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Coexistence states of a predator–prey system with non-monotonic functional response
Nonlinear Analysis: Real World Applications 8 (2007) 769 – 786 www.elsevier.com/locate/na
Coexistence states of a predator–prey system with non-monotonic functional response Wonlyul Ko, Kimun Ryu∗ Department of Information and Mathematics, Korea University, Jochiwon, Chung-Nam 339-700, South Korea Received 9 November 2005; accepted 1 March 2006
Abstract In this paper, we investigate sufficient and necessary conditions for coexistence states of a predator–prey interaction system between two species with non-monotonic functional response under Robin boundary conditions. In view of the results, there is a gap between these two conditions. In this case, we study the multiplicity, stability and some uniqueness of coexistence states depending on some parameters. 䉷 2006 Elsevier Ltd. All rights reserved. MSC: 35J60; 92D25 Keywords: Coexistence state; Monod-Haldane function; Fixed point index; Upper–lower solution; Bifurcation theory; Group defense
1. Introduction In this paper, we are interested in the following predator–prey model with non-monotonic functional response: ⎧ bv ⎪ −u = u a − u − 2 , ⎪ ⎪ u + mu + 1 ⎪ ⎪ ⎪ ⎪ ⎪ du ⎪ ⎪ ⎪ in , ⎨ −v = v c − v + u2 + mu + 1 (1.1) ⎪ ju ⎪ ⎪ 1 + u = 0, ⎪ ⎪ ⎪ j ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2 jv + v = 0 on j, j where ⊆ Rn is a bounded domain with smooth boundary j; a, b, c and d are positive constants which stand for prey intrinsic growth rate, capturing rate to predator, predator intrinsic growth rate and conversion rate of prey captured by predator, respectively. The constants > 0 and m are assumed satisfy m > − 2 , so that u2 (x) + mu(x) + 1 > 0
∗ Corresponding author. Tel.: +82 10 7368 0308.
E-mail addresses: [email protected] (W. Ko), [email protected] (K. Ryu). 1468-1218/$ - see front matter 䉷 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2006.03.003
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for u(x)0 in . Here is the Laplacian operator; 1 , 2 are non-negative constants; u and v represent the densities of preys and predators, respectively. We say that a solution (u, v) of (1.1) is a coexistence state if u(x) > 0 and v(x) > 0 for all x ∈ . If , m = 0, then (1.1) becomes the classical Lotka–Volterra predator–prey model which has been widely studied [2,4,8,17,19]. In the case of = 0 and m > 0, the term u/(1 + mu) is known as Holling type II or Michaelis–Menten functional response. This functional response was proposed by Michaelis–Menten and Holling in studying enzymatic reactions and predator–prey models. For more background, refer to [3,20]. In [3], Blat and Brown studied the existence of coexistence states by using bifurcation theory. In [7], Casal et al. also studied the same system. They found a improved necessary condition for the existence of coexistence states and showed the multiplicity and uniqueness of coexistence states when m > 0 is relatively small in the gap between the necessary and sufficient conditions. Furthermore, in this gap, Du and Lou investigated the existence, stability and number of coexistence states when the parameter m is large [11,12]. They also studied spacially inhomogeneous case when the constant m is large in [13]. In [1], Andrews proposed the Monod–Haldane functional response which is called a Holling type IV functional response. This can be explained by an inhibitory effect on the growth rate. For > 0, the Monod–Haldane function u/(u2 + mu + 1) increases to the maximum, and then decreases to zero as u → ∞. Note that ⎧ 1 ⎪ positive for 0 u < , ⎪ ⎨ d u is 1 ⎪ du u2 + mu + 1 ⎪ ⎩ negative for < u. Thus using this functional response, we can explain the situation that preys can better defend or disguise themselves when their population becomes large enough. This situation is called group defense. In population dynamics, group defense is a term used to describe the phenomenon whereby predators are decreased, or even prevented altogether, due to the increased ability of preys to better defend when their numbers are large enough. (For more biological information, refer to [14,22,27].) In this article, we investigate sufficient and necessary conditions for the existence of coexistence states to system (1.1). In view of our result (Theorem 2.7), as in the Michaelis–Menten predator–prey system, there is a gap between these two conditions. Motivated by the previous works, the multiplicity of coexistence states for (1.1) in this gap is gained; and the stability, uniqueness of coexistence states are also studied depending on some parameters. Recently, Pang and Wang showed the existence and non-existence of non-constant positive steady-state solutions for the following predator–prey model with Monod–Haldane functional response under homogeneous Neumann boundary conditions [18]: ⎧ u
uv −d u = ru 1 − − 2 , ⎪ 1 ⎨ K u +a cu ⎪ ⎩ −d2 v = v −b in , u2 + a where r, K, a, b and c are positive constants. This paper is organized as follows. In Section 2, we give sufficient and necessary conditions for the existence of coexistence states of (1.1) by using the index theory. In Section 3, by using a as a main bifurcation parameter, the multiplicity of coexistence states to (1.1) is investigated in the gap between the sufficient and necessary conditions for the existence of coexistence states which are found in Section 2. Furthermore, the local uniqueness result is studied when b and d are small. In Section 4, the multiplicity, uniqueness and stability of coexistence states of (1.1) are investigated when > 0 is large. Finally in the Appendix, we show the local existence of coexistence states for (1.1) by using the local bifurcation theory which are introduced in Section 3.
2. Existence and non-existence of coexistence states In this section, we give conditions for the existence and non-existence of coexistence states of (1.1) by using the fixed point index theory.
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For q(x) ∈ L∞ () and 0, denote the principal eigenvalue of the following problem: ⎧ ⎨ − + q(x) = in , j ⎩ +=0 on j j by 1, (q(x)) and simply denote 1, (0) by 1, . It is known that 1, (q(x)) is strictly increasing in the sense that q1 (x) q2 (x) and q1 (x) ≡ / q2 (x) implies 1, (q1 (x)) < 1, (q2 (x)). The following lemma and theorem can be found in [15]. One can also refer to [5,6]. Lemma 2.1. Let q(x) ∈ L∞ () and u 0, u ≡ / 0 in . (i) If 0 ≡ / −u + q(x)u0, then 1, (q(x)) < 0. (ii) If 0 ≡ / −u + q(x)u0, then 1, (q(x)) > 0. (iii) If −u + q(x)u ≡ 0, then 1, (q(x)) = 0. Consider the scalar equation: ⎧ ⎨ −u = uf (x, u) in , ju ⎩ +u=0 on j, j
(2.1)
where f ∈ C 1 -function in u and C -function in x for 0 < < 1. Theorem 2.2. Assume that fu < 0 for u 0 and f (x, C) < 0 for some positive constant C. (i) If 1, (−f (x, 0))0, then (2.1) has no positive solution. (ii) If 1, (−f (x, 0)) < 0, then (2.1) has a unique positive solution which satisfies u(x)C for all x ∈ . Let E be a Banach space and W a total wedge in E, i.e. W is a closed convex subset of E such that W ⊂ E for 0 and W − W = E. A wedge W is said to be a cone if W (−W ) = {0}. Let y ∈ W and define Wy = {x ∈ E : y + x ∈ W for some > 0}. Let Sy = {x ∈ W y : −x ∈ W y }. Then W y is a wedge and Sy is a closed subspace of E. We say that L has property if there is a t ∈ (0, 1) and a w ∈ W y \Sy such that w − tLw ∈ Sy . Assume A : W → W is a compact operator with fixed point y ∈ W and A is Fréchet differentiable at y. Let L = A (y) be the Fréchet derivative of A at y. Then L maps W y to itself. For an open subset N ⊂ W , index(A, N, W ) is the Leray–Schauder degree degW (I − A, N, 0), where I is the identity map. If y is an isolated fixed point of A, then the fixed point index of A at y in W is defined by index(A, y, W )= index(A, N (y), W ), where N (y) is a small open neighborhood of y in W. We denote indexW (A, y) = index(A, y, W ) and indexW (A, N ) = index(A, N, W ). The following can be obtained from the results of [10,17,21,26]. Theorem 2.3. Assume that I − L is invertible on W y . (i) If L has property , then indexW (A, y) = 0. (ii) If L does not have property , then indexW (A, y) = (−1) where is the sum of multiplicities of all the eigenvalues of L which are greater than one. Throughout this paper, denote [ ] with [ ] by the unique positive solution of the equation ⎧ ⎨ − = ( − ) in , j ⎩ + =0 on j. j
(2.2)
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if > 1, , where is a positive constant. The existence of such a solution follows from Theorem 2.2 (ii). Therefore, (1.1) has two semi-trivial solutions 1 [a] and 2 [c] if a > 1,1 and c > 1,2 which will be simply denoted by [a] and [c], respectively. Noticing that the function du/(u2 + mu + 1) has the maximum value d/(2 + m) at u = 1/ for u0, one can easily show the next lemma which give an a priori bound for coexistence states of (1.1) by using the strong maximum principle and Hopf’s lemma; its proof will be omitted. Lemma 2.4. Any coexistence state (u, v) of (1.1) has an a priori bound, i.e., u(x)a
d and v(x)c + . 2 +m
Now we introduce the following notations: (i) (ii) (iii) (iv)
E := C1 () ⊕ C2 (), where Ci () := { ∈ C() : i (j/j) + = 0 on j}; 0 (x), x ∈ }; W := K1 ⊕ K2 , where Ki = { ∈ Ci () : D := {(u, v) ∈ E : u a + 1, v c + d/(2 + m) + 1}; D := (int D) ∩ W .
For ∈ [0, 1], define a positive compact operator A : D → W via ⎞ ⎛ bv + Pu u a − u − 2 ⎟ ⎜ u + mu + 1 ⎟, A (u, v) = (− + P )−1 ⎜ ⎠ ⎝ du v c − v + 2 + Pv u + mu + 1
where P is a sufficiently large positive constant with P > max{a + b(c + d/(2 + m)), c + 2d/(2 + m)}, so that the functions u(a − u − bv/(u2 + mu + 1)) + P u and v(c − v + du/(u2 + mu + 1)) + P v are monotone increasing in u and v, respectively. Observe that (1.1) has a coexistence state in W if and only if A := A1 has a positive fixed point in D . We may assume that (0, 0), ( [a], 0) and (0, [c]) are isolated fixed points of A if exist (if not, then there must be a non-trivial fixed point in the interior of D ), and so the corresponding indexes in W are well-defined. As in [23] or [24], their fixed point indexes can be similarly calculated in the following two lemmas. Note that in [23] and [24], a certain generalized competition model and ratio-dependent predator–prey model are considered, but their calculations of the local indexes work for (1.1), and so we omit the proofs. Lemma 2.5. Assume that a > 1,1 . (i) (ii) (iii) (iv)
indexW (A, D ) = 1. indexW (A, (0, 0)) = 0. If c > 1,2 (−d [a]/( 2 [a] + m [a] + 1)), then indexW (A, ( [a], 0)) = 0. If c < 1,2 (−d [a]/( 2 [a] + m [a] + 1)), then indexW (A, ( [a], 0)) = 1.
Lemma 2.6. Assume that 1,2 < c. (i) If a > 1,1 (b [c]), then indexW (A, (0, [c])) = 0. (ii) If a < 1,1 (b [c]), then indexW (A, (0, [c])) = 1. Using Lemmas 2.5 and 2.6, we have the following theorem which gives the existence of coexistence states to system (1.1). Theorem 2.7. (i) If c > 1,2 and a > 1,1 (b [c]), then (1.1) has a coexistence state. (ii) If a > 1,1 and 1,2 (−d [a]/( 2 [a] + m [a] + 1)) < c 1,2 , then (1.1) has a coexistence state. Proof. By using Lemmas 2.5–2.6 and the additivity of the index, if (i) holds, then we have indexW (A, D ) − indexW (A, (0, 0)) − indexW (A, ( [a], 0)) − indexW (A, (0, [c])) = 1 = 0
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and indexW (A, D ) − indexW (A, (0, 0)) − indexW (A, ( [a], 0)) = 1 = 0 if (ii) holds. Hence there must be a coexistence state of (1.1) in D .
Now we give some necessary conditions for the existence of coexistence states of (1.1). Theorem 2.8. (i) If a 1,1 , then there is no coexistence state of (1.1). In addition, if c 1,2 , then there is no non-negative non-zero solution of (1.1). (ii) If c 1,2 , then the conditions a > 1,1 and c + d/(2 + m) > 1,2 are necessary for the existence of coexistence states of (1.1). Moreover, the necessary condition c + d/(2 + m) > 1,2 can be replaced by c > 1,2 (−d [a]/( 2 [a] + m [a] + 1)) when a < (1/ ). (iii) If c > 1,2 , then a > min{1,1 (b [c]), 1,1 (b [c]/( 2 [a] + m [a] + 1))} is a necessary condition for the existence of coexistence states of (1.1). Proof. (i) First assume that (u, v) is a coexistence state of (1.1), then (u, v) satisfies the equation, −u = u(a − u − bv/(u2 + mu + 1)) in and 1 ju/j + u = 0 on j, and so a = 1,1 (u + bv/(u2 + mu + 1)) by Lemma 2.1 (iii). By the comparison principle of an eigenvalue, we have a > 1,1 , a contradiction. Next assume that (u, v) is a non-negative non-zero solution of (1.1). If u ≡ / 0 and v ≡ 0, then a > 1,1 by the previous proof. One can also similarly derive / 0, which is a contradiction again. c > 1,2 when u ≡ 0 and v ≡ (ii) Assume that (u, v) is a coexistence state of (1.1). Then a > 1,1 by (i), and so the positive semi-trivial solution [a] exists. Since −u = u(a − u − bv/(u2 + mu + 1))u(a − u) in and 1 ju/j + u = 0 on j, u is a lower solution of (2.2) with ( , ) = (a, 1 ). By the uniqueness of [a], u [a]. Furthermore, since v satisfies the equation, 2 −v = v(c − v + du/(u du/(u2 + mu + + mu + 1)) in and 2 jv/j + v = 0 on j, one has 0 = 1,2 (−c + v − we have 1)) > 1,2 (−c − d/(2 + m)) by Lemma 2.1 (iii) which implies the result. In addition, if a < 1/ , then 2 2 c > 1,2 (−d [a]/( [a] + m [a] + 1)) sinceu [a] a < 1/ .(Note that the function du/(u + mu + 1) has the maximum value d/(2 + m) at u = (1/ ) for u0 and is monotone increasing in u for 0 u1/ .) (iii) Let (u, v) be a coexistence state of (1.1), then [a] exists with u [a] as in (ii). The given assumption c > 1,2 guarantees the existence of positive solution [c] of (2.2) with ( , )=(c, 2 ). Since −v =v(c −v +du/(u2 +mu+ 1)) v(c−v) in and 2 jv/j+v =0 on j, [c]v. As in the proof of (i), we have a =1,1 (u+bv/(u2 +mu+1)) which derives the desired result since the function 1/(u2 + mu + 1) has the minimum at u = 0 or u = [a] for 0 u [a]. Remark 2.9. (i) In the classical Lotka–Volterra predator–prey models, the sufficient conditions for the existence of coexistence states are also necessary [15,17]. In view of Theorems 2.7 and 2.8, the same is true if c 1,2 and a < (1/ ), but there is a gap between these two conditions in other cases. (ii) In Theorem 2.8 (iii), we can see that the condition a > 1,1 (b [c]/( 2 [a] + m [a] + 1)) is necessary for the existence of coexistence states of (1.1) when c > 1,2 and m 0 since the function bv/(u2 + mu + 1) is monotone decreasing in u when m 0. We give other necessary conditions for the existence of coexistence states to the system (1.1) under Dirichlet boundary conditions. To do this, assume that 1 , 2 = 0 in the following theorem. Theorem 2.10. If one of the following conditions holds, then (1.1) has no coexistence state: (i) a 2 + R + 1 b and a c, (ii) a 2 + R + 1 > b and a + Qb/(a 2 + R + 1)(c + Q), where
R :=
0 ma
if − 2 < m < 0 if m 0
and
d Q := . 2 +m
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Proof. (i) Suppose that there is a coexistence state (u, v) of (1.1) when a 2 + R + 1 b and a c. Recall that u(x) a by Lemma 2.4 and u2 (x) + mu(x) + 1 > 0 in . Then, we have bv 0 = 1,0 −a + u + 2 u + mu + 1 du bv du 1,0 −c + v − 2 −v+ 2 +u+ 2 u + mu + 1 u + mu + 1 u + mu + 1 du b > 1,0 −c + v − 2 − 1− 2 v u + mu + 1 u + mu + 1 du b > 1,0 −c + v − 2 − 1− 2 v u + mu + 1 a + R + 1 du 1,0 −c + v − 2 , u + mu + 1 which derives a contradiction to Lemma 2.1 (iii). (ii) In the above proof, by using the fact that v c + Q in Lemma 2.4, one can get the following inequality: bv 0 = 1,0 −a + u + 2 u + mu + 1 b du > 1,0 −c + v − 2 −a+c− 1− 2 (c + Q) u + mu + 1 a + R + 1 du 1,0 −c + v − 2 , u + mu + 1 which also derives a contradiction. The last inequality follows from the assumption a+Q b/(a 2 +R+1)(c+Q).
3. Multiplicity and some uniqueness: Part I In this section, we show that there are some ranges of the parameters involved in (1.1) for which (1.1) has at least two coexistence states by using a as a main bifurcation parameter. Furthermore, we prove the uniqueness of coexistence states when b and d are small enough. From the local bifurcation theory [9] or [25, Section 13], a branch of coexistence states of (1.1) bifurcates from (0, [c]) when a > 1,1 and c > 1,2 . (For more details, see the Appendix of this article.) More precisely, all coexistence states near a := 1,1 (b [c]) are given by {(U (s), V (s), a(s)) = (s + O(s 2 ), [c] + sd + O(s 2 ), a + a1 s + O(s 2 )) : 0 < s }, where d := d(− + 2 [c] − c)−1 ( [c]), is the principal eigenfunction of a with 2 = 1 and a1 is given by 3 a1 := + 2 (bd − bm [c]). (3.1)
By substituting (U (s), V (s), a(s)) into the first equation of (1.1), a1 can be obtained after some calculations. Using the constant a as a main bifurcation parameter, we have the following theorem that gives the multiplicity and some stability results of coexistence states for (1.1). Theorem 3.1. Assume that 3 (1 − bm [c]) < 0, a > 1,1 and c > 1,2 . Then (1.1) has at least two coexistence states for a sufficiently small d > 0. Moreover, the coexistence state (U (s), V (s)) of (1.1) with a := a(s) is nondegenerate and unstable for a ∈ ( a − , a ), where > 0 is small enough. (In other words, the coexistence state which is bifurcated from (0, [c]) is unstable.)
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Proof. First we show that any coexistence states bifurcated from (0, [c]) are non-degenerate and unstable. To this end, it suffices to show that there exists a sufficiently small > 0 such that for a ∈ ( a − , a ), any coexistence state (U (s), V (s)) of (1.1) is non-degenerate and the linearized eigenvalue problem ⎧ bU (s) bV (s)(U 2 (s) − 1) ⎪ ⎪ = , − − − a(s) − 2U (s) + ⎪ ⎪ U 2 (s) + mU (s) + 1 (U 2 (s) + mU (s) + 1)2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ dU (s) dV (s)(1 − U 2 (s)) ⎪ ⎪ − − − c − 2V (s) + = in , ⎨ U 2 (s) + mU (s) + 1 (U 2 (s) + mU (s) + 1)2 (3.2) ⎪ ⎪ j ⎪ ⎪ + = 0, 1 ⎪ ⎪ j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2 j + = 0 on j j has a unique eigenvalue ∗ such that Re(∗ ) < 0 with multiplicity one. Let {n > 0} and {dn > 0} be sequences which converge to 0 as n → ∞. Since a = a + a1 s + O(s 2 ), we can construct a ) and sn → 0 as n → ∞. For these sequences, let (Un , Vn ) be a a − n , sequences {sn > 0} and {an } such that an ∈ ( solution of (1.1). Then the corresponding linearized problem (3.2) can be written by 11 Mn Mn12 Mn = and Mn = , Mn21 Mn22 where
Mn11
: = − − an − 2Un +
Mn21 : = −
dn Vn (1 − Un2 )
bV n (Un2 − 1) (Un2 , 2
+ mU n + 1)
2
,
Mn12 :=
Mn22 := − − c − 2Vn +
(Un2 + mU n + 1)
Observe that, as n → ∞, Mn converges to M0 =
Un2
bU n , + mU n + 1
dn Un . Un2 + mU n + 1
− + b [c] − a 0 . 0 − − (c − 2 [c])
(3.3)
It is easy to see that the operator M0 has 0 as a simple eigenvalue with corresponding eigenfunction = 0 since c = 1,2 ( [c]) < 1,2 (2 [c]) and a = 1,1 (b [c]). Moreover, all the other eigenvalues are positive and stand apart from 0. Therefore, by perturbation theory [16], we know that for large n, Mn has a unique eigenvalue n which is close to zero. In addition, all the other eigenvalues of Mn have positive real parts and stand apart from 0. Note that n is simple real eigenvalue which converges to zero and we can take the corresponding eigenfunction n such that n
n n → 0 .
Once we show that n < 0 for large n, then the result follows. By multiplying to the first equation of Mn n = n
n n and integrating on , we obtain n
−
n −
n an − 2Un +
bV n (Un2 − 1) (Un2 + mU n + 1)2
+
n
bU n = n n . Un2 + mU n + 1
Moreover, multiplying the first equation of (1.1) with (u, v) = (Un , Vn ) by n and integrating, we have bV n n Un an − Un − . − n Un = Un2 + mU n + 1
(3.4)
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Since Un = sn + O(sn2 ), the above equation becomes bV n n an − Un − + O(sn2 ). − n = Un2 + mU n + 1
(3.5)
Using (3.4) and (3.5), bV n (2Un + m) bU n 2 + O(s + n Un 1 − ) = n n . n n Un2 + mU n + 1 (Un2 + mU n + 1)2 Recall that (Un , Vn ) = (sn + O(sn2 ), [c] + sn dn + O(sn2 )), and so dividing the above equation by sn and taking the limit, we have 3 (1 − bm [c]) n lim = < 0, 2 n→∞ sn which implies that n < 0 for large n. This establishes our claim. Next, to show the existence of at least two coexistence states, a contradiction argument will be used assuming that (1.1) has a unique coexistence state (u∗ , v∗ ), then this solution must be bifurcated from (0, [c]) since there exists a coexistence state near a by a local bifurcation theory, and so (u∗ , v∗ ) is non-degenerate and the corresponding linearized eigenvalue problem has a unique eigenvalue ∗ such that Re(∗ ) < 0 with multiplicity one. Using these facts, one can easily show that I − A (u∗ , v∗ ) is invertible and does not have property on W (u∗ ,v∗ ) , and therefore indexW (A, (u∗ , v∗ )) = (−1)1 = −1 by Theorem 2.3 (ii). Finally, using Lemmas 2.5–2.6 and the additivity property of the index, we have 1 = indexW (A, D ) = indexW (A, (0, 0)) + indexW (A, ( [a], 0)) + indexW (A, (0, [c])) + indexW (A, (u∗ , v∗ )) = 0 + 0 + 1 − 1 = 0, which is a contradiction. This completes the proof.
Remark 3.2. In Theorem 3.1, the multiplicity can be shown easily when m 0. Note that a1 < 0 for a sufficiently a ). Since there is no coexistence state of (1.1) if small d since 3 (1 − bm [c]) < 0, and so a = a(s) ∈ (1,1 , a 1,1 (b [c]/( 2 [a] + m [a] + 1)) by Remark 2.9 (ii) when m0 and c > 1,2 , we obtain a bifurcation diagram as in Fig. 1. Therefore we can easily conclude that there must be at least two coexistence states for a ∈ (a ∗ , a ) and some a ∗ ∈ (1,1 (b [c]/( 2 [a] + m [a] + 1)), 1,1 (b [c])). Under Dirichlet boundary conditions and a certain restricted assumption, we can find an improved condition which guarantees 3 (1 − bm [c]) < 0, and so a1 < 0 for sufficiently small d. We point out that the following corollary is also valid when 1 = 2 0.
Fig. 1. Bifurcation diagram.
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Proposition 3.3. Assume that a, c > 1,0 and b = 1. If 1/m < ((1/||) 4 [c])1/4 , then the same results in Theorem 3.1 remain true. Proof. In view of Theorem 3.1, it suffices to show that 3 (1 − bm [c]) < 0. Since [c] satisfies the equation, − [c] = [c](c − [c]) in and [c] = 0 on j, we see that c = 1,0 ( [c]) = a and = [c]/ [c] L2 . Using Hölder inequality and the given assumption, we obtain
[c]
3/4
3
[c] 4
which implies the desired one.
||
1/4
<
m 4 [c],
Now we give some conditions for the uniqueness of coexistence states to the system (1.1). Theorem 3.4. Assume that c > 1,2 and b a > 1,1 + (c + Q)
and 1 >
b (2a + |m|)(c + Q),
2
(3.6)
where := 1 − m2 /4 and Q := d/(2 + m). Then (1.1) has a unique coexistence state provided that b2 + where
d2 S a2 b 2 4 (1 + a ) + 2bd 3 + 4 2 (2a + |m|)(c + Q) 4, 2 2
(3.7)
[a] 2 [c + Q] S = max sup . , sup [c] 1 [a − b (c + Q)]
Proof. Let (u1 , v1 ) and (u2 , v2 ) be coexistence states of (1.1). Using the comparison argument for elliptic problem, it is easy to check that for i = 1, 2, b 1 a − (c + Q) ui [a] and [c]vi 2 [c + Q],
since c > 1,2 and the first condition in (3.6). Let A := u1 − u2 and B := v1 − v2 , then A and B satisfy ⎧ bv 1 ⎪ ⎪ ⎪ −A − A a − u1 − 2 ⎪ ⎪ u1 + mu1 + 1 ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ +u A + bu2 [−((u1 + u2 ) + m)v1 A + (u1 + mu1 + 1)B] = 0, ⎪ 2 ⎪ ⎪ (u21 + mu1 + 1)(u22 + mu2 + 1) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ du1 ⎪ ⎨ −B − B c − v1 + u21 + mu1 + 1 ⎪ ⎪ ⎪ dv 2 [(1 − u1 u2 )A] ⎪ ⎪ ⎪ +v2 B − =0 in , ⎪ 2 ⎪ (u1 + mu1 + 1)(u22 + mu2 + 1) ⎪ ⎪ ⎪ ⎪ ⎪ jA ⎪ ⎪ 1 + A = 0, ⎪ ⎪ ⎪ j ⎪ ⎪ ⎪ ⎪ ⎪ jB ⎪ ⎩ 2 +B =0 on j. j
(3.8)
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Since (u1 , v1 ) is a solution of (1.1), 0 is the principal eigenvalue for the following two eigenvalue problems: ⎧ bv 1 ⎪ ⎪ ⎪ ⎨ − − a − u1 − u2 + mu + 1 = in , 1 1 ⎪ ⎪ j ⎪ ⎩ 1 +=0 on j j and
⎧ ⎪ ⎪ ⎪ ⎨ − − c − v1 + ⎪ ⎪ j ⎪ ⎩ 2 +=0 j
du1 2 u1 + mu1 + 1
=
in , on j.
Therefore, using Rayleigh’s formula for the principal eigenvalue, we can see that [|∇A|2 − A2 (a − u1 − bv 1 /(u21 + mu1 + 1))]0 and [|∇B|2 − B 2 (c − v1 − du1 /(u21 + mu1 + 1))] 0. Multiplying A and B to the first and second equation of (3.8), respectively, and integrating on , we have bu2 (u21 + mu1 + 1) b((u1 + u2 ) + m)v1 2 1− A + u 2 (u21 + mu1 + 1)(u22 + mu2 + 1) (u21 + mu1 + 1)(u22 + mu2 + 1) dv 2 (1 − u1 u2 ) 2 (3.9) − AB + v2 B 0 (u21 + mu1 + 1)(u22 + mu2 + 1) by subtracting the second equation from the first one. On the other hand, the second assumption in (3.6) implies 1 − b((u1 + u2 ) + m)v1 /(u21 + mu1 + 1)(u22 + mu2 + 1) > 1 − (b/ 2 )(2a + |m|)(c + Q) > 0. Note that ui a and vi c + Q by Lemma 2.4 for i = 1, 2. Denote the discriminant for the primitive function of left hand side in (3.9) by 2 bu2 (u21 + mu1 + 1) dv 2 (1 − u1 u2 ) − I := (u21 + mu1 + 1)(u22 + mu2 + 1) (u21 + mu1 + 1)(u22 + mu2 + 1) b((u1 + u2 ) + m)v1 −4 1− u 2 v2 . (u21 + mu1 + 1)(u22 + mu2 + 1) Then
b 2 u2 d 2 v2 2 2 (u + mu + 1) + (1 − u1 u2 )2 1 1 M 2 v2 M 2 u2 bd b −2 2 (u21 + mu1 + 1)(1 − u1 u2 ) − 4 1 − ((u1 + u2 ) + m)v1 M M 2 2 2 u2 b2 2 v2 1 + u 1 u 2 u2 v2 + d v2 (u22 + mu2 + 1)2 u2 M2 4b bd +2 2 (u21 + mu1 + 1)u1 u2 + ((u1 + u2 ) + m)v1 − 4 M M 2 b2 u2 v d2 a b 2 u2 v2 2 sup + 4 (1 + 2 a 4 ) sup + 2bd 3 + 4 2 (2a + |m|)(c + Q) − 4
v2
u2
d2 S a2 b u2 v2 b2 + 2 (1 + 2 a 4 ) 2 + 2bd 3 + 4 2 (2a + |m|)(c + Q) − 4 0,
I = u2 v2
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779
where M = (u21 + mu1 + 1)(u22 + mu2 + 1). In the above derivation, note that u2i + mui + 1 > 0 for u 0 since m > − 2 . Consequently, the integral in (3.9) has a non-negative value, and thus A, B ≡ 0 which are the desired results. Remark 3.5. (i) In Theorem 3.4, the assumptions, which give the uniqueness of coexistence states of (1.1), can be satisfied when a > 1,1 and c > 1,2 for sufficiently small b and d. Therefore we may conclude that (1.1) has exactly one coexistence state when a > 1,1 and c > 1,2 for sufficiently small b and d. (ii) In the case of m 0, in (3.6) and (3.7) can be replaced by 1 since 1/M, (1 + 2 u21 u22 )/M 2 , (u21 + mu1 + 1)/M 2 1.
4. Multiplicity and some uniqueness: Part II In this section, using as a parameter, we investigate the multiplicity, stability and some uniqueness of coexistence states of (1.1). To begin with, we give some lemmas to obtain the main results of this section. These lemmas are devoted to find upper and lower solutions which do not depend on , and show the non-degeneracy at any solution of (1.1) under certain assumptions. Hereafter, we fix the constants except for , so that the upper solution [a] for u and the lower solution [c] for v do not depend on when a > 1,1 and c > 1,2 . Lemma 4.1. Assume that a > 1,1 and c > 1,2 . Then for a small > 0, there exists () such that for (), (1.1) has at least one coexistence state (u, v) which satisfies 1 [a − /2] u [a]
and [c]v 2 [c + /2].
(4.1)
Proof. It is easy to check that u(a − u − bv/(u2 + mu + 1)) and v(c − v + du/(u2 + mu + 1)) satisfy the Lipschitz condition in u, u, where u = (u, v) = ( 1 [a − /2], [c]) and u = (u, v) = ( [a], 2 [c + /2]). In view of Chapter 8 in [19] for non-quasimonotone functions, it suffices to check the following inequalities for and (u, v) ∈ u, u: ⎧ bv ⎪ ⎪ ⎪ u + u a − u − u2 + mu + 1 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ du ⎪ ⎪ 0, + v c − v + v ⎪ ⎪ ⎪ u2 + mu + 1 ⎪ ⎪ ⎪ ⎪ bv ⎪ ⎪ ⎪ a − u u 0, + u − ⎨ u2 + mu + 1 (4.2) ⎪ du ⎪ ⎪ v + v c − v + 2 0 in , ⎪ ⎪ ⎪ u + mu + 1 ⎪ ⎪ ⎪ ⎪ ⎪ ju ju ⎪ ⎪ 1 + u0 1 + u, ⎪ ⎪ j j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2 jv + v 0 2 jv + v on j. j j The first and fourth inequalities in (4.2) are trivial and one can also easily see that the equalities are satisfied on j. Recall that [ ] is the unique positive solution of the equation, − = ( − ) in and /j + = 0 on j, if > 1, , where is a positive constant. For the second and third inequalities, we have v + v c − v +
du 2 u + mu + 1
du = 2 [c + /2] − + 2 0 2 u + mu + 1
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and bv u + u a − u − 2 u + mu + 1 bv 0 = 1 [a − /2] /2 − 2 1 [a − /2] + m 1 [a − /2] + 1
for sufficiently large since du/(u2 + mu + 1), bv/( 2 1 [a − /2] + m 1 [a − /2] + 1) → 0 as → 0 which implies the result. Lemma 4.2. Assume that a > 1,1 and c > 1,2 . (i) The coexistence state of (1.1) which satisfies (4.1) converges to ( [a], [c]) as → ∞. (ii) There exists a large () such that for (), the coexistence state of (1.1) which satisfies (4.1) is non-degenerate and linearly stable. Proof. (i) The result follows clearly since the operator A(u, v) defined in Section 3 converges to the operator A(u, v) := (− + P )−1
u[a − u] + P u
v[c − v] + P v
as → ∞. (ii) Suppose the result is false. Then we can find n → ∞, n with Re(n ) 0 and (n , n ) ≡ / (0, 0) with n 2L2 + 2
n L2 = 1 such that ⎧ bun bv n (n u2n − 1) ⎪ ⎪ ⎪ −n − n a − 2un + + = n n , ⎪ 2 2 2 ⎪ n un + mun + 1 n (n un + mun + 1) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ dv n (1 − n u2n ) dun ⎪ ⎪ − c − 2vn + = n n ⎨ −n − n n u2n + mun + 1 n (n u2n + mun + 1)2 ⎪ ⎪ ⎪ ⎪ j ⎪ ⎪ 1 n + n = 0, ⎪ ⎪ j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ j ⎪ ⎩ 2 n + n = 0 j
in , (4.3)
on j,
where (un , vn ) is a coexistence state of (1.1) with = n which satisfies (4.1). From Eq. (4.3), we have n =
|∇n | −
+
|n |
2
a − 2un +
|∇n | − 2
n n
bv n (n u2n − 1) (n u2n
+ mun + 1)
dv n (1 − n u2n ) (n u2n + mun + 1)2
2
−
+
2 n u n
bun n + mun + 1 n
|n |2 c − 2vn +
dun , n u2n + mun + 1
where n and n are the complex conjugates of n and n . From the above equation, one can see that Im(n ) and Re(n ) are bounded, and so we may assume that n → with Re() 0, n → and n → . Note that n and n are also
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bounded. Taking the limit in (4.3), we obtain ⎧ − − (a − 2 [a]) = , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − − (c − 2 [c]) = in , ⎪ ⎪ ⎪ ⎨ j 1 + = 0, ⎪ ⎪ j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ j + = 0 on j 2 j
781
(4.4)
by (i), and so must be real number with 0. If ≡ / 0, then is an eigenvalue of the problem, − − (a − 2 [a]) = in and 1 j/j + = 0 on j, and thus 0 1,1 (−a + 2 [a]) which is impossible since 1,1 (−a + 2 [a]) > 1,1 (−a + [a]) = 0. Therefore ≡ 0. One can also similarly show that ≡ 0 which completes the proof. If a > 1,1 (b [c]), equivalently (a + 1,1 (b [c]))/2 > 1,1 (b [c]), then we can take a > 0 small enough such that (a + 1,1 (b [c]))/2 > 1,1 (b 2 [c + ]/(1 − )), and thus one can see that the following problem has a unique positive solution w∗ by Theorem 2.2 (ii): ⎧ a + 1 (− + b [c]) b 2 [c + ] ⎪ in , − − w∗ ⎪ ⎨ −w∗ = w∗ 2 1− (4.5) ⎪ ⎪ ⎩ 2 jw∗ + w∗ = 0 on j. j Lemma 4.3. Assume that c > 1,2 and a > 1,1 (b [c]). Then there exists a large () such that the following statements are satisfied for () and some > 0. (i) uw∗ , i.e., w∗ is a lower solution for u, where w∗ is the unique positive solution of (4.5) for a coexistence state (u, v) of (1.1) and sufficiently small positive constant with (a + 1,1 (b [c]))/2 > 1,1 (b 2 [c + ]/(1 − )). (ii) A coexistence state of (1.1) converges to ( [a], [c]) as → ∞. Moreover, it is non-degenerate and linearly stable. Proof. (i) Using Lemma 2.4 and the comparison argument, one can easily check 2 [c + d/(2 + m)]. that v Then there exists a () such that for sufficiently large () with max{d/(2 + m), m2 /4} < , bv −u = u a − u − 2 u + mu + 1 a + 1,1 (b [c]) b 2 [c + d/(2 + m)] u − −u 2 1 − (m2 /4) a + 1,1 (b [c]) b 2 [c + ] u − −u , 2 1− and so we get the desired result by the comparison argument. Note that i [ ] is increasing with respect to > 0. (ii) Since the lower solution w∗ for u does not depend on as → ∞, it can be shown similarly as in Lemma 4.2. Lemma 4.4. Assume that c ∈ (1,2 − d/(2 + m), 1,2 ] and a > 1,1 + , where := b(c + d/(2 + m))1/(1 − m2 /4). Then there exists a large such that for , (i) u 1 [a − ], (ii) a coexistence state of (1.1) is non-degenerate and linearly stable.
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Proof. (i) Using the comparison argument for elliptic problem, we have 0 v 2 [c+d/(2 +m)](c+d/(2 + m)), and so for a sufficiently large , b(c + d/(2 + m)) bv u a − −u , −u = u a − u − 2 u + mu + 1 1 − m2 /4 which implies the result. (ii) It can be shown similarly as in Theorem 3.1, and thus we just sketch the proof. Consider Eq. (4.3) with c = cn , where cn ∈ (1,2 − d/(2 + m), 1,2 ]. Since un 1 [a − n ] for n := b(cn + d/(2 n + m))1/(1 − m2 /4n ), n u2n ∞ → ∞, and so (un , vn ) → ( [a], 0). Moreover, cn → 1,2 since 1,2 − d/(2 n + m) → 1,2 . If we set vn∗ = vn / vn ∞ , then we have ⎧ dun ⎪ vn∗ + vn∗ cn − vn + = 0 in , ⎨ n u2n + mun + 1 ∗ ⎪ ⎩ 2 jvn + v ∗ = 0 on j. n j By standard elliptic regularity theory and Sobolev embedding theorems, we may assume that vn∗ → v ∗ > 0. Taking the limit, the above equation yields −v ∗ = 1,2 v ∗
on ,
2
jv ∗ + v∗ = 0 j
on j
and
v ∗ ∞ = 1.
Thus v ∗ must be the principal eigenfunction 1 corresponding to 1,2 with 1 ∞ = 1. Now we show that the linearized eigenvalue problem (4.3) has no eigenvalue n with Re(n ) 0 for large n. As n → ∞, note that (4.3) converges to ⎧ − − (a − 2 [a]) = , ⎪ ⎪ ⎪ ⎪ ⎪ − − 1,2 = in , ⎪ ⎪ ⎪ ⎨ j (4.6) 1 + = 0, ⎪ ⎪ j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2 j + = 0 on j. j It is easy to see that (4.6) has 0 as a simple eigenvalue with its corresponding eigenfunction (, ) = (0, 1 ). By the same reason as in Theorem 3.1, it suffices to show n > 0 for large n. Choose an eigenfunction (n , n ) such that (n , n ) → (0, 1 ). Using the integral equation on which can be obtained by multiplying vn to the second equation of (4.3) and the second integral equation on of (1.1) which are multiplied by n , we can derive 3 n lim = 21 > 0 n→∞ vn ∞ 1 which establishes our claim.
Now we give the multiplicity and uniqueness results of coexistence states for (1.1) by using the sequence of above lemmas. Theorem 4.5. (i) If a ∈ (1,1 , 1,1 (b [c])) and c > 1,2 , then there exists a large := () such that (1.1) has at least two coexistence states for (). (ii) If a > 1,1 (b [c]) and c > 1,2 , then there exists a large := () such that (1.1) has a unique coexistence state of (1.1) for (). (iii) If 1,2 (−d [a]/( 2 [a] + m [a] + 1)) < c 1,2 and a >1,1 + , then there exists a large such that (1.1) has a unique coexistence state for , where := b(c + d/(2 + m))1/(1 − m2 /4).
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Proof. (i) Suppose that (1.1) has a unique coexistence state (u∗ , v∗ ). Then (u∗ , v∗ ) must be contained in region (4.1) since there must be a coexistence state which satisfies (4.1). For a sufficiently large , one can easily show that I − A (u∗ , v∗ ) is invertible and has no property on W (u∗ ,v∗ ) by Lemma 4.2. In addition, A has no real eigenvalue greater than or equal to one. Thus Theorem 2.3 (ii) concludes indexW (A, (u∗ , v∗ )) = (−1)0 = 1. Using Lemma 2.5–2.6 and the additivity property of the index, we have 1 = indexW (A, D ) = indexW (A, (0, 0)) + indexW (A, ( [a], 0)) + indexW (A, (0, [c])) + indexW (A, (u∗ , v∗ )) = 0 + 0 + 1 + 1, which is a contradiction. (ii) By compactness, A has at most finitely many positive fixed points in the given region D . Let us denote them by Ui = (ui , vi ) for i = 1, . . . , k. As in the proof of (i), one can show that index(A, Ui ) = (−1)0 = 1 for i = 1, . . . , k by using Lemma 4.3 and Theorem 2.3 (ii). Finally, using the additivity property of the index, we have k=
k
indexW (A, Ui )
i=1
= indexW (A, D ) − indexW (A, (0, 0)) − indexW (A, ( [a], 0)) − indexW (A, (0, [c])) = 1 − 0 − 0 − 0 = 1. This proves the uniqueness. (iii) Note that c > 1,2 − d/(2 + m) since d [a]/( 2 [a] + m [a] + 1) d/(2 + m) by Lemma 2.4 and the semi-trivial solution (0, [c]) does not exist in this case. Then one can similarly prove as in (ii). Before closing this section, we consider the limiting behavior of coexistence states of (1.1) when → ∞. To do this, let (un , vn ) be coexistence states of (1.1) with = n , where n () and n → ∞ as n → ∞ for () defined in Theorem 4.5 (i). Theorem 4.6. Let the hypotheses in Theorem 4.5 (i) hold. Then as n → ∞, either (i) (u n , vn ) → ( [a], [c]) or (ii) ( n un , vn ) is close to a non-negative solution (U, [c]), where U is a positive solution of ⎧ b [c] ⎪ in , −U = U a − ⎪ ⎨ 1 + U2 ⎪ ⎪ ⎩ 1 jU + U = 0 on j. j
(4.7)
2 2 Proof. If n un ∞ → ∞, then the result (i) clearly follows. Therefore we only consider the case of n un ∞ < ∞. By setting n un = Un in (1.1), we have ⎧ 1 bv n ⎪ ⎪ −Un = Un a − Un − , ⎪ ⎪ ⎪ n Un2 + (m/ n )Un + 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ )v (d/ ⎪ n ⎪ in , ⎨ −vn = vn c − vn + Un2 + (m/ n )Un + 1 (4.8) ⎪ ⎪ ⎪ jUn ⎪ ⎪ ⎪ + Un = 0, 1 ⎪ ⎪ j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2 jvn + vn = 0 on j. j
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Observe that solution (Un , vn ) of (4.8) is close to a non-zero solution of the problem: ⎧ bv ⎪ ⎪ −U = U a − , ⎪ ⎪ 1 + U2 ⎪ ⎪ ⎪ ⎪ ⎪ in , ⎪ ⎨ −v = v(c − v) jU ⎪ 1 + U = 0, ⎪ ⎪ j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ jv ⎪ ⎩ 2 +v=0 on j. j as n → ∞. Since the second equation of the above problem has the unique positive solution [c], (ii) clearly holds. Remark 4.7. Using the index theory as in Theorem 4.5 (ii), one can show that the coexistence state of (1.1) which satisfies (4.1) is unique. More precisely, 1 = indexW (A, Dˆ ) = ki=1 indexW (A, (ui , vi )) = k, where (ui , vi ) are coexistence states which satisfy (4.1) and Dˆ := {(u, v) ∈ D : 21 1 [a − /2] u2 [a] and 21 [c] v 2 2 [c + /2]}. Therefore, we can see that the exact number of coexistence states is determined by the multiplicity of coexistence states of (4.8) under the same assumptions in Theorem 4.5 (i).
Appendix In this appendix, we show the existence of coexistence states of (1.1) by using the local bifurcation theory which was introduced in Section 3. Theorem A. (i) If c > 1,2 and a > 1,1 (b [c]), then the coexistence states of (1.1) bifurcate from the semi-trivial a. solution curve {(0, [c], a) : a > a := 1,1 (b [c])} if and only if a = (ii) If a > 1,1 and 1,2 (−d [a]/( 2 [a] + m [a] + 1)) < c 1,2 , then the coexistence states of (1.1) bifurcate from the semi-trivial solution curve {( [a], 0, c) : c > c := 1,2 (−d [a]/( 2 [a] + m [a] + 1))} if and only if c = c. Proof. The proofs are similar, and so we only prove (i). Define a mapping F : E × R → E by ⎞ ⎛ bv u + u a − u − 2 ⎜ u + mu + 1 ⎟ F (u, v, a) := ⎝
⎠. du v + v c − v + u2 +mu+1
By the simple calculation, for (, ) ∈ E, ⎛ ⎞ bv(u2 − 1) bu − ⎜ + a − 2u + u2 + mu + 1 ⎟ (u2 + mu + 1)2 ⎜ ⎟ F(u,v) (u, v, a) =⎜ , ⎟ ⎝ dv(1 − u2 ) ⎠ du + + c − 2v + 2 u + mu + 1 (u2 + mu + 1)2 and so F(u,v) (0, [c], a)
+ ( a − b [c]) = . + (c − 2 [c]) + d [c]
Claim 1. dim(Ker(F(u,v) (0, [c], a ))) = 1 and Ker(F(u,v) (0, [c], a )) = span{(, d )}, where is the principal eigenfunction of a = 1,1 (b [c]) and d := d(− + 2 [c] − c)−1 ( [c]).
W. Ko, K. Ryu / Nonlinear Analysis: Real World Applications 8 (2007) 769 – 786
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Proof of Claim 1. Assume that F(u,v) (0, [c], a ) = (0, 0) for (, ) ∈ E, then we have ⎧ − − ( a − b [c]) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − − (c − 2 [c]) − d [c] = 0 ⎪ ⎪ ⎨ j + = 0, 1 ⎪ ⎪ j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2 j + = 0 j
in ,
on j.
From the above equation, we see that = span{} since a = 1,1 (b [c]), and so = span{d }, where d := d(− + 2 [c] − c)−1 ( [c]). Claim 2. Codim(R(F(u,v) (0, [c], a ))) = 1. a )), then there is a (, ) ∈ E such that Proof of Claim 2. Let ( , ) ∈ R(F(u,v) (0, [c], ⎧ + ( a − b [c]) = , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + (c − 2 [c]) + d [c] = in , ⎪ ⎪ ⎨ j 1 + = 0, ⎪ ⎪ j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2 j + = 0 on j. j From the first of the equation, + ( a − b [c]) = in and 1 j/j + = 0 on j, if equation, is a solution and only if = 0. In fact, = ( + ( a − b [c])) = ( + ( a − b [c])) = 0. Therefore, we see that is uniquely determined from the second equation for such since −( + c − 2 [c]) is invertible, and thus Codim(R(F(u,v) (0, [c], a ))) = 1.
a )). / R(F(u,v) (0, [c], a ) ∈ Claim 3. F(u,v),a (0, [c], d
Proof of Claim 3. It is clear since F(u,v),a (0, [c], a ) = 0 and 2 = 0. d Finally, applying Crandall–Rabinowitz bifurcation theorem in [9] or [25, Section 13], we conclude the desired result (ii). References [1] J.F. Andrews, A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotech. Bioeng. 10 (1968) 707–723. [2] J. Blat, K.J. Brown, Bifurcation of steady-state solutions in predator-prey and competition systems, Proc. R. Soc. Edinburgh Sect. A 97 (1984) 21–34. [3] J. Blat, K.J. Brown, Global bifurcation of positive solutions in some systems of elliptic equations, SIAM J. Math. Anal. 17 (6) (1986) 1339–1353. [4] P.N. Brown, Decay to uniform states in ecological interactions, SIAM J. Appl. Math. 38 (1980) 22–37. [5] S. Cano-Casanova, Existence and structure of the set of positive solutions of a general class of sublinear elliptic non-classical mixed boundary value problems, Nonlinear Anal. Ser. A 49 (3) (2002) 361–430. [6] S. Cano-Casanova, J. López-Gómez, Properties of the principal eigenvalues of a general class of non-classical mixed boundary value problems, J. Differential Equations 178 (1) (2002) 123–211. [7] A. Casal, J.C. Eilbeck, J. López-Gómez, Existence and uniqueness of coexistence states for a predator–prey model with diffusion, Differential Integr. Equations 7 (1994) 411–439. [8] C. Cosner, A.C. Lazer, Stable coexistence states in the Volterra–Lotka competition model with diffusion, SIAM J. Appl. Math. 44 (1984) 1112–1132. [9] M.G. Crandall, P.H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971) 321–340.
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[10] E.N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl. 91 (1983) 131–151. [11] Y. Du, Y. Lou, Some uniqueness and exact multiplicity results for a predator–prey model, Trans. Am. Math. Soc. 349 (6) (1997) 2443–2475. [12] Y. Du, Y. Lou, S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator–prey model, J. Differential Equations 144 (2) (1998) 390–440. [13] Y. Du, Y. Lou, Qualitative behaviour of positive solutions of a predator–prey model: effects of saturation, Proc. R. Soc. Edinburgh Sect. A 131 (2) (2001) 321–349. [14] H.I. Freedman, G.S.K. Wolkowicz, Predator-prey systems with group defence: the paradox of enrichment revisited, Bull. Math. Biol. 48 (1986) 493–508. [15] A. Ghoreishi, R. Logan, Positive solutions of a class of biological models in a heterogeneous environment, Bull. Aust. Math. Soc. 44 (1991) 79–94. [16] T. Kato, Perturbation Theory for Linear Operators, Springer, New York, 1966. [17] L. Li, Coexistence theorems of steady states for predator–prey interacting systems, Trans. Am. Math. Soc. 305 (1998) 143–166. [18] P.Y.H. Pang, M. Wang, Non-constant positive steady states of a predator–prey system with non-monotonic functional response and diffusion, Proc. London Math. Soc. (3) 88 (1) (2004) 135–157. [19] C.V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. [20] S. Rinaldi, S. Muratori, Y.A. Kuznetsov, Multiple attractors, catastrophes and chaos in seasonally perturbed predator–prey communities, Bull. Math. Biol. 55 (1993) 15–35. [21] W. Ruan, W. Feng, On the fixed point index and multiple steady-state solutions of reaction-diffusion coefficients, Differential Integr. Equations 8 (1995) 371–391. [22] S. Ruan, D. Xiao, Global analysis in a predator–prey system with nonmonotonic functional response, SIAM J. Appl. Math. 61 (4) (2000/01) 1445–1472. [23] K. Ryu, I. Ahn, Coexistence theorem of steady states for nonlinear self-cross-diffusion systems with competitive dynamics, J. Math. Anal. Appl. 283 (2003) 46–65. [24] K. Ryu, I. Ahn, Positive solutions to ratio-dependent predator–prey interacting systems, J. Differential Equations 218 (1) (2005) 117–135. [25] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, New York, 1983. [26] M. Wang, Z.Y. Li, Q.X. Ye, Existence of positive solutions for semilinear elliptic system, in: School on Qualitative Aspects and Applications of Nonlinear Evolution Equations, Trieste, 1990, World Scientific, River Edge, NJ, 1991, pp. 256–259. [27] H. Zhu, S. Campbell, G. Wolkowicz, Bifurcation analysis of a predator–prey system with nonmonotonic functional response, SIAM J. Appl. Math. 63 (2) (2002) 636–682.
Coexistence states of a predator–prey system with non-monotonic functional response Wonlyul Ko, Kimun Ryu∗ Department of Information and Mathematics, Korea University, Jochiwon, Chung-Nam 339-700, South Korea Received 9 November 2005; accepted 1 March 2006
Abstract In this paper, we investigate sufficient and necessary conditions for coexistence states of a predator–prey interaction system between two species with non-monotonic functional response under Robin boundary conditions. In view of the results, there is a gap between these two conditions. In this case, we study the multiplicity, stability and some uniqueness of coexistence states depending on some parameters. 䉷 2006 Elsevier Ltd. All rights reserved. MSC: 35J60; 92D25 Keywords: Coexistence state; Monod-Haldane function; Fixed point index; Upper–lower solution; Bifurcation theory; Group defense
1. Introduction In this paper, we are interested in the following predator–prey model with non-monotonic functional response: ⎧ bv ⎪ −u = u a − u − 2 , ⎪ ⎪ u + mu + 1 ⎪ ⎪ ⎪ ⎪ ⎪ du ⎪ ⎪ ⎪ in , ⎨ −v = v c − v + u2 + mu + 1 (1.1) ⎪ ju ⎪ ⎪ 1 + u = 0, ⎪ ⎪ ⎪ j ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2 jv + v = 0 on j, j where ⊆ Rn is a bounded domain with smooth boundary j; a, b, c and d are positive constants which stand for prey intrinsic growth rate, capturing rate to predator, predator intrinsic growth rate and conversion rate of prey captured by predator, respectively. The constants > 0 and m are assumed satisfy m > − 2 , so that u2 (x) + mu(x) + 1 > 0
∗ Corresponding author. Tel.: +82 10 7368 0308.
E-mail addresses: [email protected] (W. Ko), [email protected] (K. Ryu). 1468-1218/$ - see front matter 䉷 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2006.03.003
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for u(x)0 in . Here is the Laplacian operator; 1 , 2 are non-negative constants; u and v represent the densities of preys and predators, respectively. We say that a solution (u, v) of (1.1) is a coexistence state if u(x) > 0 and v(x) > 0 for all x ∈ . If , m = 0, then (1.1) becomes the classical Lotka–Volterra predator–prey model which has been widely studied [2,4,8,17,19]. In the case of = 0 and m > 0, the term u/(1 + mu) is known as Holling type II or Michaelis–Menten functional response. This functional response was proposed by Michaelis–Menten and Holling in studying enzymatic reactions and predator–prey models. For more background, refer to [3,20]. In [3], Blat and Brown studied the existence of coexistence states by using bifurcation theory. In [7], Casal et al. also studied the same system. They found a improved necessary condition for the existence of coexistence states and showed the multiplicity and uniqueness of coexistence states when m > 0 is relatively small in the gap between the necessary and sufficient conditions. Furthermore, in this gap, Du and Lou investigated the existence, stability and number of coexistence states when the parameter m is large [11,12]. They also studied spacially inhomogeneous case when the constant m is large in [13]. In [1], Andrews proposed the Monod–Haldane functional response which is called a Holling type IV functional response. This can be explained by an inhibitory effect on the growth rate. For > 0, the Monod–Haldane function u/(u2 + mu + 1) increases to the maximum, and then decreases to zero as u → ∞. Note that ⎧ 1 ⎪ positive for 0 u < , ⎪ ⎨ d u is 1 ⎪ du u2 + mu + 1 ⎪ ⎩ negative for < u. Thus using this functional response, we can explain the situation that preys can better defend or disguise themselves when their population becomes large enough. This situation is called group defense. In population dynamics, group defense is a term used to describe the phenomenon whereby predators are decreased, or even prevented altogether, due to the increased ability of preys to better defend when their numbers are large enough. (For more biological information, refer to [14,22,27].) In this article, we investigate sufficient and necessary conditions for the existence of coexistence states to system (1.1). In view of our result (Theorem 2.7), as in the Michaelis–Menten predator–prey system, there is a gap between these two conditions. Motivated by the previous works, the multiplicity of coexistence states for (1.1) in this gap is gained; and the stability, uniqueness of coexistence states are also studied depending on some parameters. Recently, Pang and Wang showed the existence and non-existence of non-constant positive steady-state solutions for the following predator–prey model with Monod–Haldane functional response under homogeneous Neumann boundary conditions [18]: ⎧ u
uv −d u = ru 1 − − 2 , ⎪ 1 ⎨ K u +a cu ⎪ ⎩ −d2 v = v −b in , u2 + a where r, K, a, b and c are positive constants. This paper is organized as follows. In Section 2, we give sufficient and necessary conditions for the existence of coexistence states of (1.1) by using the index theory. In Section 3, by using a as a main bifurcation parameter, the multiplicity of coexistence states to (1.1) is investigated in the gap between the sufficient and necessary conditions for the existence of coexistence states which are found in Section 2. Furthermore, the local uniqueness result is studied when b and d are small. In Section 4, the multiplicity, uniqueness and stability of coexistence states of (1.1) are investigated when > 0 is large. Finally in the Appendix, we show the local existence of coexistence states for (1.1) by using the local bifurcation theory which are introduced in Section 3.
2. Existence and non-existence of coexistence states In this section, we give conditions for the existence and non-existence of coexistence states of (1.1) by using the fixed point index theory.
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For q(x) ∈ L∞ () and 0, denote the principal eigenvalue of the following problem: ⎧ ⎨ − + q(x) = in , j ⎩ +=0 on j j by 1, (q(x)) and simply denote 1, (0) by 1, . It is known that 1, (q(x)) is strictly increasing in the sense that q1 (x) q2 (x) and q1 (x) ≡ / q2 (x) implies 1, (q1 (x)) < 1, (q2 (x)). The following lemma and theorem can be found in [15]. One can also refer to [5,6]. Lemma 2.1. Let q(x) ∈ L∞ () and u 0, u ≡ / 0 in . (i) If 0 ≡ / −u + q(x)u0, then 1, (q(x)) < 0. (ii) If 0 ≡ / −u + q(x)u0, then 1, (q(x)) > 0. (iii) If −u + q(x)u ≡ 0, then 1, (q(x)) = 0. Consider the scalar equation: ⎧ ⎨ −u = uf (x, u) in , ju ⎩ +u=0 on j, j
(2.1)
where f ∈ C 1 -function in u and C -function in x for 0 < < 1. Theorem 2.2. Assume that fu < 0 for u 0 and f (x, C) < 0 for some positive constant C. (i) If 1, (−f (x, 0))0, then (2.1) has no positive solution. (ii) If 1, (−f (x, 0)) < 0, then (2.1) has a unique positive solution which satisfies u(x)C for all x ∈ . Let E be a Banach space and W a total wedge in E, i.e. W is a closed convex subset of E such that W ⊂ E for 0 and W − W = E. A wedge W is said to be a cone if W (−W ) = {0}. Let y ∈ W and define Wy = {x ∈ E : y + x ∈ W for some > 0}. Let Sy = {x ∈ W y : −x ∈ W y }. Then W y is a wedge and Sy is a closed subspace of E. We say that L has property if there is a t ∈ (0, 1) and a w ∈ W y \Sy such that w − tLw ∈ Sy . Assume A : W → W is a compact operator with fixed point y ∈ W and A is Fréchet differentiable at y. Let L = A (y) be the Fréchet derivative of A at y. Then L maps W y to itself. For an open subset N ⊂ W , index(A, N, W ) is the Leray–Schauder degree degW (I − A, N, 0), where I is the identity map. If y is an isolated fixed point of A, then the fixed point index of A at y in W is defined by index(A, y, W )= index(A, N (y), W ), where N (y) is a small open neighborhood of y in W. We denote indexW (A, y) = index(A, y, W ) and indexW (A, N ) = index(A, N, W ). The following can be obtained from the results of [10,17,21,26]. Theorem 2.3. Assume that I − L is invertible on W y . (i) If L has property , then indexW (A, y) = 0. (ii) If L does not have property , then indexW (A, y) = (−1) where is the sum of multiplicities of all the eigenvalues of L which are greater than one. Throughout this paper, denote [ ] with [ ] by the unique positive solution of the equation ⎧ ⎨ − = ( − ) in , j ⎩ + =0 on j. j
(2.2)
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if > 1, , where is a positive constant. The existence of such a solution follows from Theorem 2.2 (ii). Therefore, (1.1) has two semi-trivial solutions 1 [a] and 2 [c] if a > 1,1 and c > 1,2 which will be simply denoted by [a] and [c], respectively. Noticing that the function du/(u2 + mu + 1) has the maximum value d/(2 + m) at u = 1/ for u0, one can easily show the next lemma which give an a priori bound for coexistence states of (1.1) by using the strong maximum principle and Hopf’s lemma; its proof will be omitted. Lemma 2.4. Any coexistence state (u, v) of (1.1) has an a priori bound, i.e., u(x)a
d and v(x)c + . 2 +m
Now we introduce the following notations: (i) (ii) (iii) (iv)
E := C1 () ⊕ C2 (), where Ci () := { ∈ C() : i (j/j) + = 0 on j}; 0 (x), x ∈ }; W := K1 ⊕ K2 , where Ki = { ∈ Ci () : D := {(u, v) ∈ E : u a + 1, v c + d/(2 + m) + 1}; D := (int D) ∩ W .
For ∈ [0, 1], define a positive compact operator A : D → W via ⎞ ⎛ bv + Pu u a − u − 2 ⎟ ⎜ u + mu + 1 ⎟, A (u, v) = (− + P )−1 ⎜ ⎠ ⎝ du v c − v + 2 + Pv u + mu + 1
where P is a sufficiently large positive constant with P > max{a + b(c + d/(2 + m)), c + 2d/(2 + m)}, so that the functions u(a − u − bv/(u2 + mu + 1)) + P u and v(c − v + du/(u2 + mu + 1)) + P v are monotone increasing in u and v, respectively. Observe that (1.1) has a coexistence state in W if and only if A := A1 has a positive fixed point in D . We may assume that (0, 0), ( [a], 0) and (0, [c]) are isolated fixed points of A if exist (if not, then there must be a non-trivial fixed point in the interior of D ), and so the corresponding indexes in W are well-defined. As in [23] or [24], their fixed point indexes can be similarly calculated in the following two lemmas. Note that in [23] and [24], a certain generalized competition model and ratio-dependent predator–prey model are considered, but their calculations of the local indexes work for (1.1), and so we omit the proofs. Lemma 2.5. Assume that a > 1,1 . (i) (ii) (iii) (iv)
indexW (A, D ) = 1. indexW (A, (0, 0)) = 0. If c > 1,2 (−d [a]/( 2 [a] + m [a] + 1)), then indexW (A, ( [a], 0)) = 0. If c < 1,2 (−d [a]/( 2 [a] + m [a] + 1)), then indexW (A, ( [a], 0)) = 1.
Lemma 2.6. Assume that 1,2 < c. (i) If a > 1,1 (b [c]), then indexW (A, (0, [c])) = 0. (ii) If a < 1,1 (b [c]), then indexW (A, (0, [c])) = 1. Using Lemmas 2.5 and 2.6, we have the following theorem which gives the existence of coexistence states to system (1.1). Theorem 2.7. (i) If c > 1,2 and a > 1,1 (b [c]), then (1.1) has a coexistence state. (ii) If a > 1,1 and 1,2 (−d [a]/( 2 [a] + m [a] + 1)) < c 1,2 , then (1.1) has a coexistence state. Proof. By using Lemmas 2.5–2.6 and the additivity of the index, if (i) holds, then we have indexW (A, D ) − indexW (A, (0, 0)) − indexW (A, ( [a], 0)) − indexW (A, (0, [c])) = 1 = 0
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and indexW (A, D ) − indexW (A, (0, 0)) − indexW (A, ( [a], 0)) = 1 = 0 if (ii) holds. Hence there must be a coexistence state of (1.1) in D .
Now we give some necessary conditions for the existence of coexistence states of (1.1). Theorem 2.8. (i) If a 1,1 , then there is no coexistence state of (1.1). In addition, if c 1,2 , then there is no non-negative non-zero solution of (1.1). (ii) If c 1,2 , then the conditions a > 1,1 and c + d/(2 + m) > 1,2 are necessary for the existence of coexistence states of (1.1). Moreover, the necessary condition c + d/(2 + m) > 1,2 can be replaced by c > 1,2 (−d [a]/( 2 [a] + m [a] + 1)) when a < (1/ ). (iii) If c > 1,2 , then a > min{1,1 (b [c]), 1,1 (b [c]/( 2 [a] + m [a] + 1))} is a necessary condition for the existence of coexistence states of (1.1). Proof. (i) First assume that (u, v) is a coexistence state of (1.1), then (u, v) satisfies the equation, −u = u(a − u − bv/(u2 + mu + 1)) in and 1 ju/j + u = 0 on j, and so a = 1,1 (u + bv/(u2 + mu + 1)) by Lemma 2.1 (iii). By the comparison principle of an eigenvalue, we have a > 1,1 , a contradiction. Next assume that (u, v) is a non-negative non-zero solution of (1.1). If u ≡ / 0 and v ≡ 0, then a > 1,1 by the previous proof. One can also similarly derive / 0, which is a contradiction again. c > 1,2 when u ≡ 0 and v ≡ (ii) Assume that (u, v) is a coexistence state of (1.1). Then a > 1,1 by (i), and so the positive semi-trivial solution [a] exists. Since −u = u(a − u − bv/(u2 + mu + 1))u(a − u) in and 1 ju/j + u = 0 on j, u is a lower solution of (2.2) with ( , ) = (a, 1 ). By the uniqueness of [a], u [a]. Furthermore, since v satisfies the equation, 2 −v = v(c − v + du/(u du/(u2 + mu + + mu + 1)) in and 2 jv/j + v = 0 on j, one has 0 = 1,2 (−c + v − we have 1)) > 1,2 (−c − d/(2 + m)) by Lemma 2.1 (iii) which implies the result. In addition, if a < 1/ , then 2 2 c > 1,2 (−d [a]/( [a] + m [a] + 1)) sinceu [a] a < 1/ .(Note that the function du/(u + mu + 1) has the maximum value d/(2 + m) at u = (1/ ) for u0 and is monotone increasing in u for 0 u1/ .) (iii) Let (u, v) be a coexistence state of (1.1), then [a] exists with u [a] as in (ii). The given assumption c > 1,2 guarantees the existence of positive solution [c] of (2.2) with ( , )=(c, 2 ). Since −v =v(c −v +du/(u2 +mu+ 1)) v(c−v) in and 2 jv/j+v =0 on j, [c]v. As in the proof of (i), we have a =1,1 (u+bv/(u2 +mu+1)) which derives the desired result since the function 1/(u2 + mu + 1) has the minimum at u = 0 or u = [a] for 0 u [a]. Remark 2.9. (i) In the classical Lotka–Volterra predator–prey models, the sufficient conditions for the existence of coexistence states are also necessary [15,17]. In view of Theorems 2.7 and 2.8, the same is true if c 1,2 and a < (1/ ), but there is a gap between these two conditions in other cases. (ii) In Theorem 2.8 (iii), we can see that the condition a > 1,1 (b [c]/( 2 [a] + m [a] + 1)) is necessary for the existence of coexistence states of (1.1) when c > 1,2 and m 0 since the function bv/(u2 + mu + 1) is monotone decreasing in u when m 0. We give other necessary conditions for the existence of coexistence states to the system (1.1) under Dirichlet boundary conditions. To do this, assume that 1 , 2 = 0 in the following theorem. Theorem 2.10. If one of the following conditions holds, then (1.1) has no coexistence state: (i) a 2 + R + 1 b and a c, (ii) a 2 + R + 1 > b and a + Qb/(a 2 + R + 1)(c + Q), where
R :=
0 ma
if − 2 < m < 0 if m 0
and
d Q := . 2 +m
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Proof. (i) Suppose that there is a coexistence state (u, v) of (1.1) when a 2 + R + 1 b and a c. Recall that u(x) a by Lemma 2.4 and u2 (x) + mu(x) + 1 > 0 in . Then, we have bv 0 = 1,0 −a + u + 2 u + mu + 1 du bv du 1,0 −c + v − 2 −v+ 2 +u+ 2 u + mu + 1 u + mu + 1 u + mu + 1 du b > 1,0 −c + v − 2 − 1− 2 v u + mu + 1 u + mu + 1 du b > 1,0 −c + v − 2 − 1− 2 v u + mu + 1 a + R + 1 du 1,0 −c + v − 2 , u + mu + 1 which derives a contradiction to Lemma 2.1 (iii). (ii) In the above proof, by using the fact that v c + Q in Lemma 2.4, one can get the following inequality: bv 0 = 1,0 −a + u + 2 u + mu + 1 b du > 1,0 −c + v − 2 −a+c− 1− 2 (c + Q) u + mu + 1 a + R + 1 du 1,0 −c + v − 2 , u + mu + 1 which also derives a contradiction. The last inequality follows from the assumption a+Q b/(a 2 +R+1)(c+Q).
3. Multiplicity and some uniqueness: Part I In this section, we show that there are some ranges of the parameters involved in (1.1) for which (1.1) has at least two coexistence states by using a as a main bifurcation parameter. Furthermore, we prove the uniqueness of coexistence states when b and d are small enough. From the local bifurcation theory [9] or [25, Section 13], a branch of coexistence states of (1.1) bifurcates from (0, [c]) when a > 1,1 and c > 1,2 . (For more details, see the Appendix of this article.) More precisely, all coexistence states near a := 1,1 (b [c]) are given by {(U (s), V (s), a(s)) = (s + O(s 2 ), [c] + sd + O(s 2 ), a + a1 s + O(s 2 )) : 0 < s }, where d := d(− + 2 [c] − c)−1 ( [c]), is the principal eigenfunction of a with 2 = 1 and a1 is given by 3 a1 := + 2 (bd − bm [c]). (3.1)
By substituting (U (s), V (s), a(s)) into the first equation of (1.1), a1 can be obtained after some calculations. Using the constant a as a main bifurcation parameter, we have the following theorem that gives the multiplicity and some stability results of coexistence states for (1.1). Theorem 3.1. Assume that 3 (1 − bm [c]) < 0, a > 1,1 and c > 1,2 . Then (1.1) has at least two coexistence states for a sufficiently small d > 0. Moreover, the coexistence state (U (s), V (s)) of (1.1) with a := a(s) is nondegenerate and unstable for a ∈ ( a − , a ), where > 0 is small enough. (In other words, the coexistence state which is bifurcated from (0, [c]) is unstable.)
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Proof. First we show that any coexistence states bifurcated from (0, [c]) are non-degenerate and unstable. To this end, it suffices to show that there exists a sufficiently small > 0 such that for a ∈ ( a − , a ), any coexistence state (U (s), V (s)) of (1.1) is non-degenerate and the linearized eigenvalue problem ⎧ bU (s) bV (s)(U 2 (s) − 1) ⎪ ⎪ = , − − − a(s) − 2U (s) + ⎪ ⎪ U 2 (s) + mU (s) + 1 (U 2 (s) + mU (s) + 1)2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ dU (s) dV (s)(1 − U 2 (s)) ⎪ ⎪ − − − c − 2V (s) + = in , ⎨ U 2 (s) + mU (s) + 1 (U 2 (s) + mU (s) + 1)2 (3.2) ⎪ ⎪ j ⎪ ⎪ + = 0, 1 ⎪ ⎪ j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2 j + = 0 on j j has a unique eigenvalue ∗ such that Re(∗ ) < 0 with multiplicity one. Let {n > 0} and {dn > 0} be sequences which converge to 0 as n → ∞. Since a = a + a1 s + O(s 2 ), we can construct a ) and sn → 0 as n → ∞. For these sequences, let (Un , Vn ) be a a − n , sequences {sn > 0} and {an } such that an ∈ ( solution of (1.1). Then the corresponding linearized problem (3.2) can be written by 11 Mn Mn12 Mn = and Mn = , Mn21 Mn22 where
Mn11
: = − − an − 2Un +
Mn21 : = −
dn Vn (1 − Un2 )
bV n (Un2 − 1) (Un2 , 2
+ mU n + 1)
2
,
Mn12 :=
Mn22 := − − c − 2Vn +
(Un2 + mU n + 1)
Observe that, as n → ∞, Mn converges to M0 =
Un2
bU n , + mU n + 1
dn Un . Un2 + mU n + 1
− + b [c] − a 0 . 0 − − (c − 2 [c])
(3.3)
It is easy to see that the operator M0 has 0 as a simple eigenvalue with corresponding eigenfunction = 0 since c = 1,2 ( [c]) < 1,2 (2 [c]) and a = 1,1 (b [c]). Moreover, all the other eigenvalues are positive and stand apart from 0. Therefore, by perturbation theory [16], we know that for large n, Mn has a unique eigenvalue n which is close to zero. In addition, all the other eigenvalues of Mn have positive real parts and stand apart from 0. Note that n is simple real eigenvalue which converges to zero and we can take the corresponding eigenfunction n such that n
n n → 0 .
Once we show that n < 0 for large n, then the result follows. By multiplying to the first equation of Mn n = n
n n and integrating on , we obtain n
−
n −
n an − 2Un +
bV n (Un2 − 1) (Un2 + mU n + 1)2
+
n
bU n = n n . Un2 + mU n + 1
Moreover, multiplying the first equation of (1.1) with (u, v) = (Un , Vn ) by n and integrating, we have bV n n Un an − Un − . − n Un = Un2 + mU n + 1
(3.4)
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Since Un = sn + O(sn2 ), the above equation becomes bV n n an − Un − + O(sn2 ). − n = Un2 + mU n + 1
(3.5)
Using (3.4) and (3.5), bV n (2Un + m) bU n 2 + O(s + n Un 1 − ) = n n . n n Un2 + mU n + 1 (Un2 + mU n + 1)2 Recall that (Un , Vn ) = (sn + O(sn2 ), [c] + sn dn + O(sn2 )), and so dividing the above equation by sn and taking the limit, we have 3 (1 − bm [c]) n lim = < 0, 2 n→∞ sn which implies that n < 0 for large n. This establishes our claim. Next, to show the existence of at least two coexistence states, a contradiction argument will be used assuming that (1.1) has a unique coexistence state (u∗ , v∗ ), then this solution must be bifurcated from (0, [c]) since there exists a coexistence state near a by a local bifurcation theory, and so (u∗ , v∗ ) is non-degenerate and the corresponding linearized eigenvalue problem has a unique eigenvalue ∗ such that Re(∗ ) < 0 with multiplicity one. Using these facts, one can easily show that I − A (u∗ , v∗ ) is invertible and does not have property on W (u∗ ,v∗ ) , and therefore indexW (A, (u∗ , v∗ )) = (−1)1 = −1 by Theorem 2.3 (ii). Finally, using Lemmas 2.5–2.6 and the additivity property of the index, we have 1 = indexW (A, D ) = indexW (A, (0, 0)) + indexW (A, ( [a], 0)) + indexW (A, (0, [c])) + indexW (A, (u∗ , v∗ )) = 0 + 0 + 1 − 1 = 0, which is a contradiction. This completes the proof.
Remark 3.2. In Theorem 3.1, the multiplicity can be shown easily when m 0. Note that a1 < 0 for a sufficiently a ). Since there is no coexistence state of (1.1) if small d since 3 (1 − bm [c]) < 0, and so a = a(s) ∈ (1,1 , a 1,1 (b [c]/( 2 [a] + m [a] + 1)) by Remark 2.9 (ii) when m0 and c > 1,2 , we obtain a bifurcation diagram as in Fig. 1. Therefore we can easily conclude that there must be at least two coexistence states for a ∈ (a ∗ , a ) and some a ∗ ∈ (1,1 (b [c]/( 2 [a] + m [a] + 1)), 1,1 (b [c])). Under Dirichlet boundary conditions and a certain restricted assumption, we can find an improved condition which guarantees 3 (1 − bm [c]) < 0, and so a1 < 0 for sufficiently small d. We point out that the following corollary is also valid when 1 = 2 0.
Fig. 1. Bifurcation diagram.
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Proposition 3.3. Assume that a, c > 1,0 and b = 1. If 1/m < ((1/||) 4 [c])1/4 , then the same results in Theorem 3.1 remain true. Proof. In view of Theorem 3.1, it suffices to show that 3 (1 − bm [c]) < 0. Since [c] satisfies the equation, − [c] = [c](c − [c]) in and [c] = 0 on j, we see that c = 1,0 ( [c]) = a and = [c]/ [c] L2 . Using Hölder inequality and the given assumption, we obtain
[c]
3/4
3
[c] 4
which implies the desired one.
||
1/4
<
m 4 [c],
Now we give some conditions for the uniqueness of coexistence states to the system (1.1). Theorem 3.4. Assume that c > 1,2 and b a > 1,1 + (c + Q)
and 1 >
b (2a + |m|)(c + Q),
2
(3.6)
where := 1 − m2 /4 and Q := d/(2 + m). Then (1.1) has a unique coexistence state provided that b2 + where
d2 S a2 b 2 4 (1 + a ) + 2bd 3 + 4 2 (2a + |m|)(c + Q) 4, 2 2
(3.7)
[a] 2 [c + Q] S = max sup . , sup [c] 1 [a − b (c + Q)]
Proof. Let (u1 , v1 ) and (u2 , v2 ) be coexistence states of (1.1). Using the comparison argument for elliptic problem, it is easy to check that for i = 1, 2, b 1 a − (c + Q) ui [a] and [c]vi 2 [c + Q],
since c > 1,2 and the first condition in (3.6). Let A := u1 − u2 and B := v1 − v2 , then A and B satisfy ⎧ bv 1 ⎪ ⎪ ⎪ −A − A a − u1 − 2 ⎪ ⎪ u1 + mu1 + 1 ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ +u A + bu2 [−((u1 + u2 ) + m)v1 A + (u1 + mu1 + 1)B] = 0, ⎪ 2 ⎪ ⎪ (u21 + mu1 + 1)(u22 + mu2 + 1) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ du1 ⎪ ⎨ −B − B c − v1 + u21 + mu1 + 1 ⎪ ⎪ ⎪ dv 2 [(1 − u1 u2 )A] ⎪ ⎪ ⎪ +v2 B − =0 in , ⎪ 2 ⎪ (u1 + mu1 + 1)(u22 + mu2 + 1) ⎪ ⎪ ⎪ ⎪ ⎪ jA ⎪ ⎪ 1 + A = 0, ⎪ ⎪ ⎪ j ⎪ ⎪ ⎪ ⎪ ⎪ jB ⎪ ⎩ 2 +B =0 on j. j
(3.8)
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Since (u1 , v1 ) is a solution of (1.1), 0 is the principal eigenvalue for the following two eigenvalue problems: ⎧ bv 1 ⎪ ⎪ ⎪ ⎨ − − a − u1 − u2 + mu + 1 = in , 1 1 ⎪ ⎪ j ⎪ ⎩ 1 +=0 on j j and
⎧ ⎪ ⎪ ⎪ ⎨ − − c − v1 + ⎪ ⎪ j ⎪ ⎩ 2 +=0 j
du1 2 u1 + mu1 + 1
=
in , on j.
Therefore, using Rayleigh’s formula for the principal eigenvalue, we can see that [|∇A|2 − A2 (a − u1 − bv 1 /(u21 + mu1 + 1))]0 and [|∇B|2 − B 2 (c − v1 − du1 /(u21 + mu1 + 1))] 0. Multiplying A and B to the first and second equation of (3.8), respectively, and integrating on , we have bu2 (u21 + mu1 + 1) b((u1 + u2 ) + m)v1 2 1− A + u 2 (u21 + mu1 + 1)(u22 + mu2 + 1) (u21 + mu1 + 1)(u22 + mu2 + 1) dv 2 (1 − u1 u2 ) 2 (3.9) − AB + v2 B 0 (u21 + mu1 + 1)(u22 + mu2 + 1) by subtracting the second equation from the first one. On the other hand, the second assumption in (3.6) implies 1 − b((u1 + u2 ) + m)v1 /(u21 + mu1 + 1)(u22 + mu2 + 1) > 1 − (b/ 2 )(2a + |m|)(c + Q) > 0. Note that ui a and vi c + Q by Lemma 2.4 for i = 1, 2. Denote the discriminant for the primitive function of left hand side in (3.9) by 2 bu2 (u21 + mu1 + 1) dv 2 (1 − u1 u2 ) − I := (u21 + mu1 + 1)(u22 + mu2 + 1) (u21 + mu1 + 1)(u22 + mu2 + 1) b((u1 + u2 ) + m)v1 −4 1− u 2 v2 . (u21 + mu1 + 1)(u22 + mu2 + 1) Then
b 2 u2 d 2 v2 2 2 (u + mu + 1) + (1 − u1 u2 )2 1 1 M 2 v2 M 2 u2 bd b −2 2 (u21 + mu1 + 1)(1 − u1 u2 ) − 4 1 − ((u1 + u2 ) + m)v1 M M 2 2 2 u2 b2 2 v2 1 + u 1 u 2 u2 v2 + d v2 (u22 + mu2 + 1)2 u2 M2 4b bd +2 2 (u21 + mu1 + 1)u1 u2 + ((u1 + u2 ) + m)v1 − 4 M M 2 b2 u2 v d2 a b 2 u2 v2 2 sup + 4 (1 + 2 a 4 ) sup + 2bd 3 + 4 2 (2a + |m|)(c + Q) − 4
v2
u2
d2 S a2 b u2 v2 b2 + 2 (1 + 2 a 4 ) 2 + 2bd 3 + 4 2 (2a + |m|)(c + Q) − 4 0,
I = u2 v2
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where M = (u21 + mu1 + 1)(u22 + mu2 + 1). In the above derivation, note that u2i + mui + 1 > 0 for u 0 since m > − 2 . Consequently, the integral in (3.9) has a non-negative value, and thus A, B ≡ 0 which are the desired results. Remark 3.5. (i) In Theorem 3.4, the assumptions, which give the uniqueness of coexistence states of (1.1), can be satisfied when a > 1,1 and c > 1,2 for sufficiently small b and d. Therefore we may conclude that (1.1) has exactly one coexistence state when a > 1,1 and c > 1,2 for sufficiently small b and d. (ii) In the case of m 0, in (3.6) and (3.7) can be replaced by 1 since 1/M, (1 + 2 u21 u22 )/M 2 , (u21 + mu1 + 1)/M 2 1.
4. Multiplicity and some uniqueness: Part II In this section, using as a parameter, we investigate the multiplicity, stability and some uniqueness of coexistence states of (1.1). To begin with, we give some lemmas to obtain the main results of this section. These lemmas are devoted to find upper and lower solutions which do not depend on , and show the non-degeneracy at any solution of (1.1) under certain assumptions. Hereafter, we fix the constants except for , so that the upper solution [a] for u and the lower solution [c] for v do not depend on when a > 1,1 and c > 1,2 . Lemma 4.1. Assume that a > 1,1 and c > 1,2 . Then for a small > 0, there exists () such that for (), (1.1) has at least one coexistence state (u, v) which satisfies 1 [a − /2] u [a]
and [c]v 2 [c + /2].
(4.1)
Proof. It is easy to check that u(a − u − bv/(u2 + mu + 1)) and v(c − v + du/(u2 + mu + 1)) satisfy the Lipschitz condition in u, u, where u = (u, v) = ( 1 [a − /2], [c]) and u = (u, v) = ( [a], 2 [c + /2]). In view of Chapter 8 in [19] for non-quasimonotone functions, it suffices to check the following inequalities for and (u, v) ∈ u, u: ⎧ bv ⎪ ⎪ ⎪ u + u a − u − u2 + mu + 1 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ du ⎪ ⎪ 0, + v c − v + v ⎪ ⎪ ⎪ u2 + mu + 1 ⎪ ⎪ ⎪ ⎪ bv ⎪ ⎪ ⎪ a − u u 0, + u − ⎨ u2 + mu + 1 (4.2) ⎪ du ⎪ ⎪ v + v c − v + 2 0 in , ⎪ ⎪ ⎪ u + mu + 1 ⎪ ⎪ ⎪ ⎪ ⎪ ju ju ⎪ ⎪ 1 + u0 1 + u, ⎪ ⎪ j j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2 jv + v 0 2 jv + v on j. j j The first and fourth inequalities in (4.2) are trivial and one can also easily see that the equalities are satisfied on j. Recall that [ ] is the unique positive solution of the equation, − = ( − ) in and /j + = 0 on j, if > 1, , where is a positive constant. For the second and third inequalities, we have v + v c − v +
du 2 u + mu + 1
du = 2 [c + /2] − + 2 0 2 u + mu + 1
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and bv u + u a − u − 2 u + mu + 1 bv 0 = 1 [a − /2] /2 − 2 1 [a − /2] + m 1 [a − /2] + 1
for sufficiently large since du/(u2 + mu + 1), bv/( 2 1 [a − /2] + m 1 [a − /2] + 1) → 0 as → 0 which implies the result. Lemma 4.2. Assume that a > 1,1 and c > 1,2 . (i) The coexistence state of (1.1) which satisfies (4.1) converges to ( [a], [c]) as → ∞. (ii) There exists a large () such that for (), the coexistence state of (1.1) which satisfies (4.1) is non-degenerate and linearly stable. Proof. (i) The result follows clearly since the operator A(u, v) defined in Section 3 converges to the operator A(u, v) := (− + P )−1
u[a − u] + P u
v[c − v] + P v
as → ∞. (ii) Suppose the result is false. Then we can find n → ∞, n with Re(n ) 0 and (n , n ) ≡ / (0, 0) with n 2L2 + 2
n L2 = 1 such that ⎧ bun bv n (n u2n − 1) ⎪ ⎪ ⎪ −n − n a − 2un + + = n n , ⎪ 2 2 2 ⎪ n un + mun + 1 n (n un + mun + 1) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ dv n (1 − n u2n ) dun ⎪ ⎪ − c − 2vn + = n n ⎨ −n − n n u2n + mun + 1 n (n u2n + mun + 1)2 ⎪ ⎪ ⎪ ⎪ j ⎪ ⎪ 1 n + n = 0, ⎪ ⎪ j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ j ⎪ ⎩ 2 n + n = 0 j
in , (4.3)
on j,
where (un , vn ) is a coexistence state of (1.1) with = n which satisfies (4.1). From Eq. (4.3), we have n =
|∇n | −
+
|n |
2
a − 2un +
|∇n | − 2
n n
bv n (n u2n − 1) (n u2n
+ mun + 1)
dv n (1 − n u2n ) (n u2n + mun + 1)2
2
−
+
2 n u n
bun n + mun + 1 n
|n |2 c − 2vn +
dun , n u2n + mun + 1
where n and n are the complex conjugates of n and n . From the above equation, one can see that Im(n ) and Re(n ) are bounded, and so we may assume that n → with Re() 0, n → and n → . Note that n and n are also
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bounded. Taking the limit in (4.3), we obtain ⎧ − − (a − 2 [a]) = , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − − (c − 2 [c]) = in , ⎪ ⎪ ⎪ ⎨ j 1 + = 0, ⎪ ⎪ j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ j + = 0 on j 2 j
781
(4.4)
by (i), and so must be real number with 0. If ≡ / 0, then is an eigenvalue of the problem, − − (a − 2 [a]) = in and 1 j/j + = 0 on j, and thus 0 1,1 (−a + 2 [a]) which is impossible since 1,1 (−a + 2 [a]) > 1,1 (−a + [a]) = 0. Therefore ≡ 0. One can also similarly show that ≡ 0 which completes the proof. If a > 1,1 (b [c]), equivalently (a + 1,1 (b [c]))/2 > 1,1 (b [c]), then we can take a > 0 small enough such that (a + 1,1 (b [c]))/2 > 1,1 (b 2 [c + ]/(1 − )), and thus one can see that the following problem has a unique positive solution w∗ by Theorem 2.2 (ii): ⎧ a + 1 (− + b [c]) b 2 [c + ] ⎪ in , − − w∗ ⎪ ⎨ −w∗ = w∗ 2 1− (4.5) ⎪ ⎪ ⎩ 2 jw∗ + w∗ = 0 on j. j Lemma 4.3. Assume that c > 1,2 and a > 1,1 (b [c]). Then there exists a large () such that the following statements are satisfied for () and some > 0. (i) uw∗ , i.e., w∗ is a lower solution for u, where w∗ is the unique positive solution of (4.5) for a coexistence state (u, v) of (1.1) and sufficiently small positive constant with (a + 1,1 (b [c]))/2 > 1,1 (b 2 [c + ]/(1 − )). (ii) A coexistence state of (1.1) converges to ( [a], [c]) as → ∞. Moreover, it is non-degenerate and linearly stable. Proof. (i) Using Lemma 2.4 and the comparison argument, one can easily check 2 [c + d/(2 + m)]. that v Then there exists a () such that for sufficiently large () with max{d/(2 + m), m2 /4} < , bv −u = u a − u − 2 u + mu + 1 a + 1,1 (b [c]) b 2 [c + d/(2 + m)] u − −u 2 1 − (m2 /4) a + 1,1 (b [c]) b 2 [c + ] u − −u , 2 1− and so we get the desired result by the comparison argument. Note that i [ ] is increasing with respect to > 0. (ii) Since the lower solution w∗ for u does not depend on as → ∞, it can be shown similarly as in Lemma 4.2. Lemma 4.4. Assume that c ∈ (1,2 − d/(2 + m), 1,2 ] and a > 1,1 + , where := b(c + d/(2 + m))1/(1 − m2 /4). Then there exists a large such that for , (i) u 1 [a − ], (ii) a coexistence state of (1.1) is non-degenerate and linearly stable.
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Proof. (i) Using the comparison argument for elliptic problem, we have 0 v 2 [c+d/(2 +m)](c+d/(2 + m)), and so for a sufficiently large , b(c + d/(2 + m)) bv u a − −u , −u = u a − u − 2 u + mu + 1 1 − m2 /4 which implies the result. (ii) It can be shown similarly as in Theorem 3.1, and thus we just sketch the proof. Consider Eq. (4.3) with c = cn , where cn ∈ (1,2 − d/(2 + m), 1,2 ]. Since un 1 [a − n ] for n := b(cn + d/(2 n + m))1/(1 − m2 /4n ), n u2n ∞ → ∞, and so (un , vn ) → ( [a], 0). Moreover, cn → 1,2 since 1,2 − d/(2 n + m) → 1,2 . If we set vn∗ = vn / vn ∞ , then we have ⎧ dun ⎪ vn∗ + vn∗ cn − vn + = 0 in , ⎨ n u2n + mun + 1 ∗ ⎪ ⎩ 2 jvn + v ∗ = 0 on j. n j By standard elliptic regularity theory and Sobolev embedding theorems, we may assume that vn∗ → v ∗ > 0. Taking the limit, the above equation yields −v ∗ = 1,2 v ∗
on ,
2
jv ∗ + v∗ = 0 j
on j
and
v ∗ ∞ = 1.
Thus v ∗ must be the principal eigenfunction 1 corresponding to 1,2 with 1 ∞ = 1. Now we show that the linearized eigenvalue problem (4.3) has no eigenvalue n with Re(n ) 0 for large n. As n → ∞, note that (4.3) converges to ⎧ − − (a − 2 [a]) = , ⎪ ⎪ ⎪ ⎪ ⎪ − − 1,2 = in , ⎪ ⎪ ⎪ ⎨ j (4.6) 1 + = 0, ⎪ ⎪ j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2 j + = 0 on j. j It is easy to see that (4.6) has 0 as a simple eigenvalue with its corresponding eigenfunction (, ) = (0, 1 ). By the same reason as in Theorem 3.1, it suffices to show n > 0 for large n. Choose an eigenfunction (n , n ) such that (n , n ) → (0, 1 ). Using the integral equation on which can be obtained by multiplying vn to the second equation of (4.3) and the second integral equation on of (1.1) which are multiplied by n , we can derive 3 n lim = 21 > 0 n→∞ vn ∞ 1 which establishes our claim.
Now we give the multiplicity and uniqueness results of coexistence states for (1.1) by using the sequence of above lemmas. Theorem 4.5. (i) If a ∈ (1,1 , 1,1 (b [c])) and c > 1,2 , then there exists a large := () such that (1.1) has at least two coexistence states for (). (ii) If a > 1,1 (b [c]) and c > 1,2 , then there exists a large := () such that (1.1) has a unique coexistence state of (1.1) for (). (iii) If 1,2 (−d [a]/( 2 [a] + m [a] + 1)) < c 1,2 and a >1,1 + , then there exists a large such that (1.1) has a unique coexistence state for , where := b(c + d/(2 + m))1/(1 − m2 /4).
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Proof. (i) Suppose that (1.1) has a unique coexistence state (u∗ , v∗ ). Then (u∗ , v∗ ) must be contained in region (4.1) since there must be a coexistence state which satisfies (4.1). For a sufficiently large , one can easily show that I − A (u∗ , v∗ ) is invertible and has no property on W (u∗ ,v∗ ) by Lemma 4.2. In addition, A has no real eigenvalue greater than or equal to one. Thus Theorem 2.3 (ii) concludes indexW (A, (u∗ , v∗ )) = (−1)0 = 1. Using Lemma 2.5–2.6 and the additivity property of the index, we have 1 = indexW (A, D ) = indexW (A, (0, 0)) + indexW (A, ( [a], 0)) + indexW (A, (0, [c])) + indexW (A, (u∗ , v∗ )) = 0 + 0 + 1 + 1, which is a contradiction. (ii) By compactness, A has at most finitely many positive fixed points in the given region D . Let us denote them by Ui = (ui , vi ) for i = 1, . . . , k. As in the proof of (i), one can show that index(A, Ui ) = (−1)0 = 1 for i = 1, . . . , k by using Lemma 4.3 and Theorem 2.3 (ii). Finally, using the additivity property of the index, we have k=
k
indexW (A, Ui )
i=1
= indexW (A, D ) − indexW (A, (0, 0)) − indexW (A, ( [a], 0)) − indexW (A, (0, [c])) = 1 − 0 − 0 − 0 = 1. This proves the uniqueness. (iii) Note that c > 1,2 − d/(2 + m) since d [a]/( 2 [a] + m [a] + 1) d/(2 + m) by Lemma 2.4 and the semi-trivial solution (0, [c]) does not exist in this case. Then one can similarly prove as in (ii). Before closing this section, we consider the limiting behavior of coexistence states of (1.1) when → ∞. To do this, let (un , vn ) be coexistence states of (1.1) with = n , where n () and n → ∞ as n → ∞ for () defined in Theorem 4.5 (i). Theorem 4.6. Let the hypotheses in Theorem 4.5 (i) hold. Then as n → ∞, either (i) (u n , vn ) → ( [a], [c]) or (ii) ( n un , vn ) is close to a non-negative solution (U, [c]), where U is a positive solution of ⎧ b [c] ⎪ in , −U = U a − ⎪ ⎨ 1 + U2 ⎪ ⎪ ⎩ 1 jU + U = 0 on j. j
(4.7)
2 2 Proof. If n un ∞ → ∞, then the result (i) clearly follows. Therefore we only consider the case of n un ∞ < ∞. By setting n un = Un in (1.1), we have ⎧ 1 bv n ⎪ ⎪ −Un = Un a − Un − , ⎪ ⎪ ⎪ n Un2 + (m/ n )Un + 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ )v (d/ ⎪ n ⎪ in , ⎨ −vn = vn c − vn + Un2 + (m/ n )Un + 1 (4.8) ⎪ ⎪ ⎪ jUn ⎪ ⎪ ⎪ + Un = 0, 1 ⎪ ⎪ j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2 jvn + vn = 0 on j. j
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Observe that solution (Un , vn ) of (4.8) is close to a non-zero solution of the problem: ⎧ bv ⎪ ⎪ −U = U a − , ⎪ ⎪ 1 + U2 ⎪ ⎪ ⎪ ⎪ ⎪ in , ⎪ ⎨ −v = v(c − v) jU ⎪ 1 + U = 0, ⎪ ⎪ j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ jv ⎪ ⎩ 2 +v=0 on j. j as n → ∞. Since the second equation of the above problem has the unique positive solution [c], (ii) clearly holds. Remark 4.7. Using the index theory as in Theorem 4.5 (ii), one can show that the coexistence state of (1.1) which satisfies (4.1) is unique. More precisely, 1 = indexW (A, Dˆ ) = ki=1 indexW (A, (ui , vi )) = k, where (ui , vi ) are coexistence states which satisfy (4.1) and Dˆ := {(u, v) ∈ D : 21 1 [a − /2] u2 [a] and 21 [c] v 2 2 [c + /2]}. Therefore, we can see that the exact number of coexistence states is determined by the multiplicity of coexistence states of (4.8) under the same assumptions in Theorem 4.5 (i).
Appendix In this appendix, we show the existence of coexistence states of (1.1) by using the local bifurcation theory which was introduced in Section 3. Theorem A. (i) If c > 1,2 and a > 1,1 (b [c]), then the coexistence states of (1.1) bifurcate from the semi-trivial a. solution curve {(0, [c], a) : a > a := 1,1 (b [c])} if and only if a = (ii) If a > 1,1 and 1,2 (−d [a]/( 2 [a] + m [a] + 1)) < c 1,2 , then the coexistence states of (1.1) bifurcate from the semi-trivial solution curve {( [a], 0, c) : c > c := 1,2 (−d [a]/( 2 [a] + m [a] + 1))} if and only if c = c. Proof. The proofs are similar, and so we only prove (i). Define a mapping F : E × R → E by ⎞ ⎛ bv u + u a − u − 2 ⎜ u + mu + 1 ⎟ F (u, v, a) := ⎝
⎠. du v + v c − v + u2 +mu+1
By the simple calculation, for (, ) ∈ E, ⎛ ⎞ bv(u2 − 1) bu − ⎜ + a − 2u + u2 + mu + 1 ⎟ (u2 + mu + 1)2 ⎜ ⎟ F(u,v) (u, v, a) =⎜ , ⎟ ⎝ dv(1 − u2 ) ⎠ du + + c − 2v + 2 u + mu + 1 (u2 + mu + 1)2 and so F(u,v) (0, [c], a)
+ ( a − b [c]) = . + (c − 2 [c]) + d [c]
Claim 1. dim(Ker(F(u,v) (0, [c], a ))) = 1 and Ker(F(u,v) (0, [c], a )) = span{(, d )}, where is the principal eigenfunction of a = 1,1 (b [c]) and d := d(− + 2 [c] − c)−1 ( [c]).
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Proof of Claim 1. Assume that F(u,v) (0, [c], a ) = (0, 0) for (, ) ∈ E, then we have ⎧ − − ( a − b [c]) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − − (c − 2 [c]) − d [c] = 0 ⎪ ⎪ ⎨ j + = 0, 1 ⎪ ⎪ j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2 j + = 0 j
in ,
on j.
From the above equation, we see that = span{} since a = 1,1 (b [c]), and so = span{d }, where d := d(− + 2 [c] − c)−1 ( [c]). Claim 2. Codim(R(F(u,v) (0, [c], a ))) = 1. a )), then there is a (, ) ∈ E such that Proof of Claim 2. Let ( , ) ∈ R(F(u,v) (0, [c], ⎧ + ( a − b [c]) = , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + (c − 2 [c]) + d [c] = in , ⎪ ⎪ ⎨ j 1 + = 0, ⎪ ⎪ j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2 j + = 0 on j. j From the first of the equation, + ( a − b [c]) = in and 1 j/j + = 0 on j, if equation, is a solution and only if = 0. In fact, = ( + ( a − b [c])) = ( + ( a − b [c])) = 0. Therefore, we see that is uniquely determined from the second equation for such since −( + c − 2 [c]) is invertible, and thus Codim(R(F(u,v) (0, [c], a ))) = 1.
a )). / R(F(u,v) (0, [c], a ) ∈ Claim 3. F(u,v),a (0, [c], d
Proof of Claim 3. It is clear since F(u,v),a (0, [c], a ) = 0 and 2 = 0. d Finally, applying Crandall–Rabinowitz bifurcation theorem in [9] or [25, Section 13], we conclude the desired result (ii). References [1] J.F. Andrews, A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotech. Bioeng. 10 (1968) 707–723. [2] J. Blat, K.J. Brown, Bifurcation of steady-state solutions in predator-prey and competition systems, Proc. R. Soc. Edinburgh Sect. A 97 (1984) 21–34. [3] J. Blat, K.J. Brown, Global bifurcation of positive solutions in some systems of elliptic equations, SIAM J. Math. Anal. 17 (6) (1986) 1339–1353. [4] P.N. Brown, Decay to uniform states in ecological interactions, SIAM J. Appl. Math. 38 (1980) 22–37. [5] S. Cano-Casanova, Existence and structure of the set of positive solutions of a general class of sublinear elliptic non-classical mixed boundary value problems, Nonlinear Anal. Ser. A 49 (3) (2002) 361–430. [6] S. Cano-Casanova, J. López-Gómez, Properties of the principal eigenvalues of a general class of non-classical mixed boundary value problems, J. Differential Equations 178 (1) (2002) 123–211. [7] A. Casal, J.C. Eilbeck, J. López-Gómez, Existence and uniqueness of coexistence states for a predator–prey model with diffusion, Differential Integr. Equations 7 (1994) 411–439. [8] C. Cosner, A.C. Lazer, Stable coexistence states in the Volterra–Lotka competition model with diffusion, SIAM J. Appl. Math. 44 (1984) 1112–1132. [9] M.G. Crandall, P.H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971) 321–340.
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