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Positive solutions of certain nonlinear elliptic systems with self-diffusions: nondegenerate vs. degenerate diffusions
Nonlinear Analysis 63 (2005) 247 – 259 www.elsevier.com/locate/na
Positive solutions of certain nonlinear elliptic systems with self-diffusions: nondegenerate vs. degenerate diffusions Kimun Ryua,∗ , Inkyung Ahnb a School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455, USA b Department of Mathematics, Korea University, Jochiwon 339-700, Korea
Received 21 May 2004; accepted 10 May 2005
Abstract We discuss the existence of positive solutions to certain strongly-coupled nonlinear elliptic systems with self-diffusions under homogeneous Dirichlet boundary conditions. Using the global positive coexistence results for predator–prey, competition and symbiotic interactions between two species, sufficient conditions for the positive solutions of the degenerate self-diffusive systems are studied. We also investigate the local behavior, namely, the local existence, uniqueness and stability, of positive solutions for predator–prey and competition interactions. Our method is based on the decoupling technique and bifurcation theory. 䉷 2005 Elsevier Ltd. All rights reserved. MSC: 35J60 Keywords: Nonlinear elliptic system; Positive solutions; Self-diffusions; Bifurcation; Degenerate diffusions
∗ Corresponding author. Tel./fax: +1 1612 788 1290.
E-mail addresses: [email protected] (K. Ryu), [email protected] (I. Ahn). 0362-546X/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.05.010
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1. Introduction In this paper, we are concerned with the existence of positive solutions to the following strongly-coupled nonlinear elliptic system: −[(u)u] = uf (u, v), (1.1) −[(d + (v))v] = vg(u, v) in , (u, v) = (0, 0) on j, where is a bounded domain of Rn with smooth boundary j, d is a positive constant and the functions , , f, g : × R2 → R satisfy certain conditions which will be imposed later. In the biological sense, the two species are in predator–prey interaction if one of the growth rates involved is increasing in the prey while the other decreasing in the predator. Also the two species are in symbiotic interaction if each of their relative growth functions is increasing in the other, and they are in competition if these functions are decreasing in the other one. Refer [10] for more information. We say that a steady-state system has a positive solution (u, v) if u(x) > 0 and v(x) > 0 for all x ∈ . The existence of a positive solution (u, v) to the system is called a positive coexistence. The global positive coexistence used in this paper means the existence of positive solutions in terms of a whole domain. If the diffusion rate depends on the density of the own species, it is called self-diffusions. In [7], they found the positive solutions between the appropriate upper and lower solutions for the degenerate elliptic system (i.e. (0) = 0) under certain conditions of f and g: −(u) = f (x, u, v), −(v) = g(x, u, v) in , (u, v) = (0, 0) on j. In [3], the authors investigated the coexistence state (i.e., each u(x) and v(x) is nonnegative and nontrivial) for the system with degenerate diffusions (i.e., (0) = (0) = 0): −(U (x)) = U (x)h(x, U (x), V (x)), −(V (x)) = V (x)k(x, U (x), V (x)) in , U (x) = V (x) = 0 on j, using the method of a system of upper-lower solutions. One can also refer to [6] relating to the system (1.1) and the references therein. In this paper, we give sufficient conditions for the existence of positive solutions of the system (1.1) with nondegenerate diffusions, and using these facts, we study the existence of positive solutions to system (1.1) with degenerate diffusions for three different biological interactions, i.e., predator–prey, competition and symbiotic interactions. The perturbed method was employed in this study. For nondegenerate systems, the existence of positive solutions to (1.1) can be expressed in terms of the spectral property of linearized differential operators of the given system for predator–prey and competing interacting systems, and in terms of the existence of the positive equilibria of the system for symbiotic interactions. On the other hand, in case that the diffusions are degenerate (i.e. d = 0, (0) = (0) = 0 in system (1.1)), the positive coexistence depends only on the positivity of the growth rates at specific constant densities for all three interactions.
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We also consider the system of (1.1) with nondegenerate diffusions for the predator–prey and competition interactions to investigate the local behavior, namely, the local existence, uniqueness and stability, of the positive solutions. Our method is based on the decoupling technique and bifurcation theory. In our model (1.1), the functions f (u, v) and g(u, v) can be replaced by f (x, u, v) and g(x, u, v), respectively, and Dirichlet boundary conditions could be switched by Robin boundary conditions, since the corresponding results could be obtained by slight modifications of the proof provided here. For simplicity, we consider system (1.1). This article is organized as follows. In Section 2, we collect some known results and give a lemma which is useful in this article. In Section 3, we study the global positive coexistence for system (1.1) with nondegenerate diffusions. In Section 4, using the global positive coexistence results for the predator–prey, competition and symbiotic interactions, we investigate the positive coexistence of (1.1) with the degenerate diffusion pressures. In addition, an example is given as a simple application. Section 5 deals with the local behavior of positive solutions to the predator–prey and competition interactions for (1.1) with nondegenerate diffusions.
2. Preparations In this section, we collect some known results which are useful in our work. For a(x) > 0 in C 2 () and b(x) ∈ L∞ (), the eigenvalue problem: [a(x)u] + b(x)u = u in , u=0 on j
(2.1)
has a principal eigenvalue corresponding to the unique positive principal eigenfunction. (For more details, see [13].) Denote this principal eigenvalue of (2.1) by 1 (a(x) + b(x)). The following results can be found in [13]. Lemma 2.1. (i) 1 (a(x) + b(x)) is increasing in b(x). (ii) Let a(x) > 0 in C 2 (), b(x) ∈ L∞ () and u 0, u ≡ / 0 in with u = 0 on j. If (a(x) + b(x))u ≡ 0, then 1 (a(x) + b(x)) = 0. (iii) Let a(x) > 0 in C 2 (), b(x) ∈ L∞ () and M be a positive constant such that b(x) + Ma(x) > 0 for all x ∈ . If 1 (a(x) + b(x)) > 0, then r[(1/a(x))(− + M)−1 (b(x) + Ma(x))] > 1. For the first equation in system (1.1): −[(u)u] = uf (u, v(x)) in , u=0 on j, where v(x) ∈ C 2 () is fixed, the following theorem can be found in [14]. Theorem 2.2. Let (2.2) satisfy the following three assumptions: (A1) (u) is C 2 -function in u with (0) > 0 and u (u) 0 for all u 0; (A2) f (u, v) is C 1 -function in u, v with fu , fv < 0 for all (u, v) ∈ [0, ∞) × [0, ∞); (A3) f (u, 0) 0 on u ∈ [C0 , ∞) for some positive constant C0 .
(2.2)
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Assume that v(x) ∈ C 2 () is fixed. Then the following hold: (i) If 1 ((0) + f (0, v)) 0, then (2.2) has no positive solution; (ii) If 1 ((0)+f (0, v)) > 0, then (2.2) has a unique positive solution u(x) which satisfies u(x) C0 . Throughout this paper, when v(x) ≡ 0, we denote the unique positive solution of (2.2) by u0 if 1 ((0) + f (0, 0)) > 0. Similarly, denote the unique positive solution of the second equation in (1.1) by v0 when u(x) ≡ 0, if 1 ((0) + g(0, 0)) > 0. In view of Theorem 2.2, for every v ∈ C 2 (), we can define the map T : C 2 () → 2, C () for some 0 < < 1 by T v :=
uv
if 1 ((0) + f (0, v)) > 0,
0
otherwise,
where uv is the unique positive solution of (2.2). The following lemma is needed in the proof of the existence theorem. Lemma 2.3. (i) The mapping T is continuous in the sense of C 2 () → C 2, () for some 0 < < 1. (ii) If v2 v1 ≡ / v2 , then either uv2 < uv1 or uv2 ≡ uv1 ≡ 0. Proof. For the proof of (i), one can see Lemma 7 in [14] as a special case. (ii) Assume that v2 > v1 . If 1 ((0) + f (0, v1 ) 0, then 1 ((0) + f (0, v2 ) < 1 ((0) + f (0, v1 ) 0 by Lemma 2.1(i), and so uv2 ≡ uv1 ≡ 0 by Theorem 2.2(i). If 1 ((0) + f (0, v1 ) > 0, then uv1 > 0 in . Since [(uv1 )uv1 ] + uv1 f (uv1 , v2 ) [(uv1 )uv1 ] + uv1 f (uv1 , v1 ) = 0 in and uv1 = 0 on j, uv1 is a positive upper solution of the equation, −[(u)u] = uf (u, v2 ) in and u = 0 on j. By the uniqueness of the positive solution uv2 to Eq. (2.2) for v2 (x) ∈ C 2 (), we have uv2 uv1 . The strict inequality uv2 < uv1 follows from the strong maximum principle.
