The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor

In this paper we consider the diffusive competition model consisting of an invasive species with density $u$ and a native species with density $v$, in a radially symmetric setting with free boundary. We assume that $v$ undergoes diffusion and growth in $\R^N$, and $u$ exists initially in a ball $\{r<h(0)\}$, but invades into the environment with spreading front $\{r=h(t)\}$, with $h(t)$ evolving according to the free boundary condition $h'(t)=-\mu u_r(t, h(t))$, where $\mu>0$ is a given constant and $u(t,h(t))=0$. Thus the population range of $u$ is the expanding ball $\{r<h(t)\}$, while that for $v$ is $\R^N$. In the case that $u$ is a superior competitor (determined by the reaction terms), we show that a spreading-vanishing dichotomy holds, namely, as $t\to\infty$, either $h(t)\to\infty$ and $(u,v)\to (u^*,0)$, or $\lim_{t\to\infty} h(t)<\infty$ and $(u,v)\to (0,v^*)$, where $(u^*,0)$ and $(0, v^*)$ are the semitrivial steady-states of the system. Moreover, when spreading of $u$ happens, some rough estimates of the spreading speed are also given. When $u$ is an inferior competitor, we show that $(u,v)\to (0,v^*)$ as $t\to\infty$.

In this paper, we will examine the case that u invades into an environment where a native competitor already exists. This is a much more complicated situation, and we will only consider (1.1) under certain restrictions on the parameters, to be specified below.
Problem (1.1) is a variation of the diffusive Lotka-Volterra competition model, which is often considered over a bounded spatial domain with suitable boundary conditions or considered over the entire space R N ( [3,13]). For example, the dynamical behavior of the following bounded domain problem (t, x) ∈ (0, ∞) × Ω, ∂u ∂η = ∂v ∂η = 0, (t, x) ∈ (0, ∞) × ∂Ω, u(0, x) = u 0 (x) > 0, v(0, x) = v 0 (x) > 0, x ∈ Ω (1.4) is well known, where Ω is a bounded smooth domain of R N with N ≥ 1, η is the outward unit normal vector on ∂Ω. This model describes the situation that two competitors evolve in a closed habitat Ω, with no flux across the boundary ∂Ω. Therefore their competitive strengths are completely determined by the coefficients (a i , b i , c i , d i ) in the system, i = 1, 2.
Problem (1.4) admits the trivial steady state R 0 = (0, 0) and semi-trivial steady-states R 1 = (a 1 /b 1 , 0) and R 2 = (0, a 2 /c 2 ). Moreover, if b 1 /b 2 > a 1 /a 2 > c 1 /c 2 or b 1 /b 2 < a 1 /a 2 < c 1 /c 2 , the problem has a unique constant positive steady-state These are all the nonnegative constant steady-states. There may also exist non-constant positive steady-states, but they are all linearly unstable when Ω is convex ( [9]). For the constant equilibria, their roles are summarized below (see, for example, [13] page 666): (1) R 0 is always unstable; (2) when b 1 /b 2 > a 1 /a 2 > c 1 /c 2 , R * is globally asymptotically stable; (3) when a 1 /a 2 > max{b 1 /b 2 , c 1 /c 2 }, R 1 is globally asymptotically stable; (4) when a 1 /a 2 < min{c 1 /c 2 , b 1 /b 2 }, R 2 is globally asymptotically stable; (5) when b 1 /b 2 < a 1 /a 2 < c 1 /c 2 , R 1 and R 2 are locally asymptotically stable, and R * is unstable. In case (2), the competitors co-exist in the long run, and it is often referred to as the weak competition case, where no competitor wins on loses in the competition. In case (3), the competitor u wipes v out in the long run and wins the competition; so we will call u the superior competitor and v the inferior competitor. Analogously u is the inferior competitor and v is the superior competitor in case (4). Case (5) is the strong competition case, and the long-time dynamics of (1.4) is usually complicated and difficult to determine.
For the entire space problem extensive work has been done concerning the existence of traveling wave solutions in space dimension N = 1. For example, for case (3), it is shown in [8] that there exists c * > 0 such that for each c ≥ c * , (1.5) with N = 1 has a solution of the form there is no such solution when c < c * . The general long-time behavior of the Cauchy problem of (1.5), however, is still poorly understood (see Remark 3.2 below for a partial result). We end the introduction by mentioning some related research. In [11], a predator-prey model in one space dimension was considered, where the available habitat is assumed to be a bounded interval [0, l], and no-flux boundary conditions are assumed for both species, except for the predator at x = l. It is assumed that the predator satisfies a free boundary condition as in (1.1), before the free boundary x = h(t) reaches x = l, and a no-flux boundary condition at x = l is satisfied by the predator after the free boundary has reached x = l. It is shown in [11] that the free boundary always reaches l in finite time, and hence the long-time dynamical behavior of the free boundary problem is the same as the fixed boundary problem. After the first version of this paper was completed, we have learned several more closely related research. In [7], the week competition case was considered in one space dimension, but in their model, both species share the same free boundary. Such a free boundary setting was also used in [15] for the Lotka-Volterra predator-prey system in one space dimension. In [16], the Lotka-Volterra predator-prey model was considered in one space dimension, where similar to (1.1), one species (the predator) is subject to free boundary conditions, and the other is considered over the entire R 1 .
The rest of this paper is organized in the following way. In section 2, we prove some general existence and uniqueness results, which implies in particular that (1.1) has a unique solution defined for all t > 0. Moreover, some rough a priori estimates are given, as well as a rather general comparison result. These results are useful here and possibly elsewhere. In section 3, we investigate the case that u is an inferior competitor, namely the coefficients fall into case (4). Sections 4 and 5 are devoted to the case that u is a superior competitor. A spreading-vanishing dichotomy is established in section 4, and a sharp criterion to distinguish the dichotomy is also given there. In section 5, some rough estimates for the spreading speed is given for the case that spreading of u happens.

