Abstract
In this paper, the stability of traveling wave solutions to the Lotka-Volterra diffusive model is investigated. First, we convert the model 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.
1. Introduction
We are concerned here with the diffusive Lotka-Volterra competition modelwith the initial dataHere and are the population densities at time and location ; and are the diffusive coefficients; and are the net birth rates; and are the competition coefficients; and are the carrying capacities for each species. For derivation and biological interpretation of this model, we refer readers to [1, 2].
Using the transformationsthe nondimensional form of the system becomes By letting , this model can be further written as a cooperative systemwithFor our study, we will assume that and are nonnegative. The existence and uniqueness of the solution of the above problem can be easily verified by a classical argument of Picard’s iteration. Throughout this paper, we assume that the conditionis satisfied. Under this condition, equilibria to system (5) in the region are only , and . In the absence of diffusion in the system (5), it can be shown that is unstable and is stable. For the system, we are particularly interested in the traveling wave solution, connecting and , in the form where is the wave variable, is the wave speed, and is called the wavefront and satisfiessubject toThis is equivalent to studying traveling waves for the original competition system (4) that connect the boundary equilibria and .
The existence of traveling waves to the above problem is well-studied in literature. It is known that there exists so that problem (8)-(9) has a monotone solution for and no wavefront exists for ; see [3–6]. is called the minimal wave speed for this system and satisfies . When , 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 and . To this end, we use the -coordinate and to transform the -model (5) into the partial differential modelsubject 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 , and . In the case when , or the case when , 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 and is called the persistence case (or coexistence). Likewise, the nonlinearity is called the monostable case when and are satisfied, or the bistable case when and . Traveling waves to (4) have been investigated considerably. For the bistable case, please see [7, 8] for the existence of traveling waves connecting and , 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 and 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., [17–27], 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 and . To be exact, with putting , they studied the cooperative system where , and are all positive. This corresponds to the persistence case in our model (4). Under the condition , a positive equilibrium exists. They proved that the traveling wave fronts, connecting 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.
2. The Local Analysis of the Wave Profile Near the Equilibrium
In this section, we study the behavior of the traveling wave locally near the equilibrium . 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 , and . By substituting this into (8) and linearizing the equations we havewhere is given byThe system of algebraic equations (17) has a nontrivial solution if and only if . This implies , whereandIndeed, a condition so that and are reals is
For , obviously . When , we have also for all , i.e., dominates both of and . In this case, the eigenvector of corresponding to , for , is the strongly positive vector , whereIt follows thatfor or , . For the case whenthe same behavior in (23) is still true if , whereIf , then and we havefor or , . Here, is the eigenvector of corresponding to , and note that in this case. On the other hand, when behaves like (26) if . For the case when , we have Hence,for , or , . We summarize the above behaviors in Table 1.
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 and when . Our result here agrees with that in [3] when . We further study the case when .
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 .
Proof. On the contrary, assume that for some , the wavefront has the following behavior:for some . By this assumption, it follows that is a solution to the following partial differential equation:with the initial conditions We know that there exists a monotonic traveling wavefront to the system (31) for any . In particular, assume is a solution for some with the initial conditionBy a simple computation of the asymptotic behavior of this solution to (8)-(9) near , we can always obtain (by shifting if necessary) for all . From the second equation of (8), we have for all . From (31), by comparison, we getfor all . On the other hand, fix . Then is fixed, and we haveBy (34), this implies that , which is a contradiction. The proof is complete.
3. 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 and a parameter , let where and are two real functions. Substitute these formulas into (11) and linearize the system about to get the following spectrum problem:where , and are matrices given by
For in a suitable space, we shall find sign of the maximal real part to the spectrum () of the operator to determine the local stability of the traveling wave solution. To proceed, we introduce a weighted functional space ,with the norm whereis the weight function withfor some positive constants , , and to be chosen. Here, , for , is the well-known Lebesgue space of integrable functions defined on . Then we consider the operator on this new space and find its spectrum. To do this, we write in the formfor -functions and . Substituting (43) into (37) gives a new spectrum problem in the weighted space , where , and are matrices defined byand 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 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 limitsfor some constants and , are satisfied. We choosewhere is defined in (19). This makes, by using Theorem 1, and
Now, we define where and are the limits of and as , respectively. Then the essential spectrum of the operator 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)The equation has two solutions , 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 and , respectively, which are negative ifwhere , and are defined in (19)-(20). Hence, when the above condition satisfies, is on the left-half complex plane.
Similarly, we find by solving the equation , withThis gives two solutions , where From , is on the left-half complex plane.
The above analysis shows that the essential spectrum of 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 , and . 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 , then we choose for any .(2)If , then we choose and for small . In particular, we can choose and , for .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 systemwhere . The eigenpair of (37) implies a solution 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 with a strongly positive eigenvector, and all other eigenvalues must satisfy For any , we have from Theorem 1 that , , as . is an eigenvalue to the operator 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 in are all negative. We can also explain this in a simple analysis. Assume to the contrary that is an eigenpair of the eigenvalue problem (37)-(38) with and . Obviously, the one-sign function satisfies (58). For in the -space, we have essentially (or except for a set of zero measure) 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 as . Hence, by choosing sufficient large, we can have . By comparison, from the partial differential system (58), we obtain , which contradicts . This implies that for , 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.