3. Nondegenerate elliptic systems In this section, we give sufficient conditions for the positive coexistence of (1.1) with nondegenerate diffusions for three different biological interactions, i.e., predator–prey, competition and symbiotic interactions. To do this, we assume that the following hypotheses throughout this section: (H1) d is a positive constant, is C 2 -function in u with (0) > 0 and u 0 for all u 0; and is C 2 -function in v with (0) 0 and v > 0 for all v 0; (H2) f (u, v), g(u, v) are C 1 -functions in u, v with fu < 0, gv < 0 for all (u, v) ∈ [0, ∞) × [0, ∞).
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In addition, we impose the following hypotheses for three different biological interactions: I. Predator–prey model (P1) fv < 0 and gu > 0 for all (u, v) ∈ [0, ∞) × [0, ∞); (P2) there exist positive constants C1 and C2 such that f (C1 , 0) 0, g(C1 , C2 ) 0. II. Competition model (C1) fv , gu < 0 for all (u, v) ∈ [0, ∞) × [0, ∞); (C2) there exist positive constants C3 and C4 such that f (C3 , 0) 0, g(0, C4 ) 0. III. Symbiotic model (S1) fv , gu > 0 for all (u, v) ∈ [0, ∞) × [0, ∞); (S2) there exist positive constants C5 and C6 such that f (C5 , C6 ) = g(C5 , C6 ) = 0. The following lemma follows from the strong maximum principle, and so we omit the proof. Lemma 3.1. (i) The non-negative solution (u, v) of (1.1) with predator–prey interaction has an a priori bound; u(x) C1 and v(x) C2 . (ii) The non-negative solution (u, v) of (1.1) with competition interaction has an a priori bound; u(x) C3 and v(x) C4 . Consider the problem: −[(d + (v))v] = vg(T v, v)) in , v=0 on j,
(3.1)
where T is a solution operator which is defined in Section 2. Define F (d, v) := v − G−1 ◦ (−)−1 (vg(T v, v)), where G−1 (v) is the continuous inverse of the map G(v) = (d + (v))v in v. The inverse map G−1 (v) exists since jG/jv = v (v)v + d + (v) > 0 for all v 0. Then v is a positive solution of F = 0 if and only if v is a positive solution of (3.1). Lemma 3.2. Consider Eq. (1.1) with predator–prey and competition interactions and assume that 1 ((0) + f (0, 0)) > 0. If 1 ((d + (0)) + g(u0 , 0)) > 0, then F (d, v) = 0 has a positive solution for d ∈ (0, d), where the constant d is defined in the below. Proof. First we observe that the linearized operator of F (d, v) at v = 0 is Fv (d, 0) = I − (1/(d + (0)))(−)−1 (g(u0 , 0)I ). Using the variational property of the principal eigenvalue, we can find d > 0 such that 1 ((d + (0)) + g(u0 , 0)) < 0 for all d > d, where 1 ((d + (0)) + g(u0 , 0)) = 0. Since the eigenvalue problem Fv (d, 0) = in , =0 on j
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is equivalent to the equation [(d + (0))(1 − )] + g(u0 , 0) = 0 =0
in , on j,
we can see that 1 ((d + (0)) + g(u0 , 0)) = 0 if and only if 1 (Fv (d, 0)) = 0. Claim 1. (d, 0) is a bifurcation point of F (d, v) = 0. Proof. Suppose that the result is false. Then there exists a sufficiently small > 0 such that deg(F (d, ·), B , 0) is well-defined, where B is a neighborhood of 0 and is independent of d ∈ [d − , d + ]. Moreover, for those d ∈ [d − , d + such that Fv (d, 0) is invertible, deg(F (d, ·), B , 0) = (−1) d , where d = {multiplicity of negative eigenvalues ofFv (d, 0)}. Observe that 0 is a simple eigenvalue of Fv (d, 0) since 1 ((d + (0)) + g(u0 , 0)) = 0 is simple. Therefore, the difference of d+ and d− must be the multiplicity of the eigenvalue 0 of Fv (d, 0) which is 1, and so we can deduce that deg(F (d − , ·), B , 0) = −deg(F (d + , ·), B , 0) = 0. This is a contradiction to the fact that deg(F (d, ·), B , 0) is independent of d, and thus Claim 1 holds. Now, by using the similar argument of [11] or [15], we can conclude that there exists an interval (, ) ⊂ R+ such that F (d, v) = 0 has a positive solution for every d ∈ (, ). Moreover, one of the following holds: (i) (ii) (iii) (iv)
/ 0, d = < < ∞ and Fv ( , 0) = 0 for some ≡ 0 < < = d and Fv (, 0) = 0 for some ≡ / 0, (, ) = (d, ∞), (, ) = (0, d).
Claim 2. Only (iv) of (i)–(iv) is true. Proof. Since (d, 0) is a bifurcation point, there exist > 0 and C 1 -functions d(s), v(s) = s( +t (s)) with d(0)=d, t (0)=0 and F (d(s), v(s))=0 for |s| < . (For more details, see the proof of Theorem 5.1 in Section 5.) Since t (s) ∈ C 1 () and t (0)=0, v(s)=s( +t (s)) > 0 for sufficiently small s > 0. Assume that (i) holds and let v(d) be the positive solution of F = 0 for a given d. Then 1 ((d + (v(d))) + g(T v(d), v(d))) = 0 by Lemma 2.1(ii). If d → (= d), then v(d) → 0, and so 1 (( + (0)) + g(u0 , 0)) = 0 by the continuity. On the other hand, since > d, we must have 1 (( + (0)) + g(u0 , 0)) < 1 ((d + (0)) + g(u0 , 0)) = 0, which is a contradiction. Therefore (i) is excluded. Similarly, one can show that the case (ii) is not appropriate, either. Suppose (iii) holds. Then thereare sequences {dn } and {v(dn )}, where dn → ∞ as n → ∞ and v(dn ) is a corresponding positive solution of F (d, v) = 0. Thus 1 ((dn + (v(dn ))) + g(T v(dn ), v(dn ))) = 0 for
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all n. In the predator–prey case (gu > 0), we have 1 ((dn + (0)) + g(u0 , 0)) > 0 for all n, which is impossible since 1 ((dn + (0)) + g(u0 , 0)) → −∞ as n → ∞. Note that T v(dn ) u0 since −[(u0 )u0 ] = u0 f (u0 , 0) u0 f (u0 , v(d)) in for v(d) 0 by the comparison argument. One can similarly derive a contradiction in the competition case (gu < 0) by noticing that 1 ((dn + (0)) + g(0, 0)) → −∞ as n → ∞. Finally, we can conclude that only (iv) must hold for our problem. In the above proof, we used the Leray–Schauder degree, and so Lemma 3.1 is necessary in the actual calculations of degree. The following is the coexistence theorem for system (1.1) with nondegenerate self-diffusion rates. We should point out that even though the following Theorem (ii) is a known result in [12], we state here again to compare the conditions of positive coexistence among three different interactions. Alternative tool which is the bifurcation theory is employed in the proof. Theorem 3.3. Assume that 1 ((0) + f (0, 0)) > 0, 1 ((d + (0)) + g(0, 0)) > 0. (i) (predator–prey model) If 1 ((0) + f (0, v0 )) > 0, then (1.1) has a positive solution. (ii) (Competition model) If 1 ((0)+f (0, v0 )) > 0 and 1 ((d + (0))+g(u0 , 0)) > 0, then (1.1) has a positive solution. (iii) (Symbiotic model) System (1.1) has a positive solution in '0, C5 ( ⊕ '0, C6 (. Proof. (i) Since 1 ((d + (0)) + g(u0 , 0)) > 1 ((d + (0)) + g(0, 0)) > 0, F (d, v) = 0 has a positive solution v by Lemma 3.2, and so it remains to show T v > 0. Observe that (T v, v) is a non-negative solution of (1.1). If T v ≡ 0, then v ≡ v0 from Eq. (3.1) by the uniqueness of v0 , and so T v 0 > 0 since 1 ((0) + f (0, v0 )) > 0, which is a contradiction. (ii) By assumption, the positive solution v of F (d, v) = 0 exists by Lemma 3.2. To finish the proof, we need T v > 0. Since −[(d + (v0 ))v0 ] = v0 g(0, v0 ) v0 g(u, v0 ) in for u 0, we have v0 v, and thus 1 ((0) + f (0, v)) 1 ((0) + f (0, v0 )) > 0, which implies T v > 0 by the definition of T. (iii) Since 1 ((0) + f (0, v)) > 1 ((0) + f (0, 0)) > 0 by (S1) and Lemma 2.1(i), there is a positive unique solution of the equation, −[(u)u] = uf (u, v) in and u = 0 on j, by Theorem 2.2(ii). Denote it by Tv. Using the strong maximum principle, it is routine to check T C 6 C5 , and so g(T C 6 , C6 ) g(C5 , C6 ) 0. Define G−1 (v) by the continuous inverse of the map G(v) = (d + (v))v in v and Av := G−1 ◦ (− + M)−1 [(g(T v, v) + M(d + (v)))v], where M is a positive constant. Let AC 6 =: w0 , then −[(d + (w0 ))w0 ] + M(d + (w0 ))w0 = C6 g(T C 6 , C6 ) + M(d + (C6 ))C6 M(d + (C6 ))C6 and so w0 C6 by the strong maximum principle, i.e., AC 6 C6 . Since 1 ((d + (0)) + g(u0 , 0)) > 1 ((d+(0))+g(0, 0)) > 0 by (S1) and Lemma 2.1(i), one can have r(A (0))= r[(1/(d + (0)))(− + M)−1 (g(u0 , 0) + M(d + (0)))] > 1 by Lemma 2.1(iii). Consequently, there must be a positive fixed point v of A in '0, C6 ( by Theorem 7.6 in [1]. (Note that the same result can follow from Lemma 3.2.) Also since 1 ((0)+f (0, v)) > 1 ((0)+
