COMPETITION BETWEEN TWO SIMILAR SPECIES IN THE UNSTIRRED CHEMOSTAT

This paper deals with the competition between two similar species in the unstirred chemostat. Due to the strict competition of the unstirred chemostat model, the global dynamics of the system is attained by analyzing the equilibria and their stability. It turns out that the dynamics of the system essentially depends upon certain function of the growth rate. Moreover, one of the semi-trivial stationary solutions or the unique coexistence steady state is a global attractor under certain conditions. Biologically, the results indicate that it is possible for the mutant to force the extinction of resident species or to coexist with it.

This basic unstirred chemostat model has received considerable attention.The existence of positive steady state solutions is investigated in [2] by degree theory.The structure and local stability of the steady state solutions are studied in [17,19,1] by bifurcation theory.Some dynamical behaviors are considered in [6,5] by the theory of uniform persistence.However, many crucial problems still remain open.In particular, it is very difficult to determine the uniqueness and stability of positive steady-state solutions.In fact, numerical computations in [17] strongly suggest that (3) has a unique positive steady state solution, and it is globally asymptotically stable under certain conditions.But no rigorous proof has been available.In [10,3], partial results are obtained, which show that (3) has a unique positive steady-state solution when the maximal growth rates a, b are near the principal eigenvalues λ 1 , σ 1 , respectively, and the unique positive steady state solution is globally asymptotically stable under certain conditions (see [10]).
The goal of this paper is to study the uniqueness and stability of coexistence solutions of (3) and its global dynamics.It follows from Theorem 4.1 in [19] that if k 1 = k 2 = k and a > λ 0 , then (3) possesses positive steady-state solutions if and only if a = b.Moreover, (u s , v s ) = (sθ(•, a), (1 − s)θ(•, a)), 0 < s < 1 is a family of coexistence solutions in this case.Here λ 0 is the principal eigenvalue of the linear problem and θ(•, a) is the unique positive solution (see Lemma 3.2 in [19]) Our goal here is to determine how the structure of coexistence states changes under small perturbations.With this in mind, we can rewrite k 1 = k, k 2 = k+τ, b = a+βτ and consider the following perturbed version of (3): where τ > 0 is a small parameter.As mentioned earlier, if τ = 0, we observe that the two species play an identical role, and system (6) can be reduced to the single species case, i.e., (6) possesses a family of coexistence states (sθ(•, a), (1 − s)θ(•, a)), 0 < s < 1, which attracts all solutions of system (6) with nonnegative, nontrivial initial data.An interesting problem is to find out what happens when the two species are slightly different, that is, τ > 0 is small.Before stating our main results, we start by recalling some well-known results on the one-species problem (5).
Remark 2. Numerical computations suggest that the integral I A can be negative or positive.For example, we take Ω = (0, 1) and γ(x) ≡ 1 on [0,1] and choose the parameters as follows: a = 1.2, k = 0.8, β = 0.5690, we obtain that G(1.2) ≈ 0 and I A = −0.7361.Changing the parameters to a = 2.5, k = 0.8, β = 2.1260525, we get G(2.5) ≈ 0 and I A = 37.7032.We suspect that if I A is negative, it may occur that both semi-trivial steady states are locally stable and the coexistence steady state is unstable.Hence the hypothesis I A ≥ 0 is reasonable.
Remark 3. From the biological perspective, the perturbed system ( 6) is motivated by the following considerations.Suppose that random mutation produces another phenotype of species which is slightly different from the resident species.
For instance, the mutant has slightly different maximal growth rates and Michaelis-Menten constants.Two similar species might have to compete for the same limited resources.Two interesting questions arise in the study of the perturbed system (6): One is whether the mutant can invade when its initial population size is small.The other is that if the mutant does invade, whether it will drive the resident species to extinction or coexist with it.Theorem 1.2 indicates that global dynamics of the system (6) essentially depends upon the function G(a) of the growth rate.Moreover, it is possible for the mutant to force the extinction of resident species or to coexist with it.Similar problems have been studied in [7,8] and the references therein to reveal the effects of the spatial heterogeneity of environment on the invasion of the mutant and the coexistence of multiple species in classical Lotka-Volterra systems.
The method of analysis is based on Lyapunov-Schmidt reduction, stability analysis and the following well-known results on the monotone dynamical system.Lemma 1.3.[4,15] For the monotone dynamical system, (i) if there is no coexistence state, then one of the semi-trivial equilibria is unstable and the other one is the global attractor.(ii) if there is a unique coexistence state and it is stable, then it is the global attractor (in particular, both semi-trivial equilibria are unstable).(iii) if all coexistence states are asymptotically stable, then there is at most one of them.
The organization of the paper is as follows: In Section 2, the stability of semitrivial equilibria of ( 6) is investigated.Positive equilibria of ( 6) are constructed by Lyapunov-Schmidt reduction, and their stability is established by spectral analysis subsequently.Finally, Theorem 1.2 is proved by the monotone dynamical theory, which reveals the global dynamics of (6).
2. Stability of semi-trivial equilibria.The aim of this section is to study the stability of semi-trivial equilibria of (6).To this end, we first investigate the semitrivial nonnegative solutions of the steady state system ∆u + auf Clearly, it follows from Lemma 1.1 that (9) has the semi-trivial nonnegative solution (θ(•, a), 0) if a > λ 0 .In order to determine the other semi-trivial nonnegative solution, we introduce λ 0 (τ ) to be the principal eigenvalue of the following problem: By similar arguments as in Lemma 2.3 of [10,13], we have the following results.

