Coexistence of an unstirred chemostat model with B-D functional response by fixed point index theory
- Xiao-zhou Feng^{1, 2}Email authorView ORCID ID profile,
- Jian-hui Tian^{3} and
- Xiao-li Ma^{1}
https://doi.org/10.1186/s13660-016-1241-7
© Feng et al. 2016
Received: 7 July 2016
Accepted: 10 November 2016
Published: 22 November 2016
Abstract
This paper deals with an unstirred chemostat model with the Beddington-DeAngelis functional response. First, some prior estimates for positive solutions are proved by the maximum principle and the method of upper and lower solutions. Second, the calculation on the fixed point index of chemostat model is obtained by degree theory and the homotopy invariance theorem. Finally, some sufficient condition on the existence of positive steady-state solutions is established by fixed point index theory and bifurcation theory.
Keywords
1 Introduction
The chemostat is a laboratory apparatus used for the continuous culture of micro-organisms. Mathematical models of the chemostat are surprisingly amenable to analysis. Early results can be found in the articles of Levin [1] and Hsu [2]. For a general discussion of competition, Smith and Waltman [3] discussed the well-unstirred model in detail. Recently, Wu [4–6] and Nie [7, 8] studied the coexistence and asymptotic behavior of chemostat models from the viewpoint of partial differential equations theories. The response functions are mainly the Michaelis-Menten functional response \(f(S) = S/(1 + \kappa S)\), κ is a constant. Until very recently, both ecologists and mathematicians chose to base their studies on the Beddington-DeAngelis (denoted by B-D) functional response introduced by Beddington [9] and DeAngelis [10]. The Beddington-DeAngelis functional response has some of the same qualitative features as the Michaelis-Menten form but has an extra term βu in the denominator which models mutual interference between predators. It has been the source of controversy and can provide a better description of predator feeding over a range of predator-prey abundances, which are strongly supported by numerous field and laboratory experiments and observations.
In the following, we set up the fixed point index theory on this paper. Let E be a Banach Space. \(W \subset E\) is called a wedge if W is a closed convex set and \(\alpha W \subset W\) for all \(\alpha \ge 0\).
For \(y \in W\), we define \(W_{y} = \{ x \in E:\exists r = r(x) > 0, \mbox{s.t.}, y + rx \in W\}\), \(S_{y} = \{ x \in \bar{W}_{y}: - x \in \bar{W}_{y}\}\), we always assume that \(E = \overline{W - W}\). Let \(T:W_{y} \to W_{y}\) be a compact linear operator on E. We say that T has property α on \(\bar{W}_{y}\) if there exists \(t \in ( 0,1 )\) and \(\omega \in \bar{W}_{y}\backslash S_{y}\), such that \(\omega - tT\omega \in S_{y}\).
Suppose that \(F:W \to W\) is a compact operator, and \(y_{0} \in W\) is an isolated fixed point of F, such that \(Fy_{0} = y_{0}\), let \(L = F'(y_{0})\) is Fréchet differentiable at \(y_{0}\), it follows that \(L:\overline{W} \to \overline{W}\).
Proposition 1.1
[11] Dancer index theorem
- (1)
L has property α on W̄, then \(\operatorname{index}_{W}(F,y_{0}) = 0\);
- (2)
L does not have property α on W̄, then \(\operatorname{index}_{W}(F,y_{0}) = \operatorname{index}_{E}(L,\theta ) = \pm 1\).
Proposition 1.2
[11]
Proposition 1.3
[12]
- (1)
if \(Tu > u\), then \(r ( T ) > 1\);
- (2)
if \(Tu < u\), then \(r ( T ) < 1\);
- (3)
if \(Tu = u\), then \(r ( T ) = 1\).
Proposition 1.4
[13]
The organization of our paper is as follows. In Section 2, some prior estimates for positive solutions are proved by the maximum principle and the upper and lower solution method. In Section 3, the calculations on the fixed point index of chemostat model by degree theory and the homotopy invariance theorem. In Section 4, some sufficient conditions on the existence of positive steady-state solutions is established by the fixed point index theory in cone and bifurcation theory.
2 Some prior estimates for positive solutions
The main purpose of this section is to give prior upper and lower positive bounds for positive solutions of (1.3) by using the maximum principle and the upper and lower solution method.
By [9, 14, 15], 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 point wisely increasing when a is increasing.