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f (0, 0)) > 0, we have T v > 0 and it is easy to check T v C5 by using the strong maximum principle. Hence (T v, v) is a positive solution of system (1.1) in '0, C5 ( ⊕ '0, C6 (. As an example of Theorem 3.3, consider the following problem:
−[(d1 + 1 ul )u] = (a − bup − cv q )u −[(d2 + 2 v m )v] = (e − hur − kv s )v (u, v) = (0, 0)
in , on j,
(3.2)
where l, m > 1 and d1 , d2 , 1 , 2 , a, b, e, k, p, q, r, s are positive constants. Depending on the sign of c and h, (3.2) can be of predator–prey model (c 0 and h 0), of competition model (c, h 0), or of symbiotic model (c, h 0). Corollary 3.4. Consider system (3.2). (i) [predator–prey model: c 0 and h 0] If q/s a − c (ebr/p − ha r/p )/(br/p k) e , , 1 (−) < min d1 d2
then (3.2) has a positive solution. (ii) [Competition model: c, h 0] If
a − c(e/k)q/s e − h(a/b)r/p 1 (−) < min , d1 d2
,
then (3.2) has a positive solution. (iii) [Symbiotic model: c, h 0] If 1 (−) < min{a/d1 , e/d2 } and ps > qr, then (3.2) has a positive solution. Proof. Comparing system (3.2) with (1.1), note that (u)=d1 + 1 ul , (v)= 2 v m , d =d2 , f (u, v) = a − bup − cv q and g(u, v) = e − hur − kv s . From the each assumptions, we can have a > d1 1 (−) and e > d2 1 (−) which ensure the existence of positive semi-trivial solutions u0 and v0 . (i) Take C1 := (a/b)1/p and C2 := ((ebr/p − ha r/p )/(br/p k))1/s , then u0 C1 and q q v0 C2 by Lemma 3.1(i), and thus 1 (d1 + a − cv 0 ) 1 (d1 + a − cC 2 ) > 0 from the assumption. Hence (3.2) has a positive solution by Theorem 3.3(i). (ii) Take C3 := (a/b)1/p and C4 := (e/k)1/s , then we can similarly derive 1 (d1 + q a − cv 0 ) > 0 and 1 (d2 + e − hur0 ) > 0 which are the sufficient conditions for the positive coexistence in Theorem 3.3(ii). (iii) To apply Theorem 3.3(iii), we need to show that there exist positive constants C5 p q and C6 such that a − bC 5 − cC 6 = 0 and e − hC r5 − kC s6 = 0. Equivalently, p(x) = a − bx p − c((e − hx r )/k)q/s has a zero, which can be easily observed since p(0) > 0 and limx→∞ p(x) = −∞.
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4. Degenerate elliptic systems In this section, we investigate the nonlinear elliptic systems with degenerate self-diffusions. First consider the following elliptic system:
−[( + (u))u] = uf (u, v), −[( + (v))v] = vg(u, v) in , (u, v) = (0, 0) on j,
(4.1)
where is a sufficiently small positive constant. Instead of (H1), we impose the following hypothesis (D) for each interactions which can make system (4.1) degenerate: (D) is C 2 -function in u with (0) = 0 and u > 0 for all u > 0; and is C 2 -function in v with (0) = 0 and v > 0 for all v > 0. Lemma 4.1. Assume that > 0 is sufficiently small and (D), (H2) are satisfied. The following hold for system (4.1): (i) [predator–prey case: (P1)–(P2) hold] If min{f (0, C2 ), g(0, 0)} > 0, then there is a positive coexistence. (ii) [Competition case: (C1)–(C2) hold] If min{f (0, C2 ), g(C1 , 0)} > 0, then there is a positive coexistence. (iii) [Symbiotic case: (S1)–(S2) hold] If min{f (0, 0), g(0, 0)} > 0, then there is a positive coexistence. Proof. Since f (0, 0) > 0, g(0, 0) > 0 from the assumption in all cases, we have 1 ( + f (0, 0)) > 0 and 1 ( +g(0, 0)) > 0 for sufficiently small > 0, which imply the existence of the semi-trivial solutions u0 and v0 in each case. (iii) is a direct consequence of Theorem 3.3(iii). (i) By Theorem 2.2, the semi-trivial solutions satisfy u0 C1 and v0 C2 . From the given assumption, 1 ( + f (0, C2 )) > 0 and for sufficiently small > 0, and so 1 ( + f (0, v0 )) > 0 by Lemma 2.1(i). Thus Theorem 3.3(i) concludes the result. (ii) Notice that u0 C3 and v0 C4 , and by the assumptions, we have 1 ( + f (0, C2 )) > 0 and 1 ( + g(C1 , 0)) > 0 for sufficiently small > 0. Therefore 1 ( + f (0, v0 )) > 0 and 1 ( + g(u0 , 0)) > 0 by Lemma 2.1(i) again, and so the desired result follows from Theorem 3.3(ii). Observe that system (4.1) with the assumptions (D) and (H2) becomes the degenerate one when ≡ 0 since (0) = (0) = 0:
−[(u)u] = uf (u, v), −[(v)v] = vg(u, v), in , (u, v) = (0, 0) on j.
(4.2)
Theorem 4.2. Consider the degenerate system (4.2) with the assumptions (D) and (H2). Then the results (i)–(iii) in Lemma 4.1 remain true for system (4.2).
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Proof. Since the proofs are virtually same, we prove only (i). Denote the positive solution of (4.1) by (u n , v n ) for a given n , where { n } is a decreasing sequence with n → 0 as n → ∞. Then we observe that u n and v n have a priori bounds C1 and C2 ; u n C1 and v n C2 . By using Schauder estimate and the elliptic regularity theorem, one can see that {u n } and {v n } have convergent subsequences which are solutions of (4.2) as n → ∞, denote them again by {u n } and {v n }. For the positivity of these limit solutions of (4.2), it suffices to show that u n 0 and v n 0. Contrariwise, assume that u n → 0, then there exists N1 > 0 such that u n whenever n N1 for all > 0. Since lim →0+ (f ( , C2 ))/( ) = +∞, one can find a > 0 which satisfies 1 (−) < (f ( , C2 ))/( ), equivalently, 1 (( ) + f ( , C2 )) > 0. Since 1 (( n + ( )) + f ( , C2 )) → 1 (( ) + f ( , C2 )) as n → ∞, there exists a N2 > 0 such that 1 (( n + ( )) + f ( , C2 )) > 0 for all n N2 , and so if we choose N > 0 with N max{N1 , N2 }, then 1 (( N +(u N ))+f (u N , v N )) > 1 (( N + ( ))+f ( , C2 )) > 0 by Lemma 2.1(i). But this contradicts Lemma 2.1(ii) since (u N , v N ) is a positive solution of (4.1), and thus u n 0. In the above proof, if ( ) and f ( , C2 ) is replaced by ( ) and g(0, ), respectively, then the result v n 0 can be similarly checked. As a simple example of Theorem 4.2, consider the well-known degenerate Lotka–Volterra problem:
−ul = (a − bup − cv q )u, −v m = (e − hur − kv s )v in , (u, v) = (0, 0) on j,
(4.3)
where l, m > 1 and a, b, e, k, p, q, r, s are positive constants. Depending on the sign of c and h, (4.3) can be of predator–prey model (c 0 and h 0), of competition model (c, h 0), or of symbiotic model (c, h 0). Corollary 4.3. Consider system (4.3). (i) [Predator–prey model: c 0 and h 0] If min{a − c((ebr/p − ha r/p )/(br/p k))q/s , e} > 0, then (4.3) has a positive solution. (ii) [Competition model: c, h 0] If min{a − c(e/k)q/s , e − h(a/b)r/p } > 0, then (4.3) has a positive solution. (iii) [Symbiotic model: c, h 0] If min{a, e} > 0 and ps > qr, then (4.3) has a positive solution. Proof. Comparing system (4.3) with (4.2), note that (u) = ul−1 , (v) = v m−1 , f (u, v) = a − bup − cv q and g(u, v) = e − hur − kv s . 1
(i) Take C1 := (a/b) p and C2 := ((ebr/p − ha r/p )/(br/p k))1/s . (ii) Take C3 := (a/b)1/p and C4 := (e/k)1/s . (iii) The proof is virtually the same as that of Corollary 3.4(iii).