∂S
denotes the partial derivative of f (S, k) with respect to S. In view of Lemma 1.1, one can conclude that the stability of (θ, 0) is determined by the principal eigenvalue of the scalar problem Let λ 1 (τ ) be the principal eigenvalue of (11) and ψ 1 (•, τ ) be the corresponding eigenfunction such that ψ 1 > 0 on Ω and max It is easy to see that ψ 1 (•, 0) = θ, λ 1 (0) = 0. Multiplying this equation by θ, and integrating over Ω by parts, we have Expanding the eigenfunction ψ 1 in the form where G(a) is given by (7).Hence the stability of (θ(•, a), 0) is determined by the sign of G(a) when τ is small, and the following lemma holds.
By straightforward calculations, we can find that L(a, s) is given by Clearly, L(a, s) is a Fredholm operator of zero index.Moreover, where ker(L(a, s)) and R(L(a, s)) stand for the kernel and the range of L, respectively.
Let the operator P (a, s) on Y be defined by Then one can show that R(P ) = X 1 , P 2 = P and P (a, s)L(a, s) = 0, which means that P is the projection of Y onto X 1 along the range R(L(a, s)).
On the other hand, from ( 17) and ( 15), we obtain where f 2 (S, k) = ∂f ∂k (S, k) is the partial derivative of f (S, k) with respect to k. Hence It follows from ( 22), ( 23) and ( 24) that where G(a) is given by (7).
This completes the proof.
Next, we show that all positive solutions have been found by the local bifurcation analysis.
Proof.It suffices to show that if τ i → 0+, a i → a 0 , and (u i , v i ) is the nonnegative solution of ( 9) with a = a i , τ = τ i , then (a i , u i , v i ) converges to the curve Σ a0 .By Lemma 4.2 in [19], we have By standard elliptic regularity, passing to a subsequence we may assume that (u i , v i ) → (û, v) in X, where (û, v) ≥ 0 in Ω, and Clearly, (û, v) is contained on the curve Σ a0 .

