Abstract
This paper deals with an unstirred competitive chemostat model with the Beddington–DeAngelis functional response. With the help of the linear eigenvalue theory and the monotone dynamical system theory, we establish a relatively clear dynamic classification of this system in terms of the growth rates of two species. The results indicate that there exist several critical curves, which may classify the dynamics of this system into three scenarios: 1) extinction; 2) competitive exclusion; and 3) coexistence. Comparing with the classical chemostat model [26], our theoretical results reveal that under the weak–strong competition cases, the role of intraspecific competition can lead to species coexistence. Moreover, the simulations suggest that under different competitive cases, coexistence can occur for suitably small diffusion rates and some intermediate diffusion rates. These new phenomena indicate that the intraspecific competition and diffusion have a great influence on the dynamics of the unstirred chemostat model of two species competing with the Beddington–DeAngelis functional response.
1 Introduction
It is well known that the chemostat is a laboratory apparatus used for the continuous culture of microorganisms, while the chemostat models are extensively applied in ecology to simulate the growth of single-celled algal plankton in oceans and lakes [1–4]. Most of the earlier chemostat models assume the well-stirring of culture, which leads to chemostat models generally described by ordinary differential equation models (see, e.g., [2, 4, 5]). However, this idealized mixing is quite different from the real environment in which microbial populations live. Since the ability of microorganisms to move in a random fashion plays an important role in determining the survival and extinction of populations, many unstirred chemostat models have sprung up in which populations and resources are distributed in spatially variable habitats; please refer to [6–10] for small sampling of such works.
There are various types of response functions; among them, Holling types I–IV [11] are usually introduced to model the growth of microorganisms. Particularly, the various chemostat models with Holling type II functional response have been extensively studied (see, e.g., [4, 10, 12, 13]). As far as we know, for the unstirred competitive chemostat models with Holling type II functional response, So and Waltman [14] first obtained the local coexistence by standard bifurcation theorems. Later, Hsu and Waltman [6] obtained the asymptotic behavior of solutions by the theory of uniform persistence in an infinite-dimensional dynamical system and the theory of strongly order-preserving semi-dynamical system. To explore the effect of diffusion, Shi et al. [8] further studied this model and confirmed that stable coexistence solutions only occur at the intermediate diffusion rates. In addition, a diffusive predator–prey chemostat model with Holling type II functional response was studied by Nie et al. [7], and their analytical and numerical results show that a relatively small diffusion is conducive to the coexistence of species.
However, in nature, it is known that there is not only competition between two species but also mutual interference in species. Therefore, it is necessary to consider mutual interference in species. To this end, Beddington [15] and DeAngelis et al. [16] (simplified as B.–D.) proposed the following B.–D. functional response:where ki > 0 (i = 1, 2) are the Michaelis–Menten constants, S represents the density of the resources, u and v represent the density of two species, respectively, and βi > 0 (i = 1, 2) model the mutual interference between two species.
As illustrated by Harrision [17], the B.–D. functional response with intraspecific interference competition was superior to well-known Holling type II functional response in modeling the resource uptake of species. Therefore, there appear successively many works to describe the population dynamics by using the B.–D. functional response. For instance, Jiang et al. [18] discussed a competition model with the B.–D. functional response, and they applied the fixed-point index theory to obtain the sufficient conditions for the existence of positive solutions. In addition, a predator–prey model with a heterogeneous environment and the B.–D. functional response was constructed by Zhang and Wang [19], and the existence of positive stationary solutions was obtained by using the fixed-point index theory. We also refer the recent works [20–22] about population models with the B.–D. functional response.
Particularly, the unstirred chemostat models with the B.–D. functional response have also received considerable attention in the past decades. Wang et al. [23] obtained the sufficient conditions for the existence of positive steady-state solutions and studied the effect of parameter β1 on coexistence states by the fixed-point index theory, the perturbation technique, and the bifurcation theory. Meanwhile, Nie and Wu [24] studied the unstirred chemostat model with the B.–D. functional response and inhibitor, and the uniqueness, multiplicity, and stability of the coexistence solutions were obtained by the degree theory in cones, bifurcation theory, and perturbation technique. More works on chemostat models with the B.–D. functional response can be found in [25–27] and the references therein.
Mathematically speaking, these sufficient conditions for the existence of coexistence solutions are usually established in terms of the principal eigenvalues of the corresponding linearized eigenvalue problems at trivial or semi-trivial steady states (see, e.g., [
18,
19,
23,
24]). It is worth noting that these principal eigenvalues depend heavily on the model parameters, which motivates us to explore how these model parameters affect the existence of coexistence solutions and establish the dynamics classification of this system in terms of these model parameters. Moreover, it should be noted that studying the asymptotic analysis of steady states of chemostat models is non-trivial, and some new techniques need to be introduced. Overall, for the unstirred chemostat system with the B.–D. functional response, we are concerned with the following questions:
(1) How do parameters such as diffusion rates, growth rates, and intraspecific competition parameters affect the dynamics of the unstirred chemostat system with the B.–D. functional response?
(2) Can we establish a clear dynamic classification of the unstirred chemostat system with the B.–D. functional response in terms of these parameters?
(3) Will there arise a new phenomenon if one introduces the B.–D. functional response into the unstirred chemostat model?
The purpose of this paper is to address these problems. We hope that the approaches in this paper might provide some new insights on the dynamical behavior of the unstirred chemostat models.
This paper is organized as follows. In Section 2, we introduce an unstirred chemostat model with the B.–D. functional response and its corresponding limiting system. In Section 3, some preliminary results are given. In Section 4, we aim to investigate the dynamics of this limiting system and obtain a relatively clear dynamic classification of this limiting system in the m1 − m2 plane. In Section 5, the coexistence solution for this limiting system is established by a bifurcation argument. In Section 6, we study the effect of diffusion on system dynamics by numerical approaches. In Section 7, a discussion is presented from the opinion of analytic and numerical results. Finally, the proofs of some theoretical results are deferred to the Supplementary Appendix in Supplementary Section S8.
