Stationary Solutions of a Mathematical Model for

Applied Mathematics, 2015, 6, 1099-1106 Published Online June 2015 in SciRes. http://www.scirp.org/journal/am http://dx.doi.org/10.4236/am.2015.66100

Stationary Solutions of a Mathematical Model for Formation of Coral Patterns Lekam Watte Somathilake1, Janak R. Wedagedera2 1

Department of Mathematics, Faculty of Science, University of Ruhuna, Matara, Sri Lanka Simcyp-CERTARA Limited, Blades Enterprise Centre, Sheffield, UK Email: [email protected], [email protected]

2

Received 23 April 2015; accepted 7 June 2015; published 10 June 2015 Copyright © 2015 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

Abstract A reaction-diffusion type mathematical model for growth of corals in a tank is considered. In this paper, we study stationary problem of the model subject to the homogeneous Neumann boundary conditions. We derive some existence results of the non-constant solutions of the stationary problem based on Priori estimations and Topological Degree theory. The existence of non-constant stationary solutions implies the existence of spatially variant time invariant solutions for the model.

Keywords Reaction-Diffusion Equations, Stationary Solutions, Priori Estimates, Topological Degree Theory

1. Introduction Most of the corals consist of colony of polyps reside in cups like skeletal structures on stony corals called calices. Polyps of hard corals produce a stony skeleton of calcium carbonate which causes the growth of the coral reefs. Polyps’ maximum diameter is a species-specific characteristic. Once they reach this maximum diameter they divide [1]. In this way, if survive, they divide over and over and form a colony. If the coral colony does not break off, it grows as its individual polyps divide to form new polyps [2]. As new polyps are formed they build new calices to reside. This causes the growth of solid matrix of the stony corals. Various modeling approaches on coral morphogenesis processes have been reported in [1] [3]-[9]. Morphogenesis of branching corals has been described by Diffusion-Limited Aggregation (DLA) type models in [1] [6] [10]. A reaction diffusion type mathematical model for growth of corals in a tank is proposed in [11] [12] considering the nutrient polyps interaction. This model is derived based on the model appear in [8]. The non-

How to cite this paper: Somathilake, L.W. and Wedagedera, J.R. (2015) Stationary Solutions of a Mathematical Model for Formation of Coral Patterns. Applied Mathematics, 6, 1099-1106. http://dx.doi.org/10.4236/am.2015.66100

L. W. Somathilake, J .R. Wedagedera

dimensionalized version of this mathematical model takes the form:

∂u  = ∆u + 1 − u − α 2 v 2 u , x ∈ Ω ⊂  2 , t > 0  ∂t  ∂v  2 2 2 = d ∆v − λ v + α v u , x ∈ Ω ⊂  , t > 0  ∂t  ∂w  2 = λ1v, x ∈ Ω ⊂  , t > 0.  ∂t 

(1)

Here, u and v are vertically averaged nondimensionalized concentrations of dissolved nutrients (foods of coral polyps) and aggregating solid material (calcium carbonate) on the coral reefs respectively. α , d, λ and λ1 are positive constants. The local and global stabilities of the solutions of the corresponding system of ordinary differential equations

du  =1 − u − α 2 uv 2  dt  dv  = −λ v + α 2 uv 2  , dt  dw  = λ1v  dt 

(2)

are discussed in [11]. Turing type instability analysis and patterns formation behavior of the model (1) subject to the boundary conditions

∇= u ⋅ n 0, x ∈ ∂Ω,   ∇= v ⋅ n 0, x ∈ ∂Ω, 

(3)

are discussed in [12]. Here ∇ denotes the gradient operator and n denotes the outward unit normal vector to the domain boundary ∂Ω .

