EXISTENCE OF GLOBAL SOLUTION AND NONTRIVIAL STEADY STATES FOR A SYSTEM MODELING CHEMOTAXIS

Chemotaxis, the oriented movement of cells in response to ambient chemical gradients, is a prominent feature in the organization of many biological populations. Since the pioneer work of Keller and Segel [11] to propose mathematical models for chemotaxis, there has been great interest in modeling chemotaxis and in the mathematical analysis of systems like the Keller-Segel model. In this paper, motivated by the model in [15], we consider a revised model discussed in [16], that is,


Introduction
Chemotaxis, the oriented movement of cells in response to ambient chemical gradients, is a prominent feature in the organization of many biological populations.Since the pioneer work of Keller and Segel [11] to propose mathematical models for chemotaxis, there has been great interest in modeling chemotaxis and in the mathematical analysis of systems like the Keller-Segel model.In this paper, motivated by the model in [15], we consider a revised model discussed in [16], that is, db x − αbS 1 (N)N x = 0, x = 0,1, t > 0, DB x − βBS 2 (N)N x = 0, x = 0,1, t > 0, N(x,0) = N 0 (x), b(x,0) = b 0 (x), B(x,0) = B 0 (x), 0 < x < 1.
(1.1)This is the situation of two species of bacteria competing for the same nutrient, where N(x,t) is the concentration of the nutrient and b(x,t), B(x,t) are the densities of two competing species of bacteria.R i (N), i = 1,2, are the consumption rates of the nutrient per cell.−μN x , −db x , and −DB x are the random fluxes of N, b, and B, respectively, while αbS 1 (N)N x and βBS 2 (N)N x are the chemotactic fluxes of b and B, where μ > 0, d > 0, D > 0 and α ≥ 0, β ≥ 0. For definiteness, we assume that d < D. Functions S i (N), i = 1,2, the so-called sensitivity rates, are included to indicate that the sensitivity of cells to the nutrient may vary at different levels of nutrient concentration.When α = 0, β = 0, and ρ i = 1, this model reduces to the model discussed in [16].But the present model is not a trivial generalization of the model discussed in [16] because of the appearance of the chemotactic fluxes of b and B. Due to the lack of monotone structure on the system, the main tool-the comparison principle-used in [16] does not work here.In [15], the authors considered a similar model and discussed the situations when there is no positive steady state.In this paper, we will give sufficient conditions that guarantee the existence of positive solutions.The method we use here to investigate the existence of steady states is different from that used in [15].We also consider some special cases in which the sufficient conditions we will derive are not satisfied and the systems have no nontrivial steady states.The boundary conditions represent that the total fluxes of b and B at the boundary points x = 0 and x = 1 are zero.This is true for N at x = 0, but at x = 1, N is diffused into the medium.In the adjacent region, N ≡ 1, which must also be an upper bound for N inside the medium, and therefore we are only interested in solutions with 0 ≤ N ≤ 1.For this reason, we assume that 0 ≤ N 0 ≤ 1 throughout the paper.
From biological and technical considerations, we assume that R i (0) = 0, R i (N) > 0, S i (N) > 0 on [0,∞). (1. 2) The assumptions about R i guarantee the nonnegativeness of N, b, and B as long as the initial functions are nonnegative (see [13]).Therefore we will only consider nonnegative solutions of (1.1).This paper is organized as follows.In Section 2, we will prove the global existence of solutions.In Section 3, we will study the existence of steady states and some special cases.

Global existence
By standard existence theory, for example, see [3][4][5]12], it is not difficult to establish the local existence of the unique solution (N(x,t),b(x,t),B(x,t)) for 0 ≤ t < T max , where T max is determined by N 0 , b 0 , and B 0 .It is well known that local existence together with L ∞ a priori bounds ensure the global existence of classical solutions.Therefore, to establish the global existence, we need only to establish a priori estimates for (in fact, this can be proved directly by using comparison principle).Therefore we need only to establish the boundedness of b(•,t) L ∞ , and B(•,t) L ∞ .This is done by proving several lemmas.The following general imbedding result will be of use to us.

