Applied Mathematical Modelling 34 (2010) 3400–3407

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Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

Coexistence in a competitor–competitor–mutualist model

Mei Li a, Zhigui Lin b,*, Jiahong Liu b

a Dept. of Applied Mathematics, Nanjing Univ. of Finance and Economics, Nanjing 210003, Chinab School of Mathematical Science, Yangzhou Univ., Yangzhou 225002, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 20 November 2008Received in revised form 4 February 2010Accepted 12 February 2010Available online 2 March 2010

Keywords:CompetitionMutualismCross-diffusionCoexistence

0307-904X/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.apm.2010.02.029

* Corresponding author.E-mail address: zglin68@hotmail.com (Z. Lin).

This paper is concerned with a system modeling a competitor–competitor–mutualistthree-species Lotka–Volterra model. By Schauder fixed point theory, the existence of posi-tive solutions to a strongly coupled elliptic system is given. Applying the method of upperand lower solutions and its associated monotone iterations, the true solutions are con-structed and a numerical simulation is also presented. Our results show that this systempossesses at least one coexistence state if cross-diffusions and cross-reactions are weak.

� 2010 Elsevier Inc. All rights reserved.

1. Introduction

The competitor–competitor–mutualist model was initiated by Rai, et al. [1] for the ordinary differential system

ut ¼ u a1 � b1u� c1v1þr1w

� �; t > 0;

v t ¼ vða2 � b2u� c2vÞ; t > 0;

wt ¼ w a3 � b3w1þr2u

� �; t > 0;

8>>><>>>:

which governs the population densities of competitor u, competitor–mutualist v and mutualist w. The boundedness of theglobal solution and local stability or instability of the various equilibria were given. When diffusion was introduced, theabove ordinary differential system was extended by Zheng [2] to the reaction–diffusion system:

ut � D1Du ¼ u a1 � b1u� c1v1þr1w

� �; x 2 X; t > 0;

v t � D2Dv ¼ vða2 � b2u� c2vÞ; x 2 X; t > 0;

wt � D3Dw ¼ w a3 � b3w1þr2u

� �; x 2 X; t > 0;

@u=@m ¼ @v=@m ¼ @w=@m ¼ 0; x 2 @X; t > 0;uðx;0Þ ¼ g1ðxÞ; vðx;0Þ ¼ g2ðxÞ; wðx;0Þ ¼ g3ðxÞ; x 2 X;

8>>>>>>>><>>>>>>>>:

the stability and instability of the various semitrivial solutions of the corresponding steady-state problem were given byusing the monotone method and the local stability of a positive equilibrium is also discussed using spectral analysis ofthe linearized operator. There are also many papers to study the corresponding periodic systems, see for example [3,4].

. All rights reserved.

M. Li et al. / Applied Mathematical Modelling 34 (2010) 3400–3407 3401

Recently Pao [5] considered the system with time delays under Neumann conditions, and proved under a very simplecondition on the reaction rates, that for any nontrivial initial function the corresponding time-dependent solution convergesto the positive steady-state solution using the method of upper and lower solutions.

In this paper, we consider the strongly coupled elliptic system with Dirichlet boundary conditions:

�D d1 þ a11uþ a12v1þb1w

� �u

h i¼ u a1 � b1u� c1v

1þr1w

� �; x 2 X;

�D½ðd2 þ a21uþ a22vÞv � ¼ vða2 � b2u� c2vÞ; x 2 X;

�D d3 þ a31w1þb2uþ a33w

� �w

h i¼ w a3 � b3w

1þr2u

� �; x 2 X;

uðxÞ ¼ vðxÞ ¼ wðxÞ ¼ 0; x 2 @X;

8>>>>><>>>>>:

ð1:1Þ

where D is the Laplacian operator, X is a bounded domain in RN with a smooth boundary @X and di; bj; ai; bi;

cjði ¼ 1;2;3; j ¼ 1;2Þ are positive constants except for aij which may be nonnegative constants. The system represents a mod-el which involves interacting and migrating in the same habitat X among a competitor, a competitor–mutualist and a mutu-alist. The boundary condition means that the habitat X is surrounded by a hostile environment. The diffusion terms can bewritten as

div d1 þ 2a11uþ a12v1þ b1w

� �ruþ a12u

1þ b1wrv � a12b1uv

ð1þ b1wÞ2rw

( );

divfa21vruþ ðd2 þ a21uþ 2a22vÞrvg;

div�a31b2w2

ð1þ b2uÞ2ruþ d3 þ

2a31w1þ b2u

þ 2a33w� �

rw

( ):

