Asymptotic spreading of interacting species with multiple fronts I: A geometric optics approach

We establish spreading properties of the Lotka-Volterra competition-diffusion system. When the initial data vanish on a right half-line, we derive the exact spreading speeds and prove the convergence to homogeneous equilibrium states between successive invasion fronts. Our method is inspired by the geometric optics approach for Fisher-KPP equation due to Freidlin, Evans and Souganidis. Our main result settles an open question raised by Shigesada et al. in 1997, and shows that one of the species spreads to the right with a nonlocally pulled front.

1. Introduction. In this paper, we study the spreading of two competing species, modeled by the Lotka-Volterra two-species competition-diffusion system. The nondimensionalized system reads for all x ∈ R, where the positive constants d and r are the diffusion coefficient and intrinsic growth rate of v; u(t, x) and v(t, x) represent the population densities of the two competing species at time t and location x. Without loss of generality, we assume dr ≥ 1 throughout most of this paper. It is clear that (1) admits a trivial equilibrium (0, 0) and two semi-trivial equilibria (1,0) and (0,1). Throughout this paper we assume 0 < a < 1 and 0 < b < 1, so that there is a further linearly stable equilibrium: There is a vast number of mathematical results concerning the spreading of competing populations with a single interface connecting two equilibrium states, see, e.g., [27,30,31] and the references therein. By a classical result by Lewis et al., it is known that for (1), the spreading speed is closely related to the minimum wave speed of traveling wave solutions connecting the ordered pair of two equilibria of (1). Theorem 1.1 (Lewis et al. [27,29]). Let (u, v) be the solution of (1) with initial data u(0, x) = ρ 1 (x), v(0, x) = 1 − ρ 2 (x), where 0 ≤ ρ i < 1 (i = 1, 2) are compactly supported functions in R. Then there In this case, we say that the population u spreads at speed c LLW . Remark 1.2. If the initial data (u, v)(0, x) is a compact perturbation of (1, 0), then there existsc LLW ∈ [2 dr(1 − b), 2 √ dr] such that the species v spreads at speed c LLW .
Concerning the bounds of c LLW , standard linearization near the equilibrium (0, 1) shows c LLW ≥ 2 √ 1 − a. Numerical tests by Hosono [2] showed that the above equality holds only for certain values of model parameters d, r, a, b. This begs the question of if and when the equality holds, which is known as the question of linear determinacy. Recently, Huang and Han [24] rigorously demonstrated that c LLW > 2 √ 1 − a is possible via an explicit construction. On the other hand, sufficient conditions for linear determinacy are first obtained in [27] and are subsequently improved in [23]. See also [2,3] for recent development on necessary and sufficient conditions. The goal of this paper is to understand the co-invasion of two competing species for a different class of initial data (u 0 , v 0 ) ∈ C(R; [0, 1]) 2 : There exist positive constants θ 0 , x 0 such that θ 0 ≤ u 0 (x) ≤ 1 in (−∞, 0], and u 0 (x) = 0 in [x 0 , ∞). Also, v 0 (x) is non-trivial and has compact support.
In other words, we assume the right habitat is unoccupied initially. This question was raised by Shigesada and Kawasaki [36] as they considered the invasion of two or more tree species into the North American continent at the end of last ice age (approximately 16,000 years ago) [10]. An interesting scenario arises when the slower moving species invades into the (still expanding) range of the faster moving species. The numerical computations in [36,Ch. 7] illustrate that the two species set up at least two invasion fronts: The first front occurs as the faster species invades into open habitat at some speed c 1 , while the next front appears when the slower species "chases" the faster species at speed c 2 .
When the initial data u 0 and v 0 are both compactly supported, the spreading properties of (1) with a, b ∈ (0, 1) were initially studied by Lin and Li [31]. They showed that the faster species v spreads at speed c 1 = 2 √ dr and obtained an estimate of the spreading speed c 2 of the slower species u, which satisfies 2 √ 1 − a ≤ c 2 ≤ 2. In case 2 √ dr > 2 + 2 1 − a(1 − b), they obtained an improved estimate of c 2 , namely, 2 √ 1 − a ≤ c 2 ≤ 2 1 − a(1 − b).