Preliminary Results
In this section, we first prove a local existence and uniqueness result for a general free boundary problem, and then we obtain global existence results, which imply that the solution to (1.1) exists for all time t ∈ (0, ∞). Lastly, we obtain some comparison results, which will be used in the other sections.
Consider the following general free boundary problem: where f (0, v) = g(u, 0) = 0 for any u, v ∈ R, and u 0 , v 0 are as in (1.1).
Theorem 2.1. Assume that f and g are locally Lipschitz continuous in R 2 + . For any given (u 0 , v 0 ) satisfying (1.2) and any α ∈ (0, 1), there is a T > 0 such that problem (2.1) admits a unique bounded solution and the local Lipschitz coefficients of f, g.

Proof:
The proof is similar to that in [5] and [4] for the scalar problem, with some modifications. We sketch the details here for completeness. First we straighten the free boundary as in [2]. Let ζ(s) be a function in Consider the transformation which leads to the transformation As long as the above transformation x → y is a diffeomorphism from R N onto R N and the transformation s → r is also a diffeomorphism from [0, +∞) onto [0, +∞). Moreover, it changes the unknown free boundary |x| = h(t) to the fixed sphere |y| = h 0 . Now, direct calculations show that Let us also denote If we set then the free boundary problem (2.1) becomes It is not difficult to see that Γ T := W T × Z T × H T is a complete metric space with the metric Let us observe that for h 1 , h 2 ∈ H T , due to h 1 (0) = h 2 (0) = h 0 , we have . Next, we shall prove the existence and uniqueness result by using the contraction mapping theorem. Since f and g are locally Lipschitz continuous, there exists an L * depending on u 0 C([0,h 0 ]) and v 0 L ∞ ([0,+∞)) such that By standard L p theory and the Sobolev imbedding theorem [10], for any (w, z, h) ∈ Γ T , the following initial boundary value problem admits a unique bounded solution (w,z) ∈ C (1+α)/2,1+α (∆ T ) × C (1+α)/2,1+α (∆ ∞ T ) and w C (1+α)/2,1+α (∆ T ) C 1 , (2.6) where C 1 is a constant depending on α, h 0 , L * , u 0 C 2 [0,h 0 ] and v 0 C 2 [0,+∞) . The estimate (2.7) comes from the interior estimate. For any m ≥ 0, by classical parabolic regularity theory [10], one then have the estimate for some large p > 1 and α ∈ (0, 1). Now, we defineh(t)(> 0) by the fourth equation in (2.3): Now we define the map by F(w(t, s), z(t, s); h(t)) = (w(t, s),z(t, s);h(t)). It's easy to see that (w(t, s), z(t, s); h(t)) ∈ Γ T is a fixed point of F if and only if it solves (2.3). The estimates in (2.6), (2.7) and (2.9) yield }, then F maps Γ T into itself. Now we prove that for T > 0 sufficiently small, F is a contraction mapping on Γ T . Indeed, let (w i , z i , h i ) ∈ Γ T (i = 1, 2) and denote (w i ,z i ,h i ) = F(w i , z i , h i ). Then it follows from (2.6), (2.7) and (2.9) that Setting W =w 1 −w 2 , we find that W (t, s) satisfies Using the L p estimates for parabolic equations and Sobolev's imbedding theorem, we obtain where C 3 depends on C 1 , C 2 , the local Lipschitz coefficients of f, g and the functions A, B and C in the definition of the transformation (t, s) → (t, r). Similarly, we have where C 4 depends on C 1 , C 2 , the local Lipschitz coefficients of f, g and the functions A, B and C. Taking the difference of the equations for h 1 and h 2 results in Combining (2.4), (2.10), (2.11) and (2.12), and assuming T ≤ 1 we obtain , with C 5 depending on C 3 , C 4 and µ. Hence for we have This shows that for this T , F is a contraction mapping in Γ T . It follows from the contraction mapping theorem that F has a unique fixed point (w, z, h) in Γ T . In other words, (w(t, s), z(t, s); h(t)) is the solution of the problem (2.3) and therefore (u(t, r), v(t, r); h(t)) is the solution of the problem (2.1). Moreover, by using the Schauder estimates, we have additional regularity of the solution,  Suppose for contradiction that T max < ∞. Fix M * ∈ (T max , ∞). Let (U (t), V (t)) be the solution to the following ODE system: It is easy to see that Next we claim that 0 < h ′ (t) ≤ C 7 for all t ∈ (0, T max ) and some C 7 independent of T max . In fact, by the strong maximum principle and Hopf boundary lemma h ′ (t) is always positive as long as the solution exists. To derive an upper bound of h ′ (t), we define and construct an auxiliary function We will choose M so that u(t, r) ≥ u(t, r) holds over Ω.
Direct calculations show that, for (t, r) ∈ Ω, It follows that To use the maximum principle over Ω, we only have to find some M independent of T max such