The terms

d1 þ 2a11uþ a12v1þ b1w

; d2 þ a21uþ 2a22v ; d3 þ2a31w

1þ b2uþ 2a33w;

represent the ‘‘self-diffusion” and the terms

a12u1þ b1w

;�a12b1uvð1þ b1wÞ2

; a21v ;�a31b2w2

ð1þ b2uÞ2;

represent the ‘‘cross-diffusion”. Here a12u1þb1w > 0 and a21v > 0 imply that the flux of u and v in x-direction are directed toward

decreasing population of v and u respectively, i.e. the two competitors avoid each other. While �a12b1uvð1þb1wÞ2

< 0 and �a31b2w2

ð1þb2uÞ2< 0

imply that the flux of u and w in x-direction are directed toward increasing population of w and u respectively, i.e. thetwo mutualists chase each other. The above model means that, in addition to the dispersive force, the diffusion also dependson population pressure from other species. Here a solution ðu;v ;wÞ to system (1.1) is said to be positive ifðuðxÞ;vðxÞ;wðxÞÞ > ð0;0;0Þ for all x 2 X, the existence of a positive solution ðu;v ;wÞ to system (1.1) is also called a coexis-tence. We are mainly concerned with the coexistence states of system (1.1).

In the case when aij ¼ 0 for i; j ¼ 1;2;3, the above system is the classic competitor–competitor–mutualist model, while ifaij–0 for some i or j the system becomes a strongly coupled elliptic system. The strongly coupled systems of elliptic equationshave been extensively studied by many researchers [6–13]. Shigesada et al. [6] proposed the strongly coupled elliptic systemdescribing two species Lotka–Volterra competition model. For the system with Dirichlet boundary condition, positive solutionsare found by Ruan [7]. For the system with homogeneous Neumann boundary condition, the effects of diffusion, self-diffusionand cross-diffusion were investigated by Lou and Ni [8]. The multiple coexistence states for a prey–predator system with cross-diffusion was proved by Kuto and Yamada [9] by applying the bifurcation theory and Lyapunov–Schmidt procedure. The meth-od of construction of solutions for a general class of strongly coupled elliptic systems was developed by Pao [13].

The paper is organized as follows: based on the idea introduced by Pao [13], we try to obtain sufficient conditions whichguarantee the coexistence state of system (1.1) in Section 2, the true solutions of (1.1) are constructed in Section 3, a numer-ical simulation is also presented in Section 4 to illustrate the main results and Section 5 is devoted to some conclusions.

2. Coexistence

We will give a sufficient condition for that system (1.1) has a positive solution by constructing a coupled upper and lowersolutions as in [13]. We first give an equivalent form of the problem (1.1):

�D½D1ðu;v ;wÞ� ¼ f1ðu; v;wÞ; x 2 X;

�D½D2ðu;vÞ� ¼ f2ðu;vÞ; x 2 X;

�D½D3ðu;wÞ� ¼ f3ðu;wÞ; x 2 X;

uðxÞ ¼ vðxÞ ¼ wðxÞ ¼ 0; x 2 @X;

8>>><>>>:

ð2:1Þ

3402 M. Li et al. / Applied Mathematical Modelling 34 (2010) 3400–3407

where

D1ðu;v ;wÞ ¼ d1 þ a11uþ a12v1þb1w

� �u;

f1ðu;v ;wÞ ¼ u a1 � b1u� c1v1þr1w

� �;

D2ðu;vÞ ¼ ðd2 þ a21uþ a22vÞv ;f2ðu;vÞ ¼ vða2 � b2u� c2vÞ;D3ðu;wÞ ¼ d3 þ a31w

1þb2uþ a33w� �

w;

f3ðu;wÞ ¼ w a3 � b3w1þr2u

� �:

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

ð2:2Þ

Define

z1 ¼ D1ðu;v ;wÞ; z2 ¼ D2ðu;vÞ; z3 ¼ D3ðu;wÞ:

A direct calculation shows that the Jacobian J of the transformation z1; z2; z3 is given by

J ¼ @ðz1; z2; z3Þ@ðu; v;wÞ P d1d2d3 > 0 for ðu;v ;wÞP ð0;0;0Þ:

Then the inverse u ¼ g1ðz1; z2; z3Þ;v ¼ g2ðz1; z2; z3Þ;w ¼ g3ðz1; z2; z3Þ exists whenever ðz1; z2; z3ÞP ð0;0;0Þ. Hence the corre-sponding equivalent of (2.1) becomes

