}}{\beta(\delta+i\omega)})^{2}]. \end{cases} $$
(4.11)
Then we have
$$\textstyle\begin{cases} \langle q^{\ast},Q_{qq} \rangle = \langle \bigl({\scriptsize\begin{matrix}{} \frac{1}{2l\pi} \cr \frac{\beta}{2l\pi(-i\omega+\delta )\delta^{2}} \end{matrix}}\bigr), \bigl({\scriptsize\begin{matrix}{} c_{0} \cr d_{0}\end{matrix}}\bigr) \rangle\\ \phantom{\langle q^{\ast},Q_{qq} \rangle}= \frac{1}{2 }(f_{uu} +2f_{uv} b_{0}+f_{vv}b^{2}_{0}) + \frac{\beta}{2 ( i\omega+\delta)\delta^{2}}(g_{uu} +2g_{uv} b_{0}+g_{vv}b^{2}_{0}) \\ \phantom{\langle q^{\ast},Q_{qq} \rangle}= ([-1-\frac{ v_{0}(bu_{0}+1)}{(au^{2}_{0}+bu_{0}+1)^{2}}+\frac {u_{0}v_{0}(bu_{0}+2)(2au_{0}+b)}{(au^{2}_{0}+bu_{0}+1)^{3}}]+[-\frac {bu^{2}_{0}+2u_{0}}{(au^{2}_{0}+bu_{0}+1)^{2}}][\frac{\delta^{2}}{\beta(\delta+i\omega )}])\\ \phantom{\langle q^{\ast},Q_{qq} \rangle=}{}+ \frac{\beta}{2 ( i\omega+\delta)\delta^{2}}([-2\frac{\delta ^{2}}{\beta u_{0}}] +2[2\frac{\delta}{u_{0}}][\frac{\delta^{2}}{\beta(\delta +i\omega)}]^{2} ), \\ \langle q^{\ast},Q_{q\bar{q}} \rangle =\langle \bigl({\scriptsize\begin{matrix}{} \frac{1}{2l\pi} \cr \frac{\beta}{2l\pi( i\omega+\delta )\delta^{2}} \end{matrix}}\bigr) , \bigl({\scriptsize\begin{matrix}{} e_{0}\cr f_{0} \end{matrix}}\bigr) \rangle\\ \phantom{\langle q^{\ast},Q_{q\bar{q}} \rangle}= \frac{1}{2 } (f_{uu} +2f_{uv}( \overline{b}_{0}+ b_{0})+f_{vv}\vert b_{0}\vert ^{2})+ \frac{\beta}{2 ( i\omega+\delta)\delta^{2}}(g_{uu} +2g_{uv}( \overline{b}_{0}+ b_{0})+g_{vv}\vert b_{0}\vert ^{2}) \\ \phantom{\langle q^{\ast},Q_{q\bar{q}} \rangle}= \frac{1}{2 } ([-2-\frac{2v_{0}(bu_{0}+1)}{(au^{2}_{0}+bu_{0}+1)^{2}}+\frac {u_{0}v_{0}(bu_{0}+2)(4au_{0}+2b)}{(au^{2}_{0}+bu_{0}+1)^{3}}]\\ \phantom{\langle q^{\ast},Q_{q\bar{q}} \rangle=}{}+2[-\frac {bu^{2}_{0}+2u_{0}}{(au^{2}_{0}+bu_{0}+1)^{2}}][\frac{\delta^{2}}{\beta(\delta+i\omega )}+\frac{\delta^{2}}{\beta(\delta-i\omega)}])\\ \phantom{\langle q^{\ast},Q_{q\bar{q}} \rangle=}{}+\frac{\beta}{2 (i\omega +\delta)\delta^{2}}([2\frac{\delta^{2}}{\beta u_{0}}]+2[2\frac{\delta }{u_{0}}][\frac{\delta^{2}}{\beta(\delta+i\omega)}+\frac{\delta^{2}}{\beta (\delta-i\omega)}]+[-\frac{2\beta}{u_{0}}][ \frac{\delta^{2}}{\beta(\delta +i\omega)} ]^{2}), \\ \langle\overline{q^{\ast}},Q_{qq} \rangle = \langle \bigl( {\scriptsize\begin{matrix}{} \frac{1}{2l\pi} \cr \frac{\beta}{2 (-i\omega+\delta )\delta^{2}} \end{matrix}}\bigr) , {\scriptsize\begin{matrix}{} c_{0}\cr d_{0} \end{matrix}} \rangle\\ \phantom{\langle\overline{q^{\ast}},Q_{qq} \rangle}= \frac{1}{2 }(f_{uu} +2f_{uv} b_{0}+f_{vv}b^{2}_{0} )\\ \phantom{\langle\overline{q^{\ast}},Q_{qq} \rangle=}{}+ \frac{\beta}{2 (-i\omega+\delta)\delta^{2}}(g_{uu} +2g_{uv} b_{0}+g_{vv}b^{2}_{0}) \\ \phantom{\langle\overline{q^{\ast}},Q_{qq} \rangle}= ([-1-\frac{ v_{0}(bu_{0}+1)}{(au^{2}_{0}+bu_{0}+1)^{2}}+\frac {u_{0}v_{0}(bu_{0}+2)(2au_{0}+b)}{(au^{2}_{0}+bu_{0}+1)^{3}}]+[-\frac {bu^{2}_{0}+2u_{0}}{(au^{2}_{0}+bu_{0}+1)^{2}}][\frac{\delta^{2}}{\beta(\delta+i\omega )}])\\ \phantom{\langle\overline{q^{\ast}},Q_{qq} \rangle=}{}+ \frac{\beta}{2(-i\omega+\delta)\delta^{2}}([-2\frac{\delta ^{2}}{\beta