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Densitydependent effects on Turing patterns and steady state bifurcation in a Beddington–DeAngelistype predator–prey model
Boundary Value Problemsvolume 2019, Article number: 102 (2019)
Abstract
In this paper, Turing patterns and steady state bifurcation of a diffusive Beddington–DeAngelistype predator–prey model with densitydependent death rate for the predator are considered. We first investigate the stability and Turing instability of the unique positive equilibrium point for the model. Then the existence/nonexistence, the local/global structure of nonconstant positive steady state solutions, and the direction of the local bifurcation are established. Our results demonstrate that a Turing instability is induced by the densitydependent death rate under appropriate conditions, and both the general stationary pattern and Turing pattern can be observed as a result of diffusion. Moreover, some specific examples are presented to illustrate our analytical results.
Introduction
Understanding the dynamical relationship between predator and prey is a central research subject in ecology, and one significant component of the predator–prey relationship is the predator’s rate of feeding upon prey, i.e., the socalled functional response. Functional response is a double rate: It is the average number of prey killed per individual predator per unit of time. In general, the functional response can be classified into two types: Preydependent and predatordependent. Prey dependence means that the functional response is only a function of the prey’s density, while predator dependence means that the functional response is a function of both the prey’s and the predator’s densities. For the functional response functions, there are many types, such as the Holling family which are predominant in the literature [1].
Since 1959, the Holling IItype preydependent functional response has served as the basis for a very large literature on predator–prey theory [2]. However, the preydependent functional responses fail to describe the interference among predators, and have been facing challenges from the biology and physiology communities [3, 4]. Some biologists have argued that in many situations, especially when predators have to search for food (and therefore, have to share or compete for food), the functional response in a predator–prey model should be predatordependent. There is much significant evidence to suggest that predator dependence in the functional response occurs quite frequently in laboratory and natural systems [5, 6]. Given that large numbers of experiments and observations suggest that predators do indeed interfere with one another’s activities so as to result in competition effects and that prey alters its behavior under increased predatorthreat, the models with predatordependent functional response stand as reasonable alternatives to the models with preydependent functional response [2]. Starting from this argument and the traditional preydependent model, to describe mutual interference among predators, Beddington [7] and DeAngelis [8] proposed that an individual from a population of more than two predators not only allocates time in searching for and processing their prey but also takes time in encountering other predators. This result in the socalled Beddington–DeAngelis functional response \(p(u,v)=\frac{mu}{a+u+bv}\). The Beddington–DeAngelis functional response is similar to the wellknown Holling type II functional response, but it has an extra term bv in the denominator modelling mutual interference among predators, and it also has some similar qualitative features as the ratiodependent form but avoids the singular behaviors of ratiodependent models at low densities which have been the source of controversy.
We know the classical Beddington–DeAngelistype predator–prey system which has received considerable attention [9,10,11,12,13,14,15,16,17] and takes the form
A salient statistical evidence from nineteen predator–prey systems prove that Beddington–DeAngelis functional response provides better description of predator feeding over a range of predator–prey abundances [2]. In some cases, it performs even better than other functional responses. The most crucial finding of Skalski and Gilliam [2] was that predator dependence in the functional response is a nearly ubiquitous property of the published data sets. Cantrell and Cosner [10] have partially analyzed the dynamics of the system (1). Hwang [13] has solved the problem for the uniqueness of a limit cycle of the system (1). A detailed mathematical analysis of the dynamics for (1) with unlimited carrying capacity for prey population was presented in [14]. Further, Kartina [15] found that predator dependence is important at not only very high predator densities on per capita predation rate but also at low predator densities. In ecology, we should consider the predator density dependence, and we need to take into account realistic levels of predator dependence.
In this paper, we consider the following densitydependent Beddington–DeAngelistype predator–prey model:
where u and v represent prey and predator densities, respectively. q is the death rate of the predator, s is the feed concentration and δ is the densitydependent death rate. Biologically speaking, the positive densitydependent death rate δ has depressing effect on the growth rate of the predator, i.e., causes the reduction in predator growth rate [16].
In [17], the authors studied the dynamics of (2). They proved the permanence, locally and globally asymptotic stability of the positive equilibrium for the model (2) by using stability theory of differential equations and Lyapunov functions. For the permanence, they showed that the density dependence for predator gives some negative effect, compared to the models without the density dependence. In addition, the authors compared results for the model with Beddington–DeAngelis functional response on permanence, locally and globally asymptotic stability to the system with Lotka–Volterra interaction or Holling type II functional response or ratiodependent functional response.
When the densities of the prey and predator are spatially inhomogeneous in a bounded domain, and the prey and predator move randomlydescribed as Brownian random motion [18,19,20], we need consider the following reaction–diffusion model:
where Ω is a bounded domain with smooth boundary ∂Ω and ν is the outward unit normal vector of the boundary ∂Ω. The positive constants \(d_{1}\) and \(d_{2}\) are the diffusion coefficients of \(u(x,t)\) and \(v(x,t)\), respectively. △ is the Laplacian operator which describes the random moving.
In the case \(b=0\), Huang et al. in [21] derived the conditions for the existence of nonconstant steady states of the model (3) with \(\delta >0\). At the same time, they proved that the same system without the densitydependent death rate for the predators does not admit pattern formations. Hence, in the case \(b>0\), a natural question is raised: Is the densitydependent death rate δ also a decisive factor inducing Turing instability in the model (3)? We will answer this problem in this paper.
To study the stationary patterns, we need consider the steady state problem associated with (3)
The rest of this paper is organized as follows: In Sect. 2, the stability and Turing instability of the positive equilibrium point \((u^{*},v^{*})\) in (3) are discussed. In Sect. 3, we investigate the nonexistence/existence of nonconstant positive steady states. In Sect. 4, the local and global structure of nonconstant positive steady state are established, and the direction of the local bifurcation is given.