�Dz1 þ K1z1 ¼ F1ðu; v;wÞ; x 2 X;

�Dz2 þ K2z2 ¼ F2ðu; vÞ; x 2 X;

�Dz3 þ K3z3 ¼ F3ðu;wÞ; x 2 X;

u ¼ g1ðz1; z2; z3Þ; x 2 X;

v ¼ g2ðz1; z2; z3Þ; x 2 X;

w ¼ g3ðz1; z2; z3Þ; x 2 X;

ziðxÞ ¼ 0; i ¼ 1;2;3; x 2 @X;

8>>>>>>>>>>><>>>>>>>>>>>:

ð2:3Þ

where Fiðu;v ;wÞ ¼ KiDiðu;v ;wÞ þ fiðu;v ;wÞ; i ¼ 1;2;3.Now we consider the monotonicity of Fi with respect to u, v, w respectively, and also the monotonicity of gi with respect

to zj for i; j ¼ 1;2;3. First direct calculations show that

@u@z1

> 0;@u@z26 0;

@u@z3

P 0;@v@z16 0;

@v@z2

> 0;

@v@z36 0;

@w@z1

P 0;@w@z26 0;

@w@z3

> 0:

Therefore, u ¼ g1ðz1; z2; z3Þ is nondecreasing in z1; z3 and nonincreasing in z2;v ¼ g2ðz1; z2; z3Þ is nondecreasing in z2 and non-increasing in z1; z3, while w ¼ g3ðz1; z2; z3Þ is nondecreasing in z1; z3 and nonincreasing in z2 for all ðz1; z2; z3ÞP ð0;0;0Þ.

Secondly if we choose K1 ¼ b1a11;K2 ¼ c2

a22and K3 ¼ b3

a33. Then

@F1

@u¼ b1

a11d1 þ a1 þ

b1a12

a11ð1þ b1wÞ �c1

1þ r1w

� �v ;

@F1

@v ¼b1a12

a11ð1þ b1wÞ �c1

1þ r1w

� �u;

@F1

@w¼ � b1a12b1

a11ð1þ b1wÞ2þ c1r1

ð1þ r1wÞ2

!uv ;

@F2

@u¼ c2

a22a21 � b2

� �v ; @F2

@v ¼c2

a22d2 þ a2 þ

c2

a22a21 � b2

� �u;

@F2

@w¼ 0;

@F3

@u¼ � a31b3b2

ð1þ b2uÞ2a33

þ b3r2

ð1þ r2uÞ2

!w2;

@F3

@v ¼ 0;

@F3

@w¼ a3 þ d3

b3

a33þ 2ðb3 þ

a31b3

ð1þ b2uÞa33� b3

1þ r2uÞw:

Assume that

a12

a11<

c1

b1min

r1

b1;1;

b1

r1

� �;

a21

a22<

b2

c2;

a31

a33< min

r2

b2;b2

r2

� �: ð2:4Þ

Obviously, we can get @F1@v 6 0; @F1

@w P 0; @F2@u 6 0; @F3

@u P 0; @F3@w P 0 for every ðu; v;wÞP ð0;0;0Þ. Furthermore, choose

M. Li et al. / Applied Mathematical Modelling 34 (2010) 3400–3407 3403

U ¼ c2d2 þ a2a22

b2a22 � c2a21; V ¼ b1d1 þ a1a11

c1a11 � b1a12;

we can obtain that @F1@u P 0; @F2

@v P 0 when ðu;v;wÞ 2 ½0;U� � ½0;V � � ½0;1Þ. Therefore, when ðu;v ;wÞ 2 ½0;U� � ½0;V � � ½0;1Þ,the function F1 is nonincreasing in v and nondecreasing in u and w; F2 is nonincreasing in u and nondecreasing in v; F3 isnondecreasing in u and w.

