For monotone dynamical systems, the pioneering work of Weinberger et al. [55,57] (see also [43]) relates the spreading speed of the population to the minimal speed of (monostable) traveling wave solutions. Their result can be applied to the diffusive Lotka-Volterra competition system. Suitably nondimensionalized, the system is given by
with a, b ∈ (0, 1). It is clear that (1.1) admits a trivial equilibrium (0, 0), two semi-trivial equilibria (1, 0) and (0, 1), and further a linearly stable equilibrium
Theorem 1.1 (Lewis et al. [36]). Let (u, v) be the solution of (1.1) 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 exists some In this case, we say that 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 exists cLLW ∈ [2 dr(1 -b), 2 √ dr] such that the species v spreads at speed cLLW .
It is shown in [38,39] that the spreading speed c LLW (resp. cLLW ) is identical to the minimum wave speed of traveling wave solution connecting the pair of equilibria (k 1 , k 2 ) and (0, 1) (resp. (1, 0)). It is crucial for the theory that the pair of equilibria forms an ordered pair of equilibria (regarding the comparability of steady states in the theory of monotone semi-flows, see [49]).
For the weak competitive diffusive system (1.1), Tang and Fife [50] proved an additional class of traveling wave solutions connecting the positive equilibrium (k 1 , k 2 ) with the trivial equilibrium (0, 0). In this case, the equilibria (0, 0) and (k 1 , k 2 ) are un-ordered, and hence the existence of traveling wave, due to Tang and Fife [50], does not directly follow from the monotone dynamical systems framework due to Weinberger et al. [56,57] (see also [20,39]).
A natural question is whether the speed traveling wave solutions due to Tang and Fife, which connect (k 1 , k 2 ) to (0, 0), determine the spreading speed of the populations in the Cauchy problem (1.1), provided the initial data (u 0 , v 0 ) has the same asymptotics at x = ±∞ as the traveling wave solution? What happens for more general exponentially decaying initial data? Does the two species spread with different speeds?
In this paper, we continue our investigation in [41] on the spreading properties of solutions of the Cauchy problem (1.1). We are interested in determining the spreading speeds of each of the populations u and v, for the class of initial data (u 0 , v 0 ) satisfying (u 0 , v 0 )(-∞) = (1, 0), (u 0 , v 0 )(∞) = (0, 0) and such that u 0 → 0 exponentially at ∞ with rate λ u > 0; v 0 → 0 decays exponentially at ∞ (resp. -∞) with rate λ + v > 0 (resp. λ - v > 0). We introduce the Hamilton-Jacobi approach to study the spreading of twointeracting species into an open habitat, and resolve a conjecture by Shigesada [48,Ch. 7]. Inspired by the pioneering work of Freidlin [22] and of Evans and Souganidis [17] on the Fisher-KPP equation, we shall derive, via the thinfront limit, a couple of Hamilton-Jacobi equations for which solutions have to be understood in the viscosity sense. In our previous work [41], we considered the Cauchy problem (1.1) endowed with compactly supported initial data, and used the dynamics programming approach to show the uniqueness of the limiting Hamilton-Jacobi equations, and to evaluate the solution by determining the path that minimizes certain action functional. In contrast to our previous paper, we will tackle the Cauchy problem with exponentially decaying initial data using entirely PDE arguments. For this purpose, we establish a general comparison principle for discontinuous viscosity solutions associated with piecewise Lipschitz Hamiltonians, the latter arising naturally in the spreading of multiple species. The proof of the comparison result is based on combining the ideas due to Ishii [33] and Tourin [51]. With this comparison principle at our disposal, we are able to obtain large-deviation type estimates of the solutions (u, v) to the Cauchy problem (1.1) by explicit construction of simple piecewise linear superand sub-solutions.
We first recall some classical asymptotic spreading results concerning the single Fisher-KPP equation:
where d, r are positive constants. If the initial data is a Heaviside function, supported on (-∞, 0], it is shown [3,21,35] that the population, whose density is given by φ(t, x) has the spreading speed c * = 2 dr, i.e., In addition, the spreading speed c * coincides with the minimal speed of the traveling wave solutions to (1.2) in this case. If we broaden the scope of initial data φ 0 to include the class of exponentially decaying data, then the asymptotic behavior of the solution to (1.2) is sensitive to the rate of decay of φ 0 at x = ±∞ (see e.g. [30, pp.42]), which is the leading edge of the front. This is related to the fact that 0 is a saddle for (1.2), see [9,16,34,44,47].
Precisely, denoting λ * = r/ d. It is proved [34,44] that:
(i) When the initial data φ 0 (x) decays faster than exp{-λ * x} at x = ∞, then the spreading speed c * = 2 dr;
(ii) When the initial data φ 0 (x) is the form of exp{-(λ + o( 1))x} at x = ∞ with λ < λ * , then the population has the spreading speed c(λ) = dλ + r λ which is strictly greater than 2 dr.
For recent developments in asymptotic spreading of a single population in heterogeneous environments, we refer to [5,7,19] for the one-dimensional case, and to [6,8,45,56] for higher-dimensional case.
For close to three decades, researchers have been trying to extend these results to reaction-diffusion systems describing two or more interacting populations.
Motivated by the northward spreading of several tree species into the newly de-glaciated North American continent at the end of the last ice age, Shigesada et al. [48,Ch. 7] formulated the question of spreading of two or more competing species into an open habitat, i.e., one that is unoccupied by either species. In case of two competing species, it is conjectured that for large time, the solution behaves like stacked traveling fronts, i.e., it exhibits two transition layers moving at two different speeds c 1 > c 2 , connecting three homogeneous equilibrium states (0, 0), E 1 and E 2 . Here E 1 is the semi-trivial equilibrium where the faster species is present, and E 2 is either the other semi-trivial equilibrium or the coexistence equilibrium (if the latter exists). While it is not difficult to see that the spreading speed c 1 of the faster species can be predicted by the underlying single equation (since the slower species is essentially absent at the leading edge of the front), the determination of the second speed remained open over a decade. Lin and Li [40] first worked on the spreading properties of (1.1) in the weak competition case 0 < a, b < 1 with compactly supported initial condition (u 0 , v 0 ) and obtained estimates for the spreading speed c 2 of the slower species. For the strong competition case a, b > 1, Carrère [10] determined both of the spreading speeds, where c 2 is determined by the unique speed of traveling wave solutions connecting the semi-trivial steady state (1, 0) and (0, 1). The predatorprey system was considered by Ducrot et al. [15]. For cooperative systems with equal diffusion coefficients, the existence of stacked fronts for cooperative systems was also studied by [31]. In these cases, the spreading speeds of each individual species can be determined locally and is not influenced by the presence of other invasion fronts.