Stability and Turing instability of positive equilibrium point
In this section, we mainly discuss the stability and Turing instability of the positive equilibrium point of (3). For convenience, we denote
Obviously, the model (2) has a trivial equilibrium point \(E_{0}=(0,0)\), a semitrivial equilibrium point \(E_{1}=(1,0)\) and at least one positive equilibrium point \(E^{*}=(u^{*},v^{*})\) if
where
and \(v^{*}\) is the positive roots of polynomial equation
To illustrate the uniqueness of the positive equilibrium point, we first give the following lemma.
Lemma 1
(Shengjins discriminant [22])
For the equation \(x^{3}+Bx^{2}+Cx+D=0\), where \(B,C,D\in \mathbf{R}\), denote \(\mathbb{A}=B^{2}3C\), \(\mathbb{B}=BC9D\), \(\mathbb{C}=C^{2}3BD\) and \(\Delta =\mathbb{B}^{2}4\mathbb{AC}\).

(i)
The equation has three real roots if and only if \(\Delta \leq 0\).

(ii)
The equation has one real root and a pair of conjugate complex roots if and only if \(\Delta >0\).
For Eq. (5), corresponding to Lemma 1, let
and
Then we can obtained the following conclusion.
Theorem 2.1
Assume that
hold, then Eq. (5) has a unique positive root, and (2) has a unique positive equilibrium point \(E^{*}=(u^{*}, v^{*})\).
Now we discuss the stability and instability of \(E^{*}\) for the ODE model (2) and PDE model (3), respectively. By simple calculation, we can see that the Jacobian matrix of (2) evaluated at \(E^{*}\) is given by
where
The characteristic equation of \(J(E^{*})\) is
where
Obviously that \(E^{*}\) is locally asymptotically stable if \(Q<0\) and \(P>0\). Thus, we can obtain the following theorem.
Theorem 2.2
Assume \((\mathbf{H_{1}})\) hold. For the model (2), the following statements are true.

(i)
If \(mu^{*}v^{*}<(u^{*}+\delta sv^{*})(bv^{*}+a+u^{*})^{2}+bsmu ^{*}v^{*}\) and
$$ \begin{aligned} (\mathbf{H_{21}}) \quad &sv^{*}\bigl[m^{2}bu^{*}v^{*}+m \bigl(\delta v^{*}bu^{*}\bigr) \bigl(bv^{*}+a+u^{*} \bigr)^{2}\bigr] \\ &\quad < m ^{2}\bigl(a+u^{*}\bigr) \bigl(a+bv^{*}\bigr)+\delta sv^{*}\bigl(bv^{*}+a+u^{*} \bigr)^{4},\end{aligned} $$then equilibrium point \(E^{*}\) is locally asymptotically stable.

(ii)
If \(mu^{*}v^{*}>(u^{*}+\delta sv^{*})(bv^{*}+a+u^{*})^{2}+bsmu ^{*}v^{*}\) or
$$ \begin{aligned} (\mathbf{H_{22}}) \quad &sv^{*}\bigl[m^{2}bu^{*}v^{*}+m \bigl(\delta v^{*}bu^{*}\bigr) \bigl(bv^{*}+a+u^{*} \bigr)^{2}\bigr]\\&\quad >m ^{2}\bigl(a+u^{*}\bigr) \bigl(a+bv^{*}\bigr)+\delta sv^{*}\bigl(bv^{*}+a+u^{*} \bigr)^{4},\end{aligned} $$then equilibrium point \(E^{*}\) is unstable.
To consider Turing instability of \(E^{*}\) for PDE model (3), we denote \(0=\lambda _{0}<\lambda _{1}<\cdots \) , the sequence of eigenvalues for the problem
and \(\lambda _{i}\) (\(i\geq 1\)) has multiplicity \(m_{i}\geq 1\), whose corresponding normalized eigenfunctions are given by \(\phi _{ij}\), where \(j=1,2,\ldots,m_{i}\). This set of eigenfunctions form an orthogonal basis in \(L^{2}(\varOmega )\).
If \(a_{11}>0\) and
then we define \(i_{0}\) be the largest positive integer such that \(d_{1}\lambda _{i}< a_{11}\).
Clearly, if (11) is satisfied, then \(1\leq i_{0}<\infty \). In this case, let
where \(d_{2}^{i}(E^{*})\) is given by
Theorem 2.3
Assume that \((\mathbf{H_{1}})\) holds. Then the following conclusions for the model (3) are true.

(i)
If \(a_{11}<0\), then \(E^{*}\) is locally asymptotically stable.

(ii)
Let \(a_{11}>0\), \((\mathbf{H_{21}})\) and \(mu^{*}v^{*}<(u^{*}+ \delta sv^{*})(bv^{*}+a+u^{*})^{2}+bsmu^{*}v^{*}\) hold.

(ii1)
If \(d_{1}\lambda _{1}< a_{11}\) and \(0< d_{2}<\bar{d_{2}}\), then \(E^{*}\) is locally asymptotically stable.

(ii2)
If \(d_{1}\lambda _{1}< a_{11}\) and \(d_{2}>\bar{d_{2}}\), then \(E^{*}\) is unstable, and hence in the model (3) Turing instability occurs.

(ii1)
Proof
Consider the linearization operator of (3) at \(E^{*}\)
Suppose that \(\varPhi =(\varphi ,\psi )\in L^{2}(\varOmega )\times L^{2}( \varOmega )\) is an eigenfunction of L corresponding to an eigenvalue η, then
Writing \(\varphi = \sum_{0\leq i \leq \infty ,1\leq j\leq m_{i}}a_{ij}\phi _{ij}\), \(\psi = \sum_{0\leq i \leq \infty ,1\leq j\leq m_{i}}b_{ij}\phi _{ij}\), then
where
We easily see that η is the eigenvalue of L if and only if \(\det {B_{i}}=0\) for some i, which leads to
where
(i) If \(a_{11}<0\), then \(Q_{i}>0\) and \(P_{i}>0\) for all i, which implies that \(\operatorname{Re}\{\eta _{i}\}<0\) for all i, where \(\eta _{i}\) are the eigenvalues of (14). Therefore, the equilibrium point \(E^{*}\) is locally asymptotically stable.
