A COMPETITION MODEL WITH DYNAMICALLY ALLOCATED TOXIN PRODUCTION IN THE UNSTIRRED CHEMOSTAT

This paper deals with a competition model with dynamically allocated toxin production in the unstirred chemostat. First, the existence and uniqueness of positive steady state solutions of the single population model is attained by the general maximum principle, spectral analysis and degree theory. Second, the existence of positive equilibria of the two-species system is investigated by the degree theory, and the structure and stability of nonnegative equilibria of the two-species system are established by the bifurcation theory. The results show that stable coexistence solution can occur with dynamic toxin production, which cannot occur with constant toxin production. Biologically speaking, it implies that dynamically allocated toxin production is sufficiently effective in the occurrence of coexisting. Finally, numerical results illustrate that a wide variety of dynamical behaviors can be achieved for the system with dynamic toxin production, including competition exclusion, bistable attractors, stable positive equilibria and stable limit cycles, which complement the analytic results.

1. Introduction.The chemostat is a basic resource-based model for competition in an open system and a standard model for the laboratory bio-reactor, which plays an important role in the study of population dynamics and species interactions (see, e.g., [14,27]).
The study on the problem of the influence of toxicants both on the growth of one population and on the competition of two species for a critical nutrient has received considerable attention in the past decades (see, e.g., [1,2,5,11,13,21,22,18,19,20,28,30,31]).Particularly, there has been a lot of interest in the so called allelopathic competitions between species (see, e.g., [2,5,16,21,22,18,24,28]).Allelopathy can be defined as the direct or indirect harmful effect of one species on another by releasing a chemical compound into the surrounding environment [25].Allelopathic competition occurs between algal species [16], algae and bacteria [28], bacteria and bacteria [3], algae and aquatic plants [24] as well as plants and plants [2].Several experimental results concerning bacterial competition show that the production of allelopathic chemical compound depends on the concentrations of populations through a quorum sensing mechanism [3,12].As a consequence, a general mathematical model was first proposed in [5] to model such a mechanism.In [5], the basic assumption is that the chemostat is well-stirred and the weaker competitor can devote some of its resources to the dynamically allocated production of an allelopathic agent (which is also called anti-competitor toxin or just toxin).Dynamically allocated production implies that the effort devoted to toxin production can be adjusted to reflect the state of the competition.For instance, if there is no competition, there is no resource devoted to the toxin production.The numerical examples in [5] show that some new interesting dynamical behaviors occur, including stable interior rest points and stable limit cycles, in contrast to the model with constant toxin production.This suggests a possible mechanism for coexistence.Rigorous mathematical analysis of allelopathic competition models with quorum sensing in the well-stirred chemostat-like environment can be found, for example, in [1,11,13].
The production of anti-competitor toxins is of interest when the weaker competitor can produce toxins against its competitors.The introduction of the function K(u, v) is based on the assumption that the effort devoted to toxin production can be dynamically allocated as a function of the state of the system, which reflects the mechanism of quorum sensing (see [3]).In [15], Hsu and Waltman assumed K(u, v) ≡ k(contant) and studied the competition in the well-stirred chemostat when the weaker competitor produces toxins.Considering spatial heterogeneity, the system (1)- (3) with K(u, v) ≡ k(contant) was investigated in [18].The results in [15,18] indicate that coexistence cannot occur when the effort devoted to toxin production is constant even if taking into account diffusion.
The focus of this study is to investigate the dynamical behavior of the system (1)-(3) in combination with the effects of dynamically allocated toxin production and diffusion, and to explain the coexistence of two species in competition on a single resource in the unstirred chemostat.To this end, we assume that the function K(u, v) satisfies the hypotheses (H1) : K(u, v) is C 1 continuous in R + × R + , where R + = [0, +∞); (H2) : 0 ≤ K(u, v) < 1 for any u, v ∈ R + ; (H3) : K(0, 0) = 0, K(u, v) > 0 for u > 0, v > 0, and K v (0, v) ≥ 0 for any v ∈ R + .
As mentioned before, we concentrate on coexistence solutions (i.e.stable positive solutions) of the following steady state system x ∈ (0, 1), with boundary conditions The main technical difficulties in our analysis come from the basic assumption that the weaker competitor can devote some of its resources to the dynamically allocated production of anti-competitor toxins.Consequently, the usual reduction of the system to a competitive system of one order lower through the conservation of nutrient principle is lost.Thus the system with toxin production is non-monotone, and the single population model can't be reduced to a scalar system.Hence, it is hard to study the uniqueness and stability of the semitrivial nonnegative equilibria.