- (iii)
\(\lim_{a \to \frac{\lambda_{1}d}{1 - q}}\Theta = 0\) uniformly for \(x \in \bar{\Omega}\), \(\lim_{a \to \infty} \Theta = z(x)\) 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 (2.3) at Θ, then \(L_{ ( a,d )}\) is differentiable in \(C_{B}^{2}(\bar{\Omega} ) = \{ u \in C^{2} ( \bar{\Omega} ):\frac{\partial u}{\partial \nu} + ru = 0\}\), and all eigenvalues of \(L_{ ( a,d )}\) are strictly negative.
Remark 2.1
Lemma 2.2
Suppose \(a > \frac{\lambda_{1}d}{1 - q}\), there exists a unique positive solution of (2.6). Then \(0 < \upsilon < z\), and \(\theta < \upsilon < z\), when \(b > d\mu_{1}\).
Proof
Next, we will prove the existence and uniqueness of solutions. For sufficiently small \(\delta > 0\), \(\delta \phi_{1},z\) are the upper and lower solutions of (2.6). It follows from the comparison principle [16] that (2.6) exists the minimum solution \(\upsilon_{1}\) and maximum solution \(\upsilon_{2}\), satisfying \(\delta \phi {}_{1} \le \upsilon_{1} \le \upsilon_{2} \le z\).
In conclusion, we can get prior estimates on the system (1.3).
Theorem 2.3
- (i)
\(0 < u < \Theta < z\), \(0 < \upsilon \le \hat{\upsilon} < z\), \(x \in \bar{\Omega}\);
- (ii)
\(u + \upsilon < z\), \(x \in \bar{\Omega}\);
- (iii)
\(a > \frac{\lambda_{1}d}{1 - q}\).
3 Calculations of fixed point index
In this section, we will calculate the fixed point index of (1.3) by using the standard fixed point index theory in cone.
In the following, we calculate the index number of \((0,0)\) and \((0,\theta)\) by using the fixed point theory.
Lemma 3.1
- (i)
if \(a \ne \frac{\lambda_{1}d}{1 - q}\), \(b > \mu_{1}d\), 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 \(\operatorname{index}_{W}(F,(0,0)) = 1\).
Proof
(i) Suppose that \(a \ne \frac{\lambda_{1}d}{1 - q},b > \mu_{1}d\). If \(\hat{\lambda}_{1}\) is the principal eigenvalue of \(- d\Delta \psi - bg ( z,0 )\psi = \hat{\lambda} \psi\), then \(\hat{\lambda}_{1} < 0\). From Proposition 1.4, there exists \(\frac{1}{\hat{\lambda}} < 1\) is an eigenvalue of \(( - d\Delta + P)\psi = \frac{1}{\hat{\lambda}} (P + bg(z,0))\psi, \psi\) is the corresponding eigenfunction, then the \(L_{0}\) has no eigenvalue greater than 1, so \((0,\psi)\) is the corresponding eigenfunction. It follows from Proposition 1.2 that we have \(\operatorname{index}_{W} ( F, ( 0,0 ) ) = 0\).
(a) If \(\varphi \equiv 0\), then from the second equation of (3.4), we see that \(\lambda_{1}( - d\Delta - bg(z,0)) > 0\) when \(b < \mu_{1}d\), it follows from the Proposition 1.4 that (3.4) has no eigenvalues which are equal to or less than 1. This is a contradiction hypothesis.
Lemma 3.2
\(\operatorname{index}_{W} ( F,D' ) = 1\).
Proof
Taking \(\tau \in ( 0,1 )\) sufficiently small, such that \(\tau a ( 1 - q ) < \lambda_{1}d, \tau b < \lambda_{1}d\), it follows from Lemma 3.1 that \(\operatorname{index}_{W} ( F, ( 0,0 ) ) = 1\), so as \(0 < \tau << 1\), \(\deg_{W} ( I - F_{\tau},D', ( 0,0 ) ) = 1\), by the homotopy invariance, we get \(\deg_{W}(I - F_{\tau},D',(0,0)) = 1\), thus \(\operatorname{index}_{W}(F,D') = 1\). □
Lemma 3.3
- (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 (1.3) has a positive solution or \(\operatorname{index}_{W} ( F, ( 0,\theta ) ) = 1\).
Proof
From the definition of \(a \ne \frac{\hat{\lambda}_{1}d}{1 - q}\) and \(\lambda_{1}\), we can get \(\phi \equiv 0\), and taking \(\phi \equiv 0\) into the second equation, then \(d\Delta \psi + L_{b}\psi = 0\).
It follows from Lemma 2.1 that \(\psi \equiv 0\), hence \(I - L_{1}\) is inverse on \(\bar{W}_{y_{1}}\).