(ii) If (\(\mathbf{H_{21}}\)) and \(mu^{*}v^{*}<(u^{*}+\delta sv^{*})(bv ^{*}+a+u^{*})^{2}+bsmu^{*}v^{*}\) hold, then \(Q_{i}>0\) and \(d_{1}a_{22} \lambda _{i}a_{11}a_{22}+a_{12}a_{21}<0\).
(ii1) If \(a_{11}>0\), \(d_{1}\lambda _{1}< a_{11}\) and \(0< d_{2}<\bar{d _{2}}\), then \(d_{1}\lambda _{i}< a_{11}\) and \(d_{2}< d_{2}^{i}\) for all \(i\in [1,i_{0}]\). Thus,
On the other hand, if \(i>i_{0}\), then \(d_{1}\lambda _{i}>a_{11}\), and \(P_{i}>0\). The analysis yields the locally asymptotical stability of \(E^{*}\).
(ii2) If \(a_{11}>0\), \(d_{1}\lambda _{1}< a_{11}\) and \(d_{2}>\bar{d _{2}}\), then we may assume the minimum in (13) is attained at \(j\in [1,i_{0}]\). Thus \(d_{2}>d_{2}^{j}\), which implies
Hence, \(E^{*}\) is unstable in this case. The proof of Theorem 2.3 is complete. □
Example 2
We take the parameters in model (2) and (3) as
It is easy to verify that there is a unique positive equilibrium point \(E^{*}(u^{*},v^{*})=(0.22, 0.56)\).
For the ODE model (2), from Theorem 2.2, we can verify that \(E^{*}\) is stable. For the PDE model (3) in onedimensional interval \((0,\pi )\), after fixing \(d_{1}=0.015\), from Theorem 2.3, we know that if \(d_{2}>\bar{d_{2}}=0.31\), then \(E^{*}\) is Turing unstable, and model (3) exhibits Turing pattern. In Fig. 1, we show the numerical results of model (3) with different values for \(d_{2}\). Figure 1(a) shows the numerical simulations of Turing instability in model (3) with \(d_{2}=0.75>\bar{d_{2}}\). And Fig. 1(b) is for the numerical simulations of the stable positive equilibrium point of model (3) with \(d_{2}=0.20<\bar{d_{2}}\). From Fig. 2 we can observe the Turing patterns for the different values of \(d_{2}\). One can see that the model exhibits pattern formation, including a cold spots pattern in Fig. 2(a) and a spot–stripe pattern in Fig. 2(b).
Nonexistence/existence of nonconstant positive steady state
In this section, we consider the nonexistence/existence of nonconstant positive steady states of (4).
Let \(N(\lambda _{i})\) be the eigenspace corresponding to \(\lambda _{i}\) in \(H^{1}(\varOmega )\). Let \(X=[H^{1}(\varOmega )]^{2}\), \(\{\phi _{ij};j=1,\ldots,\operatorname{dim} {N(\lambda _{i})}\}\) be an orthonormal basis of \({N(\lambda _{i})}\), and \(X_{ij}=\{c\varPhi _{ij}:c\in R^{2}\}\). Then we decompose X as
A priori estimates for positive steady states
In this subsection, by using the maximum principle, we establish a priori estimates of positive steady state for (4).
Lemma 3
(Maximum principle [23]) Suppose that \(g\in C(\overline{\varOmega }\times \mathbb{R})\).

(i)
Assume that \(w \in C^{2}(\varOmega ) \cap C^{1}(\overline{\varOmega })\), and
$$ \triangle w(x)+g\bigl(x,w(x)\bigr)\geq 0, \quad x\in \varOmega , \quad \quad \frac{\partial w}{\partial \nu }\leq 0, \quad x\in \partial \varOmega . $$If \(w(x_{0})=\max_{\overline{\varOmega }} w\), then \(g(x_{0},w(x_{0})) \geq 0\).

(ii)
Assume that \(w \in C^{2}(\varOmega ) \cap C^{1}(\overline{\varOmega })\), and
$$ \triangle w(x)+g\bigl(x,w(x)\bigr)\leq 0, \quad x\in \varOmega , \quad\quad \frac{\partial w}{\partial \nu } \geq 0, \quad x\in \partial \varOmega . $$If \(w(x_{0})=\min_{\overline{\varOmega }} w\), then \(g(x_{0},w(x_{0})) \leq 0\).
Theorem 3.1
Assume \(m>(a+1)q\). Let \((u(x),v(x))\) be a positive solution of (4). If
then \((u(x),v(x))\) satisfies
where \(\alpha =\frac{m(a+1)q}{\delta (a+1)}\).
Proof
A direct application of Lemma 3 to (4) yields \(u(x)\leq 1\) and \(v(x)\leq \alpha \). To obtain the lower bound for \(u(x)\) and \(v(x)\), we let
By virtue of Lemma 3, we have
Since \(a\delta (a+1)>m(m(a+1)q)\), \(1\frac{m\alpha }{a}>0\) and \(u(x_{0})\geq 1\frac{m\alpha }{a}\).
Notice that
we have \(v(y_{0})\geq \frac{1}{\delta } (\frac{m(am\alpha )}{a(a+1+ \alpha )m\alpha }q )\). The proof is complete. □
Nonexistence of nonconstant positive steady state
In this subsection, we apply the energy method to prove the nonexistence of the nonconstant positive steady state to (4). For convenience, let \(\varGamma =\varGamma (m,a,b,s,q,\delta )\) be the set of parameters m, a, b, s, q, and δ.
