Stationary Solutions of a Mathematical Model for Formation of Coral Patterns

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.


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.
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-dimensionalized version of this mathematical model takes the form: 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 1 λ are positive constants.The local and global stabilities of the solutions of the corresponding system of ordinary differential equations are discussed in [11].Turing type instability analysis and patterns formation behavior of the model (1) subject to the boundary conditions 0, , 0, , are discussed in [12].Here ∇ denotes the gradient operator and n denotes the outward unit normal vector to the domain boundary ∂Ω .

Stationary Problem
In this article, the existence of the stationary solutions of the system (stationary problem corresponding to the system (1)): ( ) 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].

Priori Estimates
In this section we obtain estimates for the upper and lower bounds for the stationary solutions of the system (4).This boundedness property can be expressed as the following theorem: Theorem 1.Let ( ) , u v be any solution of (4) except 1 S .Then there exists a constant C such that ( ) ( ) 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 ( ) . Then applying maximum principle at 0 x we get ( ) ( ) ( ) Therefore, ( ) That is ( ) Ω Applying strong maximum principle to the above equation we get ( ) Then applying maximum principle we have ( ) max 0 x p x ∈Ω ≤ , which implies the required inequality.□ Lemma 3. Assume that ( ) on ∂Ω , the maximum principle gives the required inequality.□ Lemma 4. Let ( ) , u v be any solution for (4).Then there exist a constant From lemma (2) we get ( ) ( ) (5) we have Therefore, ( ) ( ) . Then there exist a positive constant The second equation of the system (4) can be written as 0 v Av ∆ + = in Ω , where ( ) ( ) . From lemmas (1) and (3) we get ( ) 1 u x ≤ and ( ) According to Harnack inequality [15] there exists a parameter Then applying maximum principle for the second equ- ation of (4), we have ( ) From the inequalities ( 8) and ( 9) we get min , , max ( ) , , Proof of Theorem (1): From lemma (3) we have, ( ) and, from lemma (5) we have ( ) Then we have

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: Here C is the constant defined in Equation ( 10) and ( ) , u v is any solution of the system (4).Set an auxiliary parameter ( ) , where M is a large constant to be determined.Let ( ) w denote any constant solution of the system (4).Linearizing the system (4) when . Then ( 4) and ( 11) can be written as ( ) ( ).
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)  n (14) such that 1 2 3 µ µ µ < < < .Also the discriminant D of ( 13) is given by ( ) ( ) ( ) The eigenvalues µ of ( ) By simplifying we get α λ = separately.

The Case α > 2λ
In this case there are two constant fixed points of t T in Θ which are ( ) ( ) . Now we deal with the case , u v ≡ * w . Let 2 P , 2 Q and 2 D be corresponding P value, Q value and the discriminant of ( 13) respectively.Also let 2 p and 2 q be the corresponding p and q values.

The Case
( ) The solutions for µ of the Equation ( 13) can be written as Q < then only one negative solution exists for (13).It follows that if 2 Q is negative we can find 1 m , 2 m ( ) . Let 3 P , 3 Q and 3 D be corresponding P value, Q value and the corresponding discriminant of (13).Also let 3 p and 3 q be the corresponding p and q values.In this case we can find 3 m , ( ) Therefore there are exactly one negative solutions for the corresponding Equation ( 13) when  Index , 1 The relations (17) and (18) contradict the homotopy invariance property for Thus the proof is complete.□

The Case α = 2λ
In this case the constant fixed point of t T in Θ is uniquely determined by 0 1 1 , 2 2λ . The Leray-Schauder index at this point is: where 0 σ is the number of real negative eigenvalues (counting algebraic multiplicity) of ( ) , there is exactly one negative solution for (13).No negative solutions for (13) if 2.
Then there is exactly one negative solution for (13).

Discussion
Stationary problem corresponding to a model mathematical model for formation of coral patterns is considered.
We have proved the existence of non-constant positive solutions of the stationary problem (4).Existence of nonconstant solutions to the stationary problem gives a guarantee for the existence of spatially variant time invariant solutions to the proposed reaction-diffusion system.In other words, the solution of the system reaches a steady state with spatial patterns.This is a physically important feature which gurantees the the existence of stable coral patterns of the system.
Here p and q are the trace and determinant of the matrix A respectively and m

3
then (4) has at least one positive nontrivial solution.Proof.Homotopy invariance property show that then (4) admits at least one positive non-constant solution.Proof.From the Homotopy invariance property we have