Stability of Traveling Waves to the Lotka-Volterra Competition Model

In this paper, the stability of travelingwave solutions to the Lotka-Volterra diffusivemodel is investigated. First, we convert themodel into a cooperative system by a special transformation. The local and the global stability of the traveling wavefronts are studied in a weighted functional space. For the global stability, comparison principle together with the squeezing technique is applied to derive the main results.

This is equivalent to studying traveling waves for the original competition system (4) that connect the boundary equilibria (0, 1) and (1, 0).The existence of traveling waves to the above problem is well-studied in literature.It is known that there exists  * ≥ 0 so that problem (8)-( 9) has a monotone solution (, )() for  ≥  * and no wavefront exists for  <  * ; see [3][4][5][6]. * is called the minimal wave speed for this system and satisfies  * ≥ 2√1 −  1 .When  * = 2√1 −  1 , we say that the minimal wave speed is linearly determined; see the details in [4].
We know that (, )( − ) is a special pattern that only satisfies the first two equations in (5).For the stability of this pattern, we want to know if the solution of (5) tends to (, )( − ) for given initial data  0 () and V 0 ().To this end, we use the (, )-coordinate and to transform the V-model (5) into the partial differential model subject to It is easy to see that (, )() is the steady-state to the above new system.We should mention that dynamics for ( 4) is very rich.There are always three nonnegative equilibria (0, 0), (1, 0), and (0, 1).In the case when  1 < 1,  2 < 1, or the case when  1 > 1,  2 > 1, there exists a unique positive coexistence equilibrium Based on the phase plane analysis to the ordinary differential system of (4) without diffusion terms, the nonlinearity of the model ( 4) when  1 < 1 and  2 < 1 is called the persistence case (or coexistence).Likewise, the nonlinearity is called the monostable case when  1 < 1 and  2 > 1 are satisfied, or the bistable case when  1 > 1 and  2 > 1. Traveling waves to (4) have been investigated considerably.For the bistable case, please see [7,8] for the existence of traveling waves connecting (1, 0) and (0, 1), and [9] for the uniqueness and parameter dependence of wave speeds.For the monostable case, we refer to [3,10] for the existence of traveling waves, and [11,12] for the selection of the minimal speed.For the persistence (coexistence), the existence of traveling wave connecting (0, 0) and ( φ * , ψ * ) has been studied in [13,14].When time delays are incorporated into (4) in the persistence case, Li et al. [15] and Gourley and Ruan [16] have proved the existence of traveling waves.
The stability of traveling waves to a scalar partial differential equation has been well-studied, e.g., [17][18][19][20][21][22][23][24][25][26][27], the monograph [6,28] and the survey paper [29].Indeed, the extension of this study to a general system is not trivial.As we know, when time delays are directly incorporated in the competition terms in (4), the system becomes nonmonotone and the comparison principle cannot work.Alternatively, in [30,31], the authors studied the stability of traveling waves for the so-called cooperative delayed reaction diffusion system by changing the signs of  1 and  2 .To be exact, with putting delay = 0, they studied the cooperative system where   ,   , â , and b are all positive.This corresponds to the persistence case in our model (4).Under the condition b1 b2 − â1 â2 > 0, a positive equilibrium exists.They proved that the traveling wave fronts, connecting (0, 0) and ( + ,  + ), are exponentially stable in some weighted  ∞ spaces, and obtained the decay rates by the weighted energy estimate.Despite the success in the study of the stability of traveling waves to the classical model (4) in the bistable and persistence cases, the stability of traveling wave in the monostable remains still unsolved.The purpose of this paper is to systematically study the local and the global stability of the steady-state (, )().Using the method of spectrum analysis in [32], we give the local stability.For the global stability, we construct an upper and a lower solutions to the system (11), and prove their convergence to the traveling wave (, )().In view of comparison together with the squeezing technique, we arrive at new results on the global stability of the traveling waves.We remark that our method is different from that in [30,31] where weighted energy method was applied.
The rest of the paper is organized as follows.Local analysis of the wave profile near the unstable point is studied in Section 2. In Section 3, we study the local stability of the steady-state by applying the standard linearization.The resulting spectrum problem is studied by the method in [32].A suitable weighted functional space is chosen to proceed the analysis.In Section 4, besides the weighted functional space, the upper-lower solution method together with the squeezing technique is applied to derive the global stability results.Conclusions are presented in Section 5.