However, the second speed c 2 can in general be influenced by the first front with speed c 1 , as demonstrated by the work of Holzer and Scheel [29] which applies in particular to (1.1) for the case a = 0 and b > 0. They showed that the second speed c 2 can be determined by the linear instability of the zero solution of a single equation with space-time inhomogeneous coefficient. For coupled systems, the case 0 < a < 1 < b was treated in a recently appeared paper of Girardin and the third author [26]. By deriving an explicit formula for c 2 , it is observed that c 2 can sometimes be strictly greater than the minimal speed of traveling wave connecting E 1 and E 2 , and that it depends on the first speed c 1 in a non-increasing manner. The proof in [26] is based on a delicate construction of (piecewise smooth) super-and sub-solutions for the parabolic system. In our previous paper [41], we showed that in the weak competition case 0 < a, b < 1 the formula for c 2 is exactly the same as the one in [26] but with a novel strategy of proof based on obtaining large deviation estimates via analyzing the Hamilton-Jacobi equations obtained in the thin-front limit. We also mention that coupled parabolic systems were also treated in [18,23] based on the large deviations approach, but in these papers all components spread with a single spreading speed.
In this paper, we study the spreading of two competing species into an open habitat with exponentially decaying (in space) initial data, with attention to how the spreading speeds are influenced by the exponential rates of decay at infinity.
For a function g : R → R and λ ∈ R, we say that g(x) ∼ e -λx at ∞ if
Definition for g(x) ∼ e λx at -∞ is similar. We now state our hypothesis for the initial data (u 0 , v 0 ).
) 2 is strictly positive on R, and there exist positive constants θ 0 ,
where a∧b = min{a, b} for a, b ∈ R. Here the quantity σ 1 (resp. σ 2 ) denotes the spreading speed of v (resp. u) in the absence of the competitor [34,44]. Without loss of generality, we assume σ 1 ≥ σ 2 throughout this paper. This amounts to fixing the choice of v to be the faster spreading species.
Our main result is stated as follows.
Theorem 1.3. Assume σ 1 > σ 2 . Let (u, v) be the solution of (1.1) such that the initial data satisfies (H λ ). Then there exist c 1 , c 2 , c 3 ∈ R such that c 3 < 0 < c 2 < c 1 , and for each small η > 0, the following spreading results hold:
(1.4)
Precisely, the spreading speeds c 3 < 0 < c 2 < c 1 can be determined as follows:
where c LLW (resp. cLLW ) is given in Theorem 1.1 (resp. Remark 1.2), and
with the quantity λnlp being given by
To visualize the spreading result (1.4) visually, we consider the scaling
whose asymptotic behaviors can be given in Figure 1. Note that while the spreading speed c 1 of the faster species v is entirely determined by λ + v (the exponential decay of v 0 at x ≈ ∞), and is unaffected by the slower species u, the corresponding speed c 2 of species u depends upon σ 1 and λ u (the exponential decay of u 0 at x ≈ ∞). In particular, when λ + v ≥ r d and λ u > σ1 2 , i.e., v 0 (x) and u 0 (x) decay fast enough, the speeds c 1 and c 2 are the same as that of the case of compactly supported initial data (see [41,Theorem 1.2]). Remark 1.4. We point out that the speed c 2 in Theorem 1.3 is non-increasing in both σ 1 and λ u , which follows from the following observations: (i) λnlp given by (1.7) is non-decreasing in both σ 1 and λ u ;
This fact makes intuitive sense: (i) a higher σ 1 means the region dominated by species v, which is roughly {(t, x) : c 2 t < x < σ 1 t}, is larger and thus rendering it more difficult for species u to invade; (ii) a higher λ u means there are less population at the front to pull the invasion wave, which also makes it difficult for species u to invade.
Fix σ 1 , λ u > 0 and 0 < a < 1, such that σ 1 > σ 2 holds. We shall see that the quantity ĉnlp in (1.6) can be equivalently defined by {(t, x) : w 2 (t, x) = 0} = {(t, x) : t > 0 and x ≤ ĉnlp t}, where w 2 (t, x) is the unique viscosity solution of the Hamilton-Jacobi equation
Here χ S is the indicator function of the set S ∈ (0, ∞) × R.
A further point of interest is the involvement of (0, 0) and (k 1 , k 2 ) in coinvasion process of (1.1), which happens only in the weak competition case 0 < a, b < 1. In this case, the equilibrium states (0, 0) and (k 1 , k 2 ) are un-ordered, and hence the existence of traveling wave, due to Tang and Fife [50], cannot be established by monotone dynamical systems framework due to Weinberger et al. [57] (see also [20,39]). We will see that the invasion front (k 1 , k 2 ) into (0, 0) is indeed realized in (1.1) for initial data with certain values of exponential decay rates λ u , λ + v at infinity, namely, when σ 1 = σ 2 .
Theorem 1.5. Assume σ 1 = σ 2 . Let (u, v) be the solution of (1.1) such that the initial data satisfies (H λ ). Then for each small η > 0, it holds that
where c 3 = -max{c LLW , σ 3 } and that cLLW is given in Remark 1.2.
For initial data with general exponential decay rates, Theorem 1.3 demonstrates that there are two separate monostable fronts where each of the two species invades with distinct speeds. Moreover, if the parameters of (1.1) changes in such a way that |σ 1 -σ 2 | → 0, the distance of the two fronts tends to zero. Therefore, the invasion front of (k 1 , k 2 ) transitioning directly into (0, 0), due to Tang and Fife, is in fact the special case when these two monostable fronts coincide (Theorem 1.5).
Remark 1.6. As in [17,41], 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 [48,Chapter 7].
