The effect of parameters on positive solutions and asymptotic behavior of an unstirred chemostat model with B–D functional response
- Xiaozhou Feng^{1, 2}Email author,
- Suping Sun^{1},
- Tongqian Zhang^{3, 4} and
- Xiaomin An^{1}
https://doi.org/10.1186/s13662-018-1587-x
© The Author(s) 2018
Received: 2 November 2017
Accepted: 3 April 2018
Published: 16 May 2018
Abstract
This paper deals with the effect of parameters on properties of positive solutions and asymptotic behavior of an unstirred chemostat model with the Beddington–DeAngelis (denote by B–D) functional response under the Robin boundary condition. Firstly, we establish some a priori estimates and a sufficient condition for the existence of positive solutions (see (Feng et al. in J. Inequal. Appl. 2016(1):294, 2016)). Secondly, we study the effect of the small parameter \(k_{1}\) and sufficiently large \(k_{2}\) in B–D functional response, which shows that the model has at least two positive solutions. Thirdly, we investigate the case of sufficiently large \(k_{1}\). The results show that if \(k_{1}\) is sufficiently large, then the positive solution of this model is determined by a limiting equation. Finally, we present an asymptotic behavior of solutions depending on time. The main methods used in this paper include the fixed point index theory, bifurcation theory, perturbation technique, comparison principle, and persistence theorem.
Keywords
MSC
1 Introduction
The chemostat is a very important resource-based model for the continuous culture of competition microorganisms and a standard model for the laboratory apparatus on bioreactor, which have been studied from various views such as population dynamics and species interactions [1–17]. For early works, we refer to [2–4]. The chemostat model of competition for a single-limit nutrient between plasmid-bearing and plasmid-free organisms was proposed by Stephanopoulos and Lapidus [13], who established some local results, whereas a global result was presented in [14]. The inhibition effects on plasmid populations were studied by Hsu and Waltman [15]. The chemostat model with impulsive input nutrient concentration was studied from different views in [18–22]. The above research of the chemostat model is related to the ODE model. Recently, the coexistence and stability of chemostat models were studied from the viewpoint of PDE (see Wu [5–7], Nie [8, 9, 16], Wang [17], and Zhang [23]), which can better simulate the unstirred chemostat model.
Note that Nie and Wu [16] studied the coexistence of an unstirred chemostat model with Beddington–DeAngelis functional response and inhibitor, but the parameters of B–D functional response is different from (1), and two models are essentially different. Meanwhile, Wang et al. [17] also obtained the coexistence and stability of an unstirred chemostat model with the Beddington–DeAngelis function, but their model does not include the plasmid transformation of two competition species. However, this paper deals with plasmid-bearing and plasmid-free models in the unstirred chemostat with the B–D functional response under a homogeneous Robin boundary condition.
The rest of this paper is organized as follows. In Sect. 2, some a priori estimates and a sufficient condition for the existence for positive solutions are established (see [1]). In Sect. 3, we study the effect of the small parameter \(k_{1}\) and sufficiently large \(k_{2}\) in B–D functional response, which proves that the model has at least two positive solutions. In Sect. 4, we investigate the case of sufficiently large \(k_{1}\). The results show that if \(k_{1}\) is sufficiently large, then the positive solution of this model is determined by a limiting equation. In Sect. 5, we present an asymptotic behavior of solutions depending on the change of time by comparison principle and persistence theorem. Finally, we present a brief summary of this paper.
2 Preliminaries and lemmas
By [1], for (6), we can directly get the following conclusions.
