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On a predator–prey system with cross-diffusion representing the tendency of prey to keep away from its predators
Applied Mathematics Letters 21 (2008) 1177–1183 www.elsevier.com/locate/aml
On a predator–prey system with cross-diffusion representing the tendency of prey to keep away from its predators Wonlyul Ko a , Kimun Ryu b,∗ a Department of Information and Mathematics, Korea University, Jochiwon, Chungnam 339-700, South Korea b Department of Mathematics Education, Cheongju University, Cheongju, Chungbuk 360-764, South Korea
Received 19 April 2007; received in revised form 19 November 2007; accepted 11 December 2007
Abstract In this work, we study a predator–prey system with cross-diffusion, representing the tendency of prey to keep away from its predators, under the homogeneous Dirichlet boundary condition. Using fixed point index theory, we provide some sufficient conditions for the existence of positive steady-state solutions. Furthermore, we investigate the non-existence of positive solutions. c 2008 Elsevier Ltd. All rights reserved.
Keywords: Predator–prey interaction; Cross-diffusion; Index theory
1. Introduction In this work, we study the existence and non-existence of positive solutions to the following Lotka–Volterra predator–prey system with cross-diffusion rates: −∆u − α∆v = u(a − u) − c1 uv β∆u − ∆v = −dv + c2 uv in Ω , (1.1) (u, v) = (0, 0) on ∂Ω , where Ω ⊆ R N is a bounded domain with smooth boundary ∂Ω ; and the given coefficients a, ci and d are positive constants. In system (1.1), u and v are the densities of the prey and predators, respectively; the function (a − u) represents the growth rate of the prey in the absence of its predators; c1 u is the functional response of a predator to the prey, and describes the change in the rate of exploitation of the prey by a predator as a result of a change in the prey density; cc12 is the conversion rate of predators to the prey; and d is the death rate of the predator. The positive constants α and β are cross-diffusion rates which express the respective population fluxes of the prey and predators resulting from the presence of the other species, respectively. Biologically, the induced cross-diffusion rate α in (1.1) represents the tendency of the prey to keep away from its predators; and β represents the tendency of the predator to chase its prey. ∗ Corresponding author.
E-mail addresses: [email protected] (W. Ko), [email protected] (K. Ryu). c 2008 Elsevier Ltd. All rights reserved. 0893-9659/$ - see front matter doi:10.1016/j.aml.2007.12.018
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Previously, there have been many studies on the dynamics of strongly coupled reaction–diffusion systems with competitive cross-diffusion rates, which was proposed first by Shigesada et al. in [17]. For example, refer to [5,11,12, 14,16] and the references therein. On the other hand, little attention has been given to studying the predator–prey models having cross-diffusion rates [9,14,15]. Furthermore, in these studies, the introduced cross-diffusion rates represent the tendency of predators to avoid group defense by the existence of a large number of prey species, that is, the predators diffuse away from their victims. However, in predator–prey interactions, it is more natural and realistic to consider the cases, as in (1.1), where the predators tend to diffuse in the direction where there is a higher concentration of the prey species, while the prey tends to diffuse in the direction where there is a lower concentration of its predators. For detailed biological background, one can refer to [3,4,7,13]. We point out that the diffusion terms given in (1.1) are different from the diffusion rates of the forms −∆[(1+αv)u] and −∆[(1 + βu)v], which are introduced in the previous works [9,11,12,14,16]. For under the homogeneous Neumann boundary condition, there are only a few results on the stability of non-negative constant solutions and the appearance of non-constant steady states to the predator–prey system with cross-diffusion rates of the forms −∆(u ± αv) and −∆(v ± βu). For instance, see [1,3,6,8]. Unfortunately, the existence of non-constant positive solutions has not been studied well. In the authors’ view, one of the reasons why such predator–prey systems have not been studied thoroughly is the lack of knowledge of mathematical methods to apply. This work is mainly devoted to finding some sufficient conditions for the existence of positive solutions to (1.1) by using fixed point index theory. In addition, we investigate some conditions which give the non-existence of positive solutions of (1.1). This article is organized as follows. In Section 2, to apply the fixed point index theory, we introduce another equivalent predator–prey system to (1.1), and then provide some sufficient conditions for the existence of positive steady-state solutions. Finally, in Section 3, we show the non-existence theorem for positive solutions to (1.1). 2. The existence of positive steady-state solutions In this section, by using fixed point index theory, we derive some sufficient conditions for the existence of positive solutions to (1.1). Let E be a real Banach space and W the natural positive cone of E. For y ∈ W, define W y = {x ∈ E : y + γ x ∈ W for some γ > 0} and S y = {x ∈ W y : −x ∈ W y }. Let y∗ be a fixed point of a compact operator A : W → W and L = A0 (y∗ ) be the Fr´echet derivative of A at y∗ . We say that L has property α on W y∗ if there exists a t ∈ (0, 1) and a y ∈ W y∗ \ S y∗ such that y − tLy ∈ S y∗ . For an open subset U ⊂ W, let indexW (A, U ) be the Leray–Schauder degree degW (I − A, U, 0), where I is the identity map. The fixed point index of A at y∗ in W is defined by indexW (A, y∗ ) := index(A, U (y∗ ), W), where U (y∗ ) is a small open neighborhood of y∗ in W. Then the following theorem can be obtained from the results of [2,10,18]. Theorem 2.1. Assume that I − L is invertible on W y∗ . If L has property α on W y∗ , then indexW (A, y∗ ) = 0. To begin with, consider the following coupled system which is equivalent to (1.1): 1 −∆u = [u(a − u − c1 v) − αv(−d + c2 u)] 1 + αβ 1 −∆v = [v(−d + c2 u) + βu(a − u − c1 v)] in Ω , 1 + αβ (u, v) = (0, 0) on ∂Ω .
(2.1)
Multiplying the first equation by a positive constant δ which will be determined later, and then subtracting it from the second equation in (2.1), we have 1 [u(a − u − c1 v) − αv(−d + c2 u)] −∆u = 1 + αβ 1 (2.2) −∆(v − δu) = [(1 + αδ)v(−d + c2 u) + (β − δ)u(a − u − c1 v)] in Ω , 1 + αβ (u, v) = (0, 0) on ∂Ω .
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Let w := v − δu; then the following is a system equivalent to (2.2), so (2.1) is obtained: 1 −∆u = [u (a + αdδ − (1 + δ(c1 + αc2 ))u − (c1 + αc2 )w) + αdw] 1 + αβ 1 −∆w = [w (−d(1 + αδ) + (c2 (1 + αδ) − (β − δ)c1 )u) + u(M1 + M2 u)] 1 + αβ (u, v) = (0, 0)
in Ω
(2.3)
on ∂Ω ,
where M1 := −d(1 + αδ)δ + a(β − δ) and M2 := c2 (1 + αδ)δ − (1 + c1 δ)(β − δ). Observe that, if (2.3) has a positive solution (u, w), then (1.1) has a positive solution (u, v) with v = w + δu, and thus it suffices to show that (2.3) has at least one positive solution. Theorem 2.2. Any positive solution (u, w) of (2.3) satisfies u(x) ≤ Q 1 where Q 1 :=
w(x) ≤ Q 2 ,
and
a 1+αδ 1+αδ ( 1+c1 δ
+
α c1 )
1+αδ and Q 2 := αa ( 1+c + 1δ
α c1 ).
Proof. Multiplying the first and second equations in (2.3) by 1 + αδ and α, respectively, and then adding the two equations obtained, we have −∆[(1 + αδ)u + αw] = u(a − (1 + c1 δ)u − c1 w)
in Ω .