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5. Local behavior for predator–prey and competition interactions In this section, we give the local uniqueness and stability of the positive solutions of the system (1.1) for predator–prey and competition interactions. We need the following hypotheses: (L1) is C 2 -function in u with (0) > 0 and u 0 for all u 0; and is C 2 -function in v with (0) 0 and v 0 for all v 0; (L2) f (u, v), g(u, v) are C 1 -functions in u, v with fu , fv , gv < 0 for all (u, v) ∈ [0, ∞) × [0, ∞); 3 such that f (C 1 , 0), g(0, C 3 ) 0; 1 and C (L3) there exist positive constants C 2 := C 2 (M) > 0 such that g(u, v) 0 on (L4) for each M > 0, there exists a constant C 2 . u ∈ (0, M] when v C Now we have the following: Theorem 5.1. Assume that 1 ((0) + f (0, 0)) > 0. If 1 ((d + (0)) + g(u0 , 0)) > 0 for some d > 0, then there exists a neighborhood N of (u0 , 0) in C() ⊕ C() such that there is the unique positive solution (u, v) of (1.1) in N for some range of d. This solution is stable in the case of gu > 0. Proof. Consider (d, v) := [(d + (v))v] + vg(T v, v), F v (d, 0) := (d + where the operator T is defined in Section 2. Then it is easy to note F (0))+g(u0 , 0)I . Using the variational property of the principal eigenvalue of the linearized v (d, 0), we can find d > 0 such that 1 (F v (d, 0)) < 0 for all d > d, where d operator F v (d, 0)) = 0. Let > 0 denote the principal eigenfunction of 1 (F v (d, 0)). satisfies 1 (F Local existence and uniqueness of a bifurcating solution:One can easily see that Ker v (d, 0)) is one-dimensional which is spanned by , and also ∈ Range(F v (d, 0)) if and (F v (d, 0))) = 1. Since F vd (d, 0) = and = only if = 0. Thus Codim(Range(F vd ∈ v (d, 0)). Thus we can apply Crandall–Rabinowitz − |∇ |2 < 0, we have F / Range(F bifurcation theorem in [15, Section 13] to conclude that d is a bifurcating point, that is to say, there exist > 0 and C 1 -functions d(s), v(s) = s( + t (s)) with d(0) = d, t (0) = 0 (d(s), v(s)) = 0 for |s| < . Next, we show that (T v(s), v(s)) is a positive solution of and F (1.1). Since t (s) ∈ C 1 () and t (0) = 0, we get v(s) = s( + t (s)) > 0 for sufficiently small s > 0. Also we can see that T v(s) > 0 for small s > 0 since T 0=u0 > 0 and lims→0 v(s)=0. The local uniqueness follows from the fact that the bifurcating solution is locally unique. v (d, 0))= Local / Range(F stability of a bifurcating solution in the case of gu > 0: Since ∈ { : = 0}, using Theorem 13.8 in [15], we have sd (s)1 (d) = −1, s→0 (s) lim
(5.1)
where (s) is the principal eigenvalue corresponding to the linearization of the bifurcation solution u(s). By the variational property of the principal eigenvalue, 1 (d) < 0. To deter-
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mine the sign of d (s) near at s = 0, consider the equation −[(d(s) + (v(s)))v(s)] = v(s)g(T v(s), v(s)), i.e., −[(d(s)+(s(+t (s))))s(+t (s))] = s(+t (s))g(T s(+t (s)), s(+t (s))). Dividing s on both sides and take a derivative with respect to s at s = 0, then we have −d (0) − v (0)2 − (d + (0))t (0) = t (0)g(u0 , 0) + [gu (u0 , 0)T (0) + gv (u0 , 0)]. Multiplying on both sides and take integral over , −d (0) = v (0) 2 + [gu T (0) + gv ], (5.2)
since [(d + (0))t (0) + t (0)g(u0 , 0)] = t (0)([(d + (0))] + g(u0 , 0)) = 0. Moreover from the fact v (0) 2 = −v (0) ∇ ∇ 2 = −v (0) 2|∇ |2 0, we may conclude that the right-hand side of Eq. (5.2) is negative in the predator–prey case since gu > 0, gv < 0 from the assumptions and T (v) < 0 by Lemma 2.3(ii). Thus d (0) < 0 holds. Therefore d (s) < 0 for sufficiently small s. Consequently, the function (s) in (5.1) must be negative for small positive s and thus the bifurcation solution is locally stable.
Remark 5.2. (i) Observe that no monotonicity of the function g in u is assumed in the proof of the local existence and uniqueness of a bifurcating solution, and so our results can be applied both to the so-called predator–prey model (gu > 0) and to the competition model (gu < 0) in the biological sense. (ii) For the predator–prey and competition interactions, Theorem 3.3 gives global positive coexistence results. On the other hand, Theorem 5.1 gives local positive coexistence results which depend on the diffusion rate d.
6. Conclusions In this article, we deal with certain population models with diffusions and self-diffusions between two species interacting in three different models, namely predator–prey, competing, cooperating interactions in hostile boundary environment. This article mainly concentrated on the positive coexistence of two species in a given domain, considering two cases in which the diffusions are either nondegenerate or degenerate with respect to the zero densities. In view of our results, in case that the system has nondegenerate diffusions, the coexistence of two distinct species can be affected by the shape and the size of a domain as well as the nonlinearities introduced in the equations for the predator–prey and competing models since the coexistence conditions can be expressed by the principal eigenvalue of the linearized differential operator at the marginal densities of the species, whereas cooperating species can coexist provided that there are positive equilibria of the system, without being dominated by the character of a given region (Theorem 3.3). Therefore, the coexistence problem of the model with nondegenerate diffusions has certain common features with those systems in which the diffusions = = constant (See for example, [2,4,5,8,9].) On the other hand, the model with degenerate diffusions has many different features compared with the nondegenerate systems with regard to the coexistence of species. Namely,
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two species for each of three different interactions may coexist in a region if the growth rates at specific constant densities are positive, and are unaffected by diffusions, self-diffusions and the domain undertaken (Theorem 4.2). Acknowledgements The first author was supported by Post-Doc. Grant at Korea University in 2003. The second author was supported by Grant R05-2003-000-11622-0 from the Basic Research Program of the Korea Science & Engineering Foundation. References [1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (4) (1976) 620–709. [2] J. Blat, K.J. Brown, Bifurcation of steady-state solutions in predator–prey and competition systems, Proc. Roy. Soc. Edinburgh Sect. A 97 (1984) 21–34. [3] A. Cañada, J.L. Gámez, Elliptic systems with nonlinear diffusion in population dynamics, Differential Equations Dynamic Systems 3 (2) (1995) 189–204. [4] E.N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc. 284 (2) (1984) 729–743. [5] E.N. Dancer, On positive solutions of some pairs of differential equations. II, J. Differential Equations 60 (2) (1985) 236–258. [6] M. Delgado, A. Suárez, Existence of solutions for elliptic systems with holder continuous nonlinearities, Differential Integral Equations 13 (4–6) (2000) 453–477. [7] A. Leung, G. Fan, Existence of positive solutions for elliptic systems—degenerate and nondegenerate ecological models, J. Math. Anal. Appl. 151 (2) (1990) 512–531. [8] L. Li, Coexistence theorems of steady states for predator–prey interacting systems, Trans. Amer. Math. Soc. 305 (1) (1988) 143–166. [9] L. Li, R. Logan, Positive solutions to general elliptic competition models, Differential Integral Equations 4 (4) (1991) 817–834. [10] J.D. Murray, Mathematical biology, Biomathematics, vol. 19, Springer, Berlin, 1989. [11] P.H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971) 487–513. [12] K. Ryu, I. Ahn, Positive coexistence of steady states for competitive interacting system with self-diffusion pressures, Bull. Korean Math. Soc. 38 (4) (2001) 643–655. [13] K. Ryu, I. Ahn, Positive steady-states for two interacting species models with linear self-cross diffusions, Discrete Continous Dynamic System 9 (4) (2003) 1049–1061. [14] K. Ryu, I. Ahn, On certain elliptic systems with nonlinear self-cross diffusions, Discrete Continous Syn. System An expanded vol. (2003) 752–759. [15] J. Smoller, Shock waves and reaction–diffusion equations, Grundlehren der Mathematischen Wissenschaften, second ed., vol. 258, Springer, New York, 1994.