Therefore upon choosing
we will have Applying the maximum principle to u − u over Ω gives that u(t, r) ≤ u(t, r) for (t, r) ∈ Ω, which implies that We now fix δ 0 ∈ (0, T max ). By standard parabolic regularity, we can find C 8 > 0 depending only on M * , L, C 6 and C 7 such that u(t, . It then follows from the proof of Theorem 2.1 that there exists a τ > 0 depending only on M * , L, C 6 , C 7 and C 8 such that the solution of problem (2.1) with initial time T max − τ /2 can be extended uniquely to the time T max − τ /2 + τ . This contradicts the maximality of T max .
We have the following estimates.
Theorem 2.5. Problem (1.1) admits a unique and uniformly bounded solution (u, v, h). That is, the solution is defined for all t > 0 and there exist constants M 1 and M 2 such that Moreover, there exist a constant M 3 such that (2.14) Thus we have Since Using the strong maximum principle to the equation of u we immediately obtain It remains to show that h ′ (t) ≤ M 3 for t ∈ (0, +∞) and some M 3 . The proof is similar as that of Theorem 2.4 with C 6 replaced by M 1 and M 3 = C 7 = 2M M 1 µ, we omit the details.
We next show that any solution of (1.1) is bounded, namely, there exists M > 0 such that u, v ≤ M in the range they are defined, whenever (u, v, h) is a solution to (1.1) defined in some maximal interval t ∈ (0, T ). Indeed, let U (x) be the unique boundary blow-up solution of and denoteũ(t, x) = u(t, |x|); then it is easily checked by using the comparison principle that (The existence and uniqueness of U and V is well known; see, for example, [6].) In what follows, we discuss the comparison principle for (1.1). For a given pair of functions u := (u, v) and u : is said to be quasimonotone nonincreasing if for fixed u, f is nonincreasing in v, and for fixed v, g is nonincreasing in u; this is satisfied by Lemma 2.6 (The Comparison Principle). Let (f, g) be quasimonotone nonincreasing and Lips- Let (u, v, h) be the unique bounded solution of (2.1). Then Proof: We only prove u ≤ u, v ≥ v and h ≤ h; the result involving (u, v, h) can be proved in a similar way. LetM be an upper bound of First assume that h 0 < h(0). We claim that h(t) < h(t) for all t ∈ (0, T ]. If our claim does not hold, then we can find a first t * ≤ T such that h(t) < h(t) for t ∈ (0, t * ) and h(t * ) = h(t * ). It follows that where K is sufficiently large such that K ≥ 1 + |b 11 r) are bounded, b 21 (t, r), b 12 (t, r) are bounded and nonnegative since (f, g) in (2.15) is quasimonotone nonincreasing and Lipschitz continuous.
Since the first inequality of (2.17) holds only in part of [0, ∞), we cannot use the maximum principle directly. We first prove that for any l > h(t * ), We observe that due to the inequalities satisfied by u, we can apply the maximum principle to u over the region then due to our choice of K, (U , W ) satisfies For the latter case, (W t − d 2 ∆W )(t 1 , r 1 ) ≤ 0, but Both are impossible. Therefore τ ≥ 0, that is U ≥ 0 and W ≥ 0 in [0, t * ] × [0, l], which implies that We now compare u and u over the bounded region Since Z(t, r) := u(t, r) − u(t, r) satisfies the strong maximum principle and the Hopf boundary lemma yield Z(t, r) > 0 in Ω t * , and Z r (t * , h(t * )) < 0. We then deduce that h ′ (t * ) < h ′ (t * ). But this contradicts (2.16). This proves our claim that h(t) < h(t) for all t ∈ (0, T ]. We may now apply the above procedure over . Moreover, u < u for t ∈ (0, T ] and r ∈ [0, h(t)).
If h 0 = h(0), we use approximation. For small ǫ > 0, let (u ǫ , v ǫ , h ǫ ) denote the unique solution of (1.1) with h 0 replaced by h 0 (1 − ǫ). Since the unique solution of (1.1) depends continuously on the parameters in (1.1), as ǫ → 0, (u ǫ , v ǫ , h ǫ ) converges to (u, v, h), the unique solution of (1.1). The desired result then follows by letting ǫ → 0 in the inequalities u ǫ ≤ u, v ǫ ≥ v and h ǫ < h. To see this, we only need to observe that all the arguments in the proof carry over to the new case, except that we now have to avoid r 0 = 0 or r 1 = 0, since the functions U (t, |x|) and W (t, |x|) are not C 2 in x at x = 0. However, the above conditions guarantee that U r (t, 0) < 0 and W r (t, 0) < 0. Therefore (t, 0) cannot be a minimum point of these functions. This implies that r 0 > 0 and r 1 > 0.
We next fix u 0 , v 0 , d i , a i , b i , c i and examine the dependence of the solution on µ, and we write (u µ , v µ , h µ ) to emphasize this dependence. As a consequence of Lemma 2.6, we have the following result.