Next we give the definition of coupled upper and lower solutions of (2.3) as the following:

Definition 2.1. A pair of 6-vector functions ð~u; ~zÞ ¼ ð~u; ~v ; ~w;~z1;~z2;~z3Þ; ðu; zÞ ¼ ðu; v; w; z1; z2; z3Þ in C2ðXÞ \ CðXÞ are calledcoupled upper and lower solutions of (2.3), if ~u 6 U; ~v 6 V ; ð~u; ~zÞP ðu; zÞ and if their components satisfy the relation

�D~z1 þ K1~z1 P F1ð~u; v; ~wÞ; �Dz1 þ K1z1 6 F1ðu; ~v ; wÞ; x 2 X;

�D~z2 þ K2~z2 P F2ðu; ~vÞ; �Dz2 þ K2z2 6 F2ð~u; vÞ; x 2 X;

�D~z3 þ K3~z3 P F3ð~u; ~wÞ; �Dz3 þ K3z3 6 F3ðu; wÞ; x 2 X;~u P g1ð~z1; z2;~z3Þ; u 6 g1ðz1;~z2; z3Þ; x 2 X;~v P g2ðz1;~z2; z3Þ; v 6 g2ð~z1; z2;~z3Þ; x 2 X;~w P g3ð~z1; z2;~z3Þ; w 6 g3ðz1;~z2; z3Þ; x 2 X;~ziðxÞP 0 P ziðxÞ; i ¼ 1;2;3; x 2 @X:

8>>>>>>>>>>><>>>>>>>>>>>:

ð2:5Þ

We set

S ¼ fu 2 CaðXÞ; u 6 u 6 ~ug; S� ¼ fz 2 CaðXÞ; z 6 z 6 ~zg;

where u ¼ ðu;v ;wÞ; z ¼ ðz1; z2; z3Þ; ~u ¼ ð~u; ~v; ~wÞ; u ¼ ðu; v ; wÞ and ~z ¼ ð~z1;~z2;~z3Þ; z ¼ ðz1; z2; z3Þ.For definiteness, we choose

~u ¼ g1ð~z1; z2;~z3Þ; ~v ¼ g2ðz1;~z2; z3Þ; ~w ¼ g3ð~z1; z2;~z3Þ;u ¼ g1ðz1;~z2; z3Þ; v ¼ g2ð~z1; z2;~z3Þ; w ¼ g3ðz1;~z2; z3Þ;

which is equivalent to

~z1 ¼ D1ð~u; v ; ~wÞ; ~z2 ¼ D2ðu; ~vÞ; ~z3 ¼ D3ð~u; ~wÞ;z1 ¼ D1ðu; ~v ; wÞ; z2 ¼ D2ð~u; vÞ; z3 ¼ D3ðu; wÞ:

Then the requirements of ð~u; ~v; ~wÞ; ðu; v ; wÞ in (2.5) are satisfied and those of ð~z1;~z2;~z3Þ; ðz1; z2; z3Þ are reduced to

�D½D1ð~u; v ; ~wÞ� þ K1D1ð~u; v ; ~wÞP F1ð~u; v ; ~wÞ; x 2 X;

�D½D2ðu; ~vÞ� þ K2D2ðu; ~vÞP F2ðu; ~vÞ; x 2 X;

�D½D3ð~u; ~wÞ� þ K3D3ð~u; ~wÞP F3ð~u; ~wÞ; x 2 X;

�D½D1ðu; ~v ; wÞ� þ K1D1ðu; ~v ; wÞ 6 F1ðu; ~v; wÞ; x 2 X;

�D½D2ð~u; vÞ� þ K2D2ð~u; vÞ 6 F2ð~u; vÞ; x 2 X;

�D½D3ðu; wÞ� þ K3D3ðu; wÞ 6 F3ðu; wÞ; x 2 X;~uðxÞP 0 P uðxÞ; ~vðxÞP 0 P vðxÞ; ~wðxÞP 0 P wðxÞ; x 2 @X:

8>>>>>>>>>>><>>>>>>>>>>>:

ð2:6Þ

We call the pair ð~u; ~v ; ~wÞ; ðu; v ; wÞ satisfying (2.6) and ~u 6 U; ~v 6 V ; ð~u; ~v ; ~wÞP ðu; v; wÞ are coupled upper and lower solu-tions of (1.1).

Now we seek a pair of coupled upper and lower solutions of (1.1) in the form

ð~u; ~v ; ~wÞ ¼ ðM1;M2;M3Þ; ðu; v; wÞ ¼ ðd1/; d2/; d3/Þ;

where Mi and di ði ¼ 1;2;3Þ are some positive constants with di sufficiently small, and / � /ðxÞ is the (normalized) positiveeigenfunction corresponding to k0, where k0 is the smallest eigenvalue of the Laplacian ð�DÞ under Dirichlet boundary con-dition. Indeed ðM1;M2;M3Þ; ðd1/; d2/; d3/Þ satisfy the inequalities in (2.6) if