Positive solutions of certain nonlinear elliptic systems with self-diffusions: nondegenerate vs. degenerate diffusions Kimun Ryua,∗ , Inkyung Ahnb a School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455, USA b Department of Mathematics, Korea University, Jochiwon 339-700, Korea
Received 21 May 2004; accepted 10 May 2005
Abstract We discuss the existence of positive solutions to certain strongly-coupled nonlinear elliptic systems with self-diffusions under homogeneous Dirichlet boundary conditions. Using the global positive coexistence results for predator–prey, competition and symbiotic interactions between two species, sufficient conditions for the positive solutions of the degenerate self-diffusive systems are studied. We also investigate the local behavior, namely, the local existence, uniqueness and stability, of positive solutions for predator–prey and competition interactions. Our method is based on the decoupling technique and bifurcation theory. 䉷 2005 Elsevier Ltd. All rights reserved. MSC: 35J60 Keywords: Nonlinear elliptic system; Positive solutions; Self-diffusions; Bifurcation; Degenerate diffusions
∗ Corresponding author. Tel./fax: +1 1612 788 1290.
E-mail addresses: [email protected] (K. Ryu), [email protected] (I. Ahn). 0362-546X/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.05.010
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1. Introduction In this paper, we are concerned with the existence of positive solutions to the following strongly-coupled nonlinear elliptic system: −[(u)u] = uf (u, v), (1.1) −[(d + (v))v] = vg(u, v) in , (u, v) = (0, 0) on j, where is a bounded domain of Rn with smooth boundary j, d is a positive constant and the functions , , f, g : × R2 → R satisfy certain conditions which will be imposed later. In the biological sense, the two species are in predator–prey interaction if one of the growth rates involved is increasing in the prey while the other decreasing in the predator. Also the two species are in symbiotic interaction if each of their relative growth functions is increasing in the other, and they are in competition if these functions are decreasing in the other one. Refer [10] for more information. We say that a steady-state system has a positive solution (u, v) if u(x) > 0 and v(x) > 0 for all x ∈ . The existence of a positive solution (u, v) to the system is called a positive coexistence. The global positive coexistence used in this paper means the existence of positive solutions in terms of a whole domain. If the diffusion rate depends on the density of the own species, it is called self-diffusions. In [7], they found the positive solutions between the appropriate upper and lower solutions for the degenerate elliptic system (i.e. (0) = 0) under certain conditions of f and g: −(u) = f (x, u, v), −(v) = g(x, u, v) in , (u, v) = (0, 0) on j. In [3], the authors investigated the coexistence state (i.e., each u(x) and v(x) is nonnegative and nontrivial) for the system with degenerate diffusions (i.e., (0) = (0) = 0): −(U (x)) = U (x)h(x, U (x), V (x)), −(V (x)) = V (x)k(x, U (x), V (x)) in , U (x) = V (x) = 0 on j, using the method of a system of upper-lower solutions. One can also refer to [6] relating to the system (1.1) and the references therein. In this paper, we give sufficient conditions for the existence of positive solutions of the system (1.1) with nondegenerate diffusions, and using these facts, we study the existence of positive solutions to system (1.1) with degenerate diffusions for three different biological interactions, i.e., predator–prey, competition and symbiotic interactions. The perturbed method was employed in this study. For nondegenerate systems, the existence of positive solutions to (1.1) can be expressed in terms of the spectral property of linearized differential operators of the given system for predator–prey and competing interacting systems, and in terms of the existence of the positive equilibria of the system for symbiotic interactions. On the other hand, in case that the diffusions are degenerate (i.e. d = 0, (0) = (0) = 0 in system (1.1)), the positive coexistence depends only on the positivity of the growth rates at specific constant densities for all three interactions.
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We also consider the system of (1.1) with nondegenerate diffusions for the predator–prey and competition interactions to investigate the local behavior, namely, the local existence, uniqueness and stability, of the positive solutions. Our method is based on the decoupling technique and bifurcation theory. In our model (1.1), the functions f (u, v) and g(u, v) can be replaced by f (x, u, v) and g(x, u, v), respectively, and Dirichlet boundary conditions could be switched by Robin boundary conditions, since the corresponding results could be obtained by slight modifications of the proof provided here. For simplicity, we consider system (1.1). This article is organized as follows. In Section 2, we collect some known results and give a lemma which is useful in this article. In Section 3, we study the global positive coexistence for system (1.1) with nondegenerate diffusions. In Section 4, using the global positive coexistence results for the predator–prey, competition and symbiotic interactions, we investigate the positive coexistence of (1.1) with the degenerate diffusion pressures. In addition, an example is given as a simple application. Section 5 deals with the local behavior of positive solutions to the predator–prey and competition interactions for (1.1) with nondegenerate diffusions.
2. Preparations In this section, we collect some known results which are useful in our work. For a(x) > 0 in C 2 () and b(x) ∈ L∞ (), the eigenvalue problem: [a(x)u] + b(x)u = u in , u=0 on j
(2.1)
has a principal eigenvalue corresponding to the unique positive principal eigenfunction. (For more details, see [13].) Denote this principal eigenvalue of (2.1) by 1 (a(x) + b(x)). The following results can be found in [13]. Lemma 2.1. (i) 1 (a(x) + b(x)) is increasing in b(x). (ii) Let a(x) > 0 in C 2 (), b(x) ∈ L∞ () and u 0, u ≡ / 0 in with u = 0 on j. If (a(x) + b(x))u ≡ 0, then 1 (a(x) + b(x)) = 0. (iii) Let a(x) > 0 in C 2 (), b(x) ∈ L∞ () and M be a positive constant such that b(x) + Ma(x) > 0 for all x ∈ . If 1 (a(x) + b(x)) > 0, then r[(1/a(x))(− + M)−1 (b(x) + Ma(x))] > 1. For the first equation in system (1.1): −[(u)u] = uf (u, v(x)) in , u=0 on j, where v(x) ∈ C 2 () is fixed, the following theorem can be found in [14]. Theorem 2.2. Let (2.2) satisfy the following three assumptions: (A1) (u) is C 2 -function in u with (0) > 0 and u (u) 0 for all u 0; (A2) f (u, v) is C 1 -function in u, v with fu , fv < 0 for all (u, v) ∈ [0, ∞) × [0, ∞); (A3) f (u, 0) 0 on u ∈ [C0 , ∞) for some positive constant C0 .
(2.2)
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Assume that v(x) ∈ C 2 () is fixed. Then the following hold: (i) If 1 ((0) + f (0, v)) 0, then (2.2) has no positive solution; (ii) If 1 ((0)+f (0, v)) > 0, then (2.2) has a unique positive solution u(x) which satisfies u(x) C0 . Throughout this paper, when v(x) ≡ 0, we denote the unique positive solution of (2.2) by u0 if 1 ((0) + f (0, 0)) > 0. Similarly, denote the unique positive solution of the second equation in (1.1) by v0 when u(x) ≡ 0, if 1 ((0) + g(0, 0)) > 0. In view of Theorem 2.2, for every v ∈ C 2 (), we can define the map T : C 2 () → 2, C () for some 0 < < 1 by T v :=
uv
if 1 ((0) + f (0, v)) > 0,
0
otherwise,
where uv is the unique positive solution of (2.2). The following lemma is needed in the proof of the existence theorem. Lemma 2.3. (i) The mapping T is continuous in the sense of C 2 () → C 2, () for some 0 < < 1. (ii) If v2 v1 ≡ / v2 , then either uv2 < uv1 or uv2 ≡ uv1 ≡ 0. Proof. For the proof of (i), one can see Lemma 7 in [14] as a special case. (ii) Assume that v2 > v1 . If 1 ((0) + f (0, v1 ) 0, then 1 ((0) + f (0, v2 ) < 1 ((0) + f (0, v1 ) 0 by Lemma 2.1(i), and so uv2 ≡ uv1 ≡ 0 by Theorem 2.2(i). If 1 ((0) + f (0, v1 ) > 0, then uv1 > 0 in . Since [(uv1 )uv1 ] + uv1 f (uv1 , v2 ) [(uv1 )uv1 ] + uv1 f (uv1 , v1 ) = 0 in and uv1 = 0 on j, uv1 is a positive upper solution of the equation, −[(u)u] = uf (u, v2 ) in and u = 0 on j. By the uniqueness of the positive solution uv2 to Eq. (2.2) for v2 (x) ∈ C 2 (), we have uv2 uv1 . The strict inequality uv2 < uv1 follows from the strong maximum principle.