2 The model
In this paper, we consider following the unstirred chemostat system with the B.–D. functional response:where S(x, t) is the concentration of the nutrient and u(x, t) and v(x, t) represent the population density for the two competing microorganisms with location x and time t, respectively. The positive constants m1 and m2 are corresponding to the growth rates of species u and v with nutrient concentration S. d > 0 is the diffusion rate of the nutrient and microorganisms. The initial data S0(x), u0(x), and v0(x) are non-negative non-trivial continuous functions. In the reactor, the nutrients are pumped with the rate of S0 > 0 at position x = 0, and the mixed cultures containing nutrients and microorganisms are pumped out with the rate of γ > 0 at the position x = 1, which results in the Robin boundary conditions at x = 1 [6]. Here, f1(S, u), f2(S, v) satisfying Eq. 1.1 are the nutrient uptake of species u and v at nutrient concentration S. Moreover, we redefine f1(S, u), f2(S, v) as follows [10]:
For convenience, we still denote as fi(S, u), throughout this paper.
It is worth pointing out that system (2.1) satisfies the conservation law [4]. In other words, the total biomass concentration S + u + v in the chemostat approaches asymptotically a steady state (see [14], Lemma 2.1); that is,
Hence, we apply the classical internal chain transitive theory [[28], Lemma 2.1] to reduce system (2.1) into the following limiting system:
In this paper, we are mainly concerned with the dynamics classification of system (2.2). Based on the competition relationship of two species, system (2.2) generates a strictly monotone dynamical system in the partial competitive order induced by the cone K = {(u, v) ∈ C[0, 1] × C[0, 1]: u ≥ 0, v ≤ 0} (see [27], Proposition 1.3 in Chapter 8). Since the dynamics of Eq. 2.2 are related to the stability of non-negative steady states [29], we also focus on the following steady-state system:
The contribution of this paper is to explore the effect of these model parameters on the dynamics of system (2.2). Precisely, we first apply the linear eigenvalue theory and the monotone dynamical system theory to establish the threshold dynamics of system (2.2) in terms of growth rates and intraspecific competition parameters (see Theorems 4.1, 4.2). Moreover, we give a relatively clear dynamic classification of system (2.2) in the m1 − m2 plane (Figure 2). Finally, by numerical simulations, we further investigate the effect of diffusion on the dynamics of system (2.2) (Figures 3–5). Particularly, the numerical results show that under the different competitive cases, coexistence occurs for suitably small diffusion rates and some intermediate diffusion rates, which reveals that the dynamics of system (2.2) are relatively complicated.
3 Preliminaries
In this section, some preliminary results are presented, which are helpful in the following analysis.
We first consider a linear eigenvalue problemwhere γ is a positive constant and q(x) ∈ C[0, 1]. For fixed d > 0, it is well known that problem (3.1) admits a principal eigenvalue μ1(q(x)) [29], which corresponds to a positive eigenfunction φ1(⋅, q(x)) normalized by . Furthermore, by the variation characterization of the principal eigenvalue [29], we have
Moreover, the principal eigenvalue μ1(q(x)) has the following properties.
Lemma 3.1
(See [21], Lemma 2.1).
The following statements on the principal eigenvalueμ1(
q(
x))
are true:(i) μ1(q(x)) depends continuously and differentially on parameterdin (0, + ∞), and it is strictly decreasing with respect todin (0, + ∞).
(ii) qn(x) → q(x) inC[0, 1] impliesμ1(qn(x)) → μ1(q(x)).
(iii) q1(x) ≥ q2(x) implies thatμ1(q1(x)) ≥ μ1(q2(x)), and the equality holds only ifq1(x) ≡ q2(x). Particularly,μ1(0) < 0.
Lemma 3.2
,
m,
β,
k> 0
. Letω(
x,
t)
be the solution of system(3.3)
. Then,
(i) ifm > m∗, system (3.3) admits a unique positive steady state 0 < ω∗(⋅; m, β) < ϕ(x) forx ∈ [0, 1], anduniformly on [0,1];
(ii) ifm ≤ m∗, system (3.3) has no positive steady state anduniformly on [0,1].
Lemma 3.3
suppose that
m>
m∗holds. The following statements about the positive solution
ω∗(;
m,
β) will hold.