u_{0}}]+2[2\frac{\delta}{u_{0}}][\frac{\delta^{2}}{\beta(\delta +i\omega)}]^{2} ), \\ \langle\overline{q^{\ast}},Q_{qq\bar{q}}\rangle = \langle \bigl({\scriptsize\begin{matrix}{} \frac{1}{2l\pi} \cr \frac{\beta}{2 (-i\omega+\delta )\delta^{2}} \end{matrix}}\bigr) , \bigl({\scriptsize\begin{matrix}{} g_{0}\cr h_{0} \end{matrix}}\bigr) \rangle\\ \phantom{\langle\overline{q^{\ast}},Q_{qq\bar{q}}\rangle} = \frac{1}{2} (f_{uuu} +f_{uuv}(2 b_{0}+ \overline{b}_{0})+f_{uvv}(2\vert b_{0}\vert ^{2} +b_{0}^{2} )+f_{vvv}\vert b_{0}\vert ^{2}b_{0}) \\ \phantom{\langle\overline{q^{\ast}},Q_{qq\bar{q}}\rangle=}{}+ \frac{\beta}{2 (-i\omega+\delta)\delta^{2}}(g_{uuu} +g_{uuv}(2 b_{0}+ \overline{b}_{0})+g_{uvv}(2\vert b_{0}\vert ^{2} +b_{0}^{2} )+g_{vvv}\vert b_{0}\vert ^{2}b_{0})\\ \phantom{\langle\overline{q^{\ast}},Q_{qq\bar{q}}\rangle}= \frac{1}{2 } ([-4\frac{\delta}{u^{2}_{0}}] +[-\frac{2bu_{0}+2}{(a^{2}u_{0}+bu_{0}+1)^{2}}+\frac {u_{0}(bu_{0}+2)(4au_{0}+2b)}{(au^{2}_{0}+bu_{0}+1)^{3}}][2\frac{\delta^{2}}{\beta (\delta+i\omega)}+\frac{\delta^{2}}{\beta(\delta-i\omega)}]) \\ \phantom{\langle\overline{q^{\ast}},Q_{qq\bar{q}}\rangle=}{}+ \frac{\beta}{2 (-i\omega+\delta)\delta^{2}}([6\frac{\delta^{2}}{\beta u^{2}_{0}}] +[-4\frac{\delta}{u^{2}_{0}}][ 2\frac{\delta^{2}}{\beta(\delta +i\omega)}+\frac{\delta^{2}}{\beta(\delta-i\omega)}]\\ \phantom{\langle\overline{q^{\ast}},Q_{qq\bar{q}}\rangle=}{}+[\frac{2\beta }{u^{2}_{0}}][2\vert \frac{\delta^{2}}{\beta(\delta+i\omega)}\vert ^{2}+(\frac{\delta ^{2}}{\beta(\delta+i\omega)})^{2}]) \end{cases} $$
and
$$\begin{aligned}& H_{20}= \begin{pmatrix} c_{0}\\d_{0} \end{pmatrix} -\bigl\langle q^{\ast}, Q_{qq}\bigr\rangle \begin{pmatrix} a_{0}\\b_{0} \end{pmatrix} -\bigl\langle \overline{q^{\ast}}, Q_{qq}\bigr\rangle \begin{pmatrix} \overline{a}_{0}\\ \overline{b}_{0} \end{pmatrix} =0, \\& H_{11}= \begin{pmatrix} e_{0}\\f_{0} \end{pmatrix} -\bigl\langle q^{\ast}, Q_{q\bar{q}}\bigr\rangle \begin{pmatrix} a_{0}\\b_{0} \end{pmatrix} -\bigl\langle \overline{q^{\ast}}, Q_{q\bar{q}}\bigr\rangle \begin{pmatrix} \overline{a}_{0}\\ \overline{b}_{0} \end{pmatrix} =0, \end{aligned}$$
which implies \(W_{20}=W_{11}=0\), that is to say,
$$\bigl\langle q^{\ast},Q(W_{11},q) \bigr\rangle =\bigl\langle q^{\ast},Q(W_{20},\bar{q})\bigr\rangle =0. $$
Therefore
$$\textstyle\begin{cases} c_{1}(A_{0}) = \{\frac{i}{2\omega}\langle q^{\ast },Q_{qq} \rangle \cdot \langle q^{\ast},Q_{q\bar{q}}\rangle +\frac{1}{2} \langle q^{\ast},Q_{qq\bar{q}}\rangle\} ,\\ \operatorname{Re} (c_{1}(A_{0})) =\operatorname{Re}\{\frac{i}{2\omega} \langle q^{\ast},Q_{qq} \rangle \cdot \langle q^{\ast },Q_{q\bar{q}} \rangle +\frac{1}{2} \langle q^{\ast},Q_{qq\bar{q}} \rangle \},\\ \operatorname{Im}(c_{1}(A_{0})) =\operatorname{Im}\{\frac{i}{2\omega} \langle q^{\ast},Q_{qq} \rangle \cdot \langle q^{\ast },Q_{q\bar{q}} \rangle +\frac{1}{2}\langle q^{\ast},Q_{qq\bar{q}}\rangle\},\\ T_{2} =[\operatorname{Im}(c_{1}(A_{0}))-\frac{\operatorname{Re}(c_{1}(\delta_{0}))}{\alpha^{\prime}(\delta _{0})}\omega^{\prime}(\delta_{0})], \end{cases} $$
where \(\omega^{\prime}(A_{0})=1>0\), then we have the following theorem.