Assume that (1 + αδ)u + αw attains its positive maximum at x0 ∈ Ω ; then it easily follows that −∆[(1 + αδ)u(x0 ) + αw(x0 )] = u(x0 )(a − (1 + c1 δ)u(x0 ) − c1 w(x0 )) ≥ 0. This implies a − (1 + c1 δ)u(x0 ) − c1 w(x0 ) ≥ 0, and thus a and w(x0 ) ≤ ca1 . Using these facts, it is easy to see that u(x0 ) ≤ 1+c 1δ max{(1 + αδ)u(x) + αw(x)} = (1 + αδ)u(x0 ) + αw(x0 ) ≤ (1 + αδ) x∈Ω
which yields the desired result.
a a +α , 1 + c1 δ c1
For simplicity, take δ :=
−(a + d) +
p
(a + d)2 + 4adαβ , 2αd
so that M1 = 0. Then, since β − δ =
d(1+αδ)δ , a
M2 = c2 (1 + αδ)δ − (1 + c1 δ)(β − δ) =
(1 + αδ)c1 dδ a
ac2 − d −δ . c1 d
(2.4)
1 For a positive constant P with P ≥ 1+αβ max{2(1 + c1 + αc2 )Q 1 + (c1 + αc2 )Q 2 , d(1 + αδ) + (β − δ)c1 Q 1 }, define a compact operator A by f 1 (u, w) + Pu −1 A(u, w) := (−∆ + P) , f 2 (u, w) + Pw
where 1 [u (a + αdδ − (1 + δ(c1 + αc2 ))u − (c1 + αc2 )w) + αdw] , 1 + αβ i 1 h f 2 (u, w) := w (−d(1 + αδ) + (c2 (1 + αδ) − (β − δ)c1 )u) + M2 u 2 . 1 + αβ f 1 (u, w) :=
Then, f 1 (u, w) + Pu and f 2 (u, w) + Pw are monotone increasing with respect to u and w, respectively, for all (u, w) ∈ [0, Q 1 ] × [0, Q 2 ]. For the sake of convenience, the following notation is introduced. Notation 2.3. (i) λ1 denotes the principal eigenvalue of −∆ under the homogeneous Dirichlet boundary condition. (ii) E := C D (Ω ) ⊕ C D (Ω ), where C D (Ω ) := {φ ∈ C(Ω ) : φ = 0 on ∂Ω }.
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(iii) N := N Q ⊕ N Q , where N Q := {φ ∈ C D (Ω ) : φ < max{Q 1 , Q 2 } + 1 in Ω }. (iv) W := K ⊕ K , where K := {φ ∈ C D (Ω ) : 0 ≤ φ(x), x ∈ Ω }. (v) N 0 := N ∩ W. Now, to make the operator A positive in N 0 , satisfying the following condition is imposed throughout this section: c2 a − d ≥ β. c1 d
(2.5)
> 0 and (2.4), which implies More precisely, if (2.5) holds, then M2 ≥ 0 follows from the fact that β − δ = d(1+αδ)δ a the positivity of the operator A. In addition, note that (2.3) is equivalent to (u, w) = A(u, w). Therefore, it suffices to prove that A has a positive fixed point in N 0 to show that (2.3) has a positive solution. We point out that (2.3) has no semi-trivial solutions when exactly one of the species is absent. Thus, we only need to calculate indexW (A, N 0 ) and indexW (A, (0, 0)) to investigate the existence of positive steady-state solutions. Lemma 2.4. Assume that (2.5) holds; then indexW (A, N 0 ) = 1. Proof. Define a homotopy Aθ : E → E by −1 θ ( f 1 (u, w) + Pu) Aθ (u, w) = (−∆ + P) θ ( f 2 (u, w) + Pw) for θ ∈ [0, 1]. Then, every fixed point of Aθ is in N 0 but not on ∂ N 0 . Applying the homotopy invariance and normalization properties of the index, we can conclude that indexW (A1 , N 0 ) = indexW (A0 , N 0 ) = 1. √ 2 −(a+d)+ (a+d) +4adαβ In the following lemma, recall that δ = . 2αd Lemma 2.5. Assume that (2.5) holds. If λ1 <
a+αdδ 1+αβ ,
then indexW (A, (0, 0)) = 0.