Theorem 3.2
Assume \(m>q\). Let \(\lambda _{1}\) be the smallest positive eigenvalue of the operator −△ on Ω with zeroflux boundary condition and \(d_{2}^{*}\) be a fixed positive constant satisfying \(d_{2}^{*}>\frac{s(mq)}{\lambda _{1}}\). Then there exists a positive \(d_{1}^{*}=d_{1}^{*}(\varGamma ,d_{2}^{*})\) such that model (4) has no nonconstant positive steady state provided that \(d_{1}\geq d_{1} ^{*}\), \(d_{2}\geq d_{2}^{*}\).
Proof
Let \((u,v)\) be a positive solution of (4) and denote
Then multiplying the first equation of model (4) by \((u\bar{u})\), integrating over Ω and from Theorem 3.1, we have
in a similar manner, multiplying the second equation in model (4) by \((v\bar{v})\), we have
It follows from the above and the ϵYoung inequality that
for \(L:=\frac{(b+1)m}{2b}\) and an arbitrary positive constant ϵ. It follows from the wellknown Poincaré inequality that
Since \(d_{2}^{*}\lambda _{1}>s(mq)\), from the assumption, we can choose a sufficiently small ϵ such that
Finally, by taking \(d_{1}^{*}:=\frac{1}{\lambda _{1}}(1+\frac{L}{ \epsilon })\), one can conclude that \(u=\bar{u}\) and \(v=\bar{v}\), which asserts our results. □
Existence of nonconstant positive steady state
In this subsection, by using the Leray–Schauder degree theory, we discuss the existence of nonconstant positive steady state to (4) when the diffusion coefficients \(d_{1}\) and \(d_{2}\) vary while the parameters in Γ keep fixed.
For simplicity, define \(F=(f,g)^{\top }\), where f and g are given in Sect. 2. Then the stationary problem of (4) can be written as
where \(D=\operatorname{diag}(d_{1},d_{2})\). Therefore, E solves (15) if and only if it satisfies
where \((I \Delta )^{1}\) represents the inverse of \(I \Delta \) with homogeneous Neumann boundary condition.
A straightforward computation reveals
For each \(X_{i}\), λ is an eigenvalue of \(D_{E}\widehat{f}(d _{1},d_{2};E^{*})\) on \(X_{i}\) if and only if \(\lambda (1+\lambda _{i})\) is an eigenvalue of the following matrix:
Clearly,
and \(\operatorname{tr}M_{i}=2 \lambda _{i}d_{1}^{1}a_{11}d_{2}^{1}a_{22}\). Define
Then \(\widehat{g}(d_{1},d_{2};\lambda )=d_{1}d_{2} \det M_{i}\). If
then \(\widehat{g}(d_{1},d_{2};\lambda )=0\) has two real roots:
Set
and let \(m(\lambda _{i})\) be multiplicity of \(\lambda _{i}\). In order to calculate the index of \(\widehat{f}(d_{1},d_{2};\cdot )\) at \(E^{*}\), we need the following lemma.
Lemma 4
([24])
Suppose \(\widehat{g}(d_{1},d_{2}; \lambda _{i})\neq 0\) for all \(\lambda _{i} \in S_{p}\). Then
where
In particular, if \(\widehat{g}(d_{1},d_{2};\lambda _{i}) >0\) for all \(\lambda _{i} \geq 0\), then \(\sigma =0\).
By determining the range of λ for which \(\widehat{g}(d_{1},d _{2};\lambda ) < 0\), we have the existence of nonconstant steady state to (4).
Theorem 3.3
Let \(d_{1}\), Γ be fixed and \((\mathbf{H_{1}})\), \(( \mathbf{H_{21}})\), \(a_{11}>0\) hold. If \(\frac{a_{11}}{d_{1}} \in ( \lambda _{k},\lambda _{k+1})\) for some \(k \geq 1\), and \(\sigma _{k}= \sum_{i=1}^{k} m(\lambda _{i})\) is odd, then there exists a positive constant \(d^{*}\) such that model (4) has at least one nonconstant positive steady state for all \(d_{2} \geq d^{*}\).
Proof
Notice that if \(d_{2}\) is large enough, then (17) and \(\lambda _{+}(d_{1},d_{2}) >\lambda _{}(d_{1},d_{2}) >0\) hold. Furthermore,
As \(\frac{a_{11}}{d_{1}} \in (\lambda _{k},\lambda _{k+1})\), there exists \(d_{0} \gg 1\) such that
From Theorem 3.2, we know that there exists \(\tilde{d} > d_{0}\) such that (4) with \(d_{1}=\tilde{d}\) and \(d_{2} \geq \tilde{d}\) has no nonconstant positive steady state. Let \(\tilde{d}>0\) be large enough such that \(\frac{a_{11}}{d_{1}} <\lambda _{1}\). Then there exists \(d^{*} >\tilde{d}\) such that
Now we prove that, for any \(d_{2} \geq d^{*}\), (4) has at least one nonconstant positive steady state. By way of contradiction, assume that the assertion is not true for some \(d_{2}^{*} \geq d^{*}\). By using the homotopy argument, we can derive a contradiction in the sequel.