Theorem 4.7

Suppose \(\frac{B}{\beta}-1>0\) is satisfied.

The Hopf bifurcation at \(A=A_{0}\) is backward (resp. forward);

The bifurcation periodic solution from \(A=A_{0}\) is asymptotically stable (resp. unstable) if \(\operatorname{Re}(c_{1}(A_{0})) <0\) (resp.>0);

The bifurcation periodic solution from \(A=A_{0}\) is increasing (resp. decreasing) if \(T_{2}>0\) (resp.<0).

When \(0< j\leq n^{\ast}\), for \(A_{j}\), we have the following theorem.

Theorem 4.8

Suppose \(\frac{B}{\beta}-1>0\) is satisfied.

The Hopf bifurcation at \(A=A_{j}\) for \(0< j\leq n^{\ast} \) is backward (resp.forward);

The bifurcation periodic solution from \(A=A_{j}\) is asymptotically stable (resp. unstable) if \(\operatorname{Re}(c_{1}(A_{j}))<0\) (resp.>0);

The bifurcation periodic solution from \(A=A_{j}\) is increasing (resp. decreasing) if \(T_{2}>0\) (resp.<0) for \(0< j\leq n^{\ast}\), where \(\operatorname{Re}(c_{1}(A_{j}))\) is defined in Appendix.