Lemma 2.1
- (i)
\(0<\Theta<z\);
- (ii)
Θ is continuously differentiable for \(a\in(\frac{\lambda_{1}d}{1-q},+\infty)\) and is pointwise increasing as a increases;
- (iii)
\(\lim_{a\rightarrow\frac{\lambda_{1}d}{1-q}}\Theta=0\) uniformly for \(x\in\bar{\Omega}\), and \(\lim_{a\rightarrow\infty}\Theta=z(x)\) for almost every \(x\in\Omega\);
- (iv)
Let \(L_{(a,d)}=d\Delta+a(1-q)(f(z-\Theta,\Theta)-\Theta f'_{1}(z-\Theta,\Theta)+\Theta f'_{2}(z-\Theta,\Theta))\) be the linearized operator of (6) at Θ. Then \(L_{(a,d)}\) is differentiable in \(C_{B}^{2}(\bar{\Omega}) =\{u\in C^{2}(\bar{\Omega}):\frac{\partial u}{\partial n}+ru=0\}\), and all eigenvalues of \(L_{(a,d)}\) are strictly negative.
Remark 2.1
For (7), we have the same conclusion as in Lemma 2.1. Suppose that \(b>d\mu_{1}\) and denote the unique positive solution by θ. Let \(L_{(b,d)}=d\Delta+b(g(z-\theta,\theta)-\theta g_{1}'(z-\theta,\theta)+\theta g_{2}'(z-\theta,\theta))\) be the linearized operator of (7) at θ. Then all eigenvalues of \(L_{(b,d)}\) are strictly negative.
It is easy to get the following results by the method of [1], so we omit the proof.
Lemma 2.2
Suppose \(a>\frac{\lambda_{1}d}{1-q}\). Then (9) has the unique positive solution v̂, and \(0<\hat{v}<z\). In particular, if \(b>d\mu_{1}\), then \(\theta<\hat{v}<z\).
Theorem 2.1
Suppose that \((u,v)\) is nonnegative solution of (3) and \(u\not\equiv0\), \(v\not\equiv 0\). Then (i) \(0< u<\Theta<z\), \(0< v\leq\hat{v}< z\), \(x\in\bar{\Omega}\); (ii) \(u+v< z\), \(x\in\bar{\Omega}\); (iii) \(a>\frac{\lambda_{1}d}{1-q}\).
Next, we give the fixed point index of (3) by using the standard fixed point index theory in cone.
We first set up the fixed point index theory for later use. Let E be a Banach space. A set \(W\subset E\) is called a wedge if W is a closed convex set and \(\alpha W\subset W\) for all \(\alpha\geq0\). For \(y\in W\), we define \(W_{y}=\{x\in E: \exists r=r(x)>0 \mbox{ s.t. } y+rx\in W\}\) and \(S_{y}=\{x\in\overline{W}_{y}: -x\in\overline{W}_{y}\}\), and we always assume that \(E=\overline{W-W}\). Let \(T:W_{y}\rightarrow W_{y}\) be a compact linear operator on E. We say that T has property α on \(\overline{W}_{y}\) if there exist \(t\in(0,1)\) and \(\omega\in\overline{W}_{y}\setminus S_{y}\) such that \(\omega-tT\omega\in S_{y}\).
Suppose that \(F:W\rightarrow W\) is a compact operator and \(y_{0}\in W\) is an isolated fixed point of F such that \(Fy_{0}=y_{0}\). Let \(L=F'(y_{0})\) be the Fréchet derivative at \(y_{0}\). Then \(L: \overline{W}\rightarrow\overline{W}\).
Proposition 2.1
(Dancer index theorem [25])
- (i)
If L has property α on W̅, then \(\operatorname{index}_{W}(F,y_{0})=0\);
- (ii)
If L does not have property α on W̅, then \(\operatorname{index}_{W}(F, y_{0})=\operatorname{index}_{E}(L, \theta)=(-1)^{\sigma}\), where σ is the sum of multiplicities of all eigenvalues of L greater than one.