To determine c 1 , c 2 , c 3 , we introduce large deviation approach and construct appropriate viscosity super-and sub-solutions for certain Hamilton-Jacobi equations, and then apply the comparison principle (Theorem A.1) to obtain the desired estimations. We outline the main steps leading to the determination of the nonlocally pulled spreading speed c 2 , as stated Theorem 1.3, and remark that c 1 , c 3 can be obtained by a similar even simpler argument as c 2 .
1. To estimate c 2 from below, we consider the transformation w 2 (t, x) = -log u t , x and show that the half-relaxed limits
exist, upon establishing uniform bounds in C loc (see Lemma 3.2). By constructing viscosity super-solution w 2 , which satisfies {(t, x) : w 2 (t, x) = 0} = {(t, x) : t > 0 and x ≤ ĉnlp t}, and using the comparison principle (Theorem A.1), we can show that w * 2 ≤ w 2 , and thus w 2 → 0 locally uniformly in {(t, x) : x < ĉnlp t}. One can then apply the arguments in [17,Section 4] The rest of the paper is organized as follows: In Section 2, we give upper estimates c i for i = 1, 2, 3 and c 2 ≥ c LLW . In Section 3, we give lower estimates of c 1 , c 2 . The approximate asymptotic expressions of u and v are established in Section 4, where we also determine c 2 , c 3 . In Section 5, we discuss the relation of our results with the invasion mode due to Tang and Fife [50]. In Section 6, we discuss the relation of our result with that of [26] due to Girardin and the last author. In Section 7, we prove an extension which is associated to the spreading speeds of the three-species competition systems. We conclude the article with the Appendix. Therein we give the comparison principle of Hamilton-Jacobi equation with piecewise Lipschitz continuous Hamiltonian and two other useful lemmas.
This paper concerns the Cauchy problem of a system of reaction-diffusion equations modeling two competing species. For the spreading of two species into an open habitat, we refer to [37] for an integro-difference competition model, and to [14] for a competition model with free-boundaries. See also [27,42,53,54,58] for other related results in free-boundary problems. We also note that in those works the spreading speeds are always locally determined and thus do not interact.
The concepts of maximal and minimal spreading speeds are introduced in [28, Definition 1.2] for a single species; see also [24,41]. In our setting, we define
where 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, respectively.
In this section, for initial data satisfying (H λ ), we will give some estimates of the maximal and minimal spreading speeds. The main result of this section can be precisely stated as follows.
Proposition 2.1. Let (u, v) be a solution of (1.1) with initial data satisfying (H λ ). Then the spreading speeds defined in (2.1) satisfy
where σ 1 , σ 2 , σ 3 are defined in (1.3) and c LLW , cLLW are given respectively in Theorem 1.1 and Remark 1.2. Furthermore, we have
Proof. We will complete the proof in the following order: (1)
2) holds.
Step 1. We show assertions (1), ( 2) and ( 3).
Observe that for some M > 0 the function
is a weak super-solution to the single KPP-type equation
of which u(t, x) is clearly a sub-solution. By choosing the constant M > 0 so large that u 0 (x) ≤ u(0, x) in R, it follows by comparison that
This proves c 2 ≤ σ 2 , i.e., assertion (1) holds.
Similarly, we deduce assertion (2) by comparison with
To prove assertion (3), let ṽ(t, x) = v(t, -x), we turn to consider another single KPP-type equation
Again the scalar comparison principle implies v(t, -x) = ṽ(t, x) ≥ v. By the results in [34] or [44], we have
which means c 3 ≤ -σ 3 .
Step 2. We show assertions (4) and (5).
Given any non-trivial, compactly supported function ṽ0 such that 0 ≤ ṽ0 ≤ v 0 . Then
Let (ũ LLW , ṽLLW ) be the solution to (1.1) with initial value (1, ṽ0 (x)). Then Theorem 1.1 and Remark 1.2 guarantee the existence of cLLW ≥ 2 dr(1 -b), such that
By the comparison principle for (1.1), we have
This proves c 3 ≤ -c LLW and thus assertion (4) holds.
Similarly, we can get show assertion (5), i.e., c 2 ≥ c LLW . By comparing (u, v) with the solution (u LLW , v LLW ) of (1.1) with initial condition (ũ 0 , 1), for some compactly supported ũ0 satisfying 0 ≤ ũ0 ≤ u 0 , and then using Theorem 1.1. In this way, we get
(2.6)
Step 3. We show assertion (6). In view of (2.5) and (2.6), one can deduce (2.2) from items (a) and (c) of Lemma B.1.
3 Estimating c 1 and c 2 from below
We assume σ 1 > σ 2 throughout this section. In this section, we estimate c 1 and c 2 from below via the large deviation approach and applying Theorem A.1. To this end, we introduce a small parameter via the following scaling
Under the new scaling, we rewrite the equation of u and v in (1.1) as
To obtain the asymptotic behaviors of v and u as → 0, the idea is to consider the WKB ansatz w 1 and w 2 , which are given respectively by
and satisfy, respectively, the equations
and
Proof. We first prove (a) by adapting the arguments from [17,Section 4]. Let K, K and G be given as above.
Fix an arbitrary (t 0 , x 0 ) ∈ K and define the test function
This yields
Since w 2 -ρ attains maximum over K at (t , x ), we have in particular
Recalling u (t 0 , x 0 ) = e -w 2 (t0,x0) and u (t , x ) = e -w 2 (t ,x ) , we therefore have
Since this argument is uniform for (t 0 , x 0 ) ∈ K (depends only on K, K and G), we deduce assertion (a). The proof for (b) is analogous.
Next, we will pass to the (upper and lower) limits using the half-relaxed limit method, which is due to Barles and Perthame [4]. Define
That the above are well defined is due to the following lemma: Lemma 3.2. Let w 1 and w 2 be the solutions to (3.4) and (3.5), respectively. Then there exits some Q > 0, independent of small, such that
for (t, x) ∈ [0, ∞) × R, where x + = max{x, 0} and x -= max{-x, 0}.
Proof. We only prove (3.8a) and the estimations (3.8b)-(3.8d) follow from a quite similar argument. Since v ≤ 1, we have w 1 ≥ 0 by definition. By (H λ ), there exist positive constants C 1 and C 2 such that
By definition (3.3), we have
such that
where the last inequality is due to (3.9). By comparison, (3.10) thus holds. By a similar argument, we can verify
where Q is defined by (3.11). Combining with (3.10) and (3.12) gives the desired upper bound of w 1 .