5 Global stability of equilibria

In this section, we continue to study the global stability of equilibria.

Theorem 5.1

Suppose that the hypothesis of Lemma  2.3 holds, then the positive constant equilibrium \(E_{\ast}(u_{0},v_{0})\) is globally asymptotically stable if
$$\begin{aligned}& \frac{au_{0}v_{0} }{Qp^{\ast}} -\frac{v_{0}}{[a +b +1]p^{\ast}} < 1, \end{aligned}$$
(5.1a)
$$\begin{aligned}& \frac{1}{ 1-\frac{\delta}{b\beta}} +\frac{v_{0}}{(1-\frac{\delta}{b\beta })Qp^{\ast}}-\frac{\delta^{2}}{4\beta^{2}(1-\frac{\delta}{b\beta})^{2}}- \frac {u_{0}v_{0}}{p^{\ast}}>0\quad\textit{and} \end{aligned}$$
(5.1b)
$$\begin{aligned}& \frac{\delta}{ ( 1-\frac{\delta}{b\beta})\beta}-\frac{1}{2b^{2}}>0, \end{aligned}$$
(5.1c)
where \(Q=[a(1-\frac{\delta}{b\beta})^{2}+b(1-\frac{\delta}{b\beta})+1]\).

Proof

Let \((u(x,t), v(x,t))\) be any solution of model (1.2). We introduce the Lyapunov function
$$V(u,v)= \int_{\Omega} \biggl\{ u-u_{0}-\ln\frac{u}{u_{0}} + \frac{1}{\beta }\biggl(v-v_{0}-\ln\frac{v}{v_{0}}\biggr) \biggr\} \,dx. $$
Now, we take the derivative of V with regard to t along the trajectory of model (1.2). Then
$$\dot{V} =V_{1}+V_{2}, $$
where
$$\begin{aligned} V_{1} =& d_{1} \int_{\Omega}\biggl(1-\frac{u_{0}}{u}\biggr) \Delta u \,dx+ \frac{d_{2}}{\beta }\biggl(1-\frac{v_{0}}{v}\biggr)\Delta v \,dx\\ \leq & - \int_{\Omega} \biggl\{ \frac{d_{1} u_{0} \vert \nabla u\vert ^{2}}{u^{2}} + \frac{d_{2} v_{0} \vert \nabla v\vert ^{2}}{\beta v^{2}} \biggr\} \,dx\leq0 \end{aligned}$$
and
$$V_{2}= \int_{\Omega}(u-u_{0}) \biggl(1-u-\frac{uv}{au^{2}+bu+1} \biggr)\,dx+ \int_{\Omega }(v-v_{0})\frac{1}{\beta}\biggl(\delta- \frac{\beta v}{u}\biggr)\,dx. $$
Let \(V_{2}=V_{12}+V_{21}\), where
$$\begin{aligned} V_{21} =& \int_{\Omega}(u-u_{0}) \biggl(u_{0}-u+ \frac {u_{0}v_{)}}{a(u_{0})^{2} +bu_{0}+1}-\frac{u v }{a u^{2} +bu +1}\biggr)\,dx \\ =& (u-u_{0}) \int_{\Omega}\biggl[u_{0}-u+\frac{u_{0}v_{0}(au^{2}+bu+1)-u v (au^{2}_{0}+bu_{0}+1)}{pp^{\ast}}\biggr] \,dx \\ =& \int_{\Omega} \biggl\{ -(u-u_{0})^{2}+ \frac{u-u_{0}}{pp^{\ast }}\bigl[u_{0}v_{0}\bigl(au^{2}+bu+1 \bigr)-u v \bigl(a u^{2}_{0}+bu_{0}+1\bigr)\bigr] \biggr\} \,dx \\ =& \int_{\Omega} \biggl\{ -(u-u_{0})^{2}+ \frac{1}{pp^{\ast}}\bigl[ (au_{0}v_{0}u-v_{0}) (u-u_{0})^{2}\\ &{}+\bigl(au^{2}_{0}u+buu_{0}+u \bigr) (u-u_{0}) (v-v_{0})\bigr] \biggr\} \,dx \\ =& \biggl\{ (u-u_{0})^{2}\biggl[1-\frac{au_{0}v_{0}u-v_{0}}{pp^{\ast}} \biggr]+\frac {au^{2}_{0}u+buu_{0}+u}{pp^{\ast}}(u-u_{0}) (v-v_{0}) \biggr\} \end{aligned}$$