�D½ðd1 þ a11M1 þ a12d2/1þb1M3

ÞM1�P M1ða1 � b1M1 � c1d2/1þr1M3

Þ;�D½ðd2 þ a21d1/þ a22M2ÞM2�P M2ða2 � b2d1/� c2M2Þ;�D d3 þ a31M3

1þb2M1þ a33M3

� �M3

h iP M3 a3 � b3M3

1þr2M1

� �;

�D ðd1 þ a11d1/þ a12M21þb1d3/Þd1/

h i6 d1/ a1 � b1d1/� c1M2

1þr1d3/

� �;

�D½ðd2 þ a21M1 þ a22d2/Þd2/� 6 d2/ða2 � b2M1 � c2d2/Þ;�D d3 þ a31d3/

1þb2d1/þ a33d3/� �

d3/h i

6 d3/ a3 � b3d3/1þr2d1/

� �:

8>>>>>>>>>>>><>>>>>>>>>>>>:

ð2:7Þ

Since that di ði ¼ 1;2;3Þ is sufficiently small and �D/ ¼ k0/, the inequalities in (2.7) is equivalent to

3404 M. Li et al. / Applied Mathematical Modelling 34 (2010) 3400–3407

a1 � b1M1 6 0; a2 � c2M2 6 0;a3ð1þ r2M1Þ � b3M3 6 0;ðd1 þ a12M2Þk0 < a1 � c1M2;

ðd2 þ a21M1Þk0 < a2 � b2M1;

d3k0 < a3:

8>>>>><>>>>>:

ð2:8Þ

Assume that

a1b16 M1 <

a2�d2k0k0a21þb2

; a2c26 M2 <

a1�k0d1k0a12þc1

;

M3 P a3ðb1þr2a1Þb1b3

; a3 � d3k0 > 0:

8<: ð2:9Þ

Then the requirements in (2.8) are fulfilled and also the inequalities M1 6 U;M2 6 V hold. In all, assume that

a12

a11<

c1

b1min

r1

b1;1;

b1

r1

� �;

a21

a22<

b2

c2;

a31

a33< min

r2

b2;b2

r2

� �;

a1

b1<

a2 � d2k0

k0a21 þ b2;

a2

c2<

a1 � k0d1

k0a12 þ c1; d3k0 < a3;

ð2:10Þ

there exist positive constants Mi; di ði ¼ 1;2;3Þ and / such that the pair ð~u; ~v ; ~wÞ ¼ ðM1;M2;M3Þ; ðu; v; wÞ ¼ ðd1/; d2/; d3/Þ arecoupled upper and lower solutions of problem (1.1).

Using Theorem 2.1 of [13] yields the following existence result:

Theorem 2.1. The problem (1.1) admits at least one positive solution u ¼ ðu;v ;wÞ under the condition (2.10).

Remark 2.1. It is easy to see that if k0d1 P a1 or k0d2 P a2 or k0d3 P a3, then problem (1.1) has no positive solution, see [7].Our result shows that if k0d1 < a1; k0d2 < a2 and k0d3 < a3, then problem (1.1) has at least one coexistence state provided thatcross-diffusions a12;a21;a31 and cross-reactions c1; b2 are sufficiently small.

3. Iteration

In this section, we first construct the true solutions of (1.1) based on monotone iterative schemes. Under the condition(2.10), we know that ðM1;M2;M3Þ; ðd1/; d2/; d3/Þ are coupled upper and lower solution of problem (1.1). Now we useð�uð0Þ; �v ð0Þ; �wð0ÞÞ ¼ ðM1;M2;M3Þ; ðuð0Þ;v ð0Þ;wð0ÞÞ ¼ ðd1/; d2/; d3/Þ as an initial iteration in the iteration process

�D½D1ð�uðmÞ;v ðmÞ; �wðmÞÞ� þ K1D1ð�uðmÞ;v ðmÞ; �wðmÞÞ ¼ F1ð�uðm�1Þ; v ðm�1Þ; �wðm�1ÞÞ;�D½D2ðuðmÞ; �v ðmÞÞ� þ K2D2ðuðmÞ; �v ðmÞÞ ¼ F2ðuðm�1Þ; �v ðm�1ÞÞ;�D½D3ð�uðmÞ; �wðmÞÞ� þ K3D3ð�uðmÞ; �wðmÞÞ ¼ F3ð�uðm�1Þ; �wðm�1ÞÞ;�D½D1ðuðmÞ; �v ðmÞ;wðmÞÞ� þ K1D1ðuðmÞ; �v ðmÞ;wðmÞÞ ¼ F1ðuðm�1Þ; �v ðm�1Þ;wðm�1ÞÞ;�D½D2ð�uðmÞ;v ðmÞÞ� þ K2D2ð�uðmÞ; v ðmÞÞ ¼ F2ð�uðm�1Þ;v ðm�1ÞÞ;�D½D3ðuðmÞ;wðmÞÞ� þ K3D3ðuðmÞ;wðmÞÞ ¼ F3ðuðm�1Þ;wðm�1ÞÞ; x 2 X;�uðmÞðxÞ ¼ uðmÞðxÞ ¼ 0; �v ðmÞðxÞ ¼ v ðmÞðxÞ ¼ 0; �wðmÞðxÞ ¼ wðmÞðxÞ ¼ 0; x 2 @X;