QIAN LIU, SHUANG LIU AND KING-YEUNG LAM
(b) For each small η > 0, the following spreading results hold: (2) Precisely, the spreading speeds c 1 , c 2 , c 3 can be determined as follows: where c LLW (resp.c LLW ) is the spreading speed of (k 1 , k 2 ) into (0, 1) (resp. (k 1 , k 2 ) into (1, 0)) as given in Theorem 1.1 (resp. Remark 1.2). And The above result can be abbreviated as The above result also shows that, while the spreading speed c 1 of the faster species v is the linearly determined speed of 2 √ dr and is unaffected by the slower species u, the corresponding speed c 2 of species u is a non-increasing function of c 1 . This is due to the fact that the presence of v negatively impacts the invasion of u. It is clear that, even though u 0 (x) vanishes for x ≫ 1, the spreading speed c 2 can be strictly greater than c LLW , i.e., the second front moves at an enhanced speed that is strictly greater than the minimal speed of traveling wave solutions. As we shall see, the expression (4) of c nlp coincides with that in [18,Theorem 1.1], and can be characterized by where w 1 is the unique viscosity solution of the following Hamilton-Jacobi equation with space-time inhomogeneous coefficients: where χ S is the indicator function of the set S ⊂ R 2 . (Hereafter the initial condition similar to the one in (7) is to be understood in the sense that To explain the sense in which the speed c nlp is said to be nonlocally determined, let us define c nlp for the moment by the relation (6), where w 1 is the unique viscosity solution of (7). As we will show in Lemma 3.7, w 1 (t, x) = max{J 1 (t, x), 0}, with where the infimum is taken over all curves γ ∈ H 1 loc ([0, ∞)) such that γ(0) = 0 and γ(t) = x. For each (t, x) on the front, (i.e., x = c nlp t), the minimizing path ASYMPTOTIC SPREADING OF INTERACTING SPECIES 5 γ(s) =γ t,x (s) describes how an individual located at (0, 0) arrives at the front x = c nlp t at time t. Now, when a = 0, the problem (7) is homogeneous. In this case w 1 (t, x) = t 4 x 2 t 2 − 4 , so that the front is characterized by x = 2t. Furthermore, for each (t, x) on the front, the minimizing pathγ(s) is given by the straight lineγ(s) = x t s = 2s, i.e., an individual arriving at the front x = 2t at time t has been staying at the front x = 2s for any previous time s ∈ [0, t]. Hence, we say that the spreading speed is locally determined in case a = 0.
Consider instead the problem (7) in case a ∈ (0, 1). Then the minimizing paths are not straight lines in general. In fact, if an individual finds itself at the moving front at time t, i.e., x = c nlp t, then the corresponding minimizing path is a piecewise linear curve connecting (0, 0), (τ, 2 √ drτ ), and (t, x), for some τ ∈ (0, t) (see Appendix B for details). Hence, the individual arriving at the front x = c nlp t at time t does not stay on the front in previous time. In fact, it spends a significant amount of time ahead of the front (by moving with speed 2 √ dr > c nlp ). Thus the speed c nlp is affected by the quality of habitat well ahead of the actual front, and we say that it is nonlocally determined. In fact, it is nonlocally pulled (see, e.g., [35] for the meaning of pulled versus pushed fronts).
We also mention a closely related work, due to Holzer and Scheel [21], which includes among others the special case b = 0 of (1). Their proof relies on linearization at a single moving frame y = x − 2t where the linearized problem becomes temporally constant. Such a problem was also studied by [7,15], where the complete existence and multiplicity of forced traveling waves as well as their attractivity, except for some critical cases, were obtained. In contrast, our approach can be applied to problems with coefficients depending on multiple moving frames x − c i t for several c i . This allows the treatment of the spreading of three competing species with different speeds, which will appear in our forthcoming work.
Using Theorem 1.3, which treats the case dr > 1, we can derive the following results concerning the remaining cases dr = 1 and 0 < dr < 1.
The above result can be abbreviated as Theorem 1.4 implies the invasion process from (k 1 , k 2 ) into (0, 0) does exist, which is related to the results in Tang and Fife [38] where the existence of traveling wave solutions of (1.1) connecting (k 1 , k 2 ) to (0, 0) was proved.
By switching the roles of u and v, it is not difficult to derive the following result in case dr < 1.
Precisely, the spreading speeds c 1 , c 2 , c 3 can be determined as follows: Remark 1.5. As in [14], our approach can be applied to the spreading problem of competing species in higher dimensions under minor modifications. However, we choose to focus here on the one-dimensional case to keep our exposition simple, and close to the original formulation of the conjecture in [36,Ch. 7].
Remark 1.6. We also mention here some related works concerning competition systems [11,19,32,40,41,42] with Stefan-type moving boundary conditions. Therein some estimates of asymptotic speeds of the moving boundaries were proved. In contrast to the Cauchy problem considered here, there are no far-fields effect in such moving boundary problems.
1.2. Numerical simulation of main results. The asymptotic behaviors of the solutions to (1) for the three cases: (a) dr > 1, (b) dr = 1, (c) 0 < dr < 1 are illustrated in Figure 1. Precisely, (a) with d = 1.5 shows that the solutions of (1) behave as predicted by Theorem 1.3. Therein, species v spreads faster than species u, i.e., c 1 = 2 √ dr > 2 ≥ c 2 . (b) with d = 1 corresponds to Theorem 1.4, where c 1 = c 2 = 2. Finally, (c) with d = 0.5 means that species u spreads faster than species v, i.e., c 1 = 2 > c 2 as discussed in Corollary 1. Due to the limitation of our methods, we can't get the asymptotic profiles of (1).