and
$$\begin{aligned} V_{22} =& \frac{1}{\beta} \int_{\Omega}(v-v_{0}) \biggl(\delta -\beta \frac{v}{u}\biggr)\,dx \\ =&\frac{1}{\beta} \int(v-v_{0}) \biggl(\delta-\beta\frac{v-v_{0}+v_{0}}{u}\biggr) \,dx \\ =& \frac{1}{\beta} \int_{\Omega}(v-v_{0}) \biggl(\delta-\beta \frac{v-v_{0}}{u}- \beta \frac{v_{0}}{u}\biggr)\,dx \\ =& \frac{1}{\beta} \int_{\Omega}(v-v_{0}) \biggl(\delta-\beta \frac{v-v_{0}}{u}- \beta \frac{v_{0}}{u}\biggr)\,dx \\ =& \int_{\Omega}(v-v_{0}) \biggl(\frac{\delta}{\beta}- \frac{v-v_{0}}{u}- \frac {v_{0}}{u}\biggr)\,dx \\ =& \int_{\Omega}\biggl[-\frac{(v-v_{0})^{2}}{u}+(v-v_{0}) \biggl(\frac{\delta}{\beta}-\frac {v_{0}}{u}\biggr)\biggr]\,dx \\ =& \int_{\Omega}\biggl[-\frac{(v-v_{0})^{2}}{u}+(v-v_{0}) \frac{\delta u-\beta v_{0}}{\beta u}\biggr]\,dx \\ =& \int_{\Omega}\biggl[-\frac{(v-v_{0})^{2}}{u}+(v-v_{0}) \frac{\delta u-\delta u_{0}}{\beta u}\biggr]\,dx \\ =& \int_{\Omega}\biggl[-\frac{(v-v_{0})^{2}}{u}+\frac{\delta}{\beta u}(v-v_{0}) (u-u_{0})\biggr]\,dx, \end{aligned}$$
where \(p=au^{2}+bu+1\), \(p^{\ast}=au^{2}_{0}+bu_{0}+1\), \(a_{11}= 1-\frac {au_{0}v_{0}u-v_{0}}{pp^{\ast}}\), \(a_{12}=a_{21}=\frac{1}{2} \{ \frac {au^{2}_{0} u+bu_{0}u+u}{pp^{\ast}}-\frac{\delta}{\beta u} \}\), \(a_{22}= \frac{1}{u}\).
Next, we calculate \(V_{2}\) as follows:
$$ V_{2}=- \int_{\Omega} \left\{ (u-u_{0},v-v_{0}) \begin{pmatrix} a_{11} &a_{12}\\ a_{21}& a_{22} \end{pmatrix} (u-u_{0},v-v_{0})^{T} \right\}\,dx. $$
(5.2)
It is obvious that \(\frac{dV}{dt}<0\) if and only if the matrix in integrand (5.2) is positive definite, equivalent to \(a_{11}>0 \) and \(\phi(u,v)= a_{11}a_{22}-a_{12}a_{21} >0\), where \(\phi(u,v)= a_{11}a_{22}-a_{12}a_{21}= (1-\frac{au_{0}v_{0}u-v_{0}}{pp^{\ast}}) \frac {1}{u} - \frac{1}{4} \{ (\frac{au^{2}_{0} u+bu_{0}u+u}{pp^{\ast}})^{2}-2 (\frac{au^{2}_{0} u+bu_{0}u+u}{pp^{\ast}})(\frac{\delta}{\beta u})+(\frac {\delta}{\beta u})^{2} \}>0\). By condition (5.1a)-(5.1c), we know \(a_{11}>0\) and \(\phi(u,v)>0\). Consequently, \(E_{\ast}\) is globally asymptotically stable. Thus, we complete this proof. □

Next, we give out the conditions of the global asymptotic stability of the boundary equilibrium \(E_{0}(1,0)\).