Proof. By simple calculation, we have W (0,0) = W, S(0,0) = {(0, 0)} and a + αdδ αd + P 1 + αβ 1 + αβ . L := A0 (0, 0) = (−∆ + P)−1 d(1 + αδ) 0 − +P 1 + αβ Assume that L(φ, ψ)T = (φ, ψ)T for (φ, ψ)T ∈ W, that is, a + αdδ αd φ+ ψ −∆φ = 1 + αβ 1 + αβ d(1 + αδ) ψ in Ω , −∆ψ = − 1 + αβ (φ, ψ) = (0, 0) on ∂Ω .
(2.6)
Then, the strong maximum principle obviously allows us to conclude that ψ ≡ 0 in Ω . Let ϕ > 0 be the principal eigenfunction corresponding to λ1 . Multiplying ϕ by the first equation after substituting ψ = 0 in (2.6), and then integrating it on Ω , we have Z Z Z a + αdδ a + αdδ a + αdδ 0= ϕ ∆φ + φ dx = φ ∆ϕ + ϕ dx = ϕφ −λ1 + dx. 1 + αβ 1 + αβ 1 + αβ Ω Ω Ω Since λ1 <
a+αdδ 1+αβ ,
φ ∈ K and ϕ > 0 in Ω , it follows that φ ≡ 0 in Ω . This shows that I − L is invertible on W.
Furthermore, it is easy to see that L has property α. More precisely, for t := (λ1 + P)/( a+αdδ 1+αβ + P), it is easy to check that t ∈ (0, 1), (φ, 0) ∈ W (0,0) \ S(0,0) , and (φ, 0)T − tL(φ, 0)T ∈ S(0,0) . Therefore, Theorem 2.1 leads to the conclusion that indexW (A, (0, 0)) = 0. By Lemmas 2.4 and 2.5, since indexW (A, N 0 ) 6= indexW (A, (0, 0)), we have the following theorem which provides sufficient conditions for the existence of positive solutions to (1.1).
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Theorem 2.6. Assume that c2ca−d ≥ β. Then (1.1) has at least one positive solution provided that λ1 < 1d √ 2 −(a+d)+ (a+d) +4adαβ . δ= 2αd
a+αdδ 1+αβ
for
1 Corollary 2.7. (i) If β ≤ min{ c2ca−d , a−λ αλ1 }, then (1.1) has at least one positive solution. 1d
(ii) If
a−λ1 αλ1
< β < min{ c2ca−d , 1d
(λ1 +d)(a−λ1 ) }, αλ21
then (1.1) has at least one positive solution.
a+αdδ 1 Proof. (i) Since the condition β ≤ a−λ αλ1 is equivalent to (1+αβ)λ1 ≤ a, it is easy to see that λ1 < 1+αβ is satisfied, and thus the desired result follows from Theorem 2.6. 1 (ii) Since a−λ d, so that 2λ1 (1 + αβ) − a + d > 0. In addition, since αλ1 < β, we have 2λ1 (1 + αβ) > 2a > a − p 1 λ1 (1 + αβ) − a − αdδ = 2 (2λ1 (1 + αβ) − a + d − (a + d)2 + 4adαβ) and [2λ1 (1 + αβ) − a + d]2 − (a + 1) d)2 − 4adαβ = 4(αβ + 1)(λ21 αβ − (a − λ1 )(λ1 + d)), it follows from another given assumption β < (λ1 +d)(a−λ 2
that λ1 <
a+αdδ 1+αβ .
αλ1
Therefore, Theorem 2.6 leads to the conclusion that (1.1) has at least one positive solution.