(i) For fixed d, k, β > 0, there exists positive solution ω∗(; m, β), which is continuously differentiable with respect to m in (m∗, + ∞), and it is point-wise strictly increasing in m ∈ (m∗, + ∞). Moreover,
(ii) For fixed d, k > 0 and m > m∗, there exists positive solution ω∗(⋅; m, β), which is continuously differentiable with respect to β in (0, + ∞), and it is point-wise strictly decreasing in β ∈ (0, + ∞). Moreover,
ProofFor (i), it follows from Lemma 3.2 that ω∗(⋅; m, β) exists if and only if m > m∗. Moreover, ω∗(⋅; m, β) is continuously differentiable with respect to m in (m∗, + ∞) refering to the arguments in [30], Lemma 5.4(ii). Differentiating the equation of ω∗(⋅; m, β) with respect to m and denoting Pm(x) = ∂ω∗(⋅; m, β)/∂m, Pm(x) satisfiesWe define and denote . It is easy to see that μ1(L1) < μ1(mf(ϕ − ω∗, ω∗)) = 0 by Lemma 3.1(iii). Noting that and L1(Pm) = −f(ϕ − ω∗, ω∗)ω∗ < 0, we have Pm(x) > 0 on [0,1] by the generalized maximum principle [23, Theorem 5], which implies that ω∗(⋅; m, β) is point-wise strictly increasing in m ∈ (m∗, + ∞).Since 0 < ω∗(⋅; m, β) < ϕ and , is decreasing on [0,1]. Note that the boundary conditions . Then, is uniformly bounded for x ∈ [0, 1]. It follows from the Arzela–Ascoli theorem that there exist ω1, ω2 ∈ C[0, 1] with 0 ≤ ω1 ≤ ϕ and 0 ≤ ω2 ≤ ϕ such thatTo prove ω1 = 0 on [0,1], we assume by contradiction that ω1≢0 on [0,1]. Since 0 < f(ϕ − ω∗, ω∗) < 1, the standard Lp-estimate implies that ω∗(⋅; m, β) is uniformly bounded in W2,p(0, 1) with p ∈ (1, ∞) for m ∈ (m∗, M], where M is a fixed constant larger than m∗. Therefore, weakly in W2, p (0, 1) and the convergence also holds in C1[0, 1] by the Sobolev embedding theorem. Then, ω1 satisfiesSince ω1≢0 on [0,1], we have ω1 > 0 on [0,1] by the strong maximum principle. It is easy to see that μ1(m∗f(ϕ, 0)) > μ1(m∗f(ϕ − ω1, ω1)) = 0 by Lemma 3.1(iii), a contradiction to the definition of m∗ (Eq. 3.4). Thus, ω1 = 0.We next prove ω2 = ϕ(x) on [0,1]. We recall that ω∗(⋅; m, β) satisfiesDividing the first equation of Eq. 3.9 by mω∗ and integrating over (0,1),which impliesNote in C[0, 1]. Taking m → +∞, we have , which means ω2 = ϕ(x) on [0,1] by 0 ≤ ω2 ≤ ϕ.(ii) The monotonicity of ω∗(⋅; m, β) with respect to β in (0, + ∞) can be proved by the similar arguments as in the proof of (i) and holds (see [23], Remark 1.2).It is clear that system (2.2) generates a monotone dynamical system in the partial competitive order induced by the cone K = {(u, v) ∈ C[0, 1] × C[0, 1]: u ≥ 0, v ≤ 0} (see [27], Proposition 1.3 in Chapter 8). Hence, we can recall the well-known results on the monotone dynamical system as follows.
Lemma 3.4
[
9]. For the monotone dynamical system,
(i) if two semi-trivial steady states are asymptotically stable, then it has at least one unstable coexistence steady state.
(ii) if two semi-trivial steady states are unstable, then it has at least one stable coexistence steady state. Furthermore, if its coexistence steady states are all linearly stable, then there is a unique coexistence steady state that is globally asymptotically stable.
(iii) if there is no coexistence steady state and if one semi-trivial solution is linearly unstable, the other semi-trivial solution is globally asymptotically stable.
4 The dynamics analysis of system (2.2)
As we already know, the local dynamics of system (2.2) are related to the stability of semi-trivial solutions [29]. Hence, we next establish the stability of semi-trivial solutions, including local and some global stability results. Recalling , by the similar arguments as in (3.4), we can define such that
Clearly, are dependent on ki but not on βi.
Proposition 4.1For fixed d > 0, the following statements hold:
Proof(i) Sinceit is easy to check that by k1 > k2 > 0 and Lemma 3.1(iii). Note that Eq. 4.2) is equivalent to and . Since based on k1 > k2 > 0, we have ; that is, . Therefore, . (ii) can be obtained similarly. For (iii), it is easy to obtain by the fact of k1 = k2 > 0 and .As the consequence of Lemma 3.2, system (2.2) admits the following trivial and semi-trivial solutions: trivial solution (0,0); semi-trivial solution (ω∗(⋅; m1, β1), 0) exists if and only if ; semi-trivial solution (0, ω∗(⋅; m2, β2)) exists if and only if . For convenience, we denote , , and next, we give an a priori estimate for the positive steady-state solution of system (2.2).
Lemma 4.1
Suppose that (
u(
x),
v(
x)) is a non-negative solution of system (2.2) with
u≢0 and
v≢0 on [0,1]. Then,
(i) and for x ∈ [0, 1]
(ii) 0 < u(x) + v(x) < ϕ(x) for x ∈ [0, 1]
,
k1,
k2> 0
fixed. Let(
u(
x,
t),
v(
x,
t))
be the solution of(2.2)
with any non-negative non-trivial initial condition. The following statements hold:(i)
We considerβ1,
β2> 0
fixed.(i.1) Ifand, then
(i.2) Ifand, then
(i.3) Ifand, then
(ii)
We consider,,
andβ1> 0
fixed. Then,is unstable. Moreover,(ii.1) (4.8) holds provided
(ii.2) there exists a uniquesuch thatis asymptotically stable when;is unstable when, and system (2.2) admits at least one stable coexistence steady state when.
(iii)
We consider,,
andβ2> 0
fixed. Then,
is unstable. Moreover,(iii.1) Eq. 4.9holds provided
(iii.2) there exists a uniquesuch thatis asymptotically stable when;is unstable when, and system (2.2) admits at least one stable coexistence steady state when.