To obtain the lower bound of w 1 , we may define functions
. By the same arguments as before, we can check
This completes the proof of (3.8a).
Remark 3.3. According to Lemma 3.2, by letting t = 0 and then → 0 in (3.8a) and (3.8b), we deduce that
Similarly, by setting x = 0 and then → 0 in (3.8c) and (3.8d), we have
By Proposition 2.1, c 2 ≤ σ 2 , so we deduce
(3.13) Lemma 3.4. Let (u, v) be a solution of (1.1) with initial data satisfying (H λ ). Then
where σ 2 is defined by (1.3) and λ - v , λ + v ∈ (0, ∞) are given in (H λ ). Proof. First, observe that w * 1 is upper semicontinuous (usc) by construction. By Remark 3.3, the initial and boundary conditions of (3.14) and (3.15) are satisfied.
It remains to show that w * 1 is a viscosity sub-solution of min{∂ t w +d|∂ x w| 2 + r(1 -bχ {x≤σ2t} ), w} = 0 in the domain (0, ∞) × R. According to definition of viscosity sub-solution of Hamilton-Jacobi equation, (see Appendix A), let ϕ ∈ C ∞ ((0, ∞) × R) and let (t 0 , x 0 ) be a strict local maximum point of w * 1 -ϕ such that w * 1 (t 0 , x 0 ) > 0. By passing to a sequence = k if necessary, w 1 -ϕ has a local maximum point at (t , x ) such that w 1 (t , x ) → w * 1 (t 0 , x 0 ) and (t , x ) → (t 0 , x 0 ) uniformly as → 0. At the point (t , x ), we have
By the fact that e -w 1 (t ,x )/ → 0 (as w 1 (t , x ) → w * 1 (t 0 , x 0 ) > 0), we may pass to the limit = k → 0 so that
1 is a viscosity sub-solution of (3.14) and (3.15).
Lemma 3.5. Let (u, v) be a solution of (1.1) with initial data satisfying (H λ ).
where σ 1 is defined by (1.3).
Proof. Define the function
Next, we claim that the continuous w 1 is a viscosity super-solution of (3.14). We will check the latter case of λ + v > r d as the former case can be verified analogously. Under the condition λ + v > r d , we have σ 1 = 2 √ dr. According to definition of viscosity super-solution of Hamilton-Jacobi equation (see Appendix A), let ϕ ∈ C ∞ ((0, ∞) × R) and let (t 0 , x 0 ) be a strict local minimum point of w 1 -ϕ.
If x 0 /t 0 = 2 √ dr, then w 1 is a classical solution of (3.14). If x 0 /t 0 = 2 √ dr, then w 1 (t 0 , x 0 ) = 0 by definition. Moreover,
and we must have ∂ t ϕ(t 0 , x 0 ) + 2 √ dr∂ x ϕ(t 0 , x 0 )) = 0, and hence
where the first equality follows from the fact that x 0 /t 0 = 2 √ dr = σ 1 > σ 2 . By Remark 3.3 and the expression of w 1 , we have
And recalling Lemma 3.4(a), w 1 and w * 1 is a pair of viscosity super and subsolutions of (3.14). Then, we may apply Theorem A.1 to get
which implies that
Letting → 0, we arrive at
Hence for each small η > 0, by choosing the compact sets
, we may apply Lemma 3.1(b) to deduce that
This implies c 1 ≥ σ 1 .
Corollary 3.6. Let σ 1 > σ 2 and let (u, v) be a solution of (1.1) with initial data satisfying (H λ ). Then for each small η > 0,
where σ 1 is defined by (1.3). We may then apply Lemma B.1(d) to deduce (3.17b).
By Corollary 3.6, we have
(3.18) Lemma 3.7. Let (u, v) be a solution of (1.1) with initial data satisfying (H λ ).
Then, w * 2 is a viscosity sub-solution of
where σ 1 is defined by (1.3) and λ u > 0 is given in (H λ ).
Proof. The proof is analogous to Lemma 3.4 and we omit the details.
It remains to consider cases (a) and (b). We start by defining
Suppose case (a) holds, then ĉnlp = cnlp . Define w 2 by
On the other hand, if x0 t0 = cnlp , then ∇ϕ(t 0 , x 0 ) • (1, cnlp ) = 0, and
Hence, w 2 is a viscosity super-solution of (3.19).
By Remark 3.3 and the express of w 2 , we have
And recalling that w * 2 is a viscosity sub-solution of (3.19), we may deduce by Theorem A.1 that
Now,
x ≤ ĉnlp t}. Hence, w 2 (t, x) = -log u (t, x) → 0 locally uniformly on {(t, x) : x < ĉnlp t}.
Hence for each small η > 0, by choosing the compact sets
, we may apply Lemma 3.1(a) to get
which implies c 2 ≥ ĉnlp . Finally, for case (b), then we have ĉnlp = cnlp . We define
for x t ≤ cnlp . Then one can verify that w 2 is likewise a viscosity super-solution of (3. 19), so that one can repeat the arguments for case (a) to show, again, that c 2 ≥ ĉnlp .
We assume σ 1 > σ 2 throughout this section. It remains to show c 2 ≤ max{c LLW , ĉnlp } and c 3 ≥ -max{c LLW , σ 3 }.
For δ ≥ 0, we will construct an exponent μδ depending continuously on δ such that u(t, (σ
so that we may apply Lemma B.2(a) to estimate c 2 from above.
Lemma 4.1. Let (u, v) be a solution of (1.1) with initial data satisfying (H λ ). Then w 2, * is a viscosity super-solution of
where σ 1 and σ 2 are defined in (1.3).
Proof. It follows from standard arguments as in Lemma 3.4.
Proposition 4.2. Let (u, v) be a solution of (1.1) with initial data satisfying
where c LLW and ĉnlp are defined respectively in Theorem 1.1 and 1.3.
Proof.