8>>>>>>>>>><>>>>>>>>>>:

ð3:1Þ

where m ¼ 1;2; . . . Using Lemma 3.1 of [13], we know that the sequence fð�uðmÞ; �v ðmÞ; �wðmÞÞg; fðuðmÞ;v ðmÞ;wðmÞÞg governed by(3.1) are well-defined and possess the monotone property

ðu; v; wÞ 6 ðuðm�1Þ;v ðm�1Þ;wðm�1ÞÞ 6 ðuðmÞ;v ðmÞ;wðmÞÞ 6 ð�uðmÞ; �v ðmÞ; �wðmÞÞ 6 ð�uðm�1Þ; �v ðm�1Þ; �wðm�1ÞÞ 6 ð~u; ~v ; ~wÞ

for every m ¼ 1;2; . . .

Therefore, the pointwise limits

limm!1ð�uðmÞ; �v ðmÞ; �wðmÞÞ ¼ ð�u; �v ; �wÞ; lim

m!1ðuðmÞ;v ðmÞ;wðmÞÞ ¼ ðu; v;wÞ

exist and satisfy the relation

ðu; v; wÞ 6 ðuðmÞ;v ðmÞ;wðmÞÞ 6 ðu;v ;wÞ 6 ð�u; �v; �wÞ 6 ð�uðmÞ; �v ðmÞ; �wðmÞÞ 6 ð~u; ~v ; ~wÞ

for every m ¼ 1;2; . . .

By the relation in (3.1), let m!1 and using the standard regularity argument for elliptic boundary problems show thatð�u; �v ; �wÞ and ðu;v ;wÞ satisfy the equations8

�D½D1ð�u;v ; �wÞ� þ K1D1ð�u; v; �wÞ ¼ F1ð�u;v ; �wÞ; x 2 X;�D½D2ðu; �vÞ� þ K2D2ðu; �vÞ ¼ F2ðu; �vÞ; x 2 X;�D½D3ð�u; �wÞ� þ K3D3ð�u; �wÞ ¼ F3ð�u; �wÞ; x 2 X;�D½D1ðu; �v ;wÞ� þ K1D1ðu; �v;wÞ ¼ F1ðu; �v ;wÞ; x 2 X;�D½D2ð�u;vÞ� þ K2D2ð�u; vÞ ¼ F2ð�u; vÞ; x 2 X;�D½D3ðu;wÞ� þ K3D3ðu;wÞ ¼ F3ðu;wÞ; x 2 X;�uðxÞ ¼ uðxÞ ¼ 0; �vðxÞ ¼ vðxÞ ¼ 0; �wðxÞ ¼ wðxÞ ¼ 0; x 2 @X:

>>>>>>>>>><>>>>>>>>>>:

ð3:2Þ

M. Li et al. / Applied Mathematical Modelling 34 (2010) 3400–3407 3405

By virtue of the monotonicity of the functions Fiðu;v ;wÞ and Diðu; v;wÞ, the functions Fi;Di possess the following property

F1ð�u; v; �wÞ ¼ f1ð�u;v ; �wÞ þ K1D1ð�u;v ; �wÞ; x 2 X;

F2ðu; �vÞ ¼ f2ðu; �vÞ þ K2D2ðu; �vÞ; x 2 X;

F3ð�u; �wÞ ¼ f3ð�u; �wÞ þ K3D3ð�u; �wÞ; x 2 X;

F1ðu; �v;wÞ ¼ f1ðu; �v ;wÞ þ K1D1ðu; �v ;wÞ; x 2 X;

F2ð�u; vÞ ¼ f2ð�u;vÞ þ K2D2ð�u;vÞ; x 2 X;

F3ðu;wÞ ¼ f3ðu;wÞ þ K3D3ðu;wÞ; x 2 X:

8>>>>>>>><>>>>>>>>:

ð3:3Þ

Therefore,

�D½D1ð�u;v ; �wÞ� ¼ f1ð�u; v; �wÞ; x 2 X;

�D½D2ðu; �vÞ� ¼ f2ðu; �vÞ; x 2 X;

�D½D3ð�u; �wÞ� ¼ f3ð�u; �wÞ; x 2 X;

�D½D1ðu; �v ;wÞ� ¼ f1ðu; �v;wÞ; x 2 X;

�D½D2ð�u;vÞ� ¼ f2ð�u;vÞ; x 2 X;

�D½D3ðu;wÞ� ¼ f3ðu;wÞ; x 2 X:

8>>>>>>>><>>>>>>>>:

ð3:4Þ

Then ðu; �v ;wÞ and ð�u;v ; �wÞ are true solutions of (1.1).If �u ¼ u or �v ¼ v or �w ¼ w, then ð�u; �v ; �wÞ ¼ ðu;v ;wÞð� ðu�;v�;w�ÞÞ and ðu�;v�;w�Þ is the unique solution of (1.1). To see

this, let us consider the case �v ¼ v � v�. By a subtraction of the second equation from the fifth equation in (3.3) and

D2ðu; �vÞ � D2ð�u;vÞ ¼ �a21v�ð�u� uÞ;

we obtain

D½a21v�ð�u� uÞ� ¼ �v�b2ðu� �uÞ in X:

In view of v� > 0;a21 > 0; b2 > 0, and �u� u ¼ 0 on @X, the above equation yields �u ¼ u. We can take use of the similar meth-od to obtain �w ¼ w. This shows that ð�u; �v; �wÞ ¼ ðu;v ;wÞ. Then ðu�;v�;w�Þ is the unique solution.

To summarize the above conclusions we have the following theorem:

Theorem 3.1. Under the condition (2.10) , the sequences fuðmÞ; �vðmÞ; �wðmÞg; fuðmÞ;v ðmÞ;wðmÞg obtained from (3.1) withð�uð0Þ; �v ð0Þ; �wð0ÞÞ ¼ ðM1;M2;M3Þ; ðuð0Þ;vð0Þ;wð0ÞÞ ¼ ðd1/; d2/; d3/Þ and K1 ¼ b1

a11;K2 ¼ c2

a22;K3 ¼ b3

a33, converge monotonically to

some limits ð�u; �v ; �wÞ; ðu;v ;wÞ, and ðu;v ;wÞ; ðu; v;wÞ are true solutions of (1.1) ; if either u ¼ �u or v ¼ �v or w ¼ �w, thenð�u; �v ; �wÞ ¼ ðu;v ;wÞð� ðu�;v�;w�ÞÞ andðu�;v�;w�Þ is the unique solution of problem (1.1) in S.

4. Numerical example

In what follows, we present a numerical simulation. For simplicity, we consider a special case of (1.1) in the form

�D½ðdþ auÞu� ¼ uða� buÞ; x 2 ð0;pÞ;uðxÞ ¼ 0; x ¼ 0;p:

�ð4:1Þ

The system describes a diffusive biological species, where u represents the spatial density of the species at time t and d de-notes its respective free diffusion rate. The real number a is the net birth rate of the species, b is the crowding-effect coef-ficient and a is the self-diffusion rate.

Now we use uð0Þ ¼ M;uð0Þ ¼ d1/ as an initial iteration in the iteration process

�D½ðdþ auðmÞÞuðmÞÞ� þ Kðdþ auðmÞÞuðmÞ ¼ Kðdþ auðm�1ÞÞuðm�1Þ þ uðm�1Þða� buðm�1ÞÞ;�D½ðdþ auðmÞÞuðmÞÞ� þ Kðdþ auðmÞÞuðmÞ ¼ Kðdþ auðm�1ÞÞuðm�1Þ þ uðm�1Þða� buðm�1ÞÞ;uðmÞðxÞ ¼ uðmÞðxÞ ¼ 0; x ¼ 0;p;

8><>:

where m ¼ 1;2; . . .