Theorem 5.2

The boundary equilibrium \(E_{2}\) of (1.2) is globally asymptotically stable if
$$ \frac{1}{a+1 }< 4\beta\delta+\frac{\delta}{\beta} $$
(5.3)
holds.

Proof

Define
$$V_{0}(u,v)= \int_{\Omega} \{ u-1-\ln u +v \}\,dx. $$
Taking the derivative of \(V_{0}\) with regard to t along the trajectory of model (1.2), we have
$$\dot{V}_{0} =V^{1}_{0}+V^{2}_{0}, $$
where
$$V^{1}_{0}= \int_{\Omega} \biggl\{ d_{2} \Delta v+d_{1} \biggl(1-\frac{1}{u}\biggr) \Delta u \biggr\} \,dx \leq- \int_{\Omega}\frac{d_{1} \vert \Delta u\vert ^{2}}{u^{2}}\,dx\leq0. $$
Furthermore, we calculate \(V^{2}_{0}\) and reach
$$\begin{aligned} V^{2}_{0} =& \int_{\Omega} \biggl\{ (u-1) \biggl(1-u-\frac {uv}{au^{2}+bu+1}\biggr) +v\biggl(\delta-\frac{\beta v}{u}\biggr) \biggr\} \,dx \\ =& - \int_{\Omega} \biggl\{ (u-1)^{2}+\biggl( \frac{u}{au^{2}+bu+1}-\frac{\delta}{\beta u}\biggr) (u-1)v+\frac{\beta\delta}{u}v^{2} \biggr\} \,dx. \end{aligned}$$
According to condition (5.3), \(\frac{dV_{0}}{dt}<0\) except at \(E_{0}\). By LaSalle’s invariance principle, \(E_{0}\) is globally asymptotically stable. □

6 Numerical simulations

In this section, to confirm our analytical results found in the previous section, some examples and numerical simulations are presented. We use Matlab (2012a) to simulate and plot numerical graphs.

Example 1

If we choose \(a=0.5, b=1, \delta=0.5,\beta=20, d_{1}=1, d_{2}=1\), the conditions of Theorem 3.2 are satisfied for system (1.2). We have that the axial equilibrium \(E(1, 0)\) is globally stable (see Figure 1).
Figure 1

Numerical simulations of system ( 1.2 ) for \(\pmb{a=0.5}\) , \(\pmb{\beta =1}\) , \(\pmb{d_{1}=1}\) , \(\pmb{d_{2}=100}\) , \(\pmb{b=1}\) , \(\pmb{\delta=1.2}\) . The positive equilibrium \(E^{*}(1,0)\) of (1.2) is globally asymptotically stable.

Example 2

If we choose \(a=1, b=1, \beta=2, d_{1}=1, d_{2}=10 \), \(\delta=1\), the conditions of Theorem 3.1 are satisfied for system (1.2). The positive equilibrium \(E_{0}(0.8581,0.4290)\) is globally asymptotically stable (see Figure 2).
Figure 2

Numerical simulations of system ( 1.2 ) for \(\pmb{a=1}\) , \(\pmb{\beta=2}\) , \(\pmb{d_{1}=1}\) , \(\pmb{d_{2}=10}\) , \(\pmb{b=1}\) , \(\pmb{\delta=1}\) . The positive equilibrium \(E^{\ast}\) of (1.2) is globally asymptotically stable with the initial value \((0.8, 0.4)\).