Invasion of an inferior competitor
In this section, we examine the case that u is an inferior competitor, namely The following theorem shows that the inferior invader cannot establish itself and the native species always survives the invasion. Proof: First we recall that the comparison principle gives u(t, r) ≤ u * (t) for t > 0 and Similarly, we have lim sup t→+∞ v(t, r) ≤ a 2 c 2 uniformly for r ∈ [0, ∞). Therefore for ε 1 = ( a 2 b 2 − a 1 b 1 )/2, there exists t 1 > 0 such that u(t, r) ≤ a 1 b 1 + ε 1 for t ≥ t 1 , r ∈ [0, ∞).

(3.4)
Let v * be the unique solution to It is well known ( [6]) that lim t→∞ v * (t, r) = b 2 ε 1 /c 2 uniformly in any bounded subset of [0, ∞). Therefore for any L > 0, there exists t L > t 1 such that (3.7) The system (3.7) is quasimonotone nonincreasing, which generates a monotone dynamical system with respect to the order with the initial value a 1 b 1 + ε 1 , b 2 ε 1 2c 2 an upper solution. It follows from the theory of monotone dynamical systems (see, e.g. [14] Corollary 3.6) that lim t→+∞ u(t, r) = u L (r) and lim t→+∞ v(t, r) = v L (r) uniformly in [0, L], where (u L , v L ) satisfies and is the maximal solution below ( a 1 b 1 + ε 1 , b 2 ε 1 2c 2 ) of the above problem under the order ≤ P . Next we observe that if 0 < L 1 < L 2 , then u L 1 (r) ≥ u L 2 (r) and v L 1 (r) ≤ v L 2 (r) in [0, L 1 ], which can be derived by comparing the boundary conditions and initial conditions in (3.7) for L = L 1 and L = L 2 .
We note that Theorem 3.1 gives no information on the dynamical behavior of the spreading front, and the exact behavior of (u, v) over the entire spatial range 0 ≤ r < ∞ is also unclear. Our next result provides such information, provided that the native species is already rather established at t = 0, in the sense that inf r≥0 v 0 (r) > 0.
Direct calculation gives d dt Integrating from T * to t yields which implies that h ∞ < ∞. Theorem 3.3 suggests that an inferior competitor can never penetrate deep into the habitat of a well established native species, and it dies out before its invading front reaches a certain finite limiting position.