To circle the difficulty caused by the nonlinear diffusion term, we rewrite the above iteration as:

�DwðmÞ þ KwðmÞ ¼ Kðdþ auðm�1ÞÞuðm�1Þ þ uðm�1Þða� buðm�1ÞÞ;�DwðmÞ þ KwðmÞ ¼ Kðdþ auðm�1ÞÞuðm�1Þ þ uðm�1Þða� buðm�1ÞÞ;wðmÞðxÞ ¼ wðmÞðxÞ ¼ 0; x ¼ 0;p;

uðmÞ ¼ �dþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2þ4awðmÞp

2a ;

uðmÞ ¼ �dþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2þ4awðmÞp

2a ;

8>>>>>>>><>>>>>>>>:

ð4:2Þ

where m ¼ 1;2; . . .

Table 1Numerical results for the sequences of upper and lower solutions.

Iteration N x ¼ 0:1p x ¼ 0:2p x ¼ 0:4p x ¼ 0:6p x ¼ 0:8p x ¼ 0:9p

10 u 0.1663 0.2071 0.2157 0.2156 0.2032 0.1484u 0.0036 0.0064 0.0098 0.0097 0.0059 0.0029

50 u 0.0255 0.0410 0.0534 0.0529 0.0385 0.0214u 0.0041 0.0073 0.0110 0.0108 0.0067 0.0034

150 u 0.0099 0.0171 0.0248 0.0244 0.0159 0.0082u 0.0051 0.0090 0.0135 0.0132 0.0083 0.0042

250 u 0.0077 0.0134 0.0197 0.0194 0.0124 0.0064u 0.0057 0.0101 0.0150 0.0148 0.0093 0.0047

350 u 0.0070 0.0122 0.0180 0.0177 0.0112 0.0058u 0.0061 0.0107 0.0159 0.0156 0.0098 0.0050

450 u 0.0067 0.0117 0.0173 0.0170 0.0108 0.0055u 0.0062 0.0110 0.0163 0.0160 0.0101 0.0052

550 u 0.0065 0.0114 0.0170 0.0167 0.0106 0.0054u 0.0063 0.0111 0.0165 0.0163 0.0103 0.0052

0 0.5 1 1.5 2 2.5 3 3.50

0.005

0.01

0.015

0.02

0.025

n=250

n=350

n=450

n=550

n=50

n=150

n=350

n=550

upper and Lower sequences

Upper sequences

Lower sequences

Fig. 1. Sequences of upper and lower sequences with initial values uð0Þ ¼ 1 and uð0Þ ¼ 0:01 sin x.

3406 M. Li et al. / Applied Mathematical Modelling 34 (2010) 3400–3407

Take a ¼ 2; b ¼ 3; d ¼ 1;a ¼ 0:5, then let K ¼ 4;M ¼ 1 and d ¼ 0:01 ensure that the condition (2.10) holds. We useuð0Þ ¼ 1;uð0Þ ¼ 0:01 sin x as an initial iteration and compute the respective sequences fuðmÞg; fuðmÞg. Numerical results of thesesequences are given in Table 1 and are also sketched in Fig. 1. Numerical results show that the sequences fuðmÞg; fuðmÞg ob-tained from (4.2) with uð0Þ ¼ 1 and uð0Þ ¼ 0:01 sin x, converge monotonically to a solution of the problem (4.1).

5. Concluding remarks

In this paper, we have considered a competitor–competitor–mutualist three-species model with cross-diffusion. The dif-fusion in the reaction–diffusion system usually represents the natural dispersive force of movement of an individual, and thecross-diffusion describes the mutual interferences between individuals. The system (1.1) means that, in addition to the dis-persive force, the species exercise as self-defense mechanism to protect themselves from the competitor and chase themutualist.

One of the main problems for the strongly coupled systems with homogeneous Dirichlet boundary conditions is the exis-tence of positive solutions, which is usually called coexistence. We circle the difficulty caused by the nonlinear diffusion termand obtain the sufficient conditions for the system to have a positive solution. Our results show that this system (1.1) pos-sesses at least one coexistence state if cross-diffusions and cross-reactions are weak. For the strongly coupled systems with

M. Li et al. / Applied Mathematical Modelling 34 (2010) 3400–3407 3407

homogeneous Neumann boundary conditions, the non-constant positive solution is referred as the spatial pattern, whichattracts much attention recently.

The method of upper and lower solutions and its associated monotone iterations gives an effective technique. Moreover,the true solution can be constructed. Compared to existing results such as the existence of positive solution for the stronglycoupled elliptic systems, to the best of our knowledge, there have been very few results for the long time behaviors of thecorresponding strongly coupled parabolic systems, and therefore, this model is worth further investigations.

Acknowledgement

The work is partially supported by PRC Grant NSFC 10671172 and also by ‘‘Blue Project” of Jiangsu Province.

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