ˆ d, α, ci , λ1 ) such Remark 2.8. In view of Corollary 2.7(i), we conclude that there exists a positive constant βˆ := β(a, ˆ that (1.1) has a positive solution provided that β ≤ β. Biologically, this implies that the prey and predator species may coexist when the prey can survive alone without its predator (i.e., a > λ1 ) and its corresponding intrinsic growth rate is greater than some level (i.e., a > cd2 ), provided that the cross-diffusion β which is induced on the prey by its predator is sufficiently small. 3. The non-existence of positive steady-state solutions In this section, some sufficient conditions for the non-existence of positive solutions of (1.1) are provided. Theorem 3.1. (i) If λ1 ≥ max{a, c2 a(1 +
α c1 ) − d},
then (1.1) has no positive solution.
(ii) There exists a positive constant β := β(a, d, α, λ1 ) such that (1.1) has no positive solution provided that β ≥ β. (iii) There exists a positive constant α := α(a, d, β, λ1 ) such that (1.1) has no positive solution provided that α ≥ α. Proof. (i) Let (u, v) be a positive solution of (1.1). Multiplying the first and second equations in (1.1) by u and v, respectively, and then integrating these equations on Ω , we have Z Z Z α ∇u∇vdx = u 2 (a − u − c1 v)dx − |∇u|2 dx, ΩZ ΩZ Ω Z (3.1) −β ∇u∇vdx = v 2 (−d + c2 u)dx − |∇v|2 dx. Ω
Ω
Ω
Since (1.1) is exactly equivalent to (2.3) when δ = 0, it follows from Theorem 2.2 that the positive solution (u, v) satisfies α α a u ≤ 1+ a and v ≤ 1 + . c1 c1 α Applying the Poincar´e inequality to the first equation in (3.1), we have Z Z Z 2 α ∇u∇vdx ≤ u (a − u − c1 v)dx − λ1 u 2 dx Ω Ω Ω Z Z = (a − λ1 )u 2 dx − u 2 (u + c1 v)dx < 0. Ω
Ω
The last inequality follows from the given assumption a ≤ λ1 . Therefore, using the Poincar´e inequality again and another given assumption c2 a(1 + cα1 ) − d ≤ λ1 , the following contradiction is derived from the second equation in (3.1):
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0 < −β
Z
Z ∇u∇vdx =
ZΩ
Ω
v (−d + c2 u)dx − 2
Z
|∇v|2 dx Ω
v 2 (−d + c2 u − λ1 )dx Z α − λ1 dx ≤ 0. v 2 −d + c2 a 1 + ≤ c1 Ω ≤
Ω
(ii) Suppose, by contradiction, that (1.1) has a positive solution (u, v). Multiplying the first and second equations in (1.1) by v and u, respectively, and then integrating these equations over Ω , we have Z Z Z ∇u∇vdx = |∇v|2 dx, uv(a − u − c1 v)dx − α Ω Ω Ω Z Z Z ∇u∇vdx = |∇u|2 dx. uv(−d + c2 u)dx + β Ω
Ω
Ω
The above two identities yield Z Z Z Z 2 2 α |∇v| dx + β |∇u| dx − (a + d) uvdx = − uv ((1 + c2 )u + c1 v) dx. Ω
Ω
Ω
(3.2)
Ω
Applying the Poincar´e inequality to the left-hand side of (3.2), we have Z Z Z 2 2 α |∇v| dx + β |∇u| dx − (a + d) uvdx Ω Ω ΩZ Z Z ≥ αλ1 v 2 dx + βλ1 u 2 dx − (a + d) uvdx Ω Ω Ω Z Z Z 2 v 2 u + ≥ αλ1 v 2 dx + βλ1 u 2 dx − (a + d) dx 2 Ω Ω Ω 2 Z Z (a + d) a+d 2 = αλ1 − v dx + βλ1 − u 2 dx 2 2 Ω Ω
(3.3)
1 for an arbitrary positive constant . Now, fix a positive constant 0 with 0 ≤ 2αλ a+d ; then it follows obviously that a+d . This is a contradiction to the fact that the right-hand side of (3.2) is (3.3) is non-negative for β ≥ β := 2 0 λ1 negative. (iii) This can be shown like in the proof of (ii). Thus, the proof is omitted.