Proposition 2.2
([25])
Lemma 2.3
([1])
- (i)
if \(a\neq\frac{\lambda_{1}d}{1-q}\), \(b>\mu_{1}d\), then \(\operatorname{index}_{W}(F,(0,0))=0\);
- (ii)
Suppose that \(b<\mu_{1}d\). If \(a>\frac{\lambda_{1}d}{1-q}\), then \(\operatorname{index}_{W}(F,(0,0))=0\); If \(a<\frac{\lambda_{1}d}{1-q}\), then \(\operatorname{index}_{W}(F,(0,0))=1\);
- (iii)
\(\operatorname{index}_{W}(F,D')=1\).
Lemma 2.4
([1])
- (i)
If \(a<\frac{\hat{\lambda}_{1}d}{1-q}\), then \(\operatorname{index}_{W}(F,(0,\theta))=1\); if \(a>\frac{\hat{\lambda}_{1}d}{1-q}\), then \(\operatorname{index}_{W}(F,(0,\theta))=0\);
- (ii)
If \(a=\frac{\hat{\lambda}_{1}d}{1-q}\), then either (3) has a positive solution, or \(\operatorname{index}_{W}(F,(0, \theta))=1\).
Combining with the previous lemma, according to [1], we can show the following sufficient condition for the existence of nonnegative solutions to equation (3).
Theorem 2.2
- (i)
If \(a<\frac{\lambda_{1}d}{1-q}\), \(b<\mu_{1}d\), then 0 is the only nonnegative solution of (3);
- (ii)
If \(a>\frac{\lambda_{1}d}{1-q}\), \(b<\mu_{1}d\), then (3) has at least one positive solution besides the zero solution;
- (iii)
If \(a>\frac{\hat{\lambda}_{1}d}{1-q}\), \(b>\mu_{1}d\), then (3) has at least one positive solution besides \((0,0)\) and \((0,\theta)\).
3 The effect of mutual interference between predators
In this section, we investigate the multiplicity and stability of positive solutions of system (3) under the effect of the parameters \(k_{i}\) (\(i=1,2\)) by the standard perturbation theory.
According to Theorems 2.1 and 2.2, sufficient conditions for the existence of positive solutions of (3) are \(b>\mu_{1}d\) and \(a>\frac{\hat{\lambda}_{1}d}{1-q}\), and a necessary condition is \(a>\frac{\lambda_{1}d}{1-q}\); moreover, \(\frac{\lambda_{1}d}{1-q}<\frac{\hat {\lambda}_{1}d}{1-q}\). Next, we study the multiplicity and stability of positive solutions of (3) when \(k_{1}\) is small enough and \(k_{2}\) is sufficiently large with \(q>\frac{1}{2}\).
In [26], taking a as a bifurcation parameter and using the local bifurcation theory, we get that the positive solution \((u(s),v(s))\) bifurcates from the semitrivial solution \((0,\theta)\). According to Lemma 2.4.9 in [26], we will show that system (3) has at least one positive solution besides the bifurcation solution \((u(s),v(s))\) when \(k_{1}\) is small enough and \(k_{2}\) is sufficiently large with \(q>\frac{1}{2}\). Then we can establish the following result.
Theorem 3.1
Suppose that \(b>\mu_{1}d\) and \(q>\frac{1}{2}\). If there exist sufficiently large \(K_{2}>0\) and suitably large \(D>0\) such that \(k_{2}>K_{2}\), \(d>D\), and small enough \(k_{1}\), then the local bifurcation of the positive solution \((u(s),v(s))\) is nondegenerate and unstable for \(a\in(\frac{\hat{\lambda}_{1}d}{1-q}-\varepsilon, \frac{\hat{\lambda}_{1}d}{1-q})\) with \(\varepsilon>0\); moreover, (3) has at least two positive solutions.