1.1. Constant Solutions (Steady States) There are three constant solutions (homogeneous steady sates) S1 ≡ ( us1 , vs1 ) , S2 ≡ ( us 2 , vs 2 ) and S3 ≡ ( us 3 , vs 3 ) for the system (1). Here us1 = 1 , vs1 = 0 , us 2 =

us 3 =

α + α 2 − 4λ 2 α − α 2 − 4λ 2 , vs 2 = , 2α 2αλ

α − α 2 − 4λ 2 α + α 2 − 4λ 2 and vs 3 = for α > 2λ . 2αλ 2α

1.2. Stationary Problem In this article, the existence of the stationary solutions of the system (stationary problem corresponding to the system (1)): f ( u ( x ), v ( x ) )      2 2 ∆u + 1 − u − α uv= 0, x ∈ Ω  (4)  g ( u ( x ), v ( x ) )     2 d ∆v + −λ v + α 2 uv= 0, x ∈ Ω 

(

)

(

)

subject to no-flux boundary conditions (3), is discussed. The main result presented in this article is the existence of non-constant positive solutions. These existence results are proved based on the Priori estimates and Topological Degree theory [13]-[15].

2. Priori Estimates In this section we obtain estimates for the upper and lower bounds for the stationary solutions of the system (4).

1100


This boundedness property can be expressed as the following theorem: Theorem 1. Let ( u , v ) be any solution of (4) except S1 . Then there exists a constant C such that

1 ≤ u ( x), v ( x) ≤ C C for x ∈ Ω , where Ω = Ω  ∂Ω . Our main aim here is to prove the above theorem. In order to prove this, let us first prove following results: Lemma 1. Let ( u , v ) be any nontrivial solution of (4). Then 0 ≤ u ( x ) ≤ 1 and v ( x ) ≥ 0 for x ∈ Ω . Furthermore, if ( u , v ) ≠ S1 , then v ( x ) > 0 for x ∈ Ω . Proof. Let= u 0 u= ( x0 ) min x∈Ω u ( x ) . Then applying maximum principle at x0 we get

2 2 f ( u ( x 0 ) , v ( x 0 ) ) ≤ 0 . That is, 1 − u ( x 0 ) − α u ( x 0 ) v ( x 0 ) ≤ 0 , which implies

u ( x0 ) ≥

1 > 0. 1 + α 2 v2 ( x0 )

(5)

Therefore, min x∈Ω u ( x ) > 0 . Let= u0 u= ( x0 ) max x∈Ω u ( x ) . Again applying maximum principle at x0 we 1 get f ( u ( x0 ) , v ( x0 ) ) ≥ 0 . That is, 1 − u ( x0 ) − α 2 u ( x0 ) v 2 ( x0 ) ≥ 0 , which implies u ( x0 ) ≤ ≤1. 1 + α 2 v 2 ( x0 ) That is max x∈Ω u ( x ) ≤ 1 . Since 0 ≤ u ( x ) ≤ 1 , from the second equation of (4) we have

d ∆v − λ v =−α 2 uv 2 ≤ 0 in Ω. Applying strong maximum principle to the above equation we get v ( x ) > 0 in □ Ω , provided v ( x ) ≠ 0 . The proof is complete. Lemma 2. Assume that ( u , v ) is any solution of (4). If λ > d , then u ( x ) + dv ( x ) ≤ 1 for x ∈ Ω . Proof. Let p =u + dv − 1 . Then ∆p − p =∆u + d ∆v − u − dv + 1 =−1 + u + λ v − u − dv + 1 =( λ − d ) v ≥ 0.

Also, ∇p ⋅ n =∇u ⋅ n + d ∇v ⋅ n =0 on ∂Ω . Then applying maximum principle we have max x∈Ω p ( x ) ≤ 0 , which implies the required inequality. □ Lemma 3. Assume that ( u , v ) is any solution of (4). If λ < d , then u ( x ) + dv ( x ) ≤ d λ for x ∈ Ω . Proof. Put q =u + dv − d λ , Then