Step 1. Define w 2 : [0, ∞) × R by
in case λ u > 1, and by
Then it is straightforward to verify that w 2 is a viscosity subsolution of (4.1). Since, w 2, * (0, x) = λ u max{x, 0} = w 2 (0, x) in R (by Remark 3.3), we may apply Theorem A.1 to deduce
Step 2. To show that, for each ĉ ≥ 0,
And that w 2 (1, σ 1 ) is given by
and cnlp , cnlp , λnlp are all given in Lemma 3.8. By definition of w 2, * and w 2 (t, x) = -log u (t, x), for each small > 0, by applying Step 1, we have
which implies (4.4). By the formula of w 2 , we can show (i) For σ 1 < 2λ u , we substitute (t, x) = (1, σ 1 ) in (4.2) to obtain
where
(ii) For σ 1 ≥ 2λ u , we substitute (t, x) = (1, σ 1 ) in (4.2) to obtain
Recalling the definition of λnlp in (3.20), we have
Hence, (4.7) becomes
where cnlp , λnlp are as in (3.20).
This implies (4.5) holds, which completes Step 2.
Step 3. To show c 2 ≤ max{c LLW , ĉnlp }. It follows from Proposition 2.1 and Corollary 3.6 that for ĉ ∈ (σ 2 , σ 1 ), lim t→∞ (u, v)(t, 0) = (k 1 , k 2 ) and lim t→∞ (u, v)(t, ĉt) = (0, 1).
Step 2 and observation λ LLW c LLW = λ 2 LLW + 1 -a, then we apply Lemma B.2(a) in Appendix to conclude that for ĉ ∈ (σ 2 , σ 1 ),
Letting ĉ σ 1 , (4.10) can be expressed as (denote μ = w 2 (1, σ 1 ))
and λnlp is given in Lemma 3.8. Note that
) By (4.6) and (4.9), μ = w 2 (1, σ 1 ) can be expressed as
and λ μ is as defined in (4.13). Note that G(λ) is strictly increasing on [0, σ1 2 ]. We note for later purposes that (4.14) is a quadratic equation in λ μ, so that
Since λ LLW ∈ (0, √ 1 -a], we divide our discussion into two cases: (i)
By (4.13), λnlp = λ μ < λ LLW , whence it follows from the observation
and the monotonicity of s+ 1-a s in (0, √ 1 -a] that ĉnlp ≥ c LLW . It remains to show that c σ1,μ = ĉnlp . Now, by monotonicity of G, we have
By (4.11), we have
. Hence,
where the first and second equalities follow from (4.15) and (4.14), respectively.
(ii) Case λ LLW ≤ λ μ.
By (4.13),
It follows from (4.16) that ĉnlp ≤ c LLW . It remains to show that μ ≥ G(λ LLW ), so that c σ1,μ = c LLW = max{c LLW , ĉnlp }. Indeed, one can check that λ LLW ≤ λ μ ≤ σ 1 /2, and we deduce
The proof of Proposition 4.2 is now complete.
For convenience, let ũ(t, x) = u(t, -x), ṽ(t, x) = v(t, -x), and define
Again we pass to the half-relaxed limit:
Lemma 4.3. Let (ũ, ṽ) be a solution of (1.1) such that x → (ũ(0, -x), ṽ(0, -x)) satisfies (H λ ). Then, for each small η > 0,
Proof. Let v KPP be the solution of
By choosing C to be sufficiently large, we may apply comparison principle to get 0 ≤ ṽ ≤ v KPP . Therefore, for each η > 0,
Let u KPP be the solution of
Again the scalar comparison principle implies u ≥ u KPP . By the results in [34] or [44], we have, for each small η > 0,
(4.20)
Lemma 4.4. Let (ũ, ṽ) be a solution of (1.1) such that x → (ũ(0, -x), ṽ(0, -x)) satisfies (H λ ). Then, w 3, * is a viscosity super-solution of
Proof. The proof is similar to proof of Lemma 3.4(b) and is omitted. where cLLW and σ 3 are defined in Remark 1.2 and (1.3), respectively.
Proof.
Step 1. To show
where
As in Step 1 of Proposition 4.2, one can verify that w 3 is a viscosity sub-solution of (4.21). By the expression of w 3 , Remark 3.3 and w 1, * (t, -x) = w 3, * (t, x), we have
Hence we apply Theorem A.1 to obtain (4.22).
Step 2. To show for each ĉ ≥ 0, we have ṽ(t, ĉt) ≤ exp{(w 3 (1, ĉ) + o(1))t} for t 1.
(4.23)
This can be done as in Step 2 of Proposition 4.2.
Step 3. To show c 3 ≥ -max{c LLW , σ 3 }.
We note for later purposes that μ2 is a quadratic expression in λ - v , so that
We may then apply Lemma B.2(b) to conclude
where (4.26) is used for the last inequality.
(i) For the case λ - v > λLLW , we take ĉ → ∞ in (4.28) to get μ2 ≥ λLLW (ĉ -cLLW ) , so that by (4.27), -c 3 ≤ cLLW ≤ max {c LLW , σ 3 };
. Then
and λ ĉ,μ2 ≤ λ - v (by comparing with the second part of (4.26)). Hence, we arrive at
Next, we claim that
To this end, subtract the first part of (4.26) from (4.29) to get
Dividing the above by ĉ and letting ĉ → ∞, we obtain (4.31).
By (4.31), we can take ĉ → ∞ in (4.30
This completes the proof of Proposition 4.5.
Proof of Theorem 1. Observe that the first two items of (1.4) is a direct consequence of Corollary 3.6. Next, we shall show that
Given some small η > 0, definitions of c 3 and c 1 imply the existence of
Observe that v(t, x) and δ form a pair of super-and sub-solutions to the KPPtype equation
In this section, we assume σ 1 = σ 2 and prove Theorem 1.5.
Proof of Theorem 1.5. For any small δ ∈ (0, 1), let (u δ , v δ ) and (u δ , v δ ) be respectively any solution of
and
with initial data satisfying (H λ ). By comparison, we deduce that
Notice that (u δ , v δ ) is a solution of (5.1) if and only if
where
enough. By applying Theorem 1.3 to (5.5), we deduce that the rightward and leftward spreading speeds c δ 1 and c δ 3 of V δ (which is the same as v δ ), and the rightward spreading speed c δ 2 of U δ (same as u δ ) exist. Furthermore, they can be characterized by
Precisely, c δ LLW (resp. cδ LLW ) is the spreading speed for (5.5) as given in Theorem 1.1
and moreover
where
3), we can compare with the spreading speeds c 1 , c 2 and c 3 of (u, v):
It remains to show that, assuming σ 1 = σ 2 , we have ĉδ nlp → σ 2 as δ → 0. Divide into the two cases:
which is due to
where λnlp is given in (1.7). To this end, observe that
which is a consequence of
we deduce (5.9). By (5.8) and (5.9), we have σδ 1 ≥ 2λ u and λδ nlp < 1 -a δ for δ small, so
Since we want ĉδ nlp → σ 2 , it remains to show that σ 2 = λnlp + 1-a λnlp . To this end, recall, from the definition of λnlp (1.7), that
.
Using
This implies σ 2 = λnlp + 1-a λnlp . The proof is now complete.
Hence, by the continuity of c δ LLW and cδ LLW in δ (see, e.g. [52, Theorem 4.2 of Ch. 3]), letting δ → 0 in (5.7) yields
(5.12)
By a quite similar process, we can obtain (u δ , v δ ) is a solution of (5.2) if and only if
where
By exchanging the roles of u and v in (1.1), we may follow the arguments above, and apply Theorem 1.3 once again to deduce that
Theorem 1.5 follows from combining c i ≤ c i , (5.12), (5.14) and
6 The case 0 < a < 1 < b due to Girardin and Lam
The Hamilton-Jacobi approach, which we have so far applied to study the weak competition case (0 < a, b < 1), can also be applied to tackle the case (0 < a < 1 < b), which was previously studied by Girardin and the third author [26]. This provides an alternative approach which is more transparent than the involved construction of global super-and sub-solutions for the Cauchy problem, as was done in [26]. By arguing similarly as in Theorem 1.3, one can prove the following result. Theorem 6.1. Assume 0 < a < 1 < b and σ 1 > σ 2 . Let (u, v) be the solution of (1.1) such that the initial data satisfies (H λ ). Then there exist c 1 , c 2 ∈ (0, ∞) such that c 1 > c 2 and, for each small η > 0, the following spreading results hold:
Precisely, the spreading speeds c 1 and c 2 can be determined as follows:
where σ 1 is defined in (1.3), ĉLLW denotes the minimal wave speed of (1.1) connecting (1, 0) with (0, 1) and ĉnlp is given by
By Theorem 6.1, the spreading speed c 2 is determined by σ 1 (i.e., c 1 ) and λ u . In what follows, we explore the relation of c 2 and σ 1 for fixed λ u . Define the following auxiliary functions:
where λnlp is given by (6.3). It is easily seen that f is decreasing and bijective in
], while g is decreasing and bijective in
More precisely, it follows that
.
For fixed λ u and varied λ + v (or σ 1 ), by Theorem 6.1 we can rewrite the spreading speed c 2 as follows.
(a) For g ∞ ≤ ĉLLW , we have the followings:
for σ 1 ≥ g -1 (ĉ LLW );
(b) For g ∞ > ĉLLW , we have the followings:
For the case (a) g ∞ ≤ ĉLLW , the relationship between the spreading speeds σ 1 and c 2 given by (a1)-(a3) is illustrated in Figure 2. Therein we may obtain the exact spreading speeds of (1.1), which are determined entirely by λ u , λ + v ∈ (0, ∞). Traversing all of λ u , the set of admissible speeds σ 1 and c 2 agrees with [26, , c 2 ) such that (6.4) holds.
is decreasing in λ u . Noting that c ∈ (f (c), c), we may select the unique
, c 2 ) such that (6.4) holds;
). In this case, λnlp ∈ 0, √ 1 -a and thus g(c
are also strictly monotone in λ u , so that there is the unique λ u such that (6.4) holds.
The proof is now complete.
In this section, we consider the following competition system with forcing:
where lim
We will make an observation in preparation for our forthcoming work on threespecies competition systems. Recall the definitions of σ i (i = 1, 2, 3) from (1.3).
Theorem 7.1. Let d, r, b > 0, 0 < a < 1 and σ 1 > σ 2 . Suppose that h(x, t), k(x, t) are non-negative and satisfy (7.2). Let (u, v) be the solution of (7.1) with the initial data satisfying (H λ ). Assume
Then,
where c i , c i (i = 1, 2) are defined in (2.1). Furthermore, for each small η > 0,
where σ 1 , σ 2 are defined in (1.3) and c LLW , ĉnlp are respectively given in Theorem 1.1 and 1.3.
Proof. The proof can be mimicked after that of Theorem 1.3.
Step 1. The estimates c 1 ≤ σ 1 and c 2 ≤ σ 2 can be proved by rather similar arguments as in Proposition 2.1, and the details are omitted here.
Step 2. We show that for each small η > 0, Here, we just show the first one since the proof of the second one is analogous. For the case of λ u ≥ √ 1 -a, by (7.2) and c 0 < σ 2 , the system (7.1) is approximately equal to (1.1) in {(t, x) : x ≥ Fix any c ∈ (max{
It is enough to show that there exist positive constants δ, λ1 , λ2 , T such that λ1 < λ2 and u(t, x + c t) ≥ δ 4 max e -λ1x -e -λ2x , 0 for t ≥ T, x ≥ 0.
This implies c 2 ≥ c for each c ∈ (max{
To this end, choose δ 1 > 0 small enough so that
This is possible since c < σ 2 and that s →
is monotone, so that
Next, choose T > 0 large so that
and then choose δ ∈ (0, δ 1 ] so that u(T, x) ≥ δ 4 e -λ1(x-c T ) for x ≥ c T,
where (7.7) follows from (7.2) by noting c > c 0 ; and that (7.8) holds due to u(T, x) ∼ e -λux at ∞ and λ1 > λ u (see, e.g. [52, Corollary 1 of Ch. 1]). By the choice of λ1 < λ2 , δ, δ 1 , T , it follows that u(t, x) := max δ 4 e -λ1(x-c t) -e -λ2(x-c t) , 0 ,
is a sub-solution of the KPP-type equation
where Ω := {(t, x) : t ≥ T, x ≥ c t}.
For δ ∈ (0, δ 1 ] to be specified later, define u(t, x) := max δ 4 e -λ1(x-c t) -e -λ2(x-c t) , 0 ,
where λ1 is given in (7.6) and λ2 = 1 2 c + (c ) 2 -4(1 -a -2δ) . We will choose T > 0 and δ ∈ (0, δ 1 ] so that
i.e., u and u is a pair of super-and sub-solutions of the KPP-type equation
where Ω := {(t, x) : t ≥ T, x ≥ c t}. Hence, by comparison, (7.5) holds.
To proceed further, as in Section 3, based on the scaling (3.1), we introduce the WKB ansatz w 2 , which is given by w 2 (t, x) = -log u (t, x), satisfying the equation:
Here h (t, x) = h( t , x ). By Remark 3.3, we also use the half-relaxed limit method and introduce w * 2 and w 2, * . By (7.4),
and u is moreover bounded by 1. We have then, by definitions, that w * 2 (t, (σ 2 -η)t) = w 2, * (t, (σ 2 -η)t) = 0. (7.13)
Step 3. We prove c 1 ≥ σ 1 . This follows from (7.4) and definition of c 1 .
Step 4. We prove c 2 ≥ ĉnlp .
By Step 1 and h ≥ 0, we have
In view of σ 2 > c 0 , we choose 0 < η 1 such that σ 2 -η > c 0 . We then use (7.2) to derive that lim
Based on (7.13), similar to Lemma 3.4, we can deduce that w * 2 is a viscosity sub-solution of
We then apply the same arguments developed in Lemmas 3.8 by constructing the same super-solutions, to deduce that c 2 ≥ ĉnlp .
Step 5. We show c 2 ≤ max{c LLW , ĉnlp } and (7.3). By (7.2) again, similar to Corollary 3.6, we can get lim inf
so that we may use (7.13) to deduce that w 2, * is a viscosity super-solution of
as in Lemma 4.1. Then we can get c 2 ≤ max{c LLW , ĉnlp } by the same arguments developed in Proposition 4.2. We finally deduce (7.3) by similar arguments as in the proof of Theorem 1.3, which completes the proof.
(A4) There exists some M ≥ 0 such that for each λ ∈ [0, 1) and x 0 ∈ R N , there exists constants ¯ (λ, x 0 ) > 0 and C(λ, x 0 ) > 0 such that for all p ∈ R N ,
Theorem A.1. Suppose that H satisfies the hypotheses (A1)-(A4). Let w and w be a pair of super-and sub-solutions of (A.1) such that w ≥ w on ∂ p Ω, then
where H is convex and coercive in p, and s → R(s) has bounded variation and satisfies |R(s)| ≤ M for some M ≥ 0. Then it is easy to verify that the hypotheses (A1)-(A4) hold. In particular, it applies for all our purposes in this paper.
Our condition (A3) is a quantitative version of the "local monotonicity condition" that was introduced in [11]. See [11,33] for more examples of Hamiltonians verifying the hypotheses (A1)-(A4).
Proof. Assume to the contrary that sup Ω (w -w) > 0.
(A.2)
Step 1. We may assume, without loss of generality, that M = 0 in the hypothesis (A4). Indeed, if we make the change of variables w (t, x) = w(t, x) + M t and w (t, x) = w(t, x) + M t, then w , w are, respectively, a sub-solution and a super-solution of (A.1) with L replaced by L = L + M , and H(t, x, p) replaced by H (t, x, p) = H(t, x, p) -M . This function H satisfies the hypotheses (A1)-(A4) with M = 0. Henceforth in the proof we assume that the hypothesis (A4) holds with M = 0.
Step 2. It suffices to show that w ≤ w under the additional assumption that w ≤ K for some K > 0. Indeed, if w is unbounded in Ω, then fix a constant K > 0 and take a sequence {g j } of smooth functions satisfying g j (r) min{r, K} and 0 ≤ g j (r) ≤ 1, g j (r)r ≤ r, g j (r) ≤ min{r, K} for all r ∈ R.
Then ŵ := g j (w) is a viscosity sub-solution of (A.1), since in the region {(t, x) : ŵ-Lt > 0} ⊂ {(t, x) : w -Lt > 0}, we may use the hypothesis (A4) to yield
By the stability of property of viscosity super and sub-solutions [1, Theorem 6.2], we may let j → ∞ to conclude that min{w, K} is a viscosity sub-solution of (A.1) for each K > 0. It now remains to prove Theorem A.1 for all bounded above viscosity sub-solutions, since then min{w, K} ≤ w for all K > 0 ⇒ w ≤ w.
For λ, δ ∈ (0, 1), denote
where ψ(x) = 1 2 log(|x| 2 + 1) and C = C(λ, 0) as in the hypothesis (A4).
Step 3. We choose λ 1, δ ∈ (0, ¯ (λ, 0)], R > 0 and
From (A.2) and Step 3, we may fix λ 1 and δ 0 such that
whence we may fix R 1 so that max ΩR W (t, x) = max Ω W (t, x) > 0 holds. It remains to observe that the maximum (t 0 , x 0 ) in Ω R is attained in the interior, since W (t, x) < 0 when t = T or when (t, x) ∈ ∂ p Ω.
Step 4. With x 0 as being given in Step 3, fix > 0 small enough so that C(λ, x 0 ) ≤ C(λ, 0) and δ ≤ ¯ (λ, x 0 ), (A.4) and define W (t, x) := W (t, x) -δλ ψ(x -x 0 ) -
where ψ(x) = 1 2 log (|x| 2 + 1) and C = C(λ, 0) is as before. Then, (t 0 , x 0 ) is a strict global maximum of W (t, x). Define also Ψ α,β (t, x, s, y) =λ 2 w(t, x) -w(s, y) -δ(ψ(x)
Step 5. We claim that there exists α > 0 such that if min{α, β} ≥ α, then (i) Ψ α,β has a local maximum point (t 1 , x 1 , s 1 , y 1 ) in Ω R × Ω R ;
(ii) Ψ α,β (t 1 , x 1 , s 1 , y 1 ) ≥ W (t 0 , x 0 ) = W (t 0 , x 0 ) > 0;
(iii) β|t 1 -s 1 | 2 + α|x 1 -y 1 | 2 → 0, as min{α, β} → ∞;
(iv) (t 1 , x 1 ) → (t 0 , x 0 ) and (s 1 , y 1 ) → (t 0 , x 0 ) as min{α, β} → ∞,
where Ω R = Ω ∩ [(0, T ) × B R (0)]. Since w ≥ 0 and w ≤ K by Step 2, we see that sup ΩR×ΩR Ψ α,β ≤ K independently of α and β, and has a maximum point (t 1 , x 1 , s 1 , y 1 ) ∈ Ω R × Ω R . Now, by (A.3), Ψ α,β (t 1 , x 1 , s 1 , y 1 ) ≥ max ΩR Ψ α,β (t, x, t, x) = W (t 0 , x 0 ) = W (t 0 , x 0 ).
This proves assertion (ii). Furthermore, the boundedness also yields β|t 1 -s 1 | 2 + α|x 1 -y 1 | 2 = O(1). We claim that (t 1 , x 1 ) → (t 0 , x 0 ) and (s 1 , y 1 ) → (t 0 , x 0 ). Indeed, we may pass to a subsequence to get ( t, x) such that (t 1 , x 1 ) → ( t, x) and (s 1 , y 1 ) → ( t, x) as min{α, β} → ∞. Now, by (ii) we can write
-W (t 0 , x 0 ) + ( W (t 1 , x 1 ) + w(t 1 , x 1 )) -w(s 1 , y 1 ).
Letting min{α, β} → ∞, then (t 1 , x 1 , s 1 , y 1 ) → ( t, x, t, x). Using the fact that W (t, x) + w(t, x) (which is essentially λ 2 w(t, x) up to addition of continuous functions) and -w(s, y) are both upper semi-continuous in Ω, we may take limsup as min{α, β} → ∞ and deduce that 0 ≤ lim sup α 2 |x 1 -y 1 | 2 + β 2 |t 1 -s 1 | 2 ≤ -W (t 0 , x 0 ) + W ( t, x) ≤ 0.
Since (t 0 , x 0 ) is a strict maximum point of W , we must have ( t, x) = (t 0 , x 0 ). This proves assertions (iii) and (iv). Finally, (t 1 , x 1 , s 1 , y 1 ) → (t 0 , x 0 , t 0 , x 0 ) and hence must be an interior point of Ω R × Ω R when min{α, β} is sufficiently large. This proves (i).
Step 6. We show the following inequality: δ T 2 ≤ H * (s 1 , y 1 , α(x 1 -y 1 )) -H * (t 1 , x 1 , α(x 1 -y 1 )) + |t 1 -t 0 |.
(A.6)
Observe that (t 1 , x 1 ) is an interior maximum point of the function w(t, x)ϕ(t, x), where ϕ(t, x) = 1 λ 2 [w(s 1 , y 1 ) + δ(ψ(x) + Ct + 1 T -t ) + λδ( ψ(x -x 0 ) + Ct)
Also w(t 1 , x 1 ) > 0, which is a consequence of w(s 1 , y 1 ) ≥ 0 and Ψ α (t 1 , x 1 , s 1 , y 1 ) > 0. By definition of w being a viscosity sub-solution of (A.1), we have where q = 1 λ (δD x ψ(x 1 ) + α(x 1 -y 1 )). In the view of Dψ(x 1 -x 0 ) = x1-x0 |x1-x0| 2 +1 , we may apply the hypothesis (A4) to get
where we used C(λ, x 0 ) ≤ C(λ, 0) = C (due to (A.4)) in the last inequality.
Applying the hypothesis (A4) once more, we have -δ(C + 1 T 2 + λC) -β(t 1 -s 1 ) -(t 1 -t 0 ) ≥ [H * (t 1 , x 1 , α(x 1 -y 1 )) -δC] -λδC ≥ H * (t 1 , x 1 , α(x 1 -y 1 )) -δC -λδC, and hence δ T 2 + β(t 1 -s 1 ) + (t 1 -t 0 ) + H * (t 1 , x 1 , α(x 1 -y 1 )) ≤ 0.
(A.8)
In the same way, (s 1 , y 1 ) is a interior minimum point of the function w(s, y)ψ(s, y) with ψ(s, y) =λ 2 w(t 1 , x 1 ) -δ(ψ(x 1 ) + Ct 1 + 1 T -t ) -λδ(ψ(x 1 -x 0 ) + Ct 1 )
whence β(t 1 -s 1 ) + H * (s 1 , y 1 , α(x 1 -y 1 )) ≥ 0. (A.9) Subtracting (A.8) from (A.9), we obtain (A.6) as claimed. By Step 5 (iv), we have (t 1 , x 1 ) → (t 0 , x 0 ) and (s 1 , y 1 ) → (t 0 , x 0 ) as min{α, β} → ∞. On the one hand, if (t 0 , x 0 ) / ∈ Γ, then there exists α 1 > 0 such that (t 1 , x 1 ) and (s 1 , y 1 ) enter the (δ 0 /2)-neighborhood of (t 0 , x 0 ) whenever min{α, β} ≥ α 1 . Now, fix α and let β → ∞, then after passing to a sequence, we have t 1 , s 1 → t, x 1 → x, y 1 → ȳ.
with initial data (U (0), V (0)) = (δ, 1), so that (U , V )(∞) = (k 1 , k 2 ). By comparison in the time interval [-T, 0], we reveal that for each T > 0, (û, v)(t, x) (U , V )(t + T ) for (t, x) ∈ [-T, 0] × R, so that we in particular have, for every T > 0, (û, v)(0, 0) (U , V )(T ).
Letting T → ∞, we obtain (û, v)(0, 0) (k 1 , k 2 ). In particular, we deduce that
The following result is applied to prove Proposition 4.2 and Proposition 4.5.
Lemma B.2 ( [41, Lemma 2.4]). Let ĉ > 0, t 0 > 0, and (ũ, ṽ) be a solution of
∂ t ṽ -d∂ xx ṽ = rṽ(1 -bũ -ṽ), 0 ≤ x ≤ ĉt, t > t 0 , ũ(t 0 , x) = ũ0 (x), ṽ(t 0 , x) = ṽ0 (x), 0 ≤ x ≤ ĉt 0 . , if 0 < μ < λLLW (ĉ -cLLW ).