Proof
Remark 3.1
Theorem 2.4.3 in [26] shows that, as \(a>\frac{\hat{\lambda}_{1}d}{1-q}\), the bifurcation solution extends to ∞ by a. However, Theorem 3.1 indicates that, as \(k_{1}\) is small enough and \(k_{2}\) sufficiently large, d is suitably large, and \(q>\frac{1}{2}\), then \(a=a(s)\in(\frac {\lambda_{1}d}{1-q}, \frac{\hat{\lambda}_{1}d}{1-q})\). Then there exists \(a^{*}\in(\frac{\lambda_{1}d}{1-q}, \frac{\hat{\lambda}_{1}d}{1-q})\) such that (3) has at least two solutions for \(a\in(a^{*}, \frac{\hat{\lambda}_{1}d}{1-q})\).
4 The effect of \(k_{1}\) on uniqueness and stability
In this section, we consider the effect of \(k_{1}\) on the existence, uniqueness, and stability of positive solutions of (3) as \(k_{1}\rightarrow\infty\).
Next, we investigate the uniqueness of positive solutions of (16).
Lemma 4.1
Problem (16) has one positive solution \(w_{0}\) if and only if \(a >\frac{d\hat{\lambda}_{1}}{1-q}\). In addition, the positive solution is unique and asymptotically stable.
Proof
Theorem 4.1
Suppose that \(b>d\mu_{1}\) is a fixed constant. For any small \(\varepsilon>0\) and any \(A>\frac{d\hat{\lambda}_{1}}{1-q}\), there exists sufficiently large \(K_{1}=K_{1}(\varepsilon,A)>0 \) such that any positive solution of (3) satisfies \(\|u\|_{c_{1}}+\|v-\theta\|_{c_{1}}\leq\varepsilon\) when \(k_{1}>K_{1}\). In particular, if we choose a sufficiently large \(K_{1}(\varepsilon,A)\) such that \(k_{1}>K_{1}\) and \(a \in (\frac{d\hat{\lambda}_{1}}{1-q},A]\), then \(\|k_{1}u-w_{0}\|_{c_{1}}\leq\varepsilon\), where \(w_{0}\) is the unique positive solution of (16).
Proof
Finally, we consider the existence and stability of positive solutions of (3) when parameter \(k_{1}\) is large enough.
Theorem 4.2
Proof
(i) Suppose that the conclusion is false. Then there exist \(\varepsilon_{0}>0\), \(k_{1,i}\rightarrow\infty\), and \(a_{i}\rightarrow a\in[\frac{d\lambda_{1}}{1-q}+\varepsilon_{0},\frac{d\hat {\lambda}_{1}}{1-q})\) such that \((u_{i}, v_{i})\) is a positive solution of (3) as \((a,k_{1})=(a_{i},k_{1,i})\). It follows from Lemma 4.1 that \(k_{1,i}\|u_{i}\|_{\infty}\) is uniformly bounded, Let \(w_{i}=k_{1,i}u_{i}\), then \(w_{i}\) satisfies equation (21). By the standard regularized theory and the Sobolev embedding theorem we may suppose that \(w_{i} \stackrel{C^{1}}{\rightarrow}w\). Hence, w is a nonnegative solution of (16). Since \(a\in[\frac {d\lambda_{1}}{1-q}+\varepsilon_{0},\frac{d\hat{\lambda}_{1}}{1-q})\), combining with Theorem 4.1, we get \(w\equiv0\). Applying a similar method as in the case \(a>\frac{d\hat{\lambda }_{1}}{1-q}\) in Theorem 4.1, it is easy to get a contradiction.
5 Asymptotic behavior of solutions
The goal of this section is to present some asymptotic behavior of solutions of (1) depending on the change of time by comparison principle and persistence theorem.
Lemma 5.1
Proof
By Theorem 14.2 in [27] we can get the local existence of solutions. The nonnegativity of solutions can be proved by the comparison principle of parabolic equations.