Remark 3.2. In view of Theorem 3.1, we may conclude that if the cross-diffusion rate of the prey or its predator is sufficiently large, then the prey and predator species cannot coexist. In other words, the large cross-diffusion coefficients α and β tend to mean no positive coexistence. Acknowledgements The authors thank the anonymous referee for his/her valuable comments and suggestions for improving the content of this article. The first author (W. Ko) would like to thank Professor Y. Du for his warm hospitality, mathematical advice, and encouragement in Australia. The first author’s work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2006-214-C00006). References [1] [2] [3] [4] [5] [6] [7]
J. Chattopadhyay, P.K. Tapaswi, Effect of cross-diffusion on pattern formation — a nonlinear analysis, Acta Appl. Math. 48 (1997) 1–12. E.N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl. 91 (1983) 131–151. B. Dubey, B. Das, J. Hussain, A predator–prey interaction model with self and cross-diffusion, Ecol. Modelling 141 (2001) 67–76. M. Farkas, Two ways of modelling cross-diffusion, Nonlinear Anal. 30 (2) (1997) 1224–1233. M.E. Gurtin, Some mathematical models for population dynamics that lead to segregation, Quart. Appl. Math. 32 (1974/75) 1–9. D. Horstmann, Remarks on some Lotka–Volterra type cross-diffusion models, Nonlinear Anal. RWA 8 (1) (2007) 90–117. J. Jorne, Negative ionic cross diffusion coefficients in electrolytic solutions, J. Theoret. Biol. 55 (2) 529–532.
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[8] S. Kov´acs, Turing bifurcation in a system with cross diffusion, Nonlinear Anal. 59 (2004) 567–581. [9] K. Kuto, Y. Yamada, Multiple coexistence states for a prey–predator system with cross-diffusion, J. Differential Equations 197 (2004) 315–348. [10] L. Li, Coexistence theorems of steady states for predator–prey interacting systems, Trans. Amer. Math. Soc. 305 (1) (1988) 143–166. [11] Y. Lou, W.M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations 131 (1) (1996) 79–131. [12] Y. Lou, W.M. Ni, Diffusion vs cross-diffusion: An elliptic approach, J. Differential Equations 154 (1) (1999) 157–190. [13] A. Okubo, S.A. Levin, Diffusion and ecological problems: Modern perspectives, second edn, in: Interdisciplinary Applied Mathematics, vol. 14, Springer-Verlag, New York, 2001. [14] C.V. Pao, Strongly coupled elliptic systems and applications to Lotka–Volterra models with cross-diffusion, Nonlinear Anal. 60 (2005) 1197–1217. [15] K. Ryu, I. Ahn, Positive steady-states for two interacting species models with linear self-cross diffusions, Discrete Contin. Dyn. Syst. 9 (4) (2003) 1049–1061. [16] K. Ryu, I. Ahn, Coexistence theorem of steady states for nonlinear self-cross diffusion systems with competitive dynamics, J. Math. Anal. Appl. 283 (1) (2003) 46–65. [17] N. Shigesada, K. Kawasaki, E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol. 79 (1) (1979) 83–99. [18] 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 Sci. Publishing, River Edge, NJ, 1991, pp. 256–259.
On a predator–prey system with cross-diffusion representing the tendency of prey to keep away from its predators Wonlyul Ko a , Kimun Ryu b,∗ a Department of Information and Mathematics, Korea University, Jochiwon, Chungnam 339-700, South Korea b Department of Mathematics Education, Cheongju University, Cheongju, Chungbuk 360-764, South Korea
Received 19 April 2007; received in revised form 19 November 2007; accepted 11 December 2007
Abstract In this work, we study a predator–prey system with cross-diffusion, representing the tendency of prey to keep away from its predators, under the homogeneous Dirichlet boundary condition. Using fixed point index theory, we provide some sufficient conditions for the existence of positive steady-state solutions. Furthermore, we investigate the non-existence of positive solutions. c 2008 Elsevier Ltd. All rights reserved.