∆q −

λ d

q =∆u + d ∆v − q =−1 + u + λ v −

λ

d  λ  u + dv −  =1 −  u ≥ 0. λ  d d

Since ∇q ⋅ n =0 on ∂Ω , the maximum principle gives the required inequality. Lemma 4. Let ( u , v ) be any solution for (4). Then there exist a constant C1 ( d , λ , α ) > 0 , such that u ( x ) ≥ C1 ( d , λ , α ) for x ∈ Ω . Proof. From lemma (1), we have 1 u ( x0 ) ≥ . 2 2 1 + α ( v ( x0 ) )

1− u ( x)

□

(6)

1 for all x ∈ Ω. From lemma (3) we get d d 1d  1 1 1  v ( x ) ≤  − u ( x )  ≤ . Combining these two inequalities we have v ≤ max  ,  = C * (say). Then from d λ  λ d λ  (5) we have 1 u ( x 0 ) ≥ C1 = 2 *2 . (7) 1+ α C From lemma (2) we get v ( x ) ≤

≤

□ Therefore, u ( x ) ≥ u ( x 0 ) ≥ C1 for all x ∈ Ω. Lemma 5. Assume that ( u , v ) is any solution of (4) except S1 ≡ (1, 0 ) . Then there exist a positive constant C2 such that v ( x ) ≥ C2 for all x ∈ Ω . Proof. The second equation of the system (4) can be written as ∆v + Av =0 in Ω , where

1101


1 1  d . From lemmas (1) and (3) we get u ( x ) ≤ 1 and v ( x ) ≤ max  ,  for any x ∈ Ω . Then d λ  1 1  1 1   1 1  A ( x ) ∞ ≤ λ + max  ,  . Set= µ λ + max  ,  . According to Harnack inequality [15] there d d  d λ   d λ  exists a parameter C2′ ( N , Ω, µ ) > 0 such that A ( x= )

( λ − uv )

min v ( x ) ≥ C2′ ( N , Ω, µ ) max v ( x ) . x∈Ω

x∈Ω

(8)

v ( x ) v= Denote max = ( x0 ) vˆ and max u ( x0 ) = uˆ . Then applying maximum principle for the second equx∈Ω

x∈Ω

ˆ ˆ − λ ) ≥ 0 . Since vˆ > 0 , we get ation of (4), we have vˆ ( uv

vˆ ≥

λ ≥λ uˆ

( u ( x ) ≤ 1 for all x ∈ Ω ) .

(9)

From the inequalities (8) and (9) we get v ( x ) ≥ min v ( x ) ≥ C2′ ( N , Ω, µ ) max v( x) ≥ λ C2′ ( N , Ω, µ ) for all x ∈ Ω . That is v ( x ) ≥ C2 for all x ∈ Ω , x∈Ω

x∈Ω

where C2 = λ C2′ .

□

1 1 1  Proof of Theorem (1): From lemma (3) we have, u ( x ) ≥ u ≥ C1 = 2 *2 , v ( x ) ≤ max  ,  and, from 1+ α C d λ  lemma (5) we have v ( x ) ≥ C2 for all x ∈ Ω . Set 1 1  1 1  C = max  , ,1, max  ,  .  d λ   C1 C2

Then we have

1 ≤ u ( x ) , v ( x ) ≤ C. C

(10) □

3. Existence of Non Constant Stationary Solutions In this section we investigate the existence of non-constant solutions to (4). For this, the degree theory for compact operators in Banach spaces [15] [16] are used as the main mathematical tool. Define the spaces Θ and Y as follows:

{

1   = Θ ( u , v ) ∈ C ( Ω ) × C ( Ω ) : < u , v < C  , C  

}

Y =( u , v ) ∈ C 2 ( Ω ) × C 2 ( Ω ) : ∇u ⋅ n =∇v ⋅ n =0 on ∂Ω and Y + = {( u , v ) ∈ Y : u , v > 0}. Here C is the constant defined in Equation (10) and ( u , v ) is any solution of the system (4). Set an auxiliary parameter dt = td + (1 − t ) M for t ∈ [ 0,1] , where M is a large constant to be determined. Let = S w= ( u* , v* ) denote * any constant solution of the system (4). Linearizing the system (4) when d = dt at S takes the form: ∆u + fu ( u* , v* ) u + f v ( u* , v= 0, x ∈ Ω  * )v  g (u , v ) g (u , v )  ∆v + u * * u + v * *= v 0, x ∈ Ω  dt dt  ∇u ⋅ n =∇v ⋅ n =0 x ∈ ∂Ω.