Suppose α satisfies \(0<\alpha<\eta_{0}\). Applying the maximum principle, we get that the maximum value of \(Y (T, x) \) cannot be taken on the interior and the border of the region, and hence \(Y(t,x)\leq \max_{z\in\overline{\Omega}}Y(0,x)\). Similarly, for (26), replacing Y by −Y, we have \(Y(t,x)\geq {-}\min_{z\in\overline{\Omega}}Y(0,x)\). Thus, there exists \(\hat{C}>0\) such that \(| Y(t,x)|\leq\hat{C}\), so that \(Y(t,x)\) is bounded, and the proof is complete. □
By Theorems 3.1–3.2 and the partial lemmas of [12] there are some conclusions about the persistence and extinction of the single species v.
Theorem 5.1
Suppose \(v(t,x)\) is a solution of (28). If \(b<\mu_{1}d\), then \(\lim_{t\rightarrow\infty}v(t,x)=0\); if \(b>\mu_{1}d\), then \(\lim_{t\rightarrow\infty}\sup\| v(t,\cdot)\|_{\infty}>0\).
Theorem 5.2
Suppose \(v(t,x)\) is a solution of (28). If \(b>\mu_{1}d\), then there exists a unique positive solution θ of the equilibrium equation on (28), and \(\lim_{t\rightarrow\infty}v(t,x)=\theta\).
Based on the single species conclusion, we investigate the asymptotic behavior of the solution of system (1). Similarly to Theorem 5.2, we obtain the following lemma.
Lemma 5.2
Lemma 5.3
- (i)
If \(b>\mu_{1}d\), then there exists a unique solution \(\theta^{\varsigma}\) for the equilibrium equation of (29);
- (ii)
Suppose that \(v^{\varsigma}(t,x)\) is a positive solution of (29) and θ is a unique positive solution of (7).
If \(b<\mu_{1}d\), then \(\lim_{t\rightarrow\infty,\varsigma\rightarrow0}v^{\varsigma}(t,x)=0\);
If \(b>\mu_{1}d\), then \(\lim_{t\rightarrow\infty,\varsigma\rightarrow 0}v^{\varsigma}(t,x)=\theta\).
Proof
(i) If \(b>\mu_{1}d\), by a method similar to the proof on Lemma 2.2 of [7] it is easy to get the existence and uniqueness of \(\theta^{\varsigma}\), and \(0<\theta^{\varsigma}<z\).
Combining Lemma 5.2, equation (29), and the uniqueness of the positive equilibrium solution \(\theta^{\varsigma}\), we obtain \(\lim_{t\rightarrow\infty}v^{\varsigma}(t,x)=\theta^{\varsigma}\).
Similarly to Lemma 5.3, we can establish the following lemmas.
Lemma 5.4
Lemma 5.5
- (1)
If \(b<\mu_{1}d\), then \(\lim_{t\rightarrow\infty}u(t,x)=0\) and \(\lim_{t\rightarrow\infty}v(t,x)=0\);
- (2)
If \(b>\mu_{1}d\), then \(\lim_{t\rightarrow\infty}u(t,x)=0\) and \(\lim_{t\rightarrow\infty}v(t,x)=\theta\).
Proof
Based on Lemmas 5.2–5.5, we can obtain the asymptotic behavior of solutions of system (1) as follows.
Theorem 5.3
- (i)
If \(b<\mu_{1}d\), then \(\lim_{t\rightarrow\infty}(S(t,x),u(t,x),v(t,x))=(z(x),0,0)\);
- (ii)
If \(b>\mu_{1}d\), then \(\lim_{t\rightarrow\infty }(S(t,x),u(t,x),v(t,x))=(z(x)-\theta,0,\theta)\).
Finally, we establish the uniform persistence of system (27), which suggests that two species can coexist.
Theorem 5.4
Suppose \(a>\frac{\lambda_{1}d}{1-q}\) and \(b< d\mu_{1}\). Then there exists \(\tilde{\varrho}>0\), and for any solution of (27), there exists \(\bar{t}_{0}>0\) (depending on the initial conditions) such that \(\min_{x\in\bar{\Omega}}u(t,x)>\tilde{\varrho}\) as \(t>\bar{t}_{0}\). Hence, the semidynamical system produced by (27) is strongly consistent continuous.