Example 3

If we choose \(a=10, b=0.1, \delta=0.2,\beta=0.01, d_{1}=0.1, d_{2}=10\), then we know that system (1.2) has a unique positive homogeneous equilibrium \(E(0.2484, 4.9679)\), which is asymptotically stable (see Figure 3). The system undergoes Hopf bifurcation at the equilibrium \(E^{*}\) (see Figure 4). By the formulas derived in the previous section, we get \(c_{1}(A_{0})\approx9.0339e^{+002} +8.0131e^{+002}i\) with the initial value \((0.65, 3.5)\).
Figure 3

Numerical simulations of system ( 1.2 ) for \(\pmb{a=7}\) , \(\pmb{\beta =0.01}\) , \(\pmb{d_{1}=0.1}\) , \(\pmb{d_{2}=10}\) , \(\pmb{b=0.1}\) , \(\pmb{\delta=0.2}\) . The positive equilibrium \(E^{*}\) is locally asymptotically stable with the initial value \((0.65, 3.5)\).

Figure 4

Numerical simulations of system ( 1.2 ) for \(\pmb{a=10}\) , \(\pmb{\beta=0.01}\) , \(\pmb{d_{1}=0.1}\) , \(\pmb{d_{2}=10}\) , \(\pmb{b=0.1}\) , \(\pmb{\delta=0.2}\) . The positive equilibrium \(E^{*}\) of (1.3) becomes unstable and there exist spatially homogeneous periodic solutions with the initial value \((0.65, 3.5)\).

7 Conclusions

Based on model (1.1), we propose a diffusive prey-predator model with Leslie-type sigmoidal functional response and subject to the Neumann boundary conditions which is in the form of model (1.2). The dynamic behavior of the model is investigated. Some important qualitative properties, such as the existence of a global positive solution, persistence, the local stability and global stability of the equilibria, are obtained. Hopf bifurcations are explored by analyzing the characteristic equations. Moreover, the formulas determining the direction and stability of the bifurcating periodic solutions are derived by using the center manifold and the normal form theory of partial functional differential equations. Some numerical simulations are carried out to support our theoretical analysis. In the future, some challengeable attempts to study the Hopf bifurcation, steady state solution and Turing-Hopf bifurcation of fractional diffusive in a predator-prey system with or without time delay will be proceeded.

Declarations

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 11601543), the Natural Science and Technology Foundation of Henan Province (No.15A110046), the Education Department Foundation of Henan Province (No. 15A110046 and 13A110101) and the Research Innovation Project of ZhouKou Normal University (no. ZKNUA201303).