Proof(i) can be proved by the similar arguments as in [6], Theorems 3.5, 3.6, and we omit it here. Next, we only prove (ii), since (iii) can be proved by similar arguments.Claim 1. For , , and β1 > 0 fixed, is unstable.Note that satisfieswhich implies . We recall that is asymptotically stable if and it is unstable if . Then, we conclude from and on [0,1] thatwhich means that by Lemma 3.2(iii). That is, is unstable.Claim 2. (1) For , , and β1 > 0 fixed, is asymptotically stable when , where is defined by Eq. 4.10.(2) There exists a unique such that is asymptotically stable when , and is unstable when .For (1), we recall that is asymptotically stable if and unstable if . Similarly, we can conclude from and on [0,1] thatprovided . It follows from Lemma 3.2(iii) that when . That is, is asymptotically stable when .For (2), since is point-wise strictly decreasing in β2 ∈ (0, + ∞) and uniformly on [0,1] (see Lemma 3.3(ii)), we can obtain from Lemma 3.2(iii) that is strictly increasing in β2 ∈ (0, + ∞). Furthermore,based on (Eq. 4.1). Moreover, when . Therefore, there exists a unique such thatwhich means that is asymptotically stable when , while it is unstable when .Claim 3. For , , and β1 > 0 fixed, system (2.2) has no positive steady states when .We assume by contradiction that system (2.2) admits a positive steady state , which satisfiesMultiplying the first equation of (4.13) by and the second equation by , integrating over (0,1), and then, subtracting the resulting equations, we haveSince on [0,1], we can conclude from and that the left side of (4.14) is less than 0, when . That is a contradiction.In conclusion, we can deduce that (ii.1) holds from Claim 1, Claim 2(1), Claim 3, and Lemma 3.4(iii). In addition, (ii.2) is the direct result of Claim 1, Claim 2(2), and Lemma 3.4(ii). The proof is completed.
Remark 4.1Theorem 4.1(i) implies that both species with sufficiently small growth rates are washed out, while competition exclusion occurs and the species with a sufficiently faster growth rate will finally win the competition. In particular, when both species admit sufficient fast growth rates, Theorem 4.1(ii.1) suggests that the species v with stronger growth ability ( is large) and weaker intraspecific competition (β2 is small) will finally win the competition. This is consistent with the biological intuition that the species with stronger growth ability and weaker intraspecific competition has more competitive advantages. Theorem 4.1(iii.1) illustrates the similar biological phenomenon.We next investigate the local dynamics of system (2.2). Note that the stability of is determined by the sign of , and the stability of is determined by the sign of . Clearly, depends on m1, m2, and β1, and depends on m1, m2, and β2. To this end, we define
Lemma 4.2
The principal eigenvalues
σ1(
m1,
m2,
β1) and
τ1(
m1,
m2,
β2) have the following properties:
(i) For fixed d, k1, k2 > 0 and ,
(i.1) σ1(m1, m2, β1) is strictly decreasing with respect to m1 in ,
(i.2) σ1(m1, m2, β1) is strictly increasing with respect to β1 in (0, + ∞),
(i.3) σ1(m1, m2, β1) is strictly increasing with respect to m2 in ; moreover,
(ii) For fixed d, k1, k2 > 0 and ,
(ii.1) τ1(m1, m2, β2) is strictly increasing with respect to m1 in ,
(ii.2) τ1(m1, m2, β2) is strictly increasing with respect to β2 in (0, + ∞),
(ii.3) τ1(m1, m2, β2) is strictly decreasing with respect to m2 in ; moreover,
ProofFor (i), (i.1) can be obtained by Lemma 3.1(iii) and Lemma 3.3(i). Similarly, (i.2) is followed by Lemma 3.1(iii) and Lemma 3.3(ii). To prove (i.3), it is obvious that σ1(m1, m2, β1) is strictly increasing with respect to m2 in by Lemma 3.1(iii) and by (3.2). We recall . Then, we can conclude from Lemma 3.1(ii) (iii) thatFor (ii), (ii.1) can be obtained by Lemma 3.1(iii), and (ii.2) can be proved by Lemma 3.1(iii) and Lemma 3.3(ii). We then prove (ii.3). Since is point-wise strictly increasing in by Lemma 3.3(i), it follows from Lemma 3.1(iii) that τ1(m1, m2, β2) is strictly decreasing with respect to m2 in . Noting that by (3.5), we can conclude from Lemma 3.1(ii) thatbased on (see (4.1)). Moreover, since on [0,1] (see (3.5)), we can obtain from Lemma 3.1(ii) (iii) thatThe proof is completed.Clearly, both σ1(m1, m2, β1) and τ1(m1, m2, β2) depend on m1 and m2. To investigate the local dynamics of system (2.2) in the m1 − m2 plane, we fix β1, β2 > 0 and denote them by σ1(m1, m2) and τ1(m1, m2).
Lemma 4.3
Suppose
. For fixed
d,
β1,
β2,
k1,
k2> 0, there exist two continuous critical curves
where
and
are differentially dependent on
m1and uniquely determined by
respectively. Then,
(i) the semi-trivial steady state is locally asymptotically stable if , neutrally stable if , and unstable if.
(ii) the semi-trivial steady state is locally asymptotically stable if , neutrally stable if , and unstable if .
Proof(i) By Lemma 4.2(i), we conclude that for any given, there exists a unique such thatTherefore, is locally asymptotically stable for , neutrally stable for , and unstable for .(ii) Similarly, we can conclude from Lemma 4.2(ii) that for any given, there exists a unique such thatTherefore, is unstable for , neutrally stable for , and locally asymptotically stable for . The proof is completed.Combining with Lemma 3.4 and Lemma 4.3, we obtain the following results.
Suppose
. For fixed
d,
k1,
k2,
β1,
β2> 0
,and
are defined by Eq.
4.15, respectively, and the following statements hold.
(i) Suppose . If , then is locally asymptotically stable, and is unstable if it exists. If , then is unstable, and is locally asymptotically stable. If , then and are both unstable, and system (2.2) admits at least one stable coexistence steady state.
(ii) Suppose . If , then is locally asymptotically stable, and is unstable. If , then is unstable, and is locally asymptotically stable. If , then and are both stable, and system (2.2) admits at least one unstable coexistence steady state.
(iii) Suppose . If , then is locally asymptotically stable, and is unstable. If , then is unstable, and is locally asymptotically stable.
