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] . Morpho- genesis 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] con- sidering 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., d, and 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

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

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 be any solution of (4) except. Then there exists a constant C such that

for, where.

Our main aim here is to prove the above theorem. In order to prove this, let us first prove following results:

Lemma 1. Let be any nontrivial solution of (4). Then and for. Fur-

thermore, if, then for.

Proof. Let. Then applying maximum principle at we get

. That is, , which implies

Therefore,. Let. Again applying maximum principle at we

get. That is, , which implies.

That is. Since, from the second equation of (4) we have

Applying strong maximum principle to the above equation we get in, provided. The proof is complete. □

Lemma 2. Assume that is any solution of (4). If, then for.

Proof. Let. Then

Also, on. Then applying maximum principle we have, which implies the required inequality. □

Lemma 3. Assume that is any solution of (4). If, then for.

Proof. Put, Then

Since on, the maximum principle gives the required inequality. □

Lemma 4. Let be any solution for (4). Then there exist a constant, such that for.

Proof. From lemma (1), we have

From lemma (2) we get for all From lemma (3) we get

. Combining these two inequalities we have (say). Then from (5) we have

Therefore, for all □

Lemma 5. Assume that is any solution of (4) except. Then there exist a positive constant such that for all.

Proof. The second equation of the system (4) can be written as in, where

. From lemmas (1) and (3) we get and for any. Then Set According to Harnack inequality [15] there

exists a parameter such that

Denote and. Then applying maximum principle for the second equ-

ation of (4), we have. Since, we get

From the inequalities (8) and (9) we get

for all. That is for all,

where. □

Proof of Theorem (1): From lemma (3) we have, , and, from

lemma (5) we have for all. Set

Then we have □

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 and Y as follows:

and Here C is the con-

stant defined in Equation (10) and is any solution of the system (4). Set an auxiliary parameter for, where M is a large constant to be determined. Let denote any constant solution of the system (4). Linearizing the system (4) when at S takes the form:

Denote

and

Thus,. Then (4) and (11) can be written as

respectively. Define, and That is is a compact perturbation of the identity operator. According to the definition of there is no fixed point of T on the boundary. Thus, is a positive solution of (12) if and only if So, the Leray-Schauder degree is well defined. Furthermore, we have

The index of at is defined as

where is the number of negative eigenvalues of.

Lemma 6. The eigenvalues, of are given by the equation

where and Here p and q are the trace and determinant of the matrix A respectively and are the positive eigenvalues of the eigenvalue problem

such that. Also the discriminant D of (13) is given by

Proof. The eigenvalues of satisfies

This implies

By simplifying we get

This implies

where and The discriminant of (13) is

□

Now we consider the cases and separately.

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 non- constant 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.