Positive Stationary Solutions of Cross-Diffusive Competition Model with A Protection Zone

Positive Stationary Solutions of Cross-Diffusive Competition Model with A Protection Zone

Ei Ei Kyaw

Department of Engineering Mathematics Polytechnic University (Myitkyina) Myanmar

Theint Theint Thu

Department of Engineering Mathematics Polytechnic University (Myitkyina) Myanmar

Abstract – In this paper, positive stationary solutions of a heterogeneous, and satisfies x bx 0 in 1 and cross-diffusive competition model with a protection zone for the weak competitor are examined. The asymptotic behaviour of x 1 0 and bx b 0 in \ 1 ; 0 is a constant; positive stationary solutions is obtained for any birth rate as the cross-diffusion coefficient tends to infinity. denotes the Laplacian operator on the space variable x; ux, t and vx, t represent the population densities of

Keywords – Cross-diffusion; heterogeneous environment; stationary solution

the respective competing species.

In the model, u lives in the larger habitat ,

and

1 is

INTRODUCTION

its protection zone, where u can leave and enter the

protection zone freely, while v can only live outside 1.

Competition is one of the most essential mechanisms in ecology, shaping the distribution and abundance of biological species. When two species utilize similar resources, their interaction may lead to coexistence, competitive exclusion, or complex spatial-temporal dynamics depending on environmental conditions and intrinsic population traits. To analyze such processes rigorously, mathematical models, especially those based on systems of partial differential equations, have become indispensable tools.

The effects of environmental heterogeneity with large

Thus we impose a no-flux boundary condition on 1 for v. On , a no-flux boundary condition is also assumed for both species, and no individuals cross the boundary . Throughout the paper, we write \ .

It should be noted that k x vu is the cross-diffusion term to model the habitat segregation phenomena between two competing species. From the cross-diffusion term, u diffuses to low density regions of v in their common living habitat

cross-diffusion are studied. In the following, we consider

and the coefficient k denotes the sensitivity of the

Lotka-Volterra cross-diffusive competition model with a protection

zone

ut 1 kx v u u u bx v, x , t 0,

competitor u to the population pressure from the other competitor v.

The corresponding stationary problem is

We denote by D , U and N , U the first eigenvalue

where

is a bounded domain with

of over the bounded domain U with Dirichlet and Neumann boundary conditions, respectively. We usually omit

U in the notation if U . If the potential function is

smooth boundary

, 1 is a subdomain of with smooth

omitted, then we understand 0. It is well-known that the

boundary

1

and

1 ; n is the outward unit normal

following properties hold:

vector on the boundary; positive constants and are the

1. the mapping q B q,U : L U is continuous

   intrinsic growth rates of the respective species;

   bx and d 0 are the interspecific competitive pressure on

   1

   with B D or B N;

   1. and v, respectively;

      x and b x

      are spatially

2. B q , U B q , U if q q and q q

   with

3. D q, U D q, U if U U , and

If 0, x , then 0, x , and satisfies

The usual norm of the space Lp U for p 1, is

defined by

Then, it follows that is an eigenvalue of (4), and satisfies Re N , 0. So, the eigenvalues with

1 p

u uxp dx and u max ux .

associated eigenfunctions of the form , 0 possess positive real parts.

On the other hand, if

then is an

1. Properties of Stability and A Priori Estimates

   In this section, we will show the stability of semi-trivial solutions and a priori estimates for any positive solution of (2).

   It is clear that the steady-state problem (2) admits two

   eigenvalue of the following problem:

   semi-trivial solutions , 0 and 0, in addition to the n

   trivial solution 0, 0. The stabilities of such trivial and semi- trivial solutions are shown in the following lemma.

   1. Lemma

      We have the following stability results:

      Then, if d, we see that any eigenvalue of (5) satisfies Re N d , d 0.

      Thus, the real parts of any eigenvalue of (3) are positive, , 0 is asymptotically stable.

      1. the trivial solution 0, 0 is always unstable;

         If d,

         then

         d 0

         is an eigenvalue to

      2. the semi-trivial solution , 0

         is asymptotically

         the second equation of (3) with a unique positive

         stable if d, while it is unstable if d;

         eigenfunction

         normalized as

      3. the semi-trivial solution 0,

         is asymptotically

         for denote by

         N bx

         1 kx bx ,

         stable if

         1 1 kx , 0,

         while it is unstable if

         we know that

         is an eigenvalue of (3) with an

         1 1 kx Proof:

         eigenfunction , , , 0 is unstable.

         The proof of (iii) is rather similar to that of [7].

         The proof of (i) is clear. Therefore, we start proof of (ii).

         The linearized parabolic system of (1) at

   2. Lemma

      Assume spatial dimension N 3 and U 1 kx v u,

      is

      there exists a positive constant C independent of k such that any positive solution u, v of (2) satisfies

      thus, the corresponding spectral problem is

      kx bx , x ,

      The proof of lemma is the same as in [6]. So, we omit it.

2. Asymptotic Behavior of Positive Solutions as k

    

   In this section, we study the asymptotic behavior of positive solutions of (2) for any , 0 as k , and show(3)

   the structure of the positive solution set of the limiting system.

   For the asymptotic behavior, we have the following theorem.

   1. Theorem

      Assume spatial dimension N 3,

      and

      Then by passing to a subsequence if necessary, the following conclusions hold.