(i) When \(a < \frac{\hat{\lambda}_{1}d}{1 - q}\), we can prove that \(L_{1}\) has no property α on \(\bar{W}_{y_{1}}\).
Let \(L_{1}(a,0,\theta )(\omega,\chi )^{T} = (0,0)^{T}\). From the analysis of (3.6), we see that the nuclear space of \(L_{1}(a,0,\theta )\) satisfies \(N(L_{1}(a,0,\theta )) = \operatorname{span}\{ (\hat{\phi}_{1},\chi_{1})\}\). Hence \(\operatorname{dim} N(L_{1}(a,0,\theta )) = 1\).
Let \(L_{1}^{*}(a,0,\theta )(\omega,\chi )^{T} = (0,0)^{T}\), it is easy to get \((\omega,\chi ) = (\hat{\phi}_{1},0)\), so \(N(L_{1}^{*}(a,0,\theta )) = \operatorname{span}\{ (\hat{\phi}_{1},0)\}\). According to the Fredholm theorem [18], we can obtain \(R(L_{1}(a,0,\theta )) = \{ (\omega,\chi ) \in E|\int_{\Omega} \omega \hat{\phi}_{1}\,dx = 0 \}\), so \(\operatorname{codim} R ( L ( a,0,\theta ) ) = 1\).
Owing to \(N(L_{1}^{*})\) and \(R(L_{1})\) are orthogonal, however \(\int_{\Omega} ( 1 - q )f ( z - \theta,0 )\hat{\phi}_{1}^{2} \ne 0\).
We shall discuss two possible cases as follows:
Case 1: if \(a'(s) \equiv 0\), then \(a(s) = a\), for s submitting to \(\vert s \vert < \sigma\), and \(F(a(s),u(s),\upsilon (s)) = 0\), that is, \((u(s),\upsilon (s))\) is the solution (1.3), since \(\hat{\phi}_{1} > 0\), and \(\vert s \vert \) is very small, then there exists \(\varepsilon > 0\) such that \(u(s) > 0,\upsilon (s) > 0\), when \(0 < s < \varepsilon \le \delta\). Hence, (1.3) have positive solutions.
Thanks to \(a(1 - q) = \lambda_{1}d\), then \(\lambda_{1} ( d\Delta + ( 1 - q )af ( z - \theta,0 ) - t ) < 0\), that is, \(d\Delta + ( 1 - q )af( z - \theta,0 ) - t\) is invertible, then \(\phi \equiv 0\), and from Lemma 2.1, we know that the operator L is invertible. Then, similarly, we see that \(\psi \equiv 0\). Thus, 1 is not the eigenvalue of \(L_{t}\).
4 Coexistence of the chemostat model
In this section, by using the fixed point index calculation method, combined with Lemmas 3.1-3.3, we can show that there exists the sufficient condition of existence of non-negative solutions to equation (1.3).
Theorem 4.1
- (i)
If \(a < \frac{\lambda_{1}d}{1 - q}, b < \mu_{1}d\), then the unique non-negative solution of (1.3) is zero;
- (ii)
if \(a > \frac{\lambda_{1}d}{1 - q},b < \mu_{1}d\), then (1.3) have at least one positive solution besides \((0,0)\);
- (iii)
if \(a > \frac{\lambda_{1}d}{1 - q},b > \mu_{1}d\), then (1.3) have at least one positive solution besides \((0,0), (0,\theta )\).
Proof
In the following, we show it by the upper and lower solution method.
Remark 4.1
When \(a = \frac{\lambda_{1}d}{1 - q},b > \mu_{1}d\), it follows from Lemma 3.3 that we can obtain either \(\operatorname{index}_{W} ( F, ( 0,\theta ) ) = 1\), or for (1.3) there exists a positive solution.
5 Conclusion
The coexistence of an unstirred chemostat model with B-D functional response is studied by fixed point index theory in our paper. First of all, some prior estimates for positive solutions are proved by the maximum principle and the upper and lower solution method. Second, the calculations are performed on the fixed point index of chemostat model by degree theory and the homotopy invariance theorem. Finally, some sufficient condition on the existence of positive steady-state solutions is established by fixed point index theory in cone and bifurcation theory.
Declarations
Acknowledgements
The work was partially supported by the National Natural Science Youth Fund of China (11302159; 11401356); The natural science foundation of Shaanxi Province (2013JC2-31); the president of the Xi’an Technological University Foundation (XAGDXJJ1423).
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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