The Local Analysis of the Wave Profile Near
the Equilibrium (0, 0) In this section, we study the behavior of the traveling wave (, )() locally near the equilibrium (0, 0).Assume that the solution has exponential decay as  → ∞.Indeed this claim can be easily verified by the maximum principal coupled with a comparison near the neighborhood of infinity.Therefore, we set for positive constants  1 ,  2 , and .By substituting this into (8) and linearizing the equations we have where () is given by The system of algebraic equations (17) has a nontrivial solution if and only if det() = 0.This implies  =  1,2,3 > 0, where and Indeed, a condition so that  1 and  2 are reals is For  >  0 , obviously  1 <  2 .When 0 ≤  < 1, we have also  2 <  3 for all  >  0 , i.e.,  − 1  dominates both of  − 2  and  − 3  .In this case, the eigenvector of () corresponding to   , for  = 1, 2, is the strongly positive vector ( 1 (  )  2 (  ))  , where It follows that for  1 > 0 or  1 = 0,  2 > 0. For the case when the same behavior in ( 23) is still true if  * <  ≤ ĉ, where If  > ĉ, then  1 <  3 <  2 and we have for Here, (0 1)  is the eigenvector of () corresponding to  3 , and note that  1 ( 2 ) < 0 in this case.On the other hand, when (, )() behaves like (26) if  > ĉ.For the case when  * <  < ĉ, we have  3 <  1 <  2 .Hence, for  1,3 > 0, or  1 = 0,  2,3 > 0. We summarize the above behaviors in Table 1.
Table 1: The asymptotic behavior of the wave profile (, ) near infinity.

Condition on 𝑑 Condition on 𝑐
The asymptotic behavior Kan-on in [3] derived the asymptotic behaviors of (, )() near infinity when  ≥  * .After deriving the behavior of (), he used it into the -equation to find the behavior of () when  1 ≤  2 ≤  3 and when Our result here agrees with that in [3] when  >  * .We further study the case when  1 <  3 <  2 .
Finally, we have the asymptotic behavior for the solution () when the wave speed is greater than the minimal speed  * .Theorem 1.For  >  * , the wavefront  has the following behavior: for some  1 > 0.
Then () > 0 is fixed, and we have By (34), this implies that () ≤ 0, which is a contradiction.The proof is complete.

