Blowup estimates for a mutualistic model in ecology

The cooperating two-species Lotka-Volterra model is dis- cussed. We study the blowup properties of the solution to a parabolic system with homogeneous Dirichlet boundary conditions. The upper and lower bounds of blowup rate are obtained. ut d1 u = u(a1 b1u c1v); x 2 ; t > 0; vt d2 v = v(a2 b2u c2v); x 2 ; t > 0; u(x) = v(x) = 0; x 2 @ ; t > 0; u(x; 0) = u0(x); v(x; 0) = v0(x); x 2 ;


Introduction and main results
The well-known Lotka-Volterra ecological model, which involves a coupled system of two ordinary differential equations, has been given an enormous attention in the past decades.When the effect of dispersion is taken into consideration the densities u, v of the species are governed by x ∈ Ω, t > 0, where ∆ is the Laplacian operator, Ω is a bounded domain in R N with ∂Ω uniformly C 2+α -smooth, u 0 (x) and v 0 (x) are nonnegative smooth functions with u 0 (x) = v 0 (x) = 0 on ∂Ω.d i , a i , b i and c i (i = 1, 2) are positive constants.d i represents its respective diffusion rate and the real number a i , its net birth rate.b 1 and c 2 are the coefficients of intra-specific competitions and b 2 , c 1 are that of inter-specific competitions.Here we consider the case with homogeneous Dirichlet boundary conditions, which implies that the habitat is surrounded by a totally hostile environment.
If the presence of one species encourages the growth of the other species then the system (1.1) becomes so-called mutualistic model: x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0, u(x, 0) = u 0 (x), v(x, 0) = v 0 (x), x ∈ Ω. (1.2) Because of the quasimonotone nondecreasing of reaction functions in (1.2), there is a quite different behavior of solutions compared with the solutions of (1.1).The solution of (1.1) with any nonnegative initial data is unique and global, while the blowup solutions are possible when the two species are strongly mutualistic (b 2 c 1 > b 1 c 2 ), which means that the geometric mean of the interaction coefficients exceeds that of population regulation coefficients.Here we give only the related result of Pao [20].
there exists a finite time T such that the unique solution to (1.2) exists in [0, T )×Ω and blows up in the meaning that lim t→T max(|u(x, the solution will blow up for any a 1 ≥ 0 and a 2 ≥ 0 under suitable initial data.
Based on the above result, we are chiefly interested in studying the blowup properties of the solution.We derive the upper and lower bounds of blowup rate, that is, there are positive constants c and C such that There are some related results on the blowup of solutions to nonlinear parabolic systems, see for example [19] and [24].In a recent paper, Lou etc. in [18]  gave a sufficient condition on the initial data for the solution to blow up in finite time.For the blowup estimates, as we know, no result has been given owing to the cross-coupled reactions.
For the related elliptic systems, there is an extensive literature regarding the existence and uniqueness of positive solutions, the reader can see [1,10,12,14,15,16,17,20,23] and the references therein.
The paper is arranged as follows.In §2 the comparison principles for bounded and unbounded domains are given.In §3, we derive the lower bound of blowup rate and §4 deals with its upper bound.

Comparison principles
In this section, we show the comparison principle for unbounded domains, which will be used in the sequel.For completeness, we also give the comparison principle for bounded domains.
Lemma 2.1 Let Ω be a bounded domain with smooth boundary ∂Ω. (2.1) and there exist positive constants A and γ such that Lemma 2.1 is followed by the strong Maximum principle and Lemma 2.2 is followed by the Phragman-Lindelöf principle ( see [21], [22]).
Remark 2.3 Lemmas 2.1 and 2.2 hold for the more general case.For example, for the system Lemmas 2.1 and 2.2 hold if f, g are quasi-monotone nondecreasing, i.e. f is nondecreasing with respect to the component of v and g is nondecreasing with respect to the component of u, see [21] in detail.

Lower blowup estimate
We first establish the relationship between u and v as the solution (u, v) of (1.2) near the blow-up time.
Lemma 3.1 Let (u, v) be the nonnegative solution of (1.2), which blows up at t = T .Then there exists δ such that As in [2] or [3], we argue by contradiction.Without loss of generality we may assume that there exists a sequence {t n } with t n → T as n → ∞ such that For each t n , there exists We now introduce the scaling argument inspired by [9].Let where Clearly, λ n → 0 as n → ∞ and (φ λn , ψ λn ) solves and satisfies It follows from the parabolic estimates [11] that there is a µ ∈ (0, 1) such that for any K > 0, where the constant C K is independent of n.Hence, we obtain a sequence converging to a solution (φ, ψ) of such that φ(0, 0) = 1 and φ ≤ 1, ψ ≡ 0, which leads to a contradiction.In fact, φ achieves its maximum at (0, 0); therefore where Clearly, λ n → 0 as n → ∞, Ω n approaches the halfspace H c = {y 1 > −c} as n → ∞ and (φ λn , ψ λn ) solves and satisfies then uniform Schauder's estimates for φ λn , ψ λn yield a subsequence converging to a solution (φ, ψ) of such that φ(0, 0) = 1 and φ ≤ 1, ψ ≡ 0, which leads to a contradiction as in Case (i).This prove (3.1) in Case (ii).
Remark 3.1 We claim from (3.1) that u and v blow up at the same finite time Now we first give the lower bound of the blowup rate using the integral equation.
Theorem 3.1 Let (u, v) be the nonnegative solution of (1.2), which blows up at t = T .Then there exists a constant c such that Proof: Let G i (x, t; y, τ )(i = 1, 2) be the Green's function of the parabolic operator (∂/∂t−d i ∆) in the bounded domain Ω×(0, T ] under the homogeneous Dirichlet boundary condition on ∂Ω × (0, T ].Then we have the representation formula of (1.2): u(x, t) = Ω G 1 (x, t; y, z)u(y, z)dy Noticing that Ω G i (x, t; y, τ )dy ≤ 1 and the relationship (3.1), we have EJQTDE, 2002 No. 8, p. 7 Next we use the argument as in [9].By assumption, T is the blowup time, so U (t) → +∞ as t → T − .Then we can choose z < t < T such that U (t) = 2U (z), and hence the above inequality for U becomes The proof for V (t) is similar.