For fixed d, k1, k2, β1, β2 > 0, Lemma 4.3 and Theorem 4.2 imply that there exist two critical curves Γ1 and Γ2 in the m1 − m2 plane, which divide the local dynamics of Eq. 2.2 into competitive exclusion, bi-stability, and coexistence. To further characterize classification on the dynamics of system (2.2) in the m1 − m2 plane, we next give some properties of critical curves Γ1 and Γ2. We recall
Clearly, depends on m1 and β1 and depends on m1 and β2. To emphasize these parameters, we denote and by and , respectively.
Proposition 4.2
We consider
d,
k1,
k2> 0 fixed. The critical curve
has the following properties.
(i) For any β1 > 0 given, is strictly increasing with respect to m1 in . Moreover,
(ii) For any given, is strictly decreasing with respect to β1 in (0, + ∞) and
Particularly,
(iii) For any given,
where
is the derivative of
with respect to
m1in
.
Proof(i) For any β1 > 0 given, we can conclude from Lemma 4.2(i.1) (i.3) that is strictly decreasing with respect to m1 in and strictly increasing with respect to m2 in . Then, it follows from and the implicit function theorem that is strictly increasing with respect to m1 in . Since , we can obtain based on and uniformly on [0,1] (see Eq. 3.5).(ii) For any given, it follows from Lemma 4.2(i.2) (i.3) that is strictly increasing with respect to β1 in (0, + ∞) and strictly increasing with respect to m2 in . Then, we can obtain from and the implicit function theorem that is strictly decreasing with respect to β1 in (0, + ∞). Then,Substituting β1 → 0+ in , we have by Lemma 3.1(ii). Here, u0 satisfiesThen, we can deduce from [26], Theorem 2.1 that . Substituting β1 → +∞ in , we have based on and uniformly on [0,1] (see (3.6)). The estimate of in Eq. 4.16 is obtained.Finally, when , Theorem 4.1(iii.1) shows that is globally asymptotically stable provided . Therefore, provided . Combining with Eq. 4.16, 4.17 is obtained.(iii) Let ψ1 > 0 with ‖ψ1‖∞ = 1 be the corresponding principal eigenfunction of . Then, ψ1 satisfiesBy differentiating Eq. 4.19 with respect to m1, denoting , we havewhere is the derivative of with respect to m1 in . Multiplying Eq. 4.19 by and Eq. 4.20 by ψ1, integrating over (0,1), and then, subtracting the two resulting equations,We next show that . We choose an increasing sequence {β1,n} with . Then, is the unique positive solution ofwhere . Note that and is uniformly bounded on [0,1]. By Lp estimates for p ∈ (1, + ∞) and the Sobolev embedding theorem, we may assume that in C1[0, 1] by passing to a subsequence if necessary. Since is continuously differentiable with respect to m1 in (see Lemma 3.3(i)), we differentiate (4.22) with respect to m1, denote , and obtainSince , , and are uniformly bounded on [0,1], we can use Lp estimates for p ∈ (1, + ∞) and the Sobolev embedding theorem again to assume that in C1[0, 1] by passing to a subsequence if necessary. Therefore, based on uniformly on [0,1] (see Lemma 3.3(ii)). Finally, taking β1 → +∞ in (4.21) and noting that (see (4.16)), we have . The proof is completed.
Proposition 4.3
,
k1,
k2> 0
fixed. The critical curvehas the following properties:(i) For anyβ2 > 0 given,is strictly increasing with respect tom1in. Moreover,
(ii) For anygiven,is strictly increasing with respect toβ2in (0, + ∞) and
Particularly,
(iii) For any given,
Proof(i) For any β2 > 0 given, we can conclude from Lemma 4.2(ii.1) (ii.3) that is strictly increasing with respect to m1 in and strictly decreasing with respect to m2 in . Then, is strictly increasing with respect to m1 in by and the implicit function theorem. Substituting in , we have based on . Therefore, by uniformly on [0,1] (Eq. 3.5).(ii) For any given, it follows from Lemma 4.2(ii.2) (ii.3) that is strictly increasing with respect to β2 in (0, + ∞) and strictly decreasing with respect to m2 in . Then, we can conclude from and the implicit function theorem that is strictly increasing with respect to β2 in (0, + ∞). Therefore,Similarly, substituting β2 → 0+ in , we can deduce from [8], Theorem 2.1 again that . Moreover, . Hence, the inequality about in Eq. 4.24 is obtained.When , , Theorem 4.1(ii.1) implies that is globally asymptotically stable provided . Therefore, provided . Combining with 4.24, 4.25 holds.(iii) Let φ1 with ‖φ1‖∞ = 1 be the corresponding principal eigenfunction of . Then, φ1 satisfiesBy differentiating Eq. 4.27 with respect to m1 and denoting , we havewhere is the derivative of with respect to m2 at . Multiplying (4.27) by and (4.28) by φ1, integrating over (0,1), and then, subtracting the two resulting equations,Similar to Proposition 4.2(iii), we can show based on uniformly on [0,1] (see Lemma 3.3(ii)). Substituting β2 → +∞ in (4.29), it is easy to obtain .We assume k1 > k2 > 0 without loss of generality. Then, there exist six critical curves: ,in the m1 − m2 plane (Figure 2), which classify the dynamics of system (2.2) into extinction of both species, competitive exclusion and coexistence. Clearly, it follows from Proposition 4.1(i) that line is located above line L1 and below line L2 under the assumption k1 > k2 > 0. Propositions 4.2(i) and 4.3(i) suggest that Γ1 and Γ2 are increasing with respect to , respectively. Moreover, , which implies that Γ1 and Γ2 intersect at point . In addition, due to the effect of β1 and β2, Propositions 4.2(ii) (iii) and 4.3(ii) (iii) indicate that the locations of Γ1 and Γ2 in the m1 − m2 plane have the following four occasions shown in Figures 2A–D. Here, we assume that Γ2 is always located above Γ1 in this region (Figure 2). We set Now, we are ready to illustrate the dynamical classification of system (2.2) in the m1 − m2 plane under the assumption k1 > k2 > 0, by dividing the following four cases.Case I: , (Figure 2A). Then, (4.17) holds provided ; that is, under the assumption k1 > k2 > 0. Similarly, (4.25) holds provided , which means . These suggest that both Γ1 and Γ2 lie between lines L1 and L2 (Figure 2A). It follows from Theorem 4.1(i.1) that (0,0) is g.a.s in region (m1, m2) ∈ Π0. In particular, the phase portrait graph of system (2.2) with (m1, m2) = (0.2, 0.1) ∈ Π0 is illustrated in Figure 1A, which shows that (0,0) is g.a.s in this case. Then, is g.a.s when (m1, m2) ∈ Π1 ∪ Π3 by Theorem 4.1(i.3), (iii.1). Particularly, the phase portrait graph of system (2.2) with (m1, m2) = (1, 0.1) ∈ Π1 ∪ Π3 is shown in Figure 1B. is g.a.s in region (m1, m2) ∈ Π2 ∪ Π4 by Theorem 4.1(i.2) and (ii.1). Moreover, the specific phase portrait graph with (m1, m2) = (0.2, 1) ∈ Π2 ∪ Π4 is displayed in Figure 1C. Furthermore, by Lemma 4.3, is locally asymptotically stable in region Π5 and unstable in region Π6 ∪ Π7; is locally asymptotically stable in region Π6 and unstable in region Π5 ∪ Π7. Then, we can conclude from Theorem 4.2(i) that there exist stable coexistence steady states in Π7, and the specific phase portrait graph with (m1, m2) = (1, 0.545) ∈ Π7 is presented in Figure 1D.Case II: , (Figure 2B). Then, (4.17) holds and Γ1 still lies between lines L1 and L2 when . Moreover, (4.24) holds when ; that is, under the assumption k1 > k2 > 0. This implies that Γ2 lies above lines L1 and . Note that by Proposition 4.3(i) and if k1 > k2 > 0 (see Proposition 4.1). Then, when m1 is near , which means that there exists sufficiently small ϵ > 0 such that Γ2 lies below line L2 for .Next, we illustrate two-fold that Γ2 will first intersect L2 and then lie above L2 as m1 increases (Figure 2B). On one hand, Theorem 4.1(ii) indicates that when and , there exists a unique such that is unstable when . This means that there exist regions above L2 such that is unstable when . Note that Γ2 is strictly increasing with respect to β2 in (0, + ∞) by Proposition 4.3(ii). Hence, when , we can conclude from Lemma 4.3(ii) that the critical curve Γ2 for the stability change of must intersect L2 and then lie above L2 as m1 is suitably large. This implies that there exist regions above line L2 and below Γ2 such that is unstable when . On the other hand, Proposition 4.3(iii) implies , which means that the slope of Γ2 will be bigger than 1 for suitably large β2. This, in turn, suggests that for large β2, there exists such that Γ2 must intersect L2 and then locates above L2 for all . Therefore, the region occurs for large β2. This region is denoted as in Figure 2B. Then, we deduce from Theorem 4.2(i) that there exist stable coexistence steady states in . The other regions Π0 − Π6 can be similarly defined as in Case I.Case III: , (see Figure 2C). Similar to Case II, (4.25) holds provided ; that is, line Γ2 lies between lines L1 and L2 when . Moreover, (4.16) holds when ; that is, under the assumption k1 > k2 > 0. This implies that Γ1 lies below line L2 and above line . Furthermore, note that by Proposition 4.2(i) and if k1 > k2 > 0 by Proposition 4.1. Then, , when m1 is near , which means that there exists sufficiently small ɛ > 0 such that Γ1 lies above L1 for .Similarly, we next illustrate that Γ1 will first intersect L1 and then lie below L1 as m1 increases (see Figure 2C). On one hand, Theorem 4.1(iii) suggests that when and , there exists a unique such that is unstable when . This means that there exist regions below L1 such that is unstable when . Note that Γ1 is strictly decreasing with respect to β1 in (0, + ∞) by Proposition 4.2(ii). Hence, when , we can deduce from Lemma 4.3(i) that the critical curve Γ1 for the stability change of must intersect L1 and then lie below L1 as m1 is suitably large. This implies that there exist regions below L1 and above Γ1 such that is unstable when .On the other hand, Proposition 4.2(iii) gives , which implies that the slope of Γ1 will be smaller than for suitably large β1. This, in turn, suggests that, for large β1, there exists such that Γ1 must intersect L1 and then lie below L1 for all . Therefore, the region occurs for large β1. This region is denoted as in Figure 2C. Then, we obtain from Theorem 4.2(i) that there exist stable coexistence steady states in . The other regions Π0 − Π6 can be similarly illustrated as in Case I.Case IV: , (Figure 2D). Combining the analysis of in Case II and in Case III, we know that both the regions and exist. It follows from Theorem 4.2(i) that there exist stable coexistence steady states in . The other regions Π0 − Π6 can be similarly illustrated as in Case I.We next make a comparison with the results in [8]. When the intraspecific competition is relatively weak (i.e., for the case of and ), the competitive dynamics of system (2.2) (Figure 2A) are similar to the unstirred chemostat model with Holling type II functional response (see [8], Theorems 2.1, 2.4] and Figure 1 in [8]), which suggests that the weak intraspecific competition has little effect on the competition outcomes of species.However, when the intraspecific competition becomes strong, some new phenomena may occur. For instance, for the standard unstirred chemostat models, competition exclusion always happens for the weak–strong competition of two species (see [26], Theorem 2.1), while coexistence may occur in the unstirred chemostat model with B.–D. functional response with the increase of intraspecific competition, under the weak–strong competition cases (see Theorem 4.1 and Figures 2B–D). More precisely, under the weak–strong competition case (i.e., species v has a stronger growth ability compared to species u), the competitive ability of species v becomes weak with the increase of β2, which may result in the coexistence of the two species (see Theorem 4.1(ii.2) and Figure 2B). Similarly, under the weak–strong competition case (i.e., species u has a stronger growth ability compared to species v), the competitive ability of species u becomes weak with the increase of β1, which may cause the coexistence of the two species (see Theorem 4.1(iii.2) and Figure 2C). In particular, Theorem 4.1(ii.2), (iii.2) and Figure 2D suggest that coexistence is more likely to happen, when the intraspecific competition of two species is strong.In summary, these theoretical results indicate that for the weak–strong competition cases, if the intraspecific competition parameter of the species with stronger growth ability is suitably large, we can observe different results from [8] that coexistence may occur. This new phenomenon suggests that the intraspecific competition parameters β1 and β2 have a great influence on the competitive outcomes of two species.