3. Nondegenerate elliptic systems In this section, we give sufficient conditions for the positive coexistence of (1.1) with nondegenerate diffusions for three different biological interactions, i.e., predator–prey, competition and symbiotic interactions. To do this, we assume that the following hypotheses throughout this section: (H1) d is a positive constant, is C 2 -function in u with (0) > 0 and u 0 for all u 0; and is C 2 -function in v with (0) 0 and v > 0 for all v 0; (H2) f (u, v), g(u, v) are C 1 -functions in u, v with fu < 0, gv < 0 for all (u, v) ∈ [0, ∞) × [0, ∞).
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In addition, we impose the following hypotheses for three different biological interactions: I. Predator–prey model (P1) fv < 0 and gu > 0 for all (u, v) ∈ [0, ∞) × [0, ∞); (P2) there exist positive constants C1 and C2 such that f (C1 , 0) 0, g(C1 , C2 ) 0. II. Competition model (C1) fv , gu < 0 for all (u, v) ∈ [0, ∞) × [0, ∞); (C2) there exist positive constants C3 and C4 such that f (C3 , 0) 0, g(0, C4 ) 0. III. Symbiotic model (S1) fv , gu > 0 for all (u, v) ∈ [0, ∞) × [0, ∞); (S2) there exist positive constants C5 and C6 such that f (C5 , C6 ) = g(C5 , C6 ) = 0. The following lemma follows from the strong maximum principle, and so we omit the proof. Lemma 3.1. (i) The non-negative solution (u, v) of (1.1) with predator–prey interaction has an a priori bound; u(x) C1 and v(x) C2 . (ii) The non-negative solution (u, v) of (1.1) with competition interaction has an a priori bound; u(x) C3 and v(x) C4 . Consider the problem: −[(d + (v))v] = vg(T v, v)) in , v=0 on j,
(3.1)
where T is a solution operator which is defined in Section 2. Define F (d, v) := v − G−1 ◦ (−)−1 (vg(T v, v)), where G−1 (v) is the continuous inverse of the map G(v) = (d + (v))v in v. The inverse map G−1 (v) exists since jG/jv = v (v)v + d + (v) > 0 for all v 0. Then v is a positive solution of F = 0 if and only if v is a positive solution of (3.1). Lemma 3.2. Consider Eq. (1.1) with predator–prey and competition interactions and assume that 1 ((0) + f (0, 0)) > 0. If 1 ((d + (0)) + g(u0 , 0)) > 0, then F (d, v) = 0 has a positive solution for d ∈ (0, d), where the constant d is defined in the below. Proof. First we observe that the linearized operator of F (d, v) at v = 0 is Fv (d, 0) = I − (1/(d + (0)))(−)−1 (g(u0 , 0)I ). Using the variational property of the principal eigenvalue, we can find d > 0 such that 1 ((d + (0)) + g(u0 , 0)) < 0 for all d > d, where 1 ((d + (0)) + g(u0 , 0)) = 0. Since the eigenvalue problem Fv (d, 0) = in , =0 on j
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is equivalent to the equation [(d + (0))(1 − )] + g(u0 , 0) = 0 =0
in , on j,
we can see that 1 ((d + (0)) + g(u0 , 0)) = 0 if and only if 1 (Fv (d, 0)) = 0. Claim 1. (d, 0) is a bifurcation point of F (d, v) = 0. Proof. Suppose that the result is false. Then there exists a sufficiently small > 0 such that deg(F (d, ·), B , 0) is well-defined, where B is a neighborhood of 0 and is independent of d ∈ [d − , d + ]. Moreover, for those d ∈ [d − , d + such that Fv (d, 0) is invertible, deg(F (d, ·), B , 0) = (−1) d , where d = {multiplicity of negative eigenvalues ofFv (d, 0)}. Observe that 0 is a simple eigenvalue of Fv (d, 0) since 1 ((d + (0)) + g(u0 , 0)) = 0 is simple. Therefore, the difference of d+ and d− must be the multiplicity of the eigenvalue 0 of Fv (d, 0) which is 1, and so we can deduce that deg(F (d − , ·), B , 0) = −deg(F (d + , ·), B , 0) = 0. This is a contradiction to the fact that deg(F (d, ·), B , 0) is independent of d, and thus Claim 1 holds. Now, by using the similar argument of [11] or [15], we can conclude that there exists an interval (, ) ⊂ R+ such that F (d, v) = 0 has a positive solution for every d ∈ (, ). Moreover, one of the following holds: (i) (ii) (iii) (iv)
/ 0, d = < < ∞ and Fv ( , 0) = 0 for some ≡ 0 < < = d and Fv (, 0) = 0 for some ≡ / 0, (, ) = (d, ∞), (, ) = (0, d).
Claim 2. Only (iv) of (i)–(iv) is true. Proof. Since (d, 0) is a bifurcation point, there exist > 0 and C 1 -functions d(s), v(s) = s( +t (s)) with d(0)=d, t (0)=0 and F (d(s), v(s))=0 for |s| < . (For more details, see the proof of Theorem 5.1 in Section 5.) Since t (s) ∈ C 1 () and t (0)=0, v(s)=s( +t (s)) > 0 for sufficiently small s > 0. Assume that (i) holds and let v(d) be the positive solution of F = 0 for a given d. Then 1 ((d + (v(d))) + g(T v(d), v(d))) = 0 by Lemma 2.1(ii). If d → (= d), then v(d) → 0, and so 1 (( + (0)) + g(u0 , 0)) = 0 by the continuity. On the other hand, since > d, we must have 1 (( + (0)) + g(u0 , 0)) < 1 ((d + (0)) + g(u0 , 0)) = 0, which is a contradiction. Therefore (i) is excluded. Similarly, one can show that the case (ii) is not appropriate, either. Suppose (iii) holds. Then thereare sequences {dn } and {v(dn )}, where dn → ∞ as n → ∞ and v(dn ) is a corresponding positive solution of F (d, v) = 0. Thus 1 ((dn + (v(dn ))) + g(T v(dn ), v(dn ))) = 0 for
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all n. In the predator–prey case (gu > 0), we have 1 ((dn + (0)) + g(u0 , 0)) > 0 for all n, which is impossible since 1 ((dn + (0)) + g(u0 , 0)) → −∞ as n → ∞. Note that T v(dn ) u0 since −[(u0 )u0 ] = u0 f (u0 , 0) u0 f (u0 , v(d)) in for v(d) 0 by the comparison argument. One can similarly derive a contradiction in the competition case (gu < 0) by noticing that 1 ((dn + (0)) + g(0, 0)) → −∞ as n → ∞. Finally, we can conclude that only (iv) must hold for our problem. In the above proof, we used the Leray–Schauder degree, and so Lemma 3.1 is necessary in the actual calculations of degree. The following is the coexistence theorem for system (1.1) with nondegenerate self-diffusion rates. We should point out that even though the following Theorem (ii) is a known result in [12], we state here again to compare the conditions of positive coexistence among three different interactions. Alternative tool which is the bifurcation theory is employed in the proof. Theorem 3.3. Assume that 1 ((0) + f (0, 0)) > 0, 1 ((d + (0)) + g(0, 0)) > 0. (i) (predator–prey model) If 1 ((0) + f (0, v0 )) > 0, then (1.1) has a positive solution. (ii) (Competition model) If 1 ((0)+f (0, v0 )) > 0 and 1 ((d + (0))+g(u0 , 0)) > 0, then (1.1) has a positive solution. (iii) (Symbiotic model) System (1.1) has a positive solution in '0, C5 ( ⊕ '0, C6 (. Proof. (i) Since 1 ((d + (0)) + g(u0 , 0)) > 1 ((d + (0)) + g(0, 0)) > 0, F (d, v) = 0 has a positive solution v by Lemma 3.2, and so it remains to show T v > 0. Observe that (T v, v) is a non-negative solution of (1.1). If T v ≡ 0, then v ≡ v0 from Eq. (3.1) by the uniqueness of v0 , and so T v 0 > 0 since 1 ((0) + f (0, v0 )) > 0, which is a contradiction. (ii) By assumption, the positive solution v of F (d, v) = 0 exists by Lemma 3.2. To finish the proof, we need T v > 0. Since −[(d + (v0 ))v0 ] = v0 g(0, v0 ) v0 g(u, v0 ) in for u 0, we have v0 v, and thus 1 ((0) + f (0, v)) 1 ((0) + f (0, v0 )) > 0, which implies T v > 0 by the definition of T. (iii) Since 1 ((0) + f (0, v)) > 1 ((0) + f (0, 0)) > 0 by (S1) and Lemma 2.1(i), there is a positive unique solution of the equation, −[(u)u] = uf (u, v) in and u = 0 on j, by Theorem 2.2(ii). Denote it by Tv. Using the strong maximum principle, it is routine to check T C 6 C5 , and so g(T C 6 , C6 ) g(C5 , C6 ) 0. Define G−1 (v) by the continuous inverse of the map G(v) = (d + (v))v in v and Av := G−1 ◦ (− + M)−1 [(g(T v, v) + M(d + (v)))v], where M is a positive constant. Let AC 6 =: w0 , then −[(d + (w0 ))w0 ] + M(d + (w0 ))w0 = C6 g(T C 6 , C6 ) + M(d + (C6 ))C6 M(d + (C6 ))C6 and so w0 C6 by the strong maximum principle, i.e., AC 6 C6 . Since 1 ((d + (0)) + g(u0 , 0)) > 1 ((d+(0))+g(0, 0)) > 0 by (S1) and Lemma 2.1(i), one can have r(A (0))= r[(1/(d + (0)))(− + M)−1 (g(u0 , 0) + M(d + (0)))] > 1 by Lemma 2.1(iii). Consequently, there must be a positive fixed point v of A in '0, C6 ( by Theorem 7.6 in [1]. (Note that the same result can follow from Lemma 3.2.) Also since 1 ((0)+f (0, v)) > 1 ((0)+