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

(1)
Department of Mathematics and Statistics, Zhoukou Normal University

References

  1. Kalmár-Nagy, T, Stépán, G, Moon, FC: Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations. Nonlinear Dyn. 26(2), 121-142 (2001) MathSciNetView ArticleMATHGoogle Scholar
  2. Rietkerk, M, Dekker, SC, de Ruiter, PC, van de Koppel, J: Self-organized patchiness and catastrophic shifts in ecosystems. Science 305(5692), 1926-1929 (2004) View ArticleGoogle Scholar
  3. Field, RJ, Noyes, RM: Oscillations in chemical systems. IV. Limit cycle behavior in a model of a real chemical reaction. J. Chem. Phys. 60(5), 1877-1884 (1974) View ArticleGoogle Scholar
  4. Angeli, D, Ferrell, JE, Sontag, ED: Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems. Proc. Natl. Acad. Sci. USA 101(7), 1822-1827 (2004) View ArticleGoogle Scholar
  5. Xiao, M, Cao, J: Hopf bifurcation and non-hyperbolic equilibrium in a ratio-dependent predator-prey model with linear harvesting rate. Math. Comput. Model. 50(3), 360-379 (2009) MathSciNetView ArticleMATHGoogle Scholar
  6. Huang, C, Cao, J, Xiao, M, Alsaedi, A, Alsaadi, FE: Controlling bifurcation in a delayed fractional predator-prey system with incommensurate orders. Appl. Math. Comput. 293, 293-310 (2017) MathSciNetGoogle Scholar
  7. Xu, W, Hayat, T, Cao, J, Xiao, M: Hopf bifurcation control for a fluid flow model of Internet congestion control systems via state feedback. IMA J. Math. Control Inf. 33(1), 69-93 (2016) MathSciNetView ArticleMATHGoogle Scholar
  8. Xu, W, Cao, J, Xiao, M: Bifurcation analysis of a class of \((n+1)\)-dimension Internet congestion control systems. Int. J. Bifurc. Chaos Appl. Sci. Eng. 25(2), 1550019 (2015) MathSciNetView ArticleMATHGoogle Scholar
  9. Cheng, Z, Wang, Y, Cao, J: Stability and Hopf bifurcation of a neural network model with distributed delays and strong kernel. Nonlinear Dyn. 86(1), 323-335 (2016) MathSciNetView ArticleMATHGoogle Scholar
  10. Xu, W, Cao, J, Xiao, M, Ho, DWC, Wen, G: A new framework for analysis on stability and bifurcation in a class of neural networks with discrete and distributed delays. IEEE Trans. Cybern. 45(10), 2224-2236 (2015) View ArticleGoogle Scholar
  11. Huang, C, Meng, Y, Cao, J, Alsaedi, A, Alsaadi, FE: New bifurcation results for fractional BAM neural network with leakage delay. Chaos Solitons Fractals 100, 31-44 (2017) MathSciNetView ArticleGoogle Scholar
  12. Huang, C, Cao, J, Xiao, M: Hybrid control on bifurcation for a delayed fractional gene regulatory network. Chaos Solitons Fractals 87, 19-29 (2016) MathSciNetView ArticleMATHGoogle Scholar
  13. Huang, C, Cao, J, Xiao, M, Alsaedi, A, Hayat, T: Bifurcations in a delayed fractional complex-valued neural network. Appl. Comput. Math. 292, 210-227 (2017) MathSciNetGoogle Scholar
  14. Huang, J, Ruan, S, Song, J: Bifurcations in a predator-prey system of Leslie type with generalized Holling type III functional response. J. Differ. Equ. 257(6), 1721-1752 (2014) MathSciNetView ArticleMATHGoogle Scholar
  15. Yang, Y, Xu, Y: Global stability of a diffusive and delayed virus dynamics model with Beddington-DeAngelis incidence function and CTL immune response. Comput. Math. Appl. 71(4), 922-930 (2016) MathSciNetView ArticleGoogle Scholar
  16. Yi, F, Wei, J, Shi, J: Global asymptotical behavior of the Lengyel-Epstein reaction-diffusion system. Appl. Math. Lett. 22(1), 52-55 (2009) MathSciNetView ArticleMATHGoogle Scholar
  17. Peng, R, Shi, J: Non-existence of non-constant positive steady states of two Holling type-II predator-prey systems: strong interaction case. J. Differ. Equ. 247(3), 866-886 (2009) MathSciNetView ArticleMATHGoogle Scholar
  18. Ko, W, Ryu, K: Non-constant positive steady-states of a diffusive predator-prey system in homogeneous environment. Aust. J. Math. Anal. Appl. 327(1), 539-549 (2007) View ArticleMATHGoogle Scholar
  19. Zeng, X, Liu, Z: Nonconstant positive steady states for a ratio-dependent predator-prey system with cross-diffusion. Nonlinear Anal., Real World Appl. 11(1), 372-390 (2010) MathSciNetView ArticleMATHGoogle Scholar
  20. Li, S, Wu, J, Nie, H: Steady-state bifurcation and Hopf bifurcation for a diffusive Leslie-Gower predator-prey model. Comput. Math. Appl. 70(12), 3043-3056 (2015) MathSciNetView ArticleGoogle Scholar
  21. Cholewa, JW, Dlotko, T: Global Attractors in Abstract Parabolic Problems. Cambridge University Press, Cambridge (2000) View ArticleMATHGoogle Scholar
  22. Ye, Q, Li, Z, Wang, M, Wu, Y: Introduction to Reaction-Diffusion Equations. Chinese Science Press, Beijing (1990) Google Scholar
  23. Wu, J: Theory and Applications of Partial Functional Differential Equations. Springer, New York (2012) Google Scholar
  24. Hassard, BD, Kazarinoff, ND, Wan, YH: Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge (1981) MATHGoogle Scholar
  25. Yi, F, Wei, J, Shi, J: Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system. J. Differ. Equ. 246(5), 1944-1977 (2009) MathSciNetView ArticleMATHGoogle Scholar

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