The main goal of Section 2 is to study the uniqueness and some properties of single population equilibrium by the general maximum principle, spectral analysis and degree theory.The main results are given by Theorems 2.1 and 2.2.Since the single population model ( 9) can't be reduced to a scalar system, it is much more difficult to prove Theorem 2.2 than Theorem 2.1.The crucial point of proving Theorem 2.2 is to establish Lemma 2.4, which indicates that any positive solution of ( 9) is nondegenerative and has index 1.In Section 3, the existence of positive solutions of the steady state system ( 6)-( 7) is investigated by the degree theory.The structure and stability of the nonnegative solutions of ( 6)-( 7) is established by the bifurcation theory in Section 4. Lemma 2.4 and Remark 2.1 also play a key role in verifying the main outcomes (see Theorems 3.1, 4.2 and 4.3).It turns out that stable coexistence solutions can occur with dynamic toxin production, which cannot occur with constant toxin production.Biologically speaking, it implies that dynamically allocated toxin production is sufficiently effective in the occurrence of coexisting.Finally, some numerical results illustrate the existence of coexistence solutions, bi-stable attractors or stable limit cycles, which complement the analytic results.
2. Uniqueness of single population equilibria.The goal of this section is to determine the properties of single population equilibria of (4)- (5).Mathematically, this means that u or v is set to zero in the system (4)- (5), or equivalently, the initial data u 0 (x) ≡ 0 or v 0 (x) ≡ 0, respectively.Hence, we obtain the following reduced boundary value problems To work out the properties of the solutions of the reduced boundary value problems ( 8) and ( 9), we introduce λ 1 , σ 1 as the principal eigenvalues of the problems respectively, with the associated eigenfunctions φ 1 , ψ 1 > 0 on [0, 1], normalized with max For the reduced boundary value problems (8), it is easy to see that p ≡ 0 on [0, 1], and (S, u) satisfies It follows from Theorem 2.1 in [20] that 0 is the unique nonnegative solution of ( 12) if a ≤ λ 1 , and there exists a unique positive solution of ( 12) if a > λ 1 , which is denoted by θ a .Therefore, (z, 0, 0) is the unique nonnegative solution of (8) if a ≤ λ 1 , and there exists a unique positive solution (z − θ a , θ a , 0) if a > λ 1 .Furthermore, we have the following results.
Theorem 2.1.If a ≤ λ 1 , then (z, 0, 0) is the unique nonnegative solution of the single population model (8); if a > λ 1 , then (8) has a unique positive solution (z − θ a , θ a , 0).Moreover, θ a satisfies the following properties: , and is pointwisely increasing when a increases; = 0} and all eigenvalues of L a are strictly positive, which implies that L a is a nondegenerate and positive operator in C 2 B [0, 1].Proof.By Lemmas 3.3-3.4 in [32] and Propositions 2.3-2.4 in [20], one can conclude that θ a satisfies the above properties (i)-(iii) and (v).Hence, we only need to show (iv).
Since 0 < θ a < z(x) and θ a is pointwisely increasing with respect to a ∈ (λ 1 , ∞), we only need to show that for any > 0, θ a > (1 − )z(x) provided that a is large enough.To this end, let θ ∈ C ∞ [0, 1], and (1 − )z(x) < θ < (1 − 2 )z(x).Then provided that θ xx is bounded and a is large enough.That is, for any > 0, there exists A( Clearly, z(x) is a super-solution of (12).Hence we have z(x) > θ a > θ > (1 − )z(x) by the super-and sub-solution method and the uniqueness of positive solutions to (12).Letting → 0, we obtain lim Next, we begin to study nonnegative solutions of (9).If K(0, v) ≡ 0, then it is easy to see that p ≡ 0 and S + v ≡ z(x) on [0, 1].Hence, (9) can be reduced into the scalar system By similar arguments as in Theorem 2.1, we can conclude that 0 is the unique nonnegative solution of (13) if b ≤ σ 1 , and there exists a unique positive solution of (13) if b > σ 1 , which is denoted by ϑ b .Moreover, by similar arguments as in Theorem 2.1 again, we have the following similar outcomes.