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f (0, 0)) > 0, we have T v > 0 and it is easy to check T v C5 by using the strong maximum principle. Hence (T v, v) is a positive solution of system (1.1) in '0, C5 ( ⊕ '0, C6 (. As an example of Theorem 3.3, consider the following problem:
−[(d1 + 1 ul )u] = (a − bup − cv q )u −[(d2 + 2 v m )v] = (e − hur − kv s )v (u, v) = (0, 0)
in , on j,
(3.2)
where l, m > 1 and d1 , d2 , 1 , 2 , a, b, e, k, p, q, r, s are positive constants. Depending on the sign of c and h, (3.2) can be of predator–prey model (c 0 and h 0), of competition model (c, h 0), or of symbiotic model (c, h 0). Corollary 3.4. Consider system (3.2). (i) [predator–prey model: c 0 and h 0] If q/s a − c (ebr/p − ha r/p )/(br/p k) e , , 1 (−) < min d1 d2
then (3.2) has a positive solution. (ii) [Competition model: c, h 0] If
a − c(e/k)q/s e − h(a/b)r/p 1 (−) < min , d1 d2
,
then (3.2) has a positive solution. (iii) [Symbiotic model: c, h 0] If 1 (−) < min{a/d1 , e/d2 } and ps > qr, then (3.2) has a positive solution. Proof. Comparing system (3.2) with (1.1), note that (u)=d1 + 1 ul , (v)= 2 v m , d =d2 , f (u, v) = a − bup − cv q and g(u, v) = e − hur − kv s . From the each assumptions, we can have a > d1 1 (−) and e > d2 1 (−) which ensure the existence of positive semi-trivial solutions u0 and v0 . (i) Take C1 := (a/b)1/p and C2 := ((ebr/p − ha r/p )/(br/p k))1/s , then u0 C1 and q q v0 C2 by Lemma 3.1(i), and thus 1 (d1 + a − cv 0 ) 1 (d1 + a − cC 2 ) > 0 from the assumption. Hence (3.2) has a positive solution by Theorem 3.3(i). (ii) Take C3 := (a/b)1/p and C4 := (e/k)1/s , then we can similarly derive 1 (d1 + q a − cv 0 ) > 0 and 1 (d2 + e − hur0 ) > 0 which are the sufficient conditions for the positive coexistence in Theorem 3.3(ii). (iii) To apply Theorem 3.3(iii), we need to show that there exist positive constants C5 p q and C6 such that a − bC 5 − cC 6 = 0 and e − hC r5 − kC s6 = 0. Equivalently, p(x) = a − bx p − c((e − hx r )/k)q/s has a zero, which can be easily observed since p(0) > 0 and limx→∞ p(x) = −∞.
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4. Degenerate elliptic systems In this section, we investigate the nonlinear elliptic systems with degenerate self-diffusions. First consider the following elliptic system:
−[( + (u))u] = uf (u, v), −[( + (v))v] = vg(u, v) in , (u, v) = (0, 0) on j,
(4.1)
where is a sufficiently small positive constant. Instead of (H1), we impose the following hypothesis (D) for each interactions which can make system (4.1) degenerate: (D) is C 2 -function in u with (0) = 0 and u > 0 for all u > 0; and is C 2 -function in v with (0) = 0 and v > 0 for all v > 0. Lemma 4.1. Assume that > 0 is sufficiently small and (D), (H2) are satisfied. The following hold for system (4.1): (i) [predator–prey case: (P1)–(P2) hold] If min{f (0, C2 ), g(0, 0)} > 0, then there is a positive coexistence. (ii) [Competition case: (C1)–(C2) hold] If min{f (0, C2 ), g(C1 , 0)} > 0, then there is a positive coexistence. (iii) [Symbiotic case: (S1)–(S2) hold] If min{f (0, 0), g(0, 0)} > 0, then there is a positive coexistence. Proof. Since f (0, 0) > 0, g(0, 0) > 0 from the assumption in all cases, we have 1 ( + f (0, 0)) > 0 and 1 ( +g(0, 0)) > 0 for sufficiently small > 0, which imply the existence of the semi-trivial solutions u0 and v0 in each case. (iii) is a direct consequence of Theorem 3.3(iii). (i) By Theorem 2.2, the semi-trivial solutions satisfy u0 C1 and v0 C2 . From the given assumption, 1 ( + f (0, C2 )) > 0 and for sufficiently small > 0, and so 1 ( + f (0, v0 )) > 0 by Lemma 2.1(i). Thus Theorem 3.3(i) concludes the result. (ii) Notice that u0 C3 and v0 C4 , and by the assumptions, we have 1 ( + f (0, C2 )) > 0 and 1 ( + g(C1 , 0)) > 0 for sufficiently small > 0. Therefore 1 ( + f (0, v0 )) > 0 and 1 ( + g(u0 , 0)) > 0 by Lemma 2.1(i) again, and so the desired result follows from Theorem 3.3(ii). Observe that system (4.1) with the assumptions (D) and (H2) becomes the degenerate one when ≡ 0 since (0) = (0) = 0:
−[(u)u] = uf (u, v), −[(v)v] = vg(u, v), in , (u, v) = (0, 0) on j.
(4.2)
Theorem 4.2. Consider the degenerate system (4.2) with the assumptions (D) and (H2). Then the results (i)–(iii) in Lemma 4.1 remain true for system (4.2).
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Proof. Since the proofs are virtually same, we prove only (i). Denote the positive solution of (4.1) by (u n , v n ) for a given n , where { n } is a decreasing sequence with n → 0 as n → ∞. Then we observe that u n and v n have a priori bounds C1 and C2 ; u n C1 and v n C2 . By using Schauder estimate and the elliptic regularity theorem, one can see that {u n } and {v n } have convergent subsequences which are solutions of (4.2) as n → ∞, denote them again by {u n } and {v n }. For the positivity of these limit solutions of (4.2), it suffices to show that u n 0 and v n 0. Contrariwise, assume that u n → 0, then there exists N1 > 0 such that u n whenever n N1 for all > 0. Since lim →0+ (f ( , C2 ))/( ) = +∞, one can find a > 0 which satisfies 1 (−) < (f ( , C2 ))/( ), equivalently, 1 (( ) + f ( , C2 )) > 0. Since 1 (( n + ( )) + f ( , C2 )) → 1 (( ) + f ( , C2 )) as n → ∞, there exists a N2 > 0 such that 1 (( n + ( )) + f ( , C2 )) > 0 for all n N2 , and so if we choose N > 0 with N max{N1 , N2 }, then 1 (( N +(u N ))+f (u N , v N )) > 1 (( N + ( ))+f ( , C2 )) > 0 by Lemma 2.1(i). But this contradicts Lemma 2.1(ii) since (u N , v N ) is a positive solution of (4.1), and thus u n 0. In the above proof, if ( ) and f ( , C2 ) is replaced by ( ) and g(0, ), respectively, then the result v n 0 can be similarly checked. As a simple example of Theorem 4.2, consider the well-known degenerate Lotka–Volterra problem:
−ul = (a − bup − cv q )u, −v m = (e − hur − kv s )v in , (u, v) = (0, 0) on j,
(4.3)
where l, m > 1 and a, b, e, k, p, q, r, s are positive constants. Depending on the sign of c and h, (4.3) can be of predator–prey model (c 0 and h 0), of competition model (c, h 0), or of symbiotic model (c, h 0). Corollary 4.3. Consider system (4.3). (i) [Predator–prey model: c 0 and h 0] If min{a − c((ebr/p − ha r/p )/(br/p k))q/s , e} > 0, then (4.3) has a positive solution. (ii) [Competition model: c, h 0] If min{a − c(e/k)q/s , e − h(a/b)r/p } > 0, then (4.3) has a positive solution. (iii) [Symbiotic model: c, h 0] If min{a, e} > 0 and ps > qr, then (4.3) has a positive solution. Proof. Comparing system (4.3) with (4.2), note that (u) = ul−1 , (v) = v m−1 , f (u, v) = a − bup − cv q and g(u, v) = e − hur − kv s . 1
(i) Take C1 := (a/b) p and C2 := ((ebr/p − ha r/p )/(br/p k))1/s . (ii) Take C3 := (a/b)1/p and C4 := (e/k)1/s . (iii) The proof is virtually the same as that of Corollary 3.4(iii).
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5. Local behavior for predator–prey and competition interactions In this section, we give the local uniqueness and stability of the positive solutions of the system (1.1) for predator–prey and competition interactions. We need the following hypotheses: (L1) is C 2 -function in u with (0) > 0 and u 0 for all u 0; and is C 2 -function in v with (0) 0 and v 0 for all v 0; (L2) f (u, v), g(u, v) are C 1 -functions in u, v with fu , fv , gv < 0 for all (u, v) ∈ [0, ∞) × [0, ∞); 3 such that f (C 1 , 0), g(0, C 3 ) 0; 1 and C (L3) there exist positive constants C 2 := C 2 (M) > 0 such that g(u, v) 0 on (L4) for each M > 0, there exists a constant C 2 . u ∈ (0, M] when v C Now we have the following: Theorem 5.1. Assume that 1 ((0) + f (0, 0)) > 0. If 1 ((d + (0)) + g(u0 , 0)) > 0 for some d > 0, then there exists a neighborhood N of (u0 , 0) in C() ⊕ C() such that there is the unique positive solution (u, v) of (1.1) in N for some range of d. This solution is stable in the case of gu > 0. Proof. Consider (d, v) := [(d + (v))v] + vg(T v, v), F v (d, 0) := (d + where the operator T is defined in Section 2. Then it is easy to note F (0))+g(u0 , 0)I . Using the variational property of the principal eigenvalue of the linearized v (d, 0), we can find d > 0 such that 1 (F v (d, 0)) < 0 for all d > d, where d operator F v (d, 0)) = 0. Let > 0 denote the principal eigenfunction of 1 (F v (d, 0)). satisfies 1 (F Local existence and uniqueness of a bifurcating solution:One can easily see that Ker v (d, 0)) is one-dimensional which is spanned by , and also ∈ Range(F v (d, 0)) if and (F v (d, 0))) = 1. Since F vd (d, 0) = and = only if = 0. Thus Codim(Range(F vd ∈ v (d, 0)). Thus we can apply Crandall–Rabinowitz − |∇ |2 < 0, we have F / Range(F bifurcation theorem in [15, Section 13] to conclude that d is a bifurcating point, that is to say, there exist > 0 and C 1 -functions d(s), v(s) = s( + t (s)) with d(0) = d, t (0) = 0 (d(s), v(s)) = 0 for |s| < . Next, we show that (T v(s), v(s)) is a positive solution of and F (1.1). Since t (s) ∈ C 1 () and t (0) = 0, we get v(s) = s( + t (s)) > 0 for sufficiently small s > 0. Also we can see that T v(s) > 0 for small s > 0 since T 0=u0 > 0 and lims→0 v(s)=0. The local uniqueness follows from the fact that the bifurcating solution is locally unique. v (d, 0))= Local / Range(F stability of a bifurcating solution in the case of gu > 0: Since ∈ { : = 0}, using Theorem 13.8 in [15], we have sd (s)1 (d) = −1, s→0 (s) lim
(5.1)
where (s) is the principal eigenvalue corresponding to the linearization of the bifurcation solution u(s). By the variational property of the principal eigenvalue, 1 (d) < 0. To deter-
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mine the sign of d (s) near at s = 0, consider the equation −[(d(s) + (v(s)))v(s)] = v(s)g(T v(s), v(s)), i.e., −[(d(s)+(s(+t (s))))s(+t (s))] = s(+t (s))g(T s(+t (s)), s(+t (s))). Dividing s on both sides and take a derivative with respect to s at s = 0, then we have −d (0) − v (0)2 − (d + (0))t (0) = t (0)g(u0 , 0) + [gu (u0 , 0)T (0) + gv (u0 , 0)]. Multiplying on both sides and take integral over , −d (0) = v (0) 2 + [gu T (0) + gv ], (5.2)
since [(d + (0))t (0) + t (0)g(u0 , 0)] = t (0)([(d + (0))] + g(u0 , 0)) = 0. Moreover from the fact v (0) 2 = −v (0) ∇ ∇ 2 = −v (0) 2|∇ |2 0, we may conclude that the right-hand side of Eq. (5.2) is negative in the predator–prey case since gu > 0, gv < 0 from the assumptions and T (v) < 0 by Lemma 2.3(ii). Thus d (0) < 0 holds. Therefore d (s) < 0 for sufficiently small s. Consequently, the function (s) in (5.1) must be negative for small positive s and thus the bifurcation solution is locally stable.
Remark 5.2. (i) Observe that no monotonicity of the function g in u is assumed in the proof of the local existence and uniqueness of a bifurcating solution, and so our results can be applied both to the so-called predator–prey model (gu > 0) and to the competition model (gu < 0) in the biological sense. (ii) For the predator–prey and competition interactions, Theorem 3.3 gives global positive coexistence results. On the other hand, Theorem 5.1 gives local positive coexistence results which depend on the diffusion rate d.
6. Conclusions In this article, we deal with certain population models with diffusions and self-diffusions between two species interacting in three different models, namely predator–prey, competing, cooperating interactions in hostile boundary environment. This article mainly concentrated on the positive coexistence of two species in a given domain, considering two cases in which the diffusions are either nondegenerate or degenerate with respect to the zero densities. In view of our results, in case that the system has nondegenerate diffusions, the coexistence of two distinct species can be affected by the shape and the size of a domain as well as the nonlinearities introduced in the equations for the predator–prey and competing models since the coexistence conditions can be expressed by the principal eigenvalue of the linearized differential operator at the marginal densities of the species, whereas cooperating species can coexist provided that there are positive equilibria of the system, without being dominated by the character of a given region (Theorem 3.3). Therefore, the coexistence problem of the model with nondegenerate diffusions has certain common features with those systems in which the diffusions = = constant (See for example, [2,4,5,8,9].) On the other hand, the model with degenerate diffusions has many different features compared with the nondegenerate systems with regard to the coexistence of species. Namely,
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two species for each of three different interactions may coexist in a region if the growth rates at specific constant densities are positive, and are unaffected by diffusions, self-diffusions and the domain undertaken (Theorem 4.2). Acknowledgements The first author was supported by Post-Doc. Grant at Korea University in 2003. The second author was supported by Grant R05-2003-000-11622-0 from the Basic Research Program of the Korea Science & Engineering Foundation. References [1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (4) (1976) 620–709. [2] J. Blat, K.J. Brown, Bifurcation of steady-state solutions in predator–prey and competition systems, Proc. Roy. Soc. Edinburgh Sect. A 97 (1984) 21–34. [3] A. Cañada, J.L. Gámez, Elliptic systems with nonlinear diffusion in population dynamics, Differential Equations Dynamic Systems 3 (2) (1995) 189–204. [4] E.N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc. 284 (2) (1984) 729–743. [5] E.N. Dancer, On positive solutions of some pairs of differential equations. II, J. Differential Equations 60 (2) (1985) 236–258. [6] M. Delgado, A. Suárez, Existence of solutions for elliptic systems with holder continuous nonlinearities, Differential Integral Equations 13 (4–6) (2000) 453–477. [7] A. Leung, G. Fan, Existence of positive solutions for elliptic systems—degenerate and nondegenerate ecological models, J. Math. Anal. Appl. 151 (2) (1990) 512–531. [8] L. Li, Coexistence theorems of steady states for predator–prey interacting systems, Trans. Amer. Math. Soc. 305 (1) (1988) 143–166. [9] L. Li, R. Logan, Positive solutions to general elliptic competition models, Differential Integral Equations 4 (4) (1991) 817–834. [10] J.D. Murray, Mathematical biology, Biomathematics, vol. 19, Springer, Berlin, 1989. [11] P.H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971) 487–513. [12] K. Ryu, I. Ahn, Positive coexistence of steady states for competitive interacting system with self-diffusion pressures, Bull. Korean Math. Soc. 38 (4) (2001) 643–655. [13] K. Ryu, I. Ahn, Positive steady-states for two interacting species models with linear self-cross diffusions, Discrete Continous Dynamic System 9 (4) (2003) 1049–1061. [14] K. Ryu, I. Ahn, On certain elliptic systems with nonlinear self-cross diffusions, Discrete Continous Syn. System An expanded vol. (2003) 752–759. [15] J. Smoller, Shock waves and reaction–diffusion equations, Grundlehren der Mathematischen Wissenschaften, second ed., vol. 258, Springer, New York, 1994.