      It follows that

      in .

      1.  

         In the following, we discuss the cases

         ui , kivi u, w

         positive solution of

         uniformly in

         where u, w

         is a

         vi 0 and vi in , respectively.

         in this case we first show that

      2. f limk

         Let u , v , k

         be any sequence such that ui , vi is a

         So, we see that

         i converges uniformly to some

         positive solution of (2) with k ki and ki , we further set

         nonnegative constant C2 , i.e., x viui C2 , thus viui 0

         Ui 1 kix vi ui .

         uniformly in . Since v v v du , letting

         Since U

         and v

         are uniformly bounded by,

         we see that

         and v is a nonnegative weak,

         virtue of Lemma 2.2,

         uniformly bounded for p N, we deduce that there exists a

         subsequence of k , still denoted by k

         , such that

         While v 1, it is clear that v 0 in,

         thus,

         0, it is a contradiction.

         for some nonnegative function U, v C1 C1 .

         If k v 0,

         we also set

         By Lp estimates and the Sobolev embedding theorem, and

         are uniformly bounded, we

         Thus, the Lp

         estimates and the Sobolev embedding

         i , i ,

         theorem deduce that

         can show that subject to a subsequence, i

         converges

         in C1 .

         uniformly to some nonnegative constant C1, then i

         x uivi C1 uniformly. As x 0 in 0 , we know that

         Thus,

         ui also converges uniformly to

         , and is a

         this constant C1 must be zero, i.e.,

         ui vi 0

         uniformly in

         nonnegative weak solution of the equation

         . Furthermore, as

         i , vi v

         in C1 ,

         and v is a

         nonnegative weak solution of

          

         0, x ,

         Furthermore, since

         ui , and

         wi , are uniformly

         0, x .

         bounded, the standard elliptic regularity deduces that

         n

         ui , wi u, w, where u, w is a positive smooth solution

         Hence, 0 or ,i.e., u

         i 0 or ui uniformly in

         of (6).

         1. If k v , we must have v
         • in , then

         . If ui 0,

         since

         vi vi dui dx 0, we see that for

         i i , i

         sufficiently large i, vi dui

         • 0, we derive it is a

           U

           i

            

           ui

           • 0 in C1

             .

             contradiction; if ui , then

             vi v, and v is a

             1 ki vi

             nonnegative weak solution of

             From a similar argument to that of [11], we can deduce that

             Udx 0, and U Udx 0,

             By virtue of v , 1, we know that v 0 and v 0

             Thus U2 dx Udx U Udx 0.

             in , thus v 0 in . So d, it is a contradiction.

             in C1 .

             The proof of the

             Therefore, we see that as

             vi 0, ki v

           • theorem completes.

             by passing to a subsequence if necessary. Set wi ki vi , then

             bx

             Finally, we give the positive solution set of the limiting

             system (6). Set U 1 x w u, then (6) is equivalent to

             the following system

             0, by Lemma 2.2 we can know that

         for large i. Furthermore, since

         some calculations deduce that

         By virtue of the local bifurcation theory and regarding

         as the bifurcation parameter, we give the following local bifurcation result.

         ui ui dx b ui vi dx. Then,

   2. Lemma

      Positive solutions of (7) bifurcate from

      ui , ui dx b ui vi dx

      , 0, X : 0 if and only if d. To be

      b vi , ui dx b vi , ui dx,

      precise, all positivesolutions of (7) near ,0,d X

      can be parameterized as

      which means that

      is uniformly bounded, thus we know that

      is smooth with respect to s and

      min u C min C max C max u .

   3. Theorem

Assume spatial dimension N 3, regarding as the

Furthermore, as the first equation of (7) is the same in [6],

bifurcation parameter, an unbounded branch p of positive

we can deduce that max U C1 for a large number

solutions of (7) bifurcates from the semi-trivial solution curve

C independent of .

So, from the second equation of (7), we see that

for a large positive number C independent of ,

is bounded, the elliptic regularity theory

Proof:

and the Sobolev embedding theorem deduce that

C1 is

Let p E be the maximal connected set of the local

bounded. Thus, we see that

must be unbounded.

bifurcation branch stated in Lemma 3.2 satisfying

Then, max w

is unbounded. Therefore, there exists a

U, w, E \ , 0, d :

p sequence 0,C such that w

U, w satisfies (7) with E C1 C1 ,

Setting W

i , then

Define

then we can

Since P P P , if not, there exists a sequence

such that

So, we know that W

• W, W 0, x and satisfies

together with the local bifurcation

result, we know that (8) holds, and the proof of the theorem

then for large i,

thus

completes.

ACKNOWLEDGMENT

it is a contradiction. If

The author would like to express unlimited gratitude

Since U 0, we see that U . While , 0, d is the

only bifurcation point of positive solutions of (7) bifurcates from , 0 with bifurcation parameter , we know that

d, it is a contradiction. Thus, p is contained in the set of positive solutions of (7). By a similar argument to that of [6], we can further know that p is unbounded in E .

to his foremost, respectable and noble teacher, Associate Professor Dr. Aung Zaw Myint, Department of Mathematics, University of Mandalay for his encouragements on doing research. The author acknowledges the unknown reviewers for their suggestions and patient review on this paper.

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