FIGURE 1
FIGURE 2
5 Positive solution branches of system (2.2)
We define X = W2,p(0, 1) × W2,p(0, 1) and Y = Lp(0, 1) × Lp(0, 1), where p > 1. For fixed d, k1, k2, β1, β2 > 0 and , system (2.2) admits three branches of trivial or semi-trivial solutions in the space : , where and . In this section, we will regard m2 as the bifurcation parameter and study separately positive solutions bifurcating from the semi-trivial branches Γu and Γv by the Crandall–Rabinowitz bifurcation theorem in [31].
We first show that there exists a positive solution branch that bifurcates from the semi-trivial solution . Moreover, the bifurcation of positive solutions from can only occur at by Lemma 4.3.
For fixed d, k1, k2, β1, β2 > 0 and , there is a smooth non-constant solutions curve Γ1 = {(m2(s), u(s), v(s)): s ∈ (−ϵ, ϵ)} such that (m2(s), u(s), v(s)) is a positive solution of system (2.2) for s ∈ (0, ϵ) and satisfies , , and v(s) = sψ0 + o(s). Here, ψ0 > 0 is the principal eigenfunction corresponding to the eigenvalue, which satisfiesand φ0 < 0 satisfies
Moreover,where and .
Theorem 5.1. can be proved by the similar arguments as in [35], Theorem 6.2. For completeness, we defer the proof of Theorem 5.1 to the Supplementary Appendix.
For fixed d, k1, k2, β1, β2 > 0 and , there is a smooth non-constant solutions curve Γ2 = {(m2(s), u(s), v(s)): s ∈ (−ϵ, ϵ)} such that (m2(s), u(s), v(s)) is a positive solution of (2.2) for s ∈ (0, ϵ) and satisfies ,, and. Here, is the principal eigenfunction corresponding to, which satisfiesandsatisfies
Moreover,where , , and is the derivative of with respect to m2 at .
The proof of Theorem 5.2 is similar to the arguments in Theorem 5.1, and we omit it here.
We define . The following results show that the two bifurcation continua Γ1 and Γ2 in Theorems 5.1 and 5.2 are connected.
We consider d, k1, k2, β1, β2 > 0 and fixed. Then, there exists a connected component Γ of Ω, which bifurcates from the semi-trivial solution branch Γu at and meets the other semi-trivial solution branch Γv at . In particular, system (2.3) admits a positive solution (u, v) if m2 lies between and .
The proof is motivated by the methods in [13], Theorem 6.4 (see also [8], Theorem 4.10]). For readability, the proof is given in Supplementary Appendix.
6 Numerical descriptions
In this section, we will study the effect of diffusion rates d on the dynamics of system (2.2). It follows from Eqs 4.1, 4.15 that the threshold values , , , and depend on the diffusion rates d. Since these threshold values are dependent on d and they are non-monotone in d, we cannot theoretically establish the threshold dynamics of system (2.2) in terms of d. In order to explore the effect of d on the dynamics of system (2.2), in this subsection, we resort to numerical approaches. We fix L = 1, S0 = 1, γ = 0.5, k1 = 1, k2 = 0.4, and m1 = 1 as mentioned before. Then, by presenting the bifurcation diagrams of positive equilibrium solution of system (2.2) with the bifurcation parameter d increasing, the results are divided into the following three cases.
Case I: . We take m2 = 2 such that , which means that species v has a stronger growth ability compared to species u. We can call this case the weak–strong competition [13]. To identify the effect of intraspecific competition, we fix β1 = 0.01 and let β2 change from β2 = 0.01 to β2 = 1.
First, if β1 = 0.01, β2 = 0.01, there is no coexistence and the competitive exclusion principle holds (species v with a stronger growth ability will win the competition) when d is sufficiently small (Figure 3A). As d increases, both species go extinct, which is consistent with our biological intuition that the sufficiently large diffusion rates will put species at a disadvantage. These numerical observations in Figure 3A coincide with [26], Theorem 5.4.
FIGURE 3
Second, if β1 = 0.01, β2 = 1. Clearly, under this weak–strong competition case, though species v has stronger growth ability compared to species u, the increase of β2 makes the competitive ability of v weaker. This is consistent with our biological intuition that the stronger intraspecific competition will put species at a disadvantage. Moreover, coexistence may occur when d is sufficiently small (Figure 3B), which is different from [8], Theorem 5.4. As d increases, species v wins the competition. As d further increases, the sufficiently large diffusion rates drive both species to extinction.