Stability of coexistence solutions.
To study the stability of positive solutions to (9), we consider the corresponding linear eigenvalue problem It is well-known (see [4]) that if (u, v) is a positive solution, then (28) has a principal eigenvalue µ 1 which is real, algebraically simple and all other eigenvalues have their real parts greater than µ 1 .Moreover, the principal eigenvalue µ 1 corresponds to an eigenfunction (ϕ, ψ) satisfying ϕ > 0, ψ < 0, and µ 1 is the only eigenvalue with such no sign-changing eigenfunction.The linearized stability of (u, v) is determined by the sign of the principal eigenvalue: (u, v) is stable if µ 1 > 0; it is unstable if µ 1 < 0. Hence, the next crucial step is to calculate the principal eigenvalue µ 1 .
It follows from Lemma 4.1 that the crucial step to determine the stability of positive solution (u, v) is to study the signs of the integrals Ω ϕ 3 u − ψ 3 v dx and I i (i = 1, 2, 3).It follows from (13) that for s ∈ (0, 1) and 0 < τ 1, the functions u(τ, s), v(τ, s), a(τ, s) can be expanded into where u 1 (•, s), v 1 (•, s) and a 1 (s) are smooth functions of s.At first, by using (31) and (32), it is easy to check that for 0 < τ 1, Next, we turn to investigate the signs of the integrals I 1 , I 2 and I 3 .As ϕ > 0 > ψ, we see that I 1 > 0 and I 2 > 0. It suffices to find the sign of I 3 .
Note that We have Adding the first and second equation of (38), we obtain It follows from (37) that ϕ 1 + ψ 1 = A. The proof is complete.
5. The proof of Theorem 1.2.The goal of this section is to establish Theorem 1.2.To this end, we first derive more information on the positive solutions on the curve Γ.
Lemma 5.2.Let G (a 0 ) be the derivative of G(a) at a = a 0 .Then Proof.Multiplying the first equation of ( 9) by v and the second equation of ( 9) by v, subtracting and integrating over Ω, we obtain Substituting (32) into this equation and noting that G(a 0 ) = 0, we have Differentiating both sides of ∆θ(•, a) + aθ(•, a)f (z − θ(•, a), k) = 0 with respect to a at a = a 0 , we have by some direct computation.

Remark 4.
As A, C is independent of s, one can conclude that J 1 (s), J 2 (s) and J 3 (s) are linear functions with respect to s.Thus, a 1 (s) is a linear function of s, which implies a 1 (s) must be monotone with respect to s.
It remains to prove part (iii).By Theorem 3.1 and Lemma 3.2 again, we know that for a ≈ a 0 and τ ≈ 0, all positive solutions lie on the curve Γ.Moreover, we have a(τ, s) = a 0 + τ a 1 (s) + O(τ 2 ) on the curve Γ.It follows from Lemma 5.2 and Remark 4 that a 1 (s) is a linear function of s, which implies that a 1 (s) is monotone with respect to s.Hence, a(τ, s) is also monotone with respect to s.Let a(τ, s).
The monotonicity of a(τ, s) implies that (9) has a positive solution if and only if a ∈ (a * , a * ).Moreover, the positive solution is unique for each fixed a ∈ (a * , a * ).By virtue of Lemma 4.2, positive solutions on the curve Γ are stable provided I A ≥ 0. By Lemma 1.3, we can conclude that the unique and stable positive solution is the global attractor of ( 9).The proof is complete.6. Discussion.The purpose of this paper is to study the uniqueness and stability of coexistence solutions of (3) and its global dynamics.In [10,3], partial results are obtained, which show that (3) has a unique positive steady-state solution when the maximal growth rates a, b are near the principal eigenvalues λ 1 , σ 1 , respectively, and the unique positive steady state solution is globally asymptotically stable under certain conditions (see [10]).Here we assume the random mutation can produce another phenotype of species v which is similar to the resident species u.That is, the mutant has slightly different maximal growth rates and Michaelis-Menten constants.For survival, these two species might have to compete for the same limited resources.There are two key questions.One is whether the mutant v can invade when rare.The other is if the mutant v does invade, whether it will drive the resident species u to extinction or coexist with it.Mathematically, the questions lead to the study of the perturbed version (6) of the system (3).
Biologically, the results imply there is an index that predicts the mutant and the resident species can coexist or not.More precisely, the mutant v always drives the resident species u to extinction if G(a 0 ) > 0, and the mutant can not invade if G(a 0 ) < 0. If G(a 0 ) = 0 and G (a 0 ) = 0, the mutant and the resident species can coexist when a ∈ (a * , a * ) and I A ≥ 0. Numerical computations suggest that the integral I A can be negative or positive (see Remark 2).We suspect that if I A is negative, it may occur that both semi-trivial steady states are locally stable and the coexistence steady state is unstable.Meanwhile, numerical computations also suggest that the diagram of G(a) looks like Figure 1.Hence, it is possible for the mutant to force the extinction of resident species or to coexist with it.

Figure 1 .
Figure 1.Possible diagram of the function G(a).