Spatial Segregation Limit of a Quasilinear Competition-Diffusion System

The aim of this paper is to investigate a Volterra-Lotka competition model of quasilinear parabolic equations with large interaction. Some existence, uniqueness and convergence results for the system are given. Also investigated is its spatial segregation limit when the interspecific competition rates become large. We show that the limit problem is similar to a free boundary problem.


Introduction
In this paper, we study the spatial and temporal behavior of interacting biological species.Assuming the reaction rates of competition follow the Holling-Tanner interaction mechanism, the quasilinear reaction-diffusion model under consideration can be given by , v t x stand for their population densities of the competing species at the time t and at the habitat x ∈ Ω .i a ( ) is the respective intrinsic growth rates, 1 b and 2 c represent the intra-specific competition rates, whereas 1 c and 2 b represent the inter-spe- cific competition rates.The boundary condition models the fact that species have no-flux near the boundary, where γ is the outward normal unit vector to ∂Ω . 1 σ may not be equal to 2 σ from an ecological point of view, but for the convenience of presentation, we may assume 1 2 σ σ = here.Quasilinear parabolic equations have received a great attention in recent years.We can refer to [1]- [6] and the references therein for more details.However, the main concerns in above works are for the existence of a global solution, a weak solution, periodic solutions, the existence-uniqueness of positive solutions, blow-up property of the solution, and the qualitative property of the solution including finite time extinction and large time behavior of the solution.
Our main interest is different from those of the above works, we mainly consider the spatial segregation limit of (1) when only the interspecific competition rates 1 c and 2 b are very large.To study this case, it is conve- nient to rewrite (1) as the following equivalent form: where β and k are positive constants derived from 1 c , 2 b and k is the only parameter which is large.For similar studies, here we refer [7]- [15] to the interested readers for more information.A striking difference between (2) and above relevant works is that the diffusion term in (2) is quasilinear.When ( ) and 0 σ = , the system (2) is reduced to the classical Volterra-Lotka competition model, which has been studied in [9], where Dancer et al. showed that the two competition species spatially segregate as k tends to infinity.Moreover, they proved that, for any 0 T > , there exist subsequences n k u and n k v of the k-dependent non- negative solutions converging weakly in ( ) ( ) 0, ; L T H Ω to the positive and negative parts respectively of a limit function w satisfying a scalar equation of the form where ( ) ( ) , and they also showed that the limit problem (P) turns out to be an explicit Stefan-like type free boundary problem.
Motivated by [9], our main purpose of this paper is to extend most of results of [9] to systems (2) with quasilinear diffusion terms.In addition, we will get the convergence results for the further improvement.Specifically, we have strong convergence in ( ) ( ) 0, ; L T H Ω .Note that the study of strong-competition limits in corresponding elliptic of parabolic systems is of interest not only for questions of spatial segregation and coexistence in population dynamics, as here and in [7] [9] [13] [16]- [19] but also is key to the understanding of phase separation in Hartree-Fock type approximations of systems of modelling Bose-Einstein condensates, see [10] [20] [21] [23], and reference therein.
To conclude, we observe that a couple of problems addressed and solved for family of solutions to (2) remains for further study in our general context: firstly, to develop a common regularity theory for the solutions of the system, which is independent of the competition rate 0 k > , as in [16]- [18] [21] [22]; secondly, to study the regularity of the class of limiting profiles, both in terms of the densities and in terms of the emerging free boundary problem, as in [10] [16] [23] [24]; thirdly, the precise description of the singular set in the emerging free boundary problem, as in [25] [26].These will be object of future investigation.
The outline of this paper is arranged as follows.In Section 2, we give some a prior estimates and some convergence results for solutions of problem (2).Section 3 is focused on the limit problem as k → ∞ .In Section 4, we get the further convergence results in the special case of 11 22 0 α α = = .Concluding remarks are given in the last section.

Preliminaries
In order to study the limit case as k → ∞ , we rewrite problem (2) as ( Throughout this paper, we let ( ) × Ω and suppose the initial functions 0 u and 0 v satisfy , and 0 . . in .
We say a pair ( ) 3).We now prove some basic facts of solutions for problem (3), which will be used later.
Lemma 1.The solution ( ) exists and is unique.Moreover, there exist constants Proof.The existence and uniqueness of solutions of (3) are followed from the standard parabolic equations theory [4].
By using the maximum principle, the solution is positive for 0 t > and x ∈ Ω .For the upper bound, it follows from the comparison principle that ( ) ( ) ∈ , where ( ) which is the solution of the problem ( ) ( ) Thus we have Similarly, there exists a constant 2 where 1 C is a constant which is independent of k.Proof.Integrating the equation for k u in (3) over T D and using Green's formula yield ( ) By Lemma 1 and noting that the right side of ( 7) is independent of k, we get (6).
where 2 C is a constant which is independent of k.Proof.Multiplying the equation for k u in (3) by k u , integrating over Ω and applying Green's formula, we yield where we have used Lemma 1.To get the first estimate of ( 8), we simply integrate the above inequality from 0 to T. The second inequality of ( 8) can be derived similarly.
In order to derive a free boundary problem, we also need to introduce a new function , which is related with ( ) ( ) The following result yields uniform boundedness of { } uniformly with respect to k.
Proof.Multiplying the Equation ( 9) with ( ) ( ) Ω , and integrating over T D using integration by parts, we get where , ⋅ ⋅ is the duality product between the space ( ) H Ω and ( ) 1 H − Ω .By Lemmas 1 and 3, we then have ) where M is a positive constant which is independent of k or ξ .This implies , 0, ; 0, ; , weakly in 0, ; and strongly in 0, ; 1 in 0, ; and . . in , as k → ∞ .Furthermore, by Lemma 2, we have Lemma 5. 0 uv = a.e. in T D .Below we manage to build the relations between u, v and w.Lemma 6.The subsequences k u and k v are such that ( ) 1 and in and . . in , as k → ∞ , where ( ) ( ) Proof.Let ( ) .
In order to prove the theorem, we need to divide our proof into three cases: In case ( ) ( ) , 0 a z t x > , according to the definition of limit, there exists a positive constant 1 N such that ( ) Due to Lemma 2, above inequality implies that Next we consider case ( ) ( ) , 0 b z t x < .We proceed as in the proof of case (a), then there exists a positive constant 2 N such that Recalling Lemma 2, we claim that , 0 and , , , as .
Otherwise, if there is a subsequence of k u , which we still denote by k u , such that ( ) Similarly, it is impossible to have that ( ) From the boundedness of k u and k v , it is easy to achieve convergence in ( ) ( ) 0, ; L T L Ω .To the end, we get (15) from ( 14).