Denote  f ( u, v )    Gt ( w ) =  g ( u , v )  ,  d  t  

and

1102

(11)

 fu ( u* , v* )  A =  gu ( u* , v* )  dt 


f v ( u* , v* )   g v ( u* , v* )  .  dt 

Thus, Dw Gt ( w* ) = A . Then (4) and (11) can be written as −∆ = w Gt ( w ) in Ω, = ∇w 0 on ∂Ω,

(12)

and − ∆= w A= w Dw Gt ( w* ) in Ω, ∇= w 0 on ∂Ω,

respectively. Define Tt ( w ) = ( −∆ + I ) ( Gt ( w ) + w ) , and Ft ( w = ) w − Tt ( w ) . That is Ft (.) is a compact perturbation of the identity operator. According to the definition of Θ there is no fixed point of T on the boundary ∂Θ . Thus, w is a positive solution of (12) if and only if Ft ( w ) = 0 in Y + . So, the Leray-Schauder −1 degree deg ( Ft (.) , Θ, 0 ) is well defined. Furthermore, we have Dw Ft ( w* )= I − ( −∆ + I ) ( A + I ) . The index of Ft at w* is defined as −1

Index ( Ft (.) , w* ) =

σ* (t )

( −1)

,

where σ * ( t ) is the number of negative eigenvalues of Dw Ft ( w* ) . Lemma 6. The eigenvalues, µ of Dw Ft ( w* ) are given by the equation 2 0, (1 + µm ) µ 2 + P µ + Q =

(13)

where P = (1 + µm )( p − 2µm ) and Q =µ − p µm + q. Here p and q are the trace and determinant of the matrix A respectively and µm ( m = 1, 2,) are the positive eigenvalues of the eigenvalue problem 2 m

= −∆u µ u in Ω   ∂u , = 0 on ∂Ω  ∂n 

(14)

such that µ1 < µ2 < µ3 <  . Also the discriminant D of (13) is given by

D =P 2 − 4 (1 + µm ) Q =(1 + µm ) 2

2

(p

2

)

− 4q .

Proof. The eigenvalues µ of Dw Ft ( w* ) satisfies

Dw Ft ( w* ) = µ w

µw ( I − DwTt ( w* ) ) w =

( I − ( −∆ + I )

−1

( A + I )) w = µ w

( − ( ∆ + A) ) w= µ ( −∆ + I ) w ( ( µ − 1) ∆I − ( µ I + A) ) w =0. This implies f v ( u* , v* ) (1 − µ ) µm − µ − fu ( u* , v* ) −dt−1 gu ( u* , v* ) µm (1 − µ ) − µ − dt−1 g v ( u* , v* )

= 0.

By simplifying we get

(1 + µm ) µ 2 + (1 + µm ) ( fu ( u* , v* ) + dt−1 gv ( u* , v* ) − 2µm ) µ + µm2 − dt−1 gv ( u* , v* ) + fu ( u* , v* ) + dt−1 ( fu ( u* , v* ) g v ( u* , v* ) − gu ( u* , v* ) f v ( u* , v* ) ) = 0. 2

This implies 2 0, (1 + µm ) µ 2 + P µ + Q =

where P = (1 + µm )( p − 2µm ) and Q =µm2 − p µm + q. The discriminant of (13) is

1103

(15)


(

)

2 2 2 P 2 − 4 ( µm + 1) Q =( µm + 1) ( p − 2 µm ) − 4 µm2 − p µm + q   

= ( µm + 1)

2

(p

2

)

− 4q .

□

Now we consider the cases α > 2λ and α = 2λ separately.