Invasion of a superior competitor
This section is devoted to the case that u is a superior competitor, that is Let λ 1 (R) be the principal eigenvalue of the operator −∆ in B R subject to homogeneous Dirichlet boundary conditions. It is well-known that λ 1 (R) is a strictly decreasing continuous function and lim R→0 + λ 1 (R) = +∞ and lim R→+∞ λ 1 (R) = 0.
Therefore, there exists a unique R * such that It is easy to check that R * = π/2 if N = 1.
For the sake of convenience and completeness, we first recall the following spreading-vanishing dichotomy for the radially symmetric diffusive logistic problem  If h 0 ≥ R * d a , then spreading always happens. If h 0 < R * d a , then there exists µ * > 0 depending on u 0 such that vanishing happens when µ ≤ µ * , and spreading happens when µ > µ * .
The proofs of Propositions 4.1 and 4.2 can be found in [5] for the one dimensional case and [4] for higher space dimensions.
We will show that when (4.1) holds, a similar spreading-vanishing dichotomy holds for (1.1). More precisely, we have the following results.  We prove these results by several lemmas. In the rest of this section, we always assume that (4.1) holds, and (u, v, h) is the unique solution of (1.1), with v 0 ≡ 0.
Before starting the proofs, let us note that by the symmetric positions of u and v in (1.5), we may use Remark 3.2 to conclude that when (4.1) holds, the unique solution (u, v) of the Cauchy problem of (1.5) with u(0, x), v(0, x) bounded, nonnegative and not identically zero, satisfies lim t→∞ u(t, x), v(t, x) = R 1 = a 1 b 1 , 0 locally uniformly in R N . The behavior of the free boundary problem (1.1) described by Theorems 4.3 and 4.4, however, is more complicated, where vanishing for u is possible. Therefore for Next, we prove that for any l > R * d 1 c 1 ε 2 , there exists t l > t 2 such that is the solution of the following free boundary problem discussed in [4]: Since c 1 ε 2 > d 1 ( R * l ) 2 , it follows from Proposition 4.2 that h(t) → ∞ and u l (t, r) → c 1 ε 2 b 1 as t → ∞ uniformly in [0, l], which implies that (4.5) holds for some t l . Now we know that (u, v) satisfies As in the proof of Theorem 3.1, it follows from the theory of monotone dynamical systems that lim inf t→+∞ u(t, r) ≥ u l (r) and lim sup t→+∞ v(t, r) ≤ v l (r) in [0, l], where (u l , v l ) satisfies Due to (4.1), as before, by the global dynamical behavior of the associated ODE system ( [12]), we deduce that u ∞ (r) = a 1 b 1 and v ∞ (r) = 0. We thus obtain lim inf t→+∞ u(t, r) ≥ a 1 b 1 and lim sup t→+∞ v(t, r) ≤ 0 uniform in [0, l], which implies that lim t→+∞ u(t, r) = a 1 b 1 and lim t→+∞ v(t, r) = 0 uniformly in any bounded subset of [0, ∞). Proof: Define , w(t, s) = u(t, r), z(t, s) = v(t, r).
Lemma 4.7 implies that the threshold constant µ * = 0 if h 0 ≥ R * d 1 a 1 −a 2 c 1 /c 2 . The next lemma shows that µ * > 0 if h 0 < R * d 1 a 1 . (4.14) This indicates that (u, h) is a lower solution to the problem Theorems 4.3 and 4.4 now follow directly from the conclusions proved in the above lemmas.

estimates of spreading speed
In this section we give some rough estimates on the spreading speed of h(t) for the case that spreading of u happens. We always assume that (4.1) holds.
We first recall Proposition 3.1 of [4], whose complete proof is given in [1].
the pair (u, h) is a lower solution to the problem We thus have Next we prove that Since lim sup t→∞ v ≤ a 2 c 2 uniformly for r ∈ [0, ∞) and h ∞ = ∞, for any 0 < ε < ε 0 : This implies that (u, h) is an upper solution to the problem a 1 −c 1 (a 2 /c 2 +ε) . Moreover, from [4] we have lim t→∞ t for any 0 < ε < ε 0 . Letting ε → 0 and using the continuity of k 0 with respect to its arguments, we immediately obtain the desired result.
For x 0 > 0 and L > 0 to be determined later, we definẽ x > x 0 .
Remark 5.4. It is easily seen from the above proof that if instead of (5.6), we assume that there exists T > 0 such that u(T, r) and v(T, r) satisfy the conditions in (5.6), then the conclusion of Theorem 5.3 remains valid.