In what follows, we present some numerics to illustrate the formulas of c 1 , c 2 and c 3 given in Theorem 1.3. Set a = 0.6, b = 0.5, r = 1 and d = 1.5 as in Figure 1(a), whereby the sufficient conditions for linear determinacy given by [ where, in determining c 2 , we used the facts that (i) The graphs of x i (t)/t (i = 1, 2, 3) are shown in Figure 2. They indicate that, indeed, x i (t)/t → c i as t → ∞. In fact, at t = 200, x 1 (t)/t ≈ 2.4452 comparing to the theoretical value c 1 ≈ 2.4495; x 2 (t)/t ≈ 1.3695 comparing to c 2 ≈ 1.3387; and x 3 (t)/t ≈ −1.7214 comparing to c 3 ≈ −1.7321 in Theorem 1.3. Note that we expect an error of O(t −1 log t) between the approximated value x i (t)/t and the theoretical 1. To estimate c 2 from below, we consider the transformation and show that the half-relaxed limits By the comparison principle, we show that where w 1 is the viscosity solution of the Hamilton-Jacobi equation (7). Solving w 1 explicitly by way of its variational characterization, we have x < c nlp t} locally uniformly. One can then apply the arguments in [14,Section 4] This implies that c 2 ≥ c nlp (see Proposition 4.1). 2. To estimate c 2 from above, we observe that, for some δ * > 0, Hence, together with (10) we obtain a large deviation estimate of u. Namely, forĉ = 2 √ dr − δ * , with suitable traveling wave solutions connecting (k 1 , k 2 ) with (0, 1) to control the spreading speed c 2 of u from above (Lemma 2.4).

1.4.
Organization of the paper. In Section 2, we determine c 1 , c 3 and give rough estimates of c 2 . In Section 3, we establish the approximate asymptotic expression of u and then determine c 2 in Section 4. This completes the proof of Theorem 1.3. In Section 5, Theorem 1.4 is derived as a limiting case of Theorem 1.3. To improve the exposition of ideas, we postpone the proofs of Lemma 2.4 and Proposition 3.5 to the Appendix.

2.
Preliminaries. We define the maximal and minimal spreading speeds as follows (see also [20,Definition 1.2] where related concepts were introduced for a single species): Here c 1 and c 1 (resp. c 2 and c 2 ) are the maximal and minimal rightward spreading speeds of species v (resp. species u), whereas −c 3 and −c 3 are the maximal and minimal leftward spreading speeds of v.
In this section, we will determine c 1 = c 1 and c 3 = c 3 , and give some rough estimates of c 2 and c 2 . We will also show that the solution (u, v) of (1) approaches one of the homogeneous equilibria in between successive spreading speeds. Recalling the definition of c 1 = 2 √ dr and c 3 = −c LLW in (3), the main result of this section can be precisely stated as follows.
(iii) For each small η > 0, the following spreading results hold: where c LLW ,c LLW are given in Theorem 1.1 and Remark 1.2, respectively.
Remark 2.2. Proposition 2.1 is proved in [31] under the stronger assumption Before estimating the spreading speeds of species, we first give a lemma concerning the behaviors of (u, v) between the spreading fronts.
Proof. The proof is based on classification of entire solutions of (1). For (x 1 , x 2 ) and (y 1 , y 2 ) in R 2 , we define the partial order " " so that Suppose (a) is false. Then there exists (t n , x n ) such that, as n → ∞, t n → ∞ and Passing to a subsequence, we may assume that (u n , v n ) converges to an entire solution (û,v) of (1) in C 2 loc (R× R). By construction, there exists a constant 0 < ζ 0 < 1 such that (û,v)(t, x) (1, ζ 0 ) for (t, x) ∈ R 2 . Let (U , V ) be the solution of the Lotka-Volterra system of ODEs This is a contradiction and proves (a). The other assertions follow from similar considerations.
The following lemma says that the maximal spreading speed of u (resp. v) can be estimated by the large deviation estimate of u (resp. v) along a line {(t, x) : x =ĉt}.
(a) Ifĉ > 2 and there existsμ > 0 such that Here c LLW ,c LLW are given in Theorem 1.1 and Remark 1.2, and The proof of Lemma 2.4 is based on comparison with appropriate traveling wave solutions connecting (k 1 , k 2 ) with one of the semi-trivial equilibrium points. We postpone the proof to Appendix A.
