Coexistence for a Resource-Based Growth Model with Two Resources 1

We investigate the coexistence of positive steady-state solutions to a parabolic system, which models a single species on two growth-limiting, non-reproducing resources in an un-stirred chemostat with diusion. We establish the existence of a positive steady-state solution for a range of the parameter (m; n), the bifurcation solutions and the stability of bifurcation solutions. The proof depends on the maximum principle, bifurcation theorem and perturbation theorem.

Since we are only concerned with the nonnegative solutions (S, R, u) of (1.1), we can redefine the response functions f, g for S ≤ 0, R ≤ 0 without affecting our results.The un-stirred chemostat with one resource has been considered by many authors in the past decade(see [1][2] [3]).Just as pointed out in [4], the un-stirred chemostat with two resources is more realistic and thus of interest, and the system (1.1) with equal diffusion rates is investigated in paper [5].Without the assumption of equal diffusion rates, we obtain some estimates on the size of the coexistence region near a bifurcation point in the parameter space.The existence of positive steady-state solution of the system (1.1) is established by the maximum principle and the theorem of bifurcation, which appears in [6] to study the local solutions.The stability of bifurcation solutions is also studied via the perturbation theorem.

Extinction
In this section we use the maximum principle to establish conditions under which the species become extinct.
Let λ (i) 0 > 0(i = 1, 2, 3) be the principle eigenvalue of the following problem with the eigenfunction φ Let S(x) be the unique positive solution of the following problem The existence and uniqueness of S(x) is standard, and by the maximum principle it is easy to show that S > 0 on [0, 1].Lemma 2.2.There are positive constants α i and Proof.Let ω(x, t) = S(x, t) − S(x), then ω satisfies Then, by the comparison theorem, we have ω(x, t) ≤ W (x, t), where W (x, t) is the unique solution of the linear problem In order to estimate W , let 0 < α 1 < λ (1) 0 , W (x, t) = φ (1) 0 (x)h(x, t)e −α 1 t .Then we have .
The maximum principle ( [7]) implies that and this leads to for some constants K 1 > 0. Similarly result holds for R.
As in the previous lemma, it follows that |h(x, t)| ≤ max and the lemma follows.

EJQTDE, 2003 No. 7, p. 3
In this section we consider the coexistence of the positive steady-state solutions of the system (1.1).So we consider the elliptic system with the boundary conditions and we have z First we give some estimates about the nonnegative solution of (3.1)-(3.2).The similar proof can be found in [4,8].We omit the detail here.
Let s = z − S, r = z − R, then by lemma 3.1, either 0 < s, r < z or s = r = 0, and with the boundary conditions In this subsection, we consider the case of d 2 m = d 1 n and discuss the existence of a positive solution of (3.4).
In this subsection, we discuss the existence and nonexistence of a positive solution of (3.4)(3.5).First we give a basic estimate for (s, r).
Proof.Let ω = s − r, then The following theorem shows that a positive solution of (3.4)(3.5)cannot exist if both m and n are too small.and (s, r) is a nontrivial nonnegative solution of (3.4)(3.5).Then it follows from the maximum principle that EJQTDE, 2003 No. 7, p. 5 s > 0, r > 0. If , multiplying the first equation in (3.4) by s, integrating over (0, 1) and using Green formula, we find By the variational property of the principle eigenvalue, we have , then the washout solution (z, z, 0) is the unique nontrivial nonnegative solution of (3.1)(3.2).

Bifurcation Theorem
Now, for fixed n ≤ d 2 d 3 λ 1 /(d 1 + cd 2 ), we treat m as a bifurcation parameter to obtain the local bifurcation which corresponds to the positive solution of (3.4) (3.5).