The Local Stability
To study the local stability, as usual, we add a small perturbation to the traveling wave and study the behavior of this perturbation for large time period.If this perturbation decays, then we say that the traveling wave is locally stable.
For  ≪ 1 and a parameter , let where  1 and  2 are two real functions.Substitute these formulas into (11) and linearize the system about (, ) to get the following spectrum problem: where Φ = ( 1  2 )  ,  and () are 2 × 2 matrices given by For Φ in a suitable space, we shall find sign of the maximal real part to the spectrum () of the operator L to determine the local stability of the traveling wave solution.To proceed, we introduce a weighted functional space    , with the norm where is the weight function with for some positive constants , , and  0 to be chosen.Here,   (R), for  ≥ 1, is the well-known Lebesgue space of integrable functions defined on R. Then we consider the operator L on this new space and find its spectrum.To do this, we write Φ() in the form for   -functions  1 and  2 .Substituting (43) into (37) gives a new spectrum problem in the weighted space    , where Ψ = ( 1  2 )  , () and () are 2 × 2 matrices defined by and with the -element of the matrix (),   , being given in terms of the -element of the matrix () as   = (  /  )  ; that is, ) . ( The details to find the essential spectrum of the operator L  can be finalized by using Theorem A.2 in [32] and are given below.After we choose the weight function so that the essential spectrum is on the left-half complex plane, we can determine the sign of the maximal real part of the point spectrum in the weighted space as well. First of all, to apply the method in [32], we need to choose  and  so that the matrix functions () and () are bounded; i.e., the limits for some constants  1 and  2 , are satisfied.We choose where  1 is defined in (19).This makes, by using Theorem 1,  1 = 0 and Now, we define where  ± and  ± are the limits of () and () as  → ±∞, respectively.Then the essential spectrum of the operator L  is contained in the union of regions inside or on the curves  + and  − ; see [32, pp. 140].By letting  → +∞,  + , and  + are given as (taking condition (49) into account) and The equation det(− 2  +  + +  + − ) = 0 has two solutions  =  1,2 , where This means that  + is the union of two parabolas in the complex plane which are symmetric about the real axis; namely, The most right points of these curves are  2 −  + 1 −  1 and  2 −  − , respectively, which are negative if where  1 ,  2 , and  3 are defined in ( 19)- (20).Hence, when the above condition satisfies,  + =  +,1 ∪  +,2 is on the left-half complex plane.
The above analysis shows that the essential spectrum of L  is on the left-half complex plane as long as conditions (49) and (55) are satisfied.In fact, there are many choices of  and  satisfying these conditions depending on  1 ,  2 , and  3 .We choose them by the following algorithm.Algorithm 2. Two mechanisms are valid to choose  and  so that all conditions in (49) and (55) hold: (1) If  1 <  3 , then we choose  =  for any  ∈ ( 1 , min{ 2 ,  3 }).
Finally, in order to get a local stability result, we need to check the sign of the principal eigenvalue in the point spectrum for (37)-(38).Consider the associated linear partial differential system where (, ) = ( 1 (, ),  2 (, )).The eigenpair (, Φ) of (37) implies a solution   Φ to the above system.Let   = (, , ) denote the solution semiflow of (58) for any given initial data  in   .It is easy to see   is compact and strongly positive.By the well-known Krein-Rutman theorem (see, e.g., [33]),   has a simple principal eigenvalue  max with a strongly positive eigenvector, and all other eigenvalues   must satisfy For any  >  * , we have from Theorem 1 that () ∼  1  − 1  ,  1 > 0, as  → ∞.  = 0 is an eigenvalue to the operator L defined in (37) with the one-sign (strongly positive) eigenvector (−  , −  )().By the choice of the weighted functional space    , the one-sign eigenvector (  ,   )() is not inside.Hence, the real parts of point spectrum of the operator L  in    are all negative.We can also explain this in a simple analysis.Assume to the contrary that (, Φ) is an eigenpair of the eigenvalue problem (37)-(38) with  > 0 and Φ ∈    .Obviously, the one-sign function Φ = (−  , −  )() satisfies (58).For Φ in the    -space, we have essentially (or except for a set of zero measure) Φ() > Φ() as  → ∞.On the other hand, when  → −∞, we can apply the method of asymptotic analysis and assume that the eigenfunction of (37) behaves like   for some positive values  and .By substituting it into the eigenvalue problem and using the behavior of (), we obtain that  is increasing with respect to .This implies that Φ() > Φ() as  → −∞.Hence, by choosing  sufficient large, we can have  Φ ≥ |Φ|.By comparison, from the partial differential system (58), we obtain  Φ() ≥ |Φ|  , which contradicts  > 0. This implies that for Φ ∈    , the real parts of all eigenvalues  of (37) should be nonpositive.Now we are in a position to state the local stability result.
Theorem 3.For any  >  * , the wavefront (, )() is locally stable in the weighted functional space    with the weight function () defined in ( 41)-( 42), where  and  in the formula of () are chosen by Algorithm 2.
By ( 65) and ( 67), for all  ∈ R and  ≥ 0, we have By ( 8) and (66) and using condition (C2), we can verify that  and  satisfy To study the stability in the weighted functional space  ∞  , with () defined in (41), we first let where  and  are functions in  ∞ (R) and  0 is the same number used in the weight function ().This gives where () is the same matrix defined in (18) and  = diag( − 2,  − 2).
From ( 65) and ( 67), for all  ∈ R and  ≥ 0, we have From ( 8) and (66),  and  satisfy the system with () defined in (38).By condition (C2), we have Similar to the previous analysis in the proof of Lemma 5, and making a use of the facts  <  and  < , we can prove that there exist  2 > 0 and For the choice of  0 in proof of Lemma 5, we study the stability in the weighted space  ∞  .To this end, define ( R, Ŝ)() as the solution of the system with the initial data R (0) ≥  (, 0) , Ŝ (0) ≥  (, 0) , ∀ ∈ R. (93) It is easy to see that ( R, Ŝ) is an upper solution to the system (88).The phase plane analysis shows that ( R, Ŝ)() converges origin for any initial data in the region [0, 1] × [0, 1] except the point (1,1).Similar to the previous lemma, This condition arose in the linear speed selection studies; see [36].To see that the condition (C2) can be realized for all  ∈ R, we prove the following claim.

Conclusions
The local and the global stability of traveling waves to the two-species Lotka-Volterra competition model ( 5) under the condition (C1) are investigated.Using the linearization and the essential spectrum analysis in [32], we find that the traveling wavefront is stable in some weighted functional space; see Theorem 3.Many choices of the exponential weight functions are valid; see Algorithm 2. Under some further condition (C2), we apply the upperlower solution method to obtain a global stability result.Indeed, we prove that both the upper and the lower solutions tend to the wavefront.Our main results are presented in Theorem 4.