Received 23 April 2015; accepted 7 June 2015; published 10 June 2015

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 pro- blem 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] . 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:

(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., d, and are positive constants. The local and global stabilities of the solutions of the corresponding system of ordinary differential equations

(2)

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

(3)

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

1.1. Constant Solutions (Steady States)

There are three constant solutions (homogeneous steady sates), and

for the system (1). Here, , , , and for.

1.2. Stationary Problem

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

(4)

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). 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

(5)

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

(6)

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

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

(7)

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

(8)

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

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

(9)

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

(10)

Then we have □

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 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:

(11)

Denote

and

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

(12)

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

(13)

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

(14)

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

Proof. The eigenvalues of satisfies

This implies

(15)

By simplifying we get

This implies

where and The discriminant of (13) is

□

Now we consider the cases and separately.

3.1. The Case α > 2λ

In this case there are two constant fixed points of in which are and . Now we deal with the case. Let, and be corresponding P value, Q value and the discriminant of (13) respectively. Also let and be the corresponding p and q values.

3.1.1. The Case

The solutions for of the Equation (13) can be written as

and

If then and. It can be shown that. That is, if then only one negative solution exists for (13). It follows that if is negative we can find

, such that. Therefore,

3.1.2. The Case

Next we deal with the case. Let, and be corresponding P value, Q value and the corresponding discriminant of (13). Also let and be the corresponding p and q values. In this case we can find, such that is negative when. Therefore there are exactly one neg- ative solutions for the corresponding Equation (13) when. Therefore. Also,

(16)

Theorem 2. Assume that, and are satisfied. If is even, then (4) has at least one positive nontrivial solution.

Proof. Homotopy invariance property show that

By setting as sufficiently large constant we get,. Therefore,

(17)

Also, we have

(18)

The relations (17) and (18) contradict the homotopy invariance property for,. Thus the proof is complete. □

3.2. The Case α = 2λ

In this case the constant fixed point of in is uniquely determined by. The Leray- Schauder index at this point is:

where is the number of real negative eigenvalues (counting algebraic multiplicity) of.

In this case and. Then,

and

If:

Then and. Therefore, if, then. That is if, there is

exactly one negative solution for (13). No negative solutions for (13) if

If:

In this case, Q is negative if. Then there is exactly one negative solution for (13).

Let be the number of, satisfying. Then,.

Theorem 3. Assume that. If is odd, then (4) admits at least one positive non-constant solution.

Proof. From the Homotopy invariance property we have

Suppose that (4) has no non-constant solutions if. Also

provided is sufficiently large. On the other hand

These two relations contradict the homotopy invariance property for,. Thus the proof is complete. □

4. 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 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.

References