The Limit Problem as
) In this section, we mainly consider the scalar equation First, we show that problem (18) has a weak which are defined as follows: Definition 3.1 We say that a function for all 0 T > and any test function The function defined by (15) is the unique weak solution of problem (18).Moreover, ( ) Proof.From Lemmas 1 and 3, we easily have , and Lemma 4 yields Ω is derived from by a standard regularity result (see for example [27], Theorem 3, p.287).
Multiplying ( 9) by a test function with ( ) 0 T ϕ = , and using integration by parts, we deduce Let k → ∞ along the sequence for which (12) holds.By the dominated convergence theorem and Lemma 1, we have Note that ( 16) and ( 17) yield With (20), we then have that z satisfies ( ) ( ) ( )

∫ ∫ ∫
 for all 0 T > and any test function Namely, z satisfies the differential equation in (18) as well as the homogeneous Neumann boundary condition in the sense of distributions, and the initial condition This follows easily that z is the weak solution of problem (18).It is clear from [2] that the weak solution of problem ( 18) is unique.Last, for the regularity of z, we refer to Theorems 1.1 and 1.3 in [28].
According to the above discussion, there exists a family of closed hypersurfaces .
Finally, as in [9], we rewrite a strong form of the limit problem (18), where the equations can be described a classical two-phase Stefan-like free boundary problem.
Theorem 2. Let z be a weak solution of limit problem (18), if  is smooth enough, and if the functions are smooth up to ( ) t  , then u and v satisfy ( where we suppose that 0 0 0 . . in u v a e = Ω .

Further Convergence Results
In this section, we prove that the subsequences { } , in 0, ; . and 0, ; L T H Ω .Proof.By arguments as in the proof of Theorem 1, we first obtain 0, ; Also by Lemma 7.7 in [29] and Lemma 5, we get 0, 0 . . in .
Now, multiplying the second equation in (3) by the limit u and integrating it over ( ) Integrating by parts gives Integrating (24) with respect to τ over ( ) With (4), (12) and Lemma 5, as k → ∞ , we obtain as k → ∞ .Similarly, by (12), (23) and Lemma 5, we have ( ) Next, multiplying the first equation in (3) by the limit u and integration it over ( ) Integrating above equation in τ and passing to the limit as k → ∞ yield by using ( 4), ( 12) and ( 26).Finally, multiplying the first equation in (1) again by k u and integrating it over ( ) by (28).It follows from (12) and weak lower semi-continuity that

Concluding Remarks
The study of spatial behavior of the interacting species has been attracting much attention in population ecology, in particular, in the case when the interactions are large and of competitive type.Many different models based on partial differential equations can be successfully employed to investigate the phenomenon of coexistence and exclusions of competing species.In this paper, we have attempted to study a class of quasilinear parabolic system (3) describing a Holling-Tanner's competitive interaction of two species.We prove that if inter-specific competition rates tend to infinity, then spatial segregation of the densities , u v and a scalar limit problem (21) are given.In particular, we have obtained the strong convergence results in ( ) ( ) 0, ; L T H Ω in the special case of 11 22 0 α α = = .Ecologically, our results show that competition leads to segregation.Finally, we want to mention that there are still many interesting questions to do for this kind of problem.First of all, noting that the diffusion term of the first equation in (2) can be written as describes the "self-diffusion".Naturally to ask whether our results can be extended to parabolic systems with "cross-diffusion"?Moreover, as mentioned in the introduction, we have seen that limit profiles of solutions to (2) are segregated configurations, it is then natural to define the free boundary as the nodal set ( ) { } : , : 0 x t u v Γ = = = .The regularity of the nodal set remains a challenge, and it will be the object of a forthcoming paper.

≤
With the above discussion, below we study some convergence properties.It follows from Lemmas 1 . For the convenience of presentation, we consider the special case of u v = a.e. in Ω , then up to a subsequence, ( a.e. in ( ) 0,T × Ω , we may apply Fubini theorem to obtain . The rest of the conclusions in this theorem follow consequently.
which separates the two strongly competing species.That is