Fixing \(d_{2}=d_{2}^{*}\), for \(t \in [0,1]\), we define
and consider the following problem:
Notice that E is a nonconstant positive steady state of (4) if and only if it solves (20) with \(t=1\). Evidently, \(E^{*}\) is the unique constant positive steady state of (20). For any \(t \in [0,1]\), E is a nonconstant positive steady state of (20) if and only if it is a solution of the following problem:
Form the above discussion, we know that (21) has no nonconstant positive steady state when \(t=0\), and there is no such solution for \(t=1\) at \(d_{2}=d_{2}^{*}\). Clearly, \(h(E;1)=\widehat{f}(d_{1},d_{2};E)\), \(h(E;0)=\widehat{f}(\tilde{d},d ^{*};E)\) and
Here, \(\widehat{f}(\cdot ,\cdot ;\cdot )\) is as given in (16) and \(\widetilde{D}=\operatorname{diag }(\tilde{d},d^{*})\). Form (18) and (19), we have $\mathit{\text{A}}({d}_{1},{d}_{2})\cap {S}_{p}=\{{\lambda}_{1},{\lambda}_{2},\dots ,{\lambda}_{k}\}$ and $\mathit{\text{A}}(\tilde{d},{d}^{\ast})\cap {S}_{p}=\mathrm{\varnothing}$. Since \(\sigma _{k}\) is odd, Lemma 4 yields
From Theorem 3.2, there exist positive constants \(\underline{C}=\underline{C}(\tilde{d},d_{1},d^{*},d_{2}^{*},\varGamma )\) and \(\overline{C}=\overline{C}(\tilde{d}, d^{*},\varGamma )\) such that the positive solutions of (21) satisfy \(\underline{C} < u(x),v(x)< \overline{C}\) on Ω̅ for all \(t\in [0,1]\).
Define \(\varSigma =\{E \in X:\underline{C} < u(x),v(x)< \overline{C},x \in \overline{\varOmega }\}\). Then \(h(E;t)\neq 0\) for all \(E \in \partial \varSigma \) and \(t \in [0,1]\). By virtue of the homotopy invariance of the Leray–Schauder degree, we have
Note that both equations \(h(E;0)=0\) and \(h(E;1)=1\) have the unique positive solution \(E^{*}\) in Σ, and we obtain
which contradicts (22). The proof is complete. □
Structure of nonconstant positive steady state
Let \(Y=C(\bar{\varOmega })\times C(\bar{\varOmega })\), \(X= \{(u,v)\vert u,v\in C^{2}(\bar{\varOmega }), \frac{\partial u}{\partial \nu } =\frac{ \partial v}{\partial \nu } =0, x\in \partial \varOmega \}\).
Local structure and direction of nonconstant positive steady state
In this subsection, we first study the local structure of nonconstant positive steady state for model (4). In brief, by regarding \(d_{2}\) as the bifurcation parameter, we verify the existence of positive steady state bifurcating from \((d_{2},E^{*})\). The Crandall–Rabinowitz bifurcation theorem in [25] will be applied to obtain bifurcations.
Define the map \(F: (0,\infty )\times X\rightarrow Y\) by
where f, g are given in Sect. 2. Then the solutions of boundary value problem (4) are exactly zeros of F. With \(E^{*}=(u ^{*},v^{*})\), we have
If there is a number \(d_{2}>0\) such that every neighborhood of \((d_{2},E^{*})\) contains zero of F in \((0,\infty )\times X\) not lying on the curve \((d_{2},E^{*})\), then we say that \((d_{2},E^{*})\) is a bifurcation point of the equation \(F=0\) with respect to this curve.
Theorem 4.1
Let \((\mathbf{H_{1}})\), \((\mathbf{H_{21}})\) and \(a_{11}>0\) hold. Suppose that j is a positive integer such that \(d_{1} \lambda _{j}< a_{11}\) and \(d_{2}^{k}\neq d_{2}^{j}>0\) for any integer \(k\neq j\). Then \((d_{2}^{j},E^{*})\) is a bifurcation point of \(F(d_{2},E)=0\) with respect to the curve \((d_{2},E^{*})\). There is a oneparameter family of nontrivial solution \(\varGamma _{j}(s)=(d_{2}(s),u(s),v(s))\) of the problem (4) for \(\vert s \vert \) sufficiently small, where \(d_{2}(s)\), \(u(s)\), \(v(s)\) are continuous functions, \(d_{2}(0)=d_{2}^{j}\) and
The zero set of F consists of two curves \((d_{2}, E^{*})\) and \(\varGamma _{j}(s)\) in a neighborhood of the bifurcation point \((d_{2} ^{j},E^{*})\).
Proof
It suffices to verify conditions (a)–(c) as follows [25]:

(a)
The partial derivatives \(F_{d_{2}}\), \(F_{E}\), and \(F_{d_{2}E}\) exist and are continuous.

(b)
\(\operatorname{ker}F_{E}(d_{2}^{j},E^{*})\) and \(Y/R(F_{E}(d_{2}^{j},E^{*}))\) are onedimensional.

(c)
Let \(\operatorname{ker}F_{E}(d_{2}^{j},E^{*})=\operatorname{span}\{\varPhi \}\), then \(F_{d_{2}E}(d_{2}^{j},E^{*})\varPhi \notin R( F_{E}(d_{2}^{j},E^{*}))\).
Note that
where \(a_{11}\), \(a_{12}\), \(a_{21}\) and \(a_{22}\) are given in (8). It is clear that the linear operators \(F_{E}\), \(F_{d_{2}E}\) and \(F_{d_{2}}\) are continuous, and condition (a) is verified.
Suppose \(\varPhi =(\bar{\varphi },\bar{\psi })^{\top }\in \ker L_{1}\), and write \(\bar{\varphi }=\sum_{0\leq i\leq \infty ,1\leq j\leq m_{i}} \bar{a}_{ij}\phi _{ij}\), \(\bar{\psi }= \sum_{0\leq i\leq \infty ,1\leq j\leq m_{i}} \bar{b}_{ij} \phi _{ij}\). Then
where
Since
taking \(d_{2}=d^{j}_{2}\) implies that \(\ker L_{1}=\operatorname{span}\{\varPhi _{1}\}\), where
\(\phi _{j}\) is the eigenfunction of −△. Consider the adjoint operator
In the same way as above we obtain \(\ker L^{*}_{1}=\operatorname{span}\{\varPhi _{1}^{*}\}\), where
By the Fredholm alternative theorem, we have \({R}(L_{1})= \ker (L^{*}_{1})^{\bot }\), thus
Condition (b) is also verified.