Lemma 2.2.Suppose (H1) − (H3) hold and K(0, v) ≡ 0. Then if b ≤ σ 1 , then (z, 0, 0) is the unique nonnegative solution of the single population model (9); if b > σ 1 , then (9) has a unique positive solution (z − ϑ b , ϑ b , 0).Moreover, ϑ b satisfies the following properties: , and is pointwisely increasing when b increases; ] and all eigenvalues of L b are strictly positive, which implies that L b is a non-degenerate and positive operator in (9) cannot be reduced into a scalar system, which makes it difficult to study nonnegative solutions of (9).We first consider the decoupled subsystem By similar arguments as in Lemmas 3.1-3.2(see Page 11), we establish the priori estimates for nonnegative solutions of (14).
Second, we claim that φ, ψ have at most finitely many zeros in (0,1) where φ, ψ change sign.Suppose φ(x n ) = 0 for an infinite sequence of distinct points {x n } ⊂ [0, 1], and φ changes sign at any x n .By compactness, we may assume that there is It follows from the first equation of (18) that ψ(x ∞ ) = 0.The maximum principle applied to the first equation of (18) shows that ψ must change sign in any neighborhood of x ∞ .Thus ψ x (x ∞ ) = 0.It follows from the uniqueness of the Cauchy problem associated with ( 18) that (φ, ψ) = (0, 0), which is a contradiction to (φ, ψ) = (0, 0).The same assertion holds for the zeros where ψ changes sign.
Next, assume that ψ(x i ) < 0 and φ < 0 in ( We also have two possibilities: ).The general maximum principle implies that ψ/v 0 cannot reach its nonnegative maximum in (x i , x i+1 ).By virtue of ψ(x i ) < 0, one can conclude that ψ/v 0 cannot reach its nonnegative maximum at x = x i .Assume that max By the strong maximum principle, we obtain φ > 0 in (x i , x i+1 ), a contradiction to The maximum principle applied to (21) shows that ψ < 0 in (x i , x i+1 ).Hence, By the strong maximum principle, we obtain φ > 0 in (x i , x i+1 ), a contradiction to φ < 0 in (x i , x i+1 ).Thus ψ(x i+1 ) > 0.
At last, we focus on the last interval to establish a contradiction.We have two possibility to consider: (i) φ > 0 in (x m , 1); (ii) φ < 0 in (x m , 1).
(i) The case of φ > 0 in (x m , 1).By the above arguments, we have ψ(x m ) > 0. Note that Just as above, if ).The general maximum principle implies that ψ/v 0 cannot reach its non-positive minimum in (x m , 1).By virtue of ψ(x m ) > 0, one can conclude that ψ/v 0 cannot reach its non-positive minimum at x = x m .Then min By the general maximum principle again, we have On the other hand, it is easy to see that ψ v0 x The maximum principle applied to (22) shows that ψ > 0 in (x m , 1).Hence, By the strong maximum principle, we obtain φ < 0 in (x m , 1), a contradiction to φ > 0 in (x m , 1).
Proof.(i) is a direct result of Lemma 2.3.
(ii) We first show there exists a unique positive solution of (14) provided that b > σ 1 .It suffices to show A has a unique positive fixed point in Ω 0 .It follows from Lemma 2.3 that the fixed points of A in Ω 0 are two types, which are the trivial fixed point (0, 0) and the positive fixed points (χ, v).It follows from Lemma 2.5 that any positive fixed point (χ 0 , v 0 ) of A is non-degenerative and index(A, (χ 0 , v 0 ), W 0 ) = 1.Meanwhile, by the compactness argument on the operator A and the non-degeneracy of its fixed points (including (0, 0) and positive fixed points), one knows that there are at most finitely many positive fixed points in Ω 0 .Let them be ( By the additivity property of the fixed point index and Lemma 2.4, we have That is, there exists a unique positive solution of (14) provided that b > σ 1 , which is denoted by (S * , v * ).Let p * be the unique solution to the problem It follows from the strong maximum principle that p * > 0 on [0, 1].Hence, (9) has a unique positive solution (S * , v * , p * ) provided that b > σ 1 .
Proof.(i) It follows from Lemma 3.1 that S, u, v, p > 0 and S < z on [0, 1].By Lemma A.1, we have where (ii) Assume that (S i , u i , v i , p i ) is a positive solution of ( 6)- (7) with a = a i and a i → ∞.Then it follows from the equation . Noting that a i → ∞, one can find that S i → 0 a.e. in (0, 1) as i → ∞.On the other hand, it follows from the equation , which implies b → ∞ since S i → 0 a.e. in (0, 1) as i → ∞.This is a contradiction.Hence, there exists some positive constant Λ 0 such that a < Λ 0 .