3.1. The Case α > 2λ In this case there are two constant fixed points of Tt in Θ which are w* ≡ w2 ≡ S2 ≡ ( u2 , v2 ) and w3 ≡ S3 ≡ ( u3 , v3 ) . Now we deal with the case w* ≡ ( u2 , v2 ) . Let P2 , Q2 and D2 be corresponding P value, Q value and the discriminant of (13) respectively. Also let p2 and q2 be the corresponding p and q values. 3.1.1. The Case w* ≡ ( u2 , v2 ) The solutions for µ of the Equation (13) can be written as

µ* =

− ( p2 − 2 µm ) + p22 − 4q2 2 (1 + µm )

and µ* =

− ( p2 − 2 µm ) − p22 − 4q2 2 (1 + µm )

.

(

)

If p22 − 4q2 > ( p2 − 2 µm ) then µ * > 0 and µ* < 0 . It can be shown that Q2 = ( p2 − 2 µm ) − p22 − 4q2 . That is, if Q2 < 0 then only one negative solution exists for (13). It follows that if Q2 is negative we can find σ t m −m −2 m1 , m2 ( 0 < m1 < m2 ) such that µm1 < µm < µm2 . Therefore, Index (Tt , w2 ) = ( −1) 2 ( ) = ( −1)( 2 1 ) . 2

2

3.1.2. The Case w* ≡ ( u3 , v3 ) Next we deal with the case w* = ( u3 , v3 ) . Let P3 , Q3 and D3 be corresponding P value, Q value and the corresponding discriminant of (13). Also let p3 and q3 be the corresponding p and q values. In this case we can find m3 , (1 < m3 ) such that Q3 is negative when 0 < µm < µm3 . Therefore there are exactly one negm ative solutions for the corresponding Equation (13) when 0 < µm < µm3 . Therefore Index (Tt , w3 ) = ( −1) 3 . Also, m ( m − m − 2) (16) deg ( I − Tt , Θ, 0 ) = Index (Tt , w2 ) + Index (Tt , w3 ) = ( −1) 2 1 + ( −1) 3 . Theorem 2. Assume that α > 2λ , Q2 < 0 and Q3 < 0 are satisfied. If m3 + ( m2 − m1 ) is even, then (4) has at least one positive nontrivial solution. Proof. Homotopy invariance property show that deg ( I − T0 , Θ, 0= ) deg ( I − T1 , Θ, 0 ) .

By setting d 0 = M as sufficiently large constant we get Index (T0 , w2 ) = −1 , Index (T0 , w3 ) = 1 . Therefore, deg ( I −= T0 , Θ, 0 ) Index (T0 , w2 ) + Index = (T0 , w3 ) 0.

Also, we have

(17)

deg ( I − T1 , Θ, 0 ) Index (T1 , w2 ) + Index (T1 , w3 ) = =( −1)

m2 − m1 − 2

+ ( −1)

m3

(18)

=±2

The relations (17) and (18) contradict the homotopy invariance property for deg ( I − Tt , Θ, 0 ) , Thus the proof is complete.

( 0 ≤ t ≤ 1) .

□

3.2. The Case α = 2λ 1 1  In this case the constant fixed point of Tt in Θ is uniquely determined by w0 =  ,  . The Leray 2 2λ  Schauder index at this point is: Index (T , w0 ) =

( −1)

σ0

,

where σ 0 is the number of real negative eigenvalues (counting algebraic multiplicity) of I − DwT ( w0 ) .

1104

In this case p =

λ − 2d t dt


and q = 0 . Then,  λ − 2d t  P= − 2µm  (1 + µm )   dt 

and  ( µ + 2 ) dt − λ  Q =µm2 − p µm =µm  m  . dt  

If µm = 0 :

λ − 2d t

and Q = 0 . Therefore, if dt < λ 2 , then P > 0 . That is if dt < λ 2 , there is dt exactly one negative solution for (13). No negative solutions for (13) if dt ≥ λ 2. If µm > 0 : Then P= p=

In this case, Q is negative if dt