Finally, since
and
we find \(F_{d_{2}E}(d^{j}_{2} , E^{*})\varPhi _{1}\notin R(L_{1})\), and so condition (c) is satisfied. The proof is completed. □
We investigate the direction of the steady state bifurcation of model (4) in the onedimensional interval \(\varOmega =(0,\pi )\). It is well known that the operator −△ with noflux boundary conditions has eigenvalues and eigenfunctions as follows:
for \(j=1,2,3,\ldots \) . We translate \((u^{*},v^{*})\) to the origin by the translation \((\bar{u},\bar{v})=(uu^{*},vv^{*})\). For convenience, we will denote ū, v̄ by u, v, respectively. Then we can obtain the following system:
Let
Then a straightforward calculation yields
Denote \(E=(u,v)\), then we rewrite the map \(F:\mathbb{R}^{+}\times X \rightarrow Y\) by
By Theorem 4.1, we see that \(\operatorname{dimker}F_{E}(d_{2}^{j},(0,0))= \operatorname{codim}R(F_{E}(d_{2}^{j},(0,0)))=1\) and \(\operatorname{ker}F_{E}(d _{2}^{j}, (0,0))=\operatorname{span}\{\varPhi _{1}\}\). Hence, we can decompose X and Y as
where Z is the complement of \(\operatorname{ker}F_{E}(d_{2}^{j},(0,0))\) in X and \(Z'\) is the complement of \(R(F_{E}(d_{2}^{j},(0,0)))\) in Y. Due to \(\operatorname{codim}R(F _{E}(d_{2}^{j},(0,0)))=1\), there exists \(T\in Y^{*} \) such that
where \(Y^{*}:=\operatorname{span}\{\varPhi _{1}^{*}\}\). Moreover, \(\varPhi _{1} ^{*}\) satisfies \(F_{E}(d_{2}^{j},(0,0))\varPhi _{1}^{*}=0\) by Theorem 4.1. Hence, we can define
By \(F_{{d_{2}}E}(d_{2}^{j},(0,0))\varPhi _{1}\notin R(F_{E}(d_{2}^{j},(0,0)))\) derived in Theorem 4.1, we find that
From [26], we can know that
By some calculations, we have
and
where
Hence, \(d_{2}^{\prime }(0)=0\).
Note that \(\langle F_{EE}(d_{2}^{j},(0,0))\varPhi _{1}^{2},\varPhi _{1}^{*}\rangle =0\) implies
From [26], we see that the bifurcation is supercritical (resp. subcritical) if
where θ is the solution of the following problem:
Let \(\theta =(\theta _{1},\theta _{2})\). Then θ satisfies
By direct calculation, we obtain
where
and \(b_{j}\), \(b_{j}^{*}\) are given in Sect. 4.1. Hence,
and
where
We now compute
Multiplying (25) by \(\phi _{j}^{2}\) and integrating by parts, we derive
where
Integrating (25) by parts yields
It follows from (26) that
Thus,
where
Consequently, we have
where
From the analysis above, we obtain the following results.
Theorem 4.2
Under the same hypothesis as Theorem 4.1, there exists a smooth bifurcation branch from \((d_{2}^{j},(0,0))\). Furthermore, the bifurcation is supercritical (resp. subcritical) provided that \(d_{2}^{\prime \prime }(0)>0\) (<0), where \(d_{2}^{\prime \prime }(0)\) is given by (27).
Global structure of nonconstant positive steady state
Theorem 4.1 provides no information of the bifurcating curve \(\varGamma _{j}\) far from the equilibrium point. A further study is therefore necessary in order to understand its global bifurcation. In the onedimensional interval \(\varOmega =(0,\pi )\), by using the global bifurcation theory of Rabinowitz and the Leray–Schauder degree for compact operates, we prove that \(\varGamma _{j}\) is unbounded.
Theorem 4.3
Under the same hypothesis as Theorem 4.1, the projection of the bifurcation curve \(\varGamma _{j}\) on the \(d_{2}\)axis contains \((d_{2}^{j},\infty )\).
If \(d_{2}>\bar{d}_{2}\) and \(d_{2}\neq d_{2}^{k}\) for any integer \(k>0\), then the problem (4) possesses at least one nonconstant positive steady state.
Proof
Let \(\tilde{u}=uu^{*}\), \(\tilde{v}=vv^{*}\). Then (4) is transformed into
where \(h_{1}(\tilde{u}, \tilde{v})\), \(h_{2}(\tilde{u}, \tilde{v})\) are higherorder terms of ũ and ṽ. The equilibrium point \((u^{*},v^{*})\) of (4) shifts to \((0,0)\) of this new system. Let
Then (28) is transformed into
Put \(\tilde{E}=(\tilde{u},\tilde{v})\), \(K(d_{2})\tilde{E}=(2a_{11}G_{1}( \tilde{u})+a_{12} G_{1}(\tilde{v}), a_{21}G_{2}( \tilde{u}))\) and
Recall that
Then the boundary value problem (4) can be interpreted as the equation
Note that \(K(d_{2})\) is a compact linear operator on U for any given \(d_{2}>0\) and \(H(\tilde{E})=o( \vert \tilde{E} \vert )\) for Ẽ near zero uniformly on closed \(d_{2}\) subintervals of \((0,\infty )\), and \(H(\tilde{E})\) is a compact operator on U as well.