Let χ = z − S. Then the steady state system ( 6)-( 7) is equivalent to x ∈ (0, 1), x ∈ (0, 1), Moreover, by Lemma 3.1, (S, u, v, p) is a nonnegative solution of ( 6)-( 7) if and only if (χ, u, v, p) is a nonnegative solution of (24).As mentioned before, nonnegative solutions of ( 24) can be divided into three types: (i) the trivial solution E 0 = (χ, u, v, p) = (0, 0, 0, 0), (ii) the semi-trivial solutions Next, we turn to study positive solutions of (24).To this end, we introduce the spaces It follows from standard elliptic regularity theory that A τ is compact and continuously differentiable.Let A = A 1 .By Lemma 3.1, there exists a nonnegative solution of ( 6)-( 7) (or (24) equivalently) if and only if there exists a fixed point of the operator A in Ω.Moreover, similar arguments as in Lemma 3.1 indicate that A τ has no fixed points on ∂Ω.To figure out whether there exist positive fixed points of A or not, we need to calculate the index of the trivial and semi-trivial fixed points of A firstly.
Let λ1 , σ1 be the principal eigenvalues of the problems respectively, with the corresponding eigenfunctions φ1 , ψ1 > 0 on [0, 1], normalized with max Similarly, Moreover, it follows from Theorem 2.
Proof.(i)-(ii) can be shown by similar arguments as in Lemma 2.4, and we omit it here.
(iii) To calculate index(A, E 1 , W ), we decompose X into Then U is relatively open and bounded.Choosing > 0 small enough, we have Next, we determine the spectral radius r(A 2 (E 1 )| W2 ) of the operator A 2 (E 1 )| W2 .Direct computation leads to x ∈ (0, 1), x ∈ (0, 1), Consider the eigenvalue problem Noting that we can find that the least eigenvalue η 1 < 0 of ( 27) if b > σ1 , and the least eigenvalue η 1 > 0 of ( 27) if b < σ1 .It follows from Lemma A.2 that the spectral radius −1 (M ) < 1, we can conclude that (26) has eigenvalues greater than 1 and 1 is not an eigenvalue of ( 26) corresponding to a positive eigenvector provided that b > σ1 , and ( 26) has no eigenvalues greater than or equal to 1 provided that b < σ1 .Hence, the spectral radius r(A 2 (E 1 )| W2 ) > 1 and 1 is not an eigenvalue of A 2 (E 1 )| W2 corresponding to a positive eigenvector provided that b > σ1 and the spectral radius r(A 2 (E 1 )| W2 ) < 1 provided that b < σ1 .It follows from Lemma A.4 that for > 0 small, By Leray-Schauder degree theory, deg W1 (I − A 1 | W1 , U, 0) = (−1) m , where m is the sum of the multiplicities of all eigenvalues of the Fréchet derivative A 1 (E 1 ) which are greater than one.Consider the eigenvalue problem Noting that the first eigenvalue of the eigenvalue problem is larger than 0. It follows from Lemma A.2 that the spectral radius Hence, (28) has no eigenvalues greater than or equal to 1.If Φ ≡ 0, then it is easy to see that λ < 1.Hence, A 1 (E 1 )| W1 has no eigenvalues greater than or equal to 1.It follows that for > 0 small Moreover, A 2 (χ, 0, v, p) ≡ 0 for (χ, v, p) ∈ U and A 1 (χ, 0, v, p) = (χ, v, p) for (χ, v, p) ∈ ∂U.
Proof.The proof of (i) is standard, and is omitted here.
Let b > σ 1 fixed.Then there exists a positive constant C large enough such that for c ≥ C, the positive solution branch Γ of ( 6)-( 7) is to the right, and it is stable.
In [18], the unstirred chemostat model with constant toxin production (that is, K(u, v) ≡ K 0 (constant)) has been studied.The results show that when the parameter c, which measures the effect of toxins, is large enough, the model only has unstable positive solutions.Moreover, the species v always lose the competition.However, it follows from Theorem 4.4 that the unstirred chemostat model with dynamically allocated toxin production possesses stable positive solutions(i.e.coexistence solutions).From the biological point of view, dynamically allocated toxin production has a positive effect on coexistence of species.
5. Numerical simulations and discussion.In this section, we present some results of our numerical simulations performed with Matlab, which complement the analytic results of the previous sections.
As [5], we consider two special cases that represent the extremes for reasonable functions where α, β are positive constants and chosen so that 0 < K(u, v) < 1 for u, v > 0.