Upper blowup estimate
For the upper bound of the blowup rate, we assume that Proof: From Lemma 3.1 we only need to prove that U (t) ≤ C(T − t) −1 .We use a scaling argument inspired by [8].Noticing that U (t) → ∞ as t → T , for any given t 0 ∈ ( T 2 , T ) we can define Choose λ 0 = λ(t 0 ) = U −1/2 (t 0 ) as before.We claim that where the constant D depends only N (it is independent of t 0 ).Suppose that (4.1) is not true, then there exists t n → T such that For each t n , choose (x n , tn ) as in (3.3) and let d n denote the distant of xn to ∂Ω.Similarly as in [4], we distinguish two cases: Case (i) Choose a subsequence (denoted again by {t n }) such that We introduce the scaling functions as before.Let where , Ω n := {y : λ n y + xn ∈ Ω}.Clearly, (φ λn , ψ λn ) has a sequence converging to a solution (φ, ψ) of such that φ(0, 0) = 1 and φ ≤ 1, ψ ≤ 1 δ .Moreover, since that φ achieves its maximum at (0, 0), ψ must be nontrivial as in Lemma 3.1.Therefore φ and ψ are nontrivial nonnegative bounded functions, which leads to a contradiction to the following Theorem 4.2 if N ≤ 2. This prove (4.1) in Case (i).
Step 3 of proof of Theorem 2.1 in [8] shows that (4.1) implies that To prove Theorem 4.2, it suffices to find a lower solution of (4.10) that blows up at a finite time T 0 .First we show the following three useful Lemmas: Lemma 4.1 Any nontrivial nonnegative solution of (4.10) is positive for t > 0.
Proof: If there exist x 0 ∈ R N and t 0 > 0 such that u(x 0 , t 0 ) = 0, then there exist R > 0 and T 1 with t 0 < T 1 < T such that (x 0 , t 0 ) ∈ B R × (0, T 1 ) and We find from a straightforward computation that It follows form the strong maximum principle that w ≡ 0 in B R × [0, T 1 ] or w > 0 in B R × (0, T ].It leads to a contradiction.So u(x, t) > 0 for t > 0 and also v(x, t) > 0 for t > 0 similarly.Lemma 4.2 Let w(x, t) be a nontrivial nonnegative solution of Proof: Since w satisfy the growth condition, using the comparison principle (see Lemma 2.2 for the system) and the assumptions on w(x, 0) yield w(x, t) ≥ w(x, 0) in R N × (0, T ).Using again the comparison principle gives that w(x, t + ε) ≥ w(x, t) in R N × (0, T − ε) for ε > 0 arbitrarily small.Hence w t (x, t) ≥ 0 in R N × (0, T ).
EJQTDE, 2002 No. 8, p. 10 The result that the solution is radial follows by the uniqueness and the rotation invariance of problem (4.11) in the case that w(x, 0) is radial.Furthermore, if the initial data w(x, 0) is radially nonincreasing, then the solution w(x, t) is also radially nonincreasing.

−b 1
< 0. This proves (3.1) in Case (i).Case (ii) Choose a subsequence (denoted again by {t n }) such that lim n→∞ d n λ n = c ≥ 0. Let xn ∈ ∂Ω such that d n = |x n − xn | and let R n be an orthonormal transformation in R N that maps −e 1 := (−1, 0, • • • , 0) onto the outer normal vector to ∂Ω at xn .We now introduce the new scaling.Let

b 2 c 1 > b 1 c 2 3 . 4 . 1
and N = 1.The former assumption b 2 c 1 > b 1 c 2 is the sufficient condition for the solution of (1.2) to have a finite time blowup, see Theorem 1.1 and the latter N = 1 is restriction for the solution of the related scalar problem to blow up in a finite time, see Lemma 4.Theorem Let (u, v) be the nonnegative solution of (1.2), which blows up at t = T .If b 2 c 1 > b 1 c 2 and N = 1, then there exists a constant C such that max Ω×[0,t]

Lemma 4 . 3
All nontrivial nonnegative solutions of

(4. 18 )
is nonglobal.Proof: The proof of Theorem 4.3 is similar to that of Theorem 4.2.The only difference is that in the proof of Theorem 4.2 the related scalar problem (4.12) is nonglobal if N ≤ 2 and in the proof of Theorem 4.3, the related scalar problem (4.13) is nonglobal if N = 1, see Lemma 4.3.EJQTDE, 2002 No. 8, p. 12 considered (1.2) with homogeneous Neumann boundary conditions and EJQTDE, 2002 No. 8, p. 2 .3) Since (u, v) blows up, we have that U (t n ) → ∞ as t n → T and tn → T as n → ∞.Let d n denote the distant of xn to ∂Ω.
Case (i) Choose a subsequence (denoted again by {t n }) such that lim n→∞