Theorem 5.5
Suppose \(a>\frac{\hat{\lambda}_{1}d}{1-q}\) and \(b>d\mu_{1}\). Then there exists \(\varrho>0\), and for any solution of (27), there exists \(t_{0}>0\) (depending on the initial conditions) such that \(\min_{x\in\bar{\Omega}}u(t,x)>\varrho\) as \(t>t_{0}\). Hence, the semidynamical system produced by (27) is strongly consistent continuous.
Proof
Applying the persistence theorem in [29, 30] to prove Theorems 5.4–5.5, because the proof process is similar, we only prove Theorem 5.5. Suppose that the state space of semidynamic systems produced by (1) is defined as \(Y=C^{+}(\bar{\Omega})\times C^{+}(\bar{\Omega})=X_{1} \cup X_{2}\), where \(X_{1}=\{(u,v)\in Y:\exists x_{0}\in\Omega\mbox{ s.t. } u(x_{0})>0\}\) and \(X_{2}=\{(u,v)\in Y: u\equiv0\}\). It is easy to see that \(X_{1}\subset Y\) is open invariant set, the equilibrium state \((0,0),(0,\theta)\in X_{2}\), and \(X_{2}\) is also an invariant set. By Theorem 5.3 we know that \(v(t,x)\rightarrow\theta\) (\(t\rightarrow\infty\)) as \(b>\mu_{1}d\), so \((0,\theta)\) attracts \((0,v)\) (\(v\geq,\not\equiv0\)). Hence the ω-set of orbitals starting at \(X_{2}\) is defined by \(\Omega_{2}=\{(0,0),(0,\theta)\}\). Let \(M_{1}=\{(0,0)\}\) and \(M_{2}=\{ (0,\theta)\}\). Then \(M=\{(M_{1}, M_{2})\}=\{(0,0),(0,\theta)\}\), Obviously, \(M_{1}\) can connect to \(M_{2}\), but \(M_{2}\) cannot connect to \(M_{1}\). So M is an acyclic isolated covering \(\Omega_{2}\). Next, we only prove that \(M_{i}\) (\(i=1,2\)) is weakly exclusive to \(X_{1}\) and M is isolated.
6 Conclusion
This paper deals with plasmid-bearing and plasmid-free models in the unstirred chemostat with the Beddington–DeAngelis functional response. Applying the fixed point theory, bifurcation theory, and the perturbation technique, we obtained the following result: Firstly, some a priori estimates and a sufficient condition for the existence for positive solutions are established. Secondly, we study the effect of the small parameter \(k_{1}\) and sufficiently large \(k_{2}\) in Beddington–DeAngelis functional response, and we find that the model has at least two positive solutions (Theorem 3.1). Thirdly, we investigate the case of \(k_{1}\). The results show that if \(k_{1}\) is sufficiently large, then the positive solution of this model is determined by a limiting equation (Lemma 4.1 and Theorems 4.1–4.2). Finally, in Sect. 5, we present some asymptotic behavior of solutions depending on the change of time by the comparison principle and persistence theorem (Theorems 5.3–5.5).
Declarations
Acknowledgements
This work is supported by Shandong Provincial Natural Science Foundation (No. ZR2015AQ001), National Natural Science Foundation of China (Nos. 11371230, 61102144), Shaanxi Province Department of Education Fund (14JK1353), Project for Higher Educational Science and Technology Program of Shandong Province (No. 13LI05), Research Funds for Joint Innovative Center for Safe and Effective Mining Technology and Equipment of Coal Resources by Shandong Province and SDUST Research Fund (2014TDJH102), The president of Xi’an Technological University Foundation (XAGDJJ1423, 17028).
Authors’ contributions
All authors worked together to produce the results and read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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