Case II: . We take m2 = 0.2 such that , which implies that species u has a stronger growth ability compared to species v. Similarly, we call this case the weak–strong competition case [13]. We fix β2 = 0.01 and let β1 change from β1 = 0.01 to β1 = 1. Similar to Case I, when β1 = 0.01, β2 = 0.01, results similar to those in Figure 3A are shown in Figure 4A. When β1 = 1, β2 = 0.01, the increase of β1 makes the competitive ability of u weaker. Then, we can observe that coexistence may occur when d is sufficiently small (see Figure 4B), which is quite different from [26], Theorem 5.4.
FIGURE 4
Case III: . We take m2 = 0.6 such that . We call this case the evenly matched competition [13]. Moreover, the fact of suggests that though the growth ability of the two species is evenly matched, the competitive ability of species v is still slightly better than that of species u. Then, for this evenly matched competition case, we fix β1 = 0.01 and let β2 change from β2 = 0.01 to β2 = 1.
For β1 = 0.01, β2 = 0.01, as shown in Figure 5A, the diffusion rates have a significant effect on the dynamics of system (2.2). More precisely, the dynamics of system (2.2) shift between four scenarios with the bifurcation parameter d increasing; that is, 1) competitive exclusion occurs and species v wins the competition, when d is sufficiently small; 2) coexistence occurs as d increases; 3) competitive exclusion occurs again and species u wins the competition, as d further increases; and 4) both species are washed out as d continues to increase. These suggest that system (2.2) may show a trade-off among extinction, exclusion, and coexistence as d increases. Particularly, coexistence occurs at the intermediate diffusion rates, which is in line with the theoretical results in [8].
FIGURE 5
For β1 = 0.01, β2 = 1, as stated before, the increase of β2 will make the competitive ability of v weaker. Then, we can observe from Figure 5B that both species can coexist when d is sufficiently small. As d increases, competitive exclusion happens and species u wins the competition. As d further increases, the large diffusion rates drive two species to extinction.
In shorts, for different competition Cases (I)–(III), we investigate the effect of diffusion on the dynamics of system (2.2) by taking different intraspecific competition parameters β1, β2. As shown in Figures 3–5, the impacts of diffusion and intraspecific competition on the competitive outcomes of species are complex, which further suggests that diffusion and intraspecific competition play a key role in determining the dynamics of system (2.2).
7 Discussion
In this paper, we investigate an unstirred chemostat model with the Beddington–DeAngelis functional response (see system (2.2)). The analytical and numerical results show that the intraspecific competition and diffusion have an important biological effect on the dynamics of system (2.2).
Theoretically, we first adopt a basic strategy regarding the growth rates as variable/bifurcation parameters to study the effect of growth rates on system (2.2). The results show that there exist six critical curvesin the m1 − m2 plane, which may classify the dynamics of system (2.2) into extinction of both species, competitive exclusion and coexistence (see Theorems 4.1, 4.2). To further understand the effect of βi, (i = 1, 2) on the dynamics of (2.2), we explore the properties of critical curves Γ1 and Γ2 (see Propositions 4.2 and 4.3) and get a relatively clear dynamics classification of system (2.2) in the m1 − m2 plane (Figure 2).
Numerically, since diffusion plays a key role in determining the competition outcomes of two species, we study the effect of diffusion on the dynamics of system (2.2). More precisely, for two weak–strong competition cases, due to the effect of intraspecific competition parameters β1 and β2, the coexistence may occur at sufficiently small diffusion rates (Figures 3B, 4B), while for the evenly matched competition case, the dynamics of system (2.2) shift between different scenarios (competitive exclusion, coexistence, and extinction) when β1 and β2 are small and the coexistence only occurs at the intermediate diffusion rates (Figure 5A). When β2 is larger than β1, we observe from Figure 5B that coexistence may occur at sufficiently small diffusion rates.
In conclusion, in this paper, the dynamics classification of system (2.2) in the m1 − m2 plane is established by the linear eigenvalue theory and the monotone dynamical system theory (Figure 2). Due to the effect of intraspecific competition parameters β1 and β2, the dynamics of system (2.2) are more complex than that of the unstirred chemostat model with Holling type II functional response (see Figure 1 of [8]). Numerically, we study the effect of diffusion on system (2.2) and obtain rich numerical results (Figures 3–5). These numerical observations reveal that, under different competition cases, the effects of diffusion and intraspecific competition on the dynamics of system (2.2) are complex. This, in turn, suggests that the B.–D. functional response is more biologically realistic and superior to the well-known Holling type II functional response in modeling the resource uptake of species.
Statements
Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.
Author contributions
The theory part and simulation were obtained by the first author WZ. The second and third authors HN and ZW guided the work. All authors contributed to the article and approved the submitted version.
Funding
This work was supported by the National Natural Science Foundation of China (12071270 and 12171296) and the Natural Science Basic Research Program of Shaanxi (No. 2023-JC-JQ-03).
Acknowledgments
The authors are very grateful to the referees and the handling co editor-in-chief for their kind and valuable suggestions leading to a substantial improvement of the manuscript.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
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Supplementary material
The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphy.2023.1205571/full#supplementary-material
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Summary
Keywords
unstirred chemostat model, coexistence, competitive exclusion, bifurcation, numerical simulation
Citation
Zhang W, Nie H and Wang Z (2023) Dynamics of an unstirred chemostat model with Beddington–DeAngelis functional response. Front. Phys. 11:1205571. doi: 10.3389/fphy.2023.1205571
Received
14 April 2023
Accepted
06 July 2023
Published
27 July 2023
Volume
11 - 2023
Edited by
Xiaoming Zheng, Central Michigan University, United States
Reviewed by
Xueyong Zhou, Xinyang Normal University, China
Raid Naji, University of Baghdad, Iraq
Updates
Copyright
© 2023 Zhang, Nie and Wang.
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*Correspondence: Zhiguo Wang, zgwang@snnu.edu.cn
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.