In order to apply Rabinowitz’s global bifurcation theorem, we first verify that 1 is an eigenvalue of \(K(d^{j}_{2})\) of algebraic multiplicity one. From the argument in the proof of Theorem 4.1 it is seen that \(\ker (K(d^{j}_{2})I)=\ker L_{1}= \operatorname{span}\{\varPhi _{1}\}\), so 1 is indeed an eigenvalue of \(K=K(d^{j}_{2})\), and \(\dim \ker (KI)=1\). As the algebraic multiplicity of the eigenvalue 1 is the dimension of the generalized null space \(\bigcup_{i=1}^{\infty }\ker (KI)^{i}\), we need to verify that \(\ker (KI)=\ker (KI)^{2}\), or \(\ker (KI)\cap R(KI)={0}\).
We now compute \(\ker (K^{*}I)\) following the calculation in [27], where \(K^{*}\) is the adjoint of K. Let \((\hat{\varphi },\hat{\psi })\in \ker (K^{*}I)\). Then
By the definition of \(G_{1}\) and \(G_{2}\) we obtain
where
Write \(\hat{\varphi }=\sum_{0\leq i\leq \infty ,1\leq j\leq m_{i}} \hat{a}_{ij}\phi _{ij}\), \(\hat{\psi }= \sum_{0\leq i\leq \infty ,1\leq j\leq m_{i}} \hat{b}_{ij}\phi _{ij}\). Then
where
By a straightforward calculation one can check that \(\det \hat{B}_{i} =a_{12}\det \bar{B}_{i}\), where \(\bar{B}_{i}\) is given in (23) by replacing \(d_{2}\) with \(d^{j}_{2}\). Thus \(\det \bar{B}_{i}=0\) only for \(i=j\), and \(\ker (K^{*}I)=\operatorname{span}\{\hat{\varPhi }\}\), where \(\hat{\varPhi }=(\frac{d_{1}\lambda _{i}+a_{11}}{a_{12}},1)^{\top }\phi _{j}\). Since \((\varPhi _{1},\hat{\varPhi })_{Y}=\frac{2d_{1}\lambda _{j}}{a _{12}}\neq 0\), \(\varPhi _{1}\notin (\ker (K^{*}I))^{\bot }=R((KI))\), so \(\ker (KI)\cap R(KI)={0}\) and the eigenvalue 1 has algebraic multiplicity one.
If \(0< d_{2}\neq d^{j}_{2}\) is in a small neighborhood of \(d^{j}_{2}\), then the linear operator \(IK(d_{2}):U\rightarrow U\) is a bijection and 0 is an isolated solution of (29) for this fixed \(d_{2}\). The index of this isolated zero of \(IK(d_{2})H\) is given by
where B is a sufficiently small ball with center at 0, and p is the sum of the algebraic multiplicities of the eigenvalues of \(K(d_{2})\) which are larger than 1. For our bifurcation analysis, it is also necessary to verify that this index changes as \(d_{2}\) crosses \(d^{j}_{2}\), that is, for \(\epsilon >0\) sufficiently small,
Indeed, if μ is an eigenvalue of \(K(d_{2})\) with an eigenfunction \((\tilde{\varphi },\tilde{\psi })\), then
By the definition of \(G_{1}\), \(G_{2}\) and
we have
where
Thus the set of eigenvalues of \(K(d_{2})\) consists of all μ that solve the characteristic equation
In particular, for \(d_{2}=d^{j}_{2}\), if \(\mu =1\) is a root of (31), then a simple calculation leads to \(d^{j}_{2}=d^{i}_{2}\), and \(j=i\) by the assumption. For \(i=j\) in (31), we let \(\mu _{1}(d_{2}^{j})\), \(\mu _{2}(d_{2}^{j})\) denote the two roots. First we find that
Now for \(d_{2}\) close to \(d^{j}_{2}\), the root of (31) is given by
And \(\mu _{2}(d_{2})\) is an increasing function of \(d_{2}\), there is a small \(\epsilon >0\) such that
Consequently, \(K(d^{j}_{2}+\epsilon )\) has exactly one more eigenvalues that are larger than 1 than \(K(d^{j}_{2}\epsilon )\) does, and by a similar argument to above we can show this eigenvalue has algebraic multiplicity one. This verifies (30).
With the help of (30), we can use the argument in [25] to conclude that \(\varGamma _{j}\) either meets infinity in \(R\times U\) or meets \((d_{2}^{k},0)\) for some \(k\neq j\), \(d_{2}^{k}>0\). We now show that the first alternative must occur, following the idea of [28] and [29]. Indeed, if \(\varGamma _{j}\) is bounded, then it is compact, and \(\varGamma _{j}\) meets some other bifurcation points. Let k be such that \(\varGamma _{j}\) meets \((d_{2}^{k},0)\), but not \((d_{2}^{i},0)\) for any \(i>k\). Consider the problem (4) on the interval \((0,\pi )\) subject to the boundary condition
We first note that if Ē solves (4) and (32), then one can construct a solution E of (4) by a reflective and periodic extension: Let \(x_{n}=n\pi \), \(n=0,1,\ldots,k\), and define
It is easy see that \((d_{2}^{k},0)\) is also a bifurcation point of the problem (4) and (32). Let \(\varLambda _{k}\) denote the bifurcation branch of this new problem that meets infinity or meets \((d_{2}^{k},0)\), then use the same argument above it is clear that it either meets infinity or meets \((d_{2}^{k'},0)\) for some \(k'>k\). If the second case occurs, then by the above extension one sees that \(\varGamma _{j}\) meets \((d_{2}^{k'},0)\), which violates the definition of k, hence \(\varLambda _{k}\) meets infinity, and then by the extension again \(\varGamma _{j}\) meets infinity too. It then follows that the projection of \(\varGamma _{j}\) on the \(d_{2}\) interval must be unbounded, since the solutions u, v are bounded by constants independent of \(d_{2}\). It also follows from the a priori estimates that any solutions on the curve \(\varGamma _{j}\) must be positive. And the proof is complete. □
References
 1.
Murray, J.D.: Mathematical Biology I: An Introduction. Springer, New York (2013)
 2.
Skalski, G.T., Gilliam, J.F.: Functional responses with predator interference: viable alternatives to the Holling type II model. Ecology 82(11), 3083–3092 (2001)
 3.
Arditi, R., Ginzburg, L.R.: Coupling in predator–prey dynamics: ratio dependence. J. Theor. Biol. 139(3), 311–326 (1989)
 4.
Arditi, R., Saiah, H.: Empirical evidence of the role of heterogeneity in ratiodependent consumption. Ecology 73(5), 1544–1551 (1992)
 5.
Dolman, P.M.: The intensity of interference varies with resource density: evidence from a field study with snow buntings, plectrophenax nivalis. Oecologia 102(4), 511–514 (1995)
 6.
Jos, C., Ellner, S.P.: Testing for predator dependence in predator–prey dynamics: a nonparametric approach. Proc. Royal Soc., Biol. Sci. 267(1453), 1611–1620 (2000)
 7.
Beddington, J.R.: Mutual interference between parasites or predators and its effect on searching efficiency. J. Anim. Ecol. 44(1), 331–340 (1975)
 8.
Deangelis, D.L., Goldstein, R.A., O’Neill, R.V.: A model for tropic interaction. Ecology 56(4), 881–892 (1975)
 9.
Cantrell, R.S., Cosner, C.: Spatial Ecology Via Reaction–Diffusion Equations, vol. 41, p. xvi. Wiley, Chichester (2003)
 10.
Cantrell, R.S., Cosner, C.: On the dynamics of predator–prey models with the Beddington–DeAngelis functional response. J. Math. Anal. Appl. 257, 206–222 (2001)
 11.
Hwang, T.W.: Uniqueness of limit cycles of the predator–prey system with Beddington–DeAngelis functional response. J. Math. Anal. Appl. 290(1), 113–122 (2004)
 12.
Cantrell, R.S., Cosner, C.: Effects of domain on the persistence of populations in a diffusive foodchain model with Beddington–DeAngelis functional response. Nat. Resour. Model. 14(3), 335–367 (2001)
 13.
Hwang, T.W.: Global analysis of the predator–prey system with Beddington–DeAngelis functional response. J. Math. Anal. Appl. 281(1), 395–401 (2003)
 14.
Dimitrov, D., Kojouharov, H.: Complete mathematical analysis of predator–prey models with linear prey growth and Beddington–DeAngelis functional response. Appl. Math. Comput. 162(2), 523–538 (2005)
 15.
Kratina, P., Vos, M., Bateman, A., Anholt, B.R.: Functional responses modified by predator density. Oecologia 159(2), 425–433 (2009)
 16.
Stover, J.P., Kendall, B.E., Fox, G.A.: Demographic heterogeneity impacts densitydependent population dynamics. Theor. Ecol. 5, 297–309 (2012)
 17.
Li, H.Y., Takeuchi, Y.: Dynamics of the density dependent predator–prey system with Beddington–DeAngelis functional response. J. Math. Anal. Appl. 372(2), 644–654 (2011)
 18.
Ghergu, M., Radulescu, V.: Turing patterns in general reaction–diffusion systems of Brusselator type. Commun. Contemp. Math. 12(4), 661–679 (2010)
 19.
Ghergu, M., Radulescu, V.: Nonlinear PDEs. Mathematical Models in Biology, Chemistry and Population Genetics. Springer Monographs in Mathematics. Springer, Heidelberg (2012)
 20.
Fall, M.M.: Periodic patterns for a model involving shortrange and longrange interactions. Nonlinear Anal. 175, 73–107 (2018)
 21.
Huang, K.G., Cai, Y.L., Rao, F., Fu, S.M., Wang, W.M.: Positive steady states of a densitydependent predator–prey model with diffusion. Discrete Contin. Dyn. Syst., Ser. B 23(8), 3087–3107 (2018)
 22.
Fan, S.: A new extracting formula and a new distinguishing means on the one variable cubic equation. Natur. Sci. J. Hainan Teach. Coll. 2, 91–98 (1989) (in Chinese)
 23.
Lou, Y., Ni, W.M.: Diffusion vs crossdiffusion: an elliptic approach. J. Differ. Equ. 154, 157–190 (1999)
 24.
Pang, P.Y.H., Wang, M.X.: Strategy and stationary pattern in a threespecies predator–prey model. J. Differ. Equ. 200, 245–273 (2004)
 25.
Crandall, M.G., Rabinowitz, P.H.: Bifurcation from simple eigenvalues. J. Funct. Anal. 8, 321–340 (1971)
 26.
Shi, J.P.: Persistence and bifurcation of degenerate solutions. J. Funct. Anal. 169, 494–531 (1999)
 27.
Ni, W.M., Tang, M.: Turing patterns in the Lengyel–Epstein system for the CIMA reaction. Trans. Am. Math. Soc. 357, 3953–3969 (2005)
 28.
Ni, W.M.: Diffusion, crossdiffusion and their spikelayer steady states. Not. Am. Math. Soc. 45, 9–18 (1998)
 29.
Takagi, I.: Pointcondensation for a reaction–diffusion system. J. Differ. Equ. 61, 208–249 (1986)
Acknowledgements
The authors would like to thank the editors and reviewers for their useful suggestions which have significantly improved the paper.
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This research was supported by the National Natural Science Foundation of China (Nos. 11761063, 11361055, 11661051).
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DOI
MSC
 92D25
 35K57
 35B35
Keywords
 Predator–prey model
 Densitydependent
 Turing instability
 Bifurcation
 Steady state