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Density-dependent effects on Turing patterns and steady state bifurcation in a Beddington–DeAngelis-type predator–prey model
- Hongwu Xu^{1} and
- Shengmao Fu^{2}Email authorView ORCID ID profile
- Received: 29 January 2019
- Accepted: 21 May 2019
- Published: 29 May 2019
Abstract
In this paper, Turing patterns and steady state bifurcation of a diffusive Beddington–DeAngelis-type predator–prey model with density-dependent 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 density-dependent 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.
Keywords
- Predator–prey model
- Density-dependent
- Turing instability
- Bifurcation
- Steady state
MSC
- 92D25
- 35K57
- 35B35
1 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 so-called 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: Prey-dependent and predator-dependent. 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 II-type prey-dependent functional response has served as the basis for a very large literature on predator–prey theory [2]. However, the prey-dependent 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 predator-dependent. 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 predator-threat, the models with predator-dependent functional response stand as reasonable alternatives to the models with prey-dependent functional response [2]. Starting from this argument and the traditional prey-dependent 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 so-called Beddington–DeAngelis functional response \(p(u,v)=\frac{mu}{a+u+bv}\). The Beddington–DeAngelis functional response is similar to the well-known 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 ratio-dependent form but avoids the singular behaviors of ratio-dependent models at low densities which have been the source of controversy.
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 ratio-dependent functional response.
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 density-dependent death rate for the predators does not admit pattern formations. Hence, in the case \(b>0\), a natural question is raised: Is the density-dependent death rate δ also a decisive factor inducing Turing instability in the model (3)? We will answer this problem in this paper.
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.
2 Stability and Turing instability of positive equilibrium point
To illustrate the uniqueness of the positive equilibrium point, we first give the following lemma.
Lemma 1
(Shengjins discriminant [22])
- (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 \(\Delta >0\).
Theorem 2.1
Theorem 2.2
- (i)If \(mu^{*}v^{*}<(u^{*}+\delta sv^{*})(bv^{*}+a+u^{*})^{2}+bsmu ^{*}v^{*}\) andthen equilibrium point \(E^{*}\) is locally asymptotically stable.$$ \begin{aligned} (\mathbf{H_{21}}) \quad &sv^{*}\bigl[m^{2}bu^{*}v^{*}+m \bigl(\delta v^{*}-bu^{*}\bigr) \bigl(bv^{*}+a+u^{*} \bigr)^{2}\bigr] \\ &\quad < m ^{2}\bigl(a+u^{*}\bigr) \bigl(a+bv^{*}\bigr)+\delta sv^{*}\bigl(bv^{*}+a+u^{*} \bigr)^{4},\end{aligned} $$
- (ii)If \(mu^{*}v^{*}>(u^{*}+\delta sv^{*})(bv^{*}+a+u^{*})^{2}+bsmu ^{*}v^{*}\) orthen equilibrium point \(E^{*}\) is unstable.$$ \begin{aligned} (\mathbf{H_{22}}) \quad &sv^{*}\bigl[m^{2}bu^{*}v^{*}+m \bigl(\delta v^{*}-bu^{*}\bigr) \bigl(bv^{*}+a+u^{*} \bigr)^{2}\bigr]\\&\quad >m ^{2}\bigl(a+u^{*}\bigr) \bigl(a+bv^{*}\bigr)+\delta sv^{*}\bigl(bv^{*}+a+u^{*} \bigr)^{4},\end{aligned} $$
Theorem 2.3
- (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.
- (ii-1)
If \(d_{1}\lambda _{1}< a_{11}\) and \(0< d_{2}<\bar{d_{2}}\), then \(E^{*}\) is locally asymptotically stable.
- (ii-2)
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.
- (ii-1)
Proof
(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 \(d_{1}a_{22} \lambda _{i}-a_{11}a_{22}+a_{12}a_{21}<0\).
Example 2
3 Nonexistence/existence of nonconstant positive steady state
In this section, we consider the nonexistence/existence of nonconstant positive steady states of (4).
3.1 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
- (i)Assume that \(w \in C^{2}(\varOmega ) \cap C^{1}(\overline{\varOmega })\), andIf \(w(x_{0})=\max_{\overline{\varOmega }} w\), then \(g(x_{0},w(x_{0})) \geq 0\).$$ \triangle w(x)+g\bigl(x,w(x)\bigr)\geq 0, \quad x\in \varOmega , \quad \quad \frac{\partial w}{\partial \nu }\leq 0, \quad x\in \partial \varOmega . $$
- (ii)Assume that \(w \in C^{2}(\varOmega ) \cap C^{1}(\overline{\varOmega })\), andIf \(w(x_{0})=\min_{\overline{\varOmega }} w\), then \(g(x_{0},w(x_{0})) \leq 0\).$$ \triangle w(x)+g\bigl(x,w(x)\bigr)\leq 0, \quad x\in \varOmega , \quad\quad \frac{\partial w}{\partial \nu } \geq 0, \quad x\in \partial \varOmega . $$
Theorem 3.1
Proof
3.2 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 zero-flux boundary condition and \(d_{2}^{*}\) be a fixed positive constant satisfying \(d_{2}^{*}>\frac{s(m-q)}{\lambda _{1}}\). 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
3.3 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.
Lemma 4
([24])
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
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.
4 Structure of nonconstant positive steady state
Let \(Y=C(\bar{\varOmega })\times C(\bar{\varOmega })\), \(X= \{(u,v)\vert u,v\in C^{2}(\bar{\varOmega }), \frac{\partial u}{\partial \nu } =\frac{ \partial v}{\partial \nu } =0, x\in \partial \varOmega \}\).
4.1 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.
Theorem 4.1
Proof
- (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 one-dimensional.
- (c)
Let \(\operatorname{ker}F_{E}(d_{2}^{j},E^{*})=\operatorname{span}\{\varPhi \}\), then \(F_{d_{2}E}(d_{2}^{j},E^{*})\varPhi \notin R( F_{E}(d_{2}^{j},E^{*}))\).
From the analysis above, we obtain the following results.
4.2 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 one-dimensional 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
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 \(\dim \ker (K-I)=1\). As the algebraic multiplicity of the eigenvalue 1 is the dimension of the generalized null space \(\bigcup_{i=1}^{\infty }\ker (K-I)^{i}\), we need to verify that \(\ker (K-I)=\ker (K-I)^{2}\), or \(\ker (K-I)\cap R(K-I)={0}\).
Declarations
Acknowledgements
The authors would like to thank the editors and reviewers for their useful suggestions which have significantly improved the paper.
Availability of data and materials
Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.
Funding
This research was supported by the National Natural Science Foundation of China (Nos. 11761063, 11361055, 11661051).
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Competing interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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