Proof.
Let Obviously, b(t) ≥ 0 and B(t) ≥ 0. Therefore, to prove the lemma, we need only to prove that there exists a constant M > 0 such that for 0 ≤ t < T max , (2.6) In fact, by adding the b-equation and B-equation in (1.1) and using the boundary conditions, we have where R = max{ρ 1 R 1 (1),ρ 2 R 2 (1)}.This implies (2.6).
Lemma 2.6.There exists a positive constant M > 0 such that for (2.36) The proof is similar to that of Lemma 4.7 in [14] and therefore is omitted.Thus we have the following global existence and boundedness theorem.

Existence of steady states
In this section, we study the existence of steady states of (1.1).Basically, we study the existence of nontrivial steady state solutions of (1.1) in the framework of [9].But in [9], the author made several assumptions about the reaction terms.Unfortunately, in our model the reaction functions do not satisfy all these assumptions.This fact causes difficulties in using the theory developed in [9].Therefore, we must do some careful and technical analysis for our model.The steady states of (1.1) satisfy Obviously, (1,0,0) is a solution of (3.1), that is, it is a steady state of (1.1).For this, we have the following theorem.Proof.To prove this theorem, we use the definition of instability (e.g., see [6]).That is, if O is a neighborhood of (1,0,0) consisting of (N,b,B) such that we can show that for a small > 0, the solution (N(x,t),b(x,t),B(x,t)) always leaves O in finite time no matter how close the initial values (N 0 ,b 0 ,B 0 ) are to (1,0,0).In fact, for > 0 small, we have Then by integrating the b-equation in (1.1), we have This implies that (N(x,t),b(x,t),B(x,t)) must leave O in finite time.
Since N (x) is increasing, we must have N (x) < γ for 0 ≤ x ≤ 1.Also, from the Nboundary condition at x = 1, we have N(1) 1. Observe that N(x) is also increasing, hence for 0 ≤ x ≤ 1, N(x) < 1.By the comparison principle and the condition R i (0) = 0, we have N(x) > 0 for x > 0.
Integrating the b-equation in (3.1) from 0 to 1 and using the boundary conditions, we have Therefore, (3.12) It follows that where R = max{ρ 1 R 1 (1),ρ 2 R 2 (1)}.This implies that there is a constant K such that In turn, this implies that (3.7) is true.
and z satisfies Assume that z(x) has its maximum at x 1 .Then z (x 1 ) = 0 and z (x 1 ) ≤ 0. From the above equation, we have, at x 1 , Integrating the b-equation from 0 to x and using the b-boundary condition at x = 0, we have We also have 2 e (α/d) 1 0 S1(y)dy . (3.24) Now, let We can prove a similar estimate for |B (x)|.Therefore (3.8) is true.
Corollary 3.3.For any ν ∈ (0,1), there is a positive constant K such that for any nontrivial solution (N,b,B) of (3.1), What we are interested in is whether (3.1) has any nontrivial solutions.The case N ≡ 0 is excluded by the boundary conditions.Therefore we need only to consider the possibilities of the existence of following two types of solutions: (i) semitrivial solutions: (N,b,0), (N,0,B); (ii) positive solutions: (N,b,B), where the components N > 0, b > 0, B > 0. In what follows, we use the theory of fixed point index on cones in a Banach space to study the existence of solutions of these types.First we study the existence of semitrivial solutions.

Existence of semitrivial solutions.
From the symmetry of b and B, we need only to study the existence of solutions of the form (N,b,0).For the convenience of notations, we write N and b as u 0 and u 1 , respectively, omit the subscripts of R 1 , S 1 , and ρ 1 , and consider the system (3.28)For ν ∈ (0,1), let then E is an ordered Banach space with positive cone C. For V = (v 0 ,v 1 ) ∈ C, let u 0 = A 0 (V ) be the solution of With u 0 = u 0 (v 0 ,v 1 ) given, define operators Φ 1 and Γ 1 as follows.
)), the Banach space of bounded linear maps from C ν ([0,1]) to itself, is defined by the following. For and define where P is a positive constant such that , where P 1 = 2ρR(1)e (α/d) 1 0 S(y)dy .Then system (3.28) can be written as a fixed point equation U = A(U), where U = A(V ) is given by It is easily seen that Γ 1 satisfies, Now we prove the following lemmas.
Lemma 3.4.The operator A(V ) = (A 0 (V ),A 1 (V )) : Ω → C is a well-defined completely continuous operator, where Moreover, fixed points of A in C are nonnegative solutions of (3.28).