4. The Global Stability
We study here the global stability of the steady-state in a special choice of the weighted functional space . Let and define the norm , for some weight function . Assume . By Algorithm 2, we choose . Specifically, let , for small positive number . Also, we assume that the functions and satisfy the condition
Theorem 4. Suppose , , and conditions - hold true. Assume that the initial data and satisfy and Then the solution to (11) exists globally with and converges to the steady-state exponentially in the sense of for positive constants and .
To prove Theorem 4, we will find an upper and a lower solution to the partial differential equations system (11). For , define It is easy to see that the following inequalities are true:Denote and as the solutions to the system (11) with the initial data and , respectively; that is,By the comparison principle, one gets
In the following lemmas we shall prove the convergence of and to the wavefront . Then we apply the squeezing theorem to obtain the result in Theorem 4.
Lemma 5. Under the conditions in Theorem 4, converges to .
Proof. For , defineThese functions, and , satisfy the initial value conditionsBy (65) and (67), for all and , we haveBy (8) and (66) and using condition , we can verify that and satisfyTo study the stability in the weighted functional space , with defined in (41), we first let where and are functions in and is the same number used in the weight function . This giveswhere is the same matrix defined in (18) and .
Define and as for some constants to be chosen and is the eigenvector of the matrix associated with the eigenvalue . Simple computations give which are positive for small and . Since the initial values and are in the space , we can choose . Direct computations and using condition show that both of and are negative. This allows choosing a positive value to so that the inequalityholds. Hence, since and by comparison on unbounded domain, see, e.g., [34, Proposition 2.1], In particular, this is true when , for any fixed .
Now, we introduce the weight function defined in (41)-(42) with . By the above analysis, we need to prove the convergence of to for . Note that the full system of can be expressed asHere, is the same matrix defined in (38). Let be chosen so thatfor some given small , when . This is equivalent to require that is close to for all . Define as the solution of the autonomous systemwith the initial dataThen is an upper solution to system (78).
Now we need to prove the convergence of to as . The Jacobian matrix of system (80) at the fixed point has two eigenvalues, . By the phase plane analysis, there exists so that the flow in the space converges to origin for any initial data in the box . Hence, we conclude that for positive constant and is the eigenvector of corresponding to . For the maximal possible choice of the constant so that we have the convergence result inside the box ; see Remark 6.
We can choose large and so that, at the boundary , we have Hence, by comparison on the domain , see, e.g., [35, Lemma 3.2],This completes the proof.
Remark 6. The maximal possible value of the constant , which could be , depends on the location of the fourth fixed point to the system (80) near or inside the box . See Figure 1 for all possible different cases. In Figure 1(a), the positive fixed point is far away from the box and does not affect the flow. This happens when . Hence we set . Figure 1(b) shows the effect of the positive fixed point on the flow, which still outside the box. The maximal choice of for this case exists in the interval . The number is the positive -intercept of the nullcline . A fixed point exists inside the box in Figure 1(c), where becomes close to the value .
(a) and We choose
(b) and The maximal possible value of is in
(c) and The maximal possible value of becomes close to
Lemma 7. Under the conditions in Theorem 4, converges to .
Proof. For , define These functions, and , satisfy the initial value conditionsFrom (65) and (67), for all and , we have From (8) and (66), and satisfy the systemwith defined in (38). By condition , we haveSimilar to the previous analysis in the proof of Lemma 5, and making a use of the facts and , we can prove that there exist and so thatFor the choice of in proof of Lemma 5, we study the stability in the weighted space . To this end, define as the solution of the systemwith the initial dataIt is easy to see that is an upper solution to the system (88). The phase plane analysis shows that converges to origin for any initial data in the region except the point . Similar to the previous lemma, for some positive constants and . This completes the proof.
Now, we are ready to give the proof of Theorem 4.
Proof of Theorem 4. From (67), for all , we haveBy Lemmas 5 and 7 and the squeezing theorem, it follows that there exist and so that for all . This proves the desired result.
Condition is used in the previous analysis to construct the upper solutions in the proof of Lemmas 5 and 7. It implies that, at and , and it can be guaranteed by This condition arose in the linear speed selection studies; see [36]. To see that the condition can be realized for all , we prove the following claim.
Proof. In the case when , the equation can be written in the formSince , we need to prove for all . Assume, for contrary, this is not true for some . By (99), is increasing at the neighborhood of . Since is a decreasing function, we have , for some . Similarly, we can show that is increasing for all , which contradicts the fact . This implies that condition holds true.
5. Conclusions
The local and the global stability of traveling waves to the two-species Lotka-Volterra competition model (5) under the condition 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 , we apply the upper-lower 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.
Data Availability
We have used Maple codes to do the figure in the paper. They are available from the corresponding author upon request. No other data were used in this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work is partially supported by the NSERC Discovery Grant.