Proof of Proposition 2.1. It follows directly from definition that c i ≤ c i for i = 1, 2, 3. We will complete the proof in the following order: After that, we establish (11a)-(11d) by applying Lemma 2.3. Our proof adapts the ideas of [12] and [18,Proposition 3.1], and can be skipped by the motivated reader.
It can be verified that v(t, x) and v c (t, x) are respectively super-and sub-solutions of the equation By the comparison principle, we deduce that Hence, c 1 ≥ c. Letting c ր 2 √ dr, we have c 1 ≥ 2 √ dr.
Step 6. We show that Fix a small η > 0. By definition of c 2 > 0, there exists c ′ 2 ∈ (c 2 − η, c 2 ) and Observe also that v ≤ 1 and thus u is a super-solution to where θ 0 > 0 is given by (H ∞ ). It follows from the classical results in [16,26] that, for some Since By (19) and (21), we deduce that δ := min{ inf t≥T2 u(t, c ′ 2 t), 1−a 2 , inf x≤c ′ 2 T2 u(T 2 , x)} is positive, then u is a super-solution to the KPP-type equation x) ≥ δ on the parabolic boundary. Therefore, we deduce u(t, x) ≥ δ in Ω ′ and (18) follows.
To this end, choose K > 0 such that v 0 (x) ≤ χ [−K,∞) , then the right hand side of (23) defines a weak super-solution of the KPP-type equation Step 10. We finally show c 3 ≥ −c LLW and establish (11d).
This was accomplished in [18] for the case a < 1 < b by a delicate construction of global super-and sub-solutions, in the sense that they are defined and respect the differential inequalities for (t, x) ∈ (0, ∞) × R.
In this section, we shall derive the exponential estimate (26) by the ideas of large deviations. Using this method, one can obtain an exponential estimate of u without constructing global super-and sub-solutions for system (1).
Moreover, for each τ > 0, w ǫ (τ, x) is finite for all x ∈ R. Since , we obtain by comparison that This completes the proof of the local bounds of w ǫ .
Having established the C loc bounds, we will pass to the (upper and lower) limits of w ǫ by using the half-relaxed limit method, which is due to Barles and Perthame [5]. We begin with the following definition: Remark 3.3. By (30), it follows that for any δ ∈ (0, 1) and ǫ small, Sending first ǫ → 0 then δ → 0, we deduce w * (0, x) = w * (0, x) = 0 for all x ≤ 0.
Lemma 3.4. Assume that (25) holds for some c 1 >c 1 ≥ 2. Then (i) w * is upper semicontinuous and is a viscosity sub-solution of
To study the limits w * and w * of w ǫ , we introduce the auxiliary functions w i (i = 1, 2) as follows. and where for i = 1, 2, L i (t, x, q) is the Legendre transformation of H i (t, x, p), i.e., We state the following calculus lemma, whose proof is postponed to Appendix B.
Finally, we verify c 1 >c nlp , which implies that the ranges in the statement of the lemma are well-defined and lie within P = {(t, x) : J 1 (t, x) > 0}. Indeed, this follows from the direct calculation: as c 1 > 2. Hence, the formulas of w 1 follow from those of J 1 given in (40).
Proof. By definitions of w * and w * in (32), it is obvious that w * ≥ w * . It remains to prove w * ≤ w * in {(t, x) : x ≥ (c 1 − δ * )t}. By Lemmas 3.6 and 3.7, we have By Proposition 3.5(c), there exists δ * > 0 such that This yields the desired conclusion.

4.
Estimating c 2 and c 2 . In this section, we apply results in Section 3 with c 1 = 2 √ dr andc 1 = 2 to determine the spreading speeds c 2 and c 2 .
where c nlp is given in (4) in the statement of Theorem 1.3.
Proof. By Lemma 3.7, We claim that it is enough to show that i.e., c 2 ≥ c for all c < c nlp , so that c 2 ≥ c nlp .
Proof of Theorem 1.3. Let c nlp be as given in (4)  Proof of Theorem 1.4. Let (u, v) be a solution of (1) with initial data (u 0 , v 0 ) satisfying (H ∞ ). For any small δ ∈ (0, 1), let (u δ , v δ ) and (u δ , v δ ) be respectively the solutions of and with the same initial data (u 0 , v 0 ). By comparison, we deduce that Notice that (u δ , v δ ) is a solution of (61) if and only if is a solution of
Therefore, for each δ > 0, we arrive at Since the above is true for all δ > 0, we deduce that Thus (75) holds.
Appendix B. Proof of Proposition 3.5. This section is devoted to the proof of Proposition 3.5. Let c 1 >c 1 ≥ 2 be given and let J i (t, x) be given by (38), we may equivalently write where L i is given in (39), and Y t, Proof of Proposition 3.5(a). We divide the proof into several steps.
Step 1 is thereby completed.