Lemma 3.5.
There is an M > 0 such that where We use the homotopy invariance property to H(η,U).
is given by It is easy to verify that H(η,U) is completely continuous and there is a constant K such that for the solution of U = H(η,U), that is, U = A(ηU), we have U E ≤ K. Therefore for M > K, U = A(ηU) has no solution satisfying U E = M.This implies that for 0 Let then we have is the partial derivative of Γ 1 (v 0 ,v 1 ) with respect to v 1 .An easy computation shows that the operator T 1 ((1,0)) :

is the solution of the boundary value problem
Now we cite the following theorem.
Theorem 3.6 (see [7]).Let ᏸ(y) = a 2 (x)y + a 1 (x)y + a 0 (x)y be a linear differential operator with no singular points in [x 1 ,x 2 ], and suppose that f Assume also that (A 1 ,A 2 ) = (0,0) and (B 1 ,B 2 ) = (0,0).Then the BVP has a unique solution if and only if the associated homogeneous problem ᏸ(y) = 0 with the same boundary conditions has only the trivial solution.
It is easily seen that the homogeneous problem associated with (3.46) has only the trivial solution when P > 0. Therefore, from the theorem, we know that for any v ∈ C 1 , (3.46) has unique solution and by maximum principle, we have u Obviously, 1 is not an eigenvalue of (3.49) corresponding to a positive eigenfunction.In fact, the eigenvalues of (3.49) are and the associated eigenfunctions are ψ n = cos(nπx).Therefore we can see that the eigenvalue that corresponds to the positive eigenfunction is λ = ((ρ/d)R(1) + P)/P > 1.This implies that the spectral radius of T 1 ((1,0)) is greater than 1 and therefore, from [9, Theorem 3.1], we have ind(A,Δ {0} ) = 0. From Lemma 3.5, we know that for some M > 0, the set of fixed points of A is in B M .Therefore, from Lemma 3.5, we have For ν ∈ (0,1), let (3.52) With u 0 given, define operators Φ i and Γ i , i = 1,2, as where P is a positive constant such that d +P > 0 for 0≤v 0 ≤ 2, 0 ≤ v 1 ≤ P 1 , and 0≤v 2 ≤ P 2 , where P 1 = 2ρ 1 R 1 (1)e (α/d) 1 0 S1(y)dy and P 2 = 2ρ 2 R 2 (1)e (β/D) 1 0 S2(y)dy .Then system (3.1) can be written as a fixed point equation U = A(U), where U = A(V ) is given by It is easily seen that Γ i satisfies Similar to the proofs of Lemmas 3.4 and 3.5, we can prove the following two lemmas.
Lemma 3.9.There is an M > 0 such that where B M = {U ∈ C : U E < M}.

Zhenbu Zhang 21
Now we can prove the following theorem.(3.80) The largest eigenvalues of the two eigenvalue problems above are greater than 1 if and only if the largest eigenvalues of the following two eigenvalue problems are (3.82) A special case is α/d = β/D.For this case, it is easily seen that the largest eigenvalue of (3.81) is λ 1 = d/D < 1 and the associated eigenfunction is φ = ǔ1 .The largest eigenvalue of (3.82) is λ 1 = D/d > 1 and the associated eigenfunction is ψ = û2 .Therefore, neither (Ꮽ 1 ) nor (Ꮽ 2 ) is satisfied.In fact, from [15], we know that (3.78) has no positive solutions for this situation.

From [ 9 ,Theorem 3 . 7 . 3 . 2 .
Theorem 3.1]  mentioned above, we know that (3.28) has at least one positive solution.This implies that (3.1) has solutions of the form (N,b,0) with N > 0 and b > 0. Similarly, we know that (3.1) has solutions of the form (N,0,B) with N > 0 and B > 0. Summarizing the analysis above, we have the following theorem.System (3.1) has solutions of the form (N,b,0) and (N,0,B) with N > 0, b > 0, and B > 0. Existence of positive solutions.Now we study the existence of positive solutions of (3.1).As before, we write N, b, and B as u 0 , u 1 , and u 2 , respectively, and write system (3.1) in the form of a fixed point equation as follows.
.71)We denote the spectral radius of operator ᏸ by Υ(ᏸ).It is well known that Υ(ᏸ 0 ) > 1 if and only if Υ(ᏸ P ) > 1 for all P ≥ 0. Therefore we know that the largest eigenvalues of both (3.69) and (3.70) are greater than 1 if and only if the largest eigenvalues of the following two eigenvalue problems are greater than 1: