Asymptotic spreading speed for the weak competition system with a free boundary

This paper is concerned with a diffusive Lotka-Volterra type competition system with a free boundary in one space dimension. Such a system may be used to describe the invasion of a new species into the habitat of a native competitor. We show that the longtime dynamical behavior of the system is determined by a spreading-vanishing dichotomy, and provide sharp criteria for spreading and vanishing of the invasive species. Moreover, we determine the asymptotic spreading speed of the invasive species when its spreading is successful, which involves two systems of traveling wave type equations, and is highly nontrivial to establish.

The cases (I) and (II) are usually called the weak-strong competition case, while (III) and (IV) are known as the weak and strong competition cases, respectively. A number of variations of (1.2) (or (1.1)) have been used to model the spreading of a new or invasive species. For example, to describe the invasion of a new species into the habitat of a native competitor, Du and Lin [9] considered the following free boundary problem , t), t > 0, where x = g(t) is usually called a free boundary, which is to be determined together with u and v.
The initial functions satisfy This model describes how a new species with population density u invades into the habitat of a native competitor v. It is assumed that the species u exists initially in the range 0 < x < g 0 , invades into new territory through its invading front x = g(t). The native species v undergoes diffusion and growth in the available habitat 0 < x < ∞. Both u and v obey a no-flux boundary condition at x = 0. The equation g ′ (t) = −γu x (g(t), t) means that the invading speed is proportional to the gradient of the population density of u at the invading front, which coincides with the well-known Stefan free bounary condition. All parameters d, k, h, r, g 0 and γ are assumed to be positive. For more biological background, we refer to [1,5,8,9].
The work [9] considers the weak-strong competition case only. It is shown in [9] that when the invasive species u is the inferior competitor (k > 1 > h), if the resident species v is already well established initially (i.e., v 0 satisfies the conditions in (1.4)), then u can never invade deep into the underlying habitat, and it dies out before its invading front reaches a certain finite limiting position, whereas if the invasive species u is superior (h > 1 > k), a spreading-vanishing dichotomy holds for u (see Theorem 4.4 in [9]). Moreover, when spreading of u happens, the precise asymptotic spreading speed has been given by Du, Wang and Zhou [14]; it concludes that the spreading speed of u has an asymptotic limit as time goes to infinity, which is determined by a certain traveling wave type system.
We will show that a similar spreading-vanishing dichotomy holds for the invasive species u, but in sharp contrast to the weak-strong competition case (h > 1 > k) in [9], where when u spreads successfully, v vanishes eventually (namely (u, v) → (1, 0) as t → ∞), here in the weak competition case, when u spreads successfully, the two populations converge to the co-existence steady state (u * , v * ) as time goes to infinity. In fact, our results here indicate that the native competitor v always survives the invasion of u. Moreover, we also determine the precise spreading speed of u when the invasion is successful, which turns out to be the most difficult part of this work and consititutes the main body of the paper. We would like to stress that while the main steps in the approach here are similar in spirit to those in [9] and [14], highly nontrivial changes are needed in the detailed techniques, due to the different nature of the dynamical behavior of the system under the current weak competition assumption.
We now state our main results more precisely. From [9] we know that (1.3) has a unique solution, which is defined for all t > 0. Our aim here is to determine its long-time behavior.
The next theorem provides a sharp criterion for the above spreading-vanishing dichotomy. When case (i) happens in Theorem 1.1, the spreading speed of u is asymptotically linear, as indicated by the following theorem. As usual, the positive constant c 0 in Theorem 1.3 is called the asymptotic spreading speed of u. The key in the proof of this theorem is to find a way to determine c 0 . It turns out that two systems of traveling wave type equations are needed in order to determine c 0 . The first one is obtained by looking for traveling wave solutions of (1.2), namely (1.5) (The second system is (1.6) below.) Clearly, if (Φ(s), Ψ(s)) solves (1.5), then is a solution of (1.2), which is often called a traveling wave solution with speed c. By Theorem 4.2 and Example 4.2 in [18], we have the following result on (1.5): Proposition 1.4. Assume h, k ∈ (0, 1). Then there exists a critical speed c * ≥ 2 √ 1 − k such that (1.5) has a solution when c ≥ c * and it has no solution when c < c * .
We can further show that c * ≤ 2 √ u * . Making use of c * , we have the following result on the system below which gives traveling wave type solutions to (1.3): (1.6) We will show that the asymptotic spreading speed of u in Theorem 1.3 is given by If c = c γ , then we have additionally We call (φ c , ψ c ) with c = c γ the semi-wave associated with (1.3). This pair of functions (and its suitable variations) will play a crucial role in the proof of Theorem 1.3, and they also provide upper and lower bounds for the solution pair (u, v) (see the proof of Lemmas 5.1 and 5.2 for details).
The rest of this paper is organized as follows. In section 2, we present some basic results including the existence of solutions to (1.3), and the existence of solutions to a more general system than (1.6). In section 3, we prove Theorem 1.5 based on an upper and lower solution result (Proposition 2.5) established in section 2, and many other techniques. In section 4, we investigate the long time behavior of the solution to (1.3) and prove the spreading-vanishing dichotomy, and obtain sharp criteria for spreading and vanishing, which finish the proof of Theorems 1.1 and 1.2. In section 5, we complete the proof of Theorem 1.3 by making use of Theorem 1.5. Section 6 consists of the proof of Proposition 2.5 stated in section 2.

Preliminary results
In this section, we collect some basic facts which will be needed in our proof of the main results. We first note that (1.3) always has a unique solution. Indeed, by Theorems 2.4 and 2.5 in [9], we have the following results on the solution of system (1.3).
Next we recall a comparison result for the free boundary problem, which is a special case of Lemma 2.6 in [9].
Lastly in this section, we modify some well known upper and lower solution technique to show the existence of a solution for a general coorperative system of the form where ϕ = (ϕ 1 , ϕ 2 ), c ≥ 0, d i > 0, and f i : R 2 → R (i = 1, 2) satisfy the following conditions: We are particularly interested in solutions ϕ of (2.1) that satisfy the asymptotic boundary conditions Indeed, solving (2.1) and (2.2) will supply the main step for solving (1.6). On the other hand, our method here to solve the more general problems (2.1) and (2.2) may have other applications.
2) has a pair of upper and lower solutions associated with R satisfying ϕ 1 (s) ≡ 0 and sup t≤s ϕ(t) ≤ ϕ(s) for s ∈ R.
Then (2.1) has a monotone non-decreasing solution ϕ satisfying (2.2). Proposition 2.5 will play an important role in the proof of Theorem 1.5 in the next section. The proof of Proposition 2.5 is based on some upper and lower solution arguments and involves the Schauder fixed point theorem. Our proof is similar in spirit to that in several works on various different traveling wave problems (see, for example, [20,21,28,31]), but the detailed techniques are rather different. Since the proof is long, it is postponed to Section 6 at the end of the paper.

Semi-wave solutions
The aim of this section is to prove Theorem 1.5. Firstly, we recall some known results for the Fisher-KPP equation  , the solution χ(s) of (3.1) is strictly increasing and has the following asymptotic behavior where b χ is a positive constant.
The conclusion (i) can be found in [8,1], and the proof of (ii) is standard (see, for example, [23]).
Lemma 3.7. The solution of (3.5) is unique.
Proof. Let (φ, ψ) and (φ 1 , ψ 1 ) be two arbitrary solutions of (3.5). We are going to show that Note that if we are able to prove (3.13), then the same argument can also be used to show (φ 1 , ψ 1 ) ≥ (φ, ψ). Hence uniqueness will follow if we can show (3.13).
Proof. Let (Φ 0 , Ψ 0 ) be a solution of (3.21) with c = c * . It is easily checked that (0, 0, 0, 0) is a saddle equilibrium point of the ODE system satisfied by Following the idea in the proof of Lemma 3.6, we rewrite the equation satisfied by Φ 0 as and view it as a perturbed linear equation to Using the fundamental solutions of this latter equation we see that, as s → −∞, the asymptotic behaviour of (Φ 0 , Φ ′ 0 ) is given by Next, we prove that system (3.5) has a solution for c = c * − ǫ. To this end, we will treat the cases c * > 2 √ 1 − k and c * = 2 √ 1 − k separately.
where the positive constant ǫ 1 will be determined later.

The spreading-vanishing dichotomy
We prove Theorems 1.1 and 1.2 in this section. Let us recall that for the problem the following result holds.  Proof. The proof is similar to the proof of Lemma 4.6 in [9]. For readers' convenience, we give the details here. Define By direct calculation, Henceû satisfies  By Proposition 2.1, there exists M > 0 such that Since g 0 ≤ g(t) < g ∞ < ∞, the differential operator is uniformly parabolic. Therefore we can apply standard L p theory to obtain, for any p > 1, where C 1 is a constant depending on p, g 0 , M and u 0 C 1+α [0,g 0 ] . For each T ≥ 1, we can apply the partial interior-boundary estimate over [0, g 0 ] × [T, T + 2] to obtain û W 2,1 p ([0,g 0 ]×[T +1/2,T +2]) ≤ C 2 for some constant C 2 depending on α, g 0 , M and u 0 C 1+α [0,g 0 ] , but independent of T . Therefore, we can use the Sobolev imbedding theorem to obtain, for any α ∈ (0, 1), where C 3 is a constant depending on α, g 0 , M and u 0 C 1+α [0,g 0 ] . Similarly we may use interior estimates to the equation ofv to obtain where C 4 is a constant depending on α, g 0 , M and v 0 C 1+α [0,g∞+1] . Since it follows that there exists a constantC depending on α, γ, g 0 , (u 0 , v 0 ) C 1+α [0,g 0 ] and g ∞ such that For contradiction, we assume that lim sup t→+∞ u(·, t) C([0,g(t)]) = δ > 0.
Using this fact and a simple comparison argument we easily deduce lim t→∞ v(·, t) = 1 uniformly in any compact subset of [0, ∞).  Proof. We define Then define inductively for n ≥ 1, It is easily checked that {u n } and {v n } are decreasing, {u n } and {v n } are increasing, and lim n→∞ (u n , v n ) = lim n→∞ (u n , v n ) = (u * , v * ). (4.4) We claim that, for every n ≥ 1, uniformly in any compact subset of [0, ∞). The conclusion of the Lemma clearly follows directly from (4.5) and (4.4). So it suffices to prove (4.5). We do that by an induction argument.
This completes the proof of Step 1.
Step 2. If (4.5) holds for n = j ≥ 1, then it holds for n = j + 1. Since (4.5) holds for n = j, for any small ǫ > 0 and large l > max g 0 , π It follows from the comparison principle that u( It is well known that this problem has a unique positive steady-state solutionû * (x) and lim t→∞û ( It follows, since ǫ > 0 can be arbitrarily small, that lim sup t→∞ u(x, t) ≤ 1 − kv j = u j+1 locally uniformly in [0, ∞).
Analogously, from the comparison principle we obtain u(x, t) ≥ u(x, t) for x ∈ [0, l] and t > t 2 , where u(x, t) satisfies from which we can deduce The proof for is similar, and we omit the details.  Proof. Assume for contradiction that π 2 √ 1−k < g ∞ < ∞. Then there exists T 1 > 0 such that for ǫ sufficiently small. By a simple comparison consideration, there exists which implies that (u, g) is an upper solution to the problem .
Hence (u, g) is an upper solution to the problem (4.8) The comparison principle infers g(t) ≥ĝ(t) for t > t 3 . Applying Lemma 4.1 to (4.8) we see that there exists γ ≥ 0 depending on g(t 3 ) and u(x, t 3 ) (which are uniquely determined by u 0 and v 0 ) such that spreading happens for (4.8) when γ > γ. Thus lim t→∞ g(t) = ∞ when γ > γ, and by Lemma 4.3, spreading happens to (1.3) for such γ.
Proof. Clearly, (u, g) satisfies That is, (u, g) is a lower solution to the problem It follows from the comparison principle that g(t) ≤ḡ(t). Since g 0 < π/2, by Lemma 4.1 there exists Theorem 1.2 now follows directly from Lemmas 4.4-4.7.

Asymptotic spreading speed
We prove Theorem 1.3 in this section. Proof. Let V (t) be the unique solution of Then a simple comparison consideration yields v(x, t) ≤ V (t) for x ≥ 0 and t > 0. Since lim t→∞ V (t) = 1, we can find We now consider the auxiliary problem where δ > 0 is small. We claim that there exists a unique c δ γ > 0 such that (5.2) has a unique solution (φ δ , ψ δ ) when c = c δ γ ; moreover, lim Indeed, if we define , then a direct calculation shows that (c, φ δ , ψ δ ) solves (5.2) if and only if (c,φ δ ,ψ δ ) satisfies (1.6) and γφ ′ (0) =c when (r, h, k, u * , v * ) in (1.6) is replaced by (r δ ,h δ ,k δ , u * δ , v * δ ). So the claim follows directly from Theorem 1.5 and Lemma 3.14, and the continuous dependence of the unique solution on the parameters.
Since ψ δ (−∞) = 1 + 2δ, ψ δ (+∞) = v * δ > v * and ψ ′ δ < 0, there exists L > 1 such that By the spreading assumption, we have Hence, in view of u * > u * δ and v * < v * δ , there exists T 0 > T ′ 0 large such that for x ∈ [0, L + 1] and t ≥ T 0 , Then g(T 0 ) = 1 < g(T 0 ), and in view of (5.3) and (5.1), we also have Let us also note that Furthermore, Hence, we can use Proposition 2.2 and Remark 2.3 to conclude that It follows that lim inf t→∞ t ≥ c δ γ , which yields the required inequality by letting δ → 0. Proof. For small τ > 0 we consider the auxiliary problem As in the proof of Lemma 5.1 we can use a change of variable trick to reduce (5.4) to (1.6), and then apply Theorem 1.5 and Lemma 3.14 to conclude that there exists a unique c τ γ > 0 such that (5.4) has a unique solution (φ τ , ψ τ ) when c = c τ γ , and moreover, Let us also observe that For clarity we divide the analysis below into three steps.
Step 1. We prove that for any small τ > 0, we can find T ′ 0 > 0 such that for each T ≥ T ′ 0 , there exists L(T ) > 0 having the following property: v(x, T ) ≥ 1 − τ for x ≥ L(T ).

By (1.4) we haveṽ
It is well known that the solution of (5.5) satisfies where w * is the unique solution of Moreover, w * has the property that w ′ * > 0 and w * (∞) = 1. Therefore, there exist positive constants L 1 , T ′ 0 large enough such that Applying the maximum principle to the equation satisfied by w x (x, t), we deduce w x (x, t) ≥ 0 for x > 0 and t > 0. It follows that we can use the comparison principle to deduce v(x, t) ≥w(x, t) = w(x − g(T ), t) for x > g(T ) and 0 < t ≤ T.
This completes the proof of Step 1.
Step 2. We prove that for any small τ > 0, there exists T ′ 1 > 0 such that We prove the claimed inequalities in (5.6) by a comparison argument involving the following ODE system     ǔ Indeed, by the comparison principle for coorporative system we easily obtain But it is well known (for example, see [15]) that lim t→∞ (ǔ(t),v(t)) = (u * , v * ).
Step 3. We complete the proof of the lemma by constructing a suitable comparison function triple (u(x, t), v(x, t), g(t)), and applying the comparison principle.
We fix T 0 := max{T ′ 0 , T ′ 1 }. Then by the conclusions in Steps 1 and 2 we obtain Choose S > L(T 0 ) > g(T 0 ) large so that φ τ (x) > u * τ /2 , ψ τ (x) < v * τ /2 for x ≥ S − L(T 0 ), and then define Clearly g(T 0 ) = S > g(T 0 ) and Finally, direct calculations show that Hence, we can use Proposition 2.2 to conclude that It follows that lim sup t→∞ t ≤ c τ γ , which gives the required inequality by letting τ → 0. Although we follow some standard steps in the proof of Proposition 2.5, since the first equation of (2.1) is only satisfied for s > 0, nontrivial changes are needed. We break the rather long proof into several lemmas.
We start with a second order ODE of the following form d 1 y ′′ − cy ′ − βy + f (s) = 0, s > 0, where the constants c and β are positive, and the nonlinear function f is specified below.
If y(s) is given by (6.2), then it is easy to check that y(s) satisfies (6.1) and y(s) = O(e αs ) as s → ∞.
Define the operators H 1 : C R (R + , R 2 ) → C(R + , R) and H 2 : C R (R, R 2 ) → C(R, R) by where the positive constant β is large enough such that H i (ϕ) is nondecreasing with respect to ϕ 1 and ϕ 2 , for (ϕ 1 (s), ϕ 2 (s)) ∈ R = [0, Let F 1 : C R (R, R) → C(R, R) be given by where K i (ξ, s) is given by (6.3). By Lemma 6.1, it is easy to see that the operator F 1 is well defined and Let µ 1 = c − c 2 + 4βd 2 2d 2 , µ 2 = c + c 2 + 4βd 2 2d 2 be the two roots of It is easy to show that the operator F 2 is well defined and satisfies Clearly, ϕ is a fixed point of the operator F in C R (R, R 2 ) if and only if it is a solution of (2.1) in C R (R, R 2 ).
Clearly Γ is a nonempty, bounded, closed, convex subset of the Banach space B σ (R, R 2 ).
Proof. We show that F 1 satisfies (i) and (ii) stated in the lemma.
Proof. From the hypothesis (A 3 ), it is easy to see that, for some L > 0 and allφ,φ ∈ Γ, By a direct calculation, we have which clearly implies F 1 : Γ → B σ (R, R 2 ) is continuous. Similarly we can show F 2 : Γ → B σ (R, R 2 ) is continuous. Hence F is continuous on Γ.
Thus {F (ϕ)(s) : ϕ ∈ Γ} is a family of equi-continuous functions of s ∈ R. Let Φ j be a sequence of Γ and υ j = F (Φ j ). Then the sequence υ j is equi-continuous. It follows from Lemma 6.2(ii) that υ j (s) is nondecreasing in s ∈ R. Noting that Γ is bounded in L ∞ (R, R 2 ), by the Arzela-Ascoli theorem, we conclude that for any R > 0, there exists a convergent subsequence of υ j | [−R,R] in C([−R, R], R 2 ). Using a standard diagonal selection scheme, we can extract a subsequence υ j k that converges in C([−R, R], R 2 ) for every R > 0. Without loss of generality, we assume that the sequence υ j itself converges in each C([−R, R], R 2 ). From this, it follows easily that υ j is Cauchy in B σ (R, R 2 ), and hence it is convergent. This proves the precompactness of F (Γ).
Since Γ is a bounded closed convex set of B σ (R, R 2 ), by Lemmas 6.3, 6.4 and 6.5, we can apply Schauder's fixed point theorem to conclude that F has a fixed point ϕ in Γ, which is a non-decreasing solution of (2.1). To complete the proof of Proposition 2.5, it remains to prove the following result.