Stationary Solutions of a Mathematical Model for Formation of Coral Patterns ()
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.