(43) is monotone increasing in v while (44) is monotone increasing in u.These reflect two opposite strategies.For (43), if v is large, it devotes more of its resources to producing the toxin, which guards against invasion.For (44), if u is large, v increases the toxin production since it is already losing the competition and facing extinction, which is a desperation strategy.The advantage of this strategy is that if there is no competition, no resource is wasted on toxin production.The numerical simulations show that a wide variety of dynamical behaviors can be achieved for the system with dynamically allocated toxin production, including competition exclusion, bistable attractors, stable positive equilibria and stable limit cycles.The most interesting numerical results are stable positive equilibria and stable limit cycles, which cannot occur in the system with constant toxin production.Stable positive equilibria and limit cycles provide coexistence, which suggest a possible mechanism to explain coexistence phenomena.In all of our figures except Figure 4(c)(d), the L 1 norms of the solutions (S(•, t), u(•, t), v(•, t), p(•, t)) to ( 4)-( 5) are plotted versus the temporal variable.In Figure 4(c)(d), two positive equilibria of (4)-( 5) are plotted versus the spatial variable.5.1.Numerical results with K(u, v) = αv β+u+v .At first, we choose the basic parameters of the species to be a = 1.17, b = 1.17, k 1 = 0.017, k 2 = 0.025 and ν = 0.6.Namely, we assume that u is the better competitor in the absence of toxins.Taking the parameters d = 0.1, α = 0.2, β = 0.01, and varying the parameter values of c, we observe competitive exclusion independent of initial conditions and competitive exclusion that depends on initial conditions (bistability).
More precisely, for small c, the species u can competitively exclude the species v independent of initial conditions (see Figure 1 depending on their initial conditions (see Figure 1(b)(c)).Increasing c eventually causes the species v can competitively exclude the species u independent of initial conditions (see Figure 1(d)).Biologically speaking, the numerical results show that toxins can help the weaker competitor to win in the competition.Secondly, we assume that v is the better competitor in the absence of toxins and take the basic parameters of the species to be a = 1.17, b = 1.17, k 1 = 1.7, k 2 = 0.025 and ν = 0.6.Taking the parameters α = 0.8, β = 0.001, c = 0.2, and varying the diffusion rate d, we observe competitive exclusion independent of initial conditions, stable positive equilibria and stable limit cycles.
More precisely, for small d, the species v can competitively exclude the species u independent of initial conditions (see Figure 2(a)).Increasing the diffusion rate d can destabilize the system and cause it to switch to a stable limit cycle.Moreover, the amplitude decreases when d increases(see Figure 2(b)(c)(d)).If one continues to increase the diffusion rate d, the system generates a stable positive equilibrium (see Figure 2(e)).Stable positive equilibria and stable limit cycles provide coexistence, which can be called diffusion-driven coexistence.Increasing d eventually causes the system to converge to the washout solution.That is, all species including two competitors u, v and the toxin p go to zero eventually (see Figure 2(f)).As mentioned before, diffusion-driven coexistence can not occur when K(u, v) is constant.Hence our numerical results imply that dynamically allocated toxin production is sufficiently effective in the occurrence of coexisting.More precisely, for small c, the species u can competitively exclude the species v independent of initial conditions (see Figure 3   where γ(x) ∈ C(∂Ω) and γ(x) ≥ 0. Then all eigenvalues of (A.1) can be listed in order 0 < λ 1 (c, q) < λ 2 (c, q) ≤ λ 3 (c, q) ≤ • • • → ∞, and λ 1 (c, q) = inf ϕ Ω (|∇ϕ| 2 + c(x)ϕ 2 )dx + ∂Ω γ(x)ϕ 2 ds Ω q(x)ϕ 2 dx is a simple eigenvalue with the associated eigenfunction ϕ 1 > 0 on Ω, which is called the principal eigenvalue.Moreover, λ i (c, q)(i = 1, 2, • • • ) is continuous with respect to c and q, and the following comparison principles hold: (i) λ i (c 1 , q) ≤ λ i (c 2 , q) if c 1 ≤ c 2 on Ω and the strict inequality holds if c 1 ≡ c 2 , (ii) λ i (c, q 1 ) ≥ λ i (c, q 2 ) if q 1 ≤ q 2 on Ω and the strict inequality holds if q 1 ≡ q 2 .

ψ1 = 1 .
It follows from Lemma 3.1 and Lemma A.1 that

Figure 3 .
Figure 3.We further fix d = 0.1, α = 0.2, β = 1, and observe the effects of the parameter c: coexistence in the form of equilibria is observed in (b)(c) when c = 0.2, 0.3 respectively; competitive exclusion occurs in (a)(d) when c = 0.01, 0.6 respectively.