Keywords: Predator–prey interaction; Cross-diffusion; Index theory
1. Introduction In this work, we study the existence and non-existence of positive solutions to the following Lotka–Volterra predator–prey system with cross-diffusion rates: −∆u − α∆v = u(a − u) − c1 uv β∆u − ∆v = −dv + c2 uv in Ω , (1.1) (u, v) = (0, 0) on ∂Ω , where Ω ⊆ R N is a bounded domain with smooth boundary ∂Ω ; and the given coefficients a, ci and d are positive constants. In system (1.1), u and v are the densities of the prey and predators, respectively; the function (a − u) represents the growth rate of the prey in the absence of its predators; c1 u is the functional response of a predator to the prey, and describes the change in the rate of exploitation of the prey by a predator as a result of a change in the prey density; cc12 is the conversion rate of predators to the prey; and d is the death rate of the predator. The positive constants α and β are cross-diffusion rates which express the respective population fluxes of the prey and predators resulting from the presence of the other species, respectively. Biologically, the induced cross-diffusion rate α in (1.1) represents the tendency of the prey to keep away from its predators; and β represents the tendency of the predator to chase its prey. ∗ Corresponding author.
E-mail addresses: [email protected] (W. Ko), [email protected] (K. Ryu). c 2008 Elsevier Ltd. All rights reserved. 0893-9659/$ - see front matter doi:10.1016/j.aml.2007.12.018
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Previously, there have been many studies on the dynamics of strongly coupled reaction–diffusion systems with competitive cross-diffusion rates, which was proposed first by Shigesada et al. in [17]. For example, refer to [5,11,12, 14,16] and the references therein. On the other hand, little attention has been given to studying the predator–prey models having cross-diffusion rates [9,14,15]. Furthermore, in these studies, the introduced cross-diffusion rates represent the tendency of predators to avoid group defense by the existence of a large number of prey species, that is, the predators diffuse away from their victims. However, in predator–prey interactions, it is more natural and realistic to consider the cases, as in (1.1), where the predators tend to diffuse in the direction where there is a higher concentration of the prey species, while the prey tends to diffuse in the direction where there is a lower concentration of its predators. For detailed biological background, one can refer to [3,4,7,13]. We point out that the diffusion terms given in (1.1) are different from the diffusion rates of the forms −∆[(1+αv)u] and −∆[(1 + βu)v], which are introduced in the previous works [9,11,12,14,16]. For under the homogeneous Neumann boundary condition, there are only a few results on the stability of non-negative constant solutions and the appearance of non-constant steady states to the predator–prey system with cross-diffusion rates of the forms −∆(u ± αv) and −∆(v ± βu). For instance, see [1,3,6,8]. Unfortunately, the existence of non-constant positive solutions has not been studied well. In the authors’ view, one of the reasons why such predator–prey systems have not been studied thoroughly is the lack of knowledge of mathematical methods to apply. This work is mainly devoted to finding some sufficient conditions for the existence of positive solutions to (1.1) by using fixed point index theory. In addition, we investigate some conditions which give the non-existence of positive solutions of (1.1). This article is organized as follows. In Section 2, to apply the fixed point index theory, we introduce another equivalent predator–prey system to (1.1), and then provide some sufficient conditions for the existence of positive steady-state solutions. Finally, in Section 3, we show the non-existence theorem for positive solutions to (1.1). 2. The existence of positive steady-state solutions In this section, by using fixed point index theory, we derive some sufficient conditions for the existence of positive solutions to (1.1). Let E be a real Banach space and W the natural positive cone of E. For y ∈ W, define W y = {x ∈ E : y + γ x ∈ W for some γ > 0} and S y = {x ∈ W y : −x ∈ W y }. Let y∗ be a fixed point of a compact operator A : W → W and L = A0 (y∗ ) be the Fr´echet derivative of A at y∗ . We say that L has property α on W y∗ if there exists a t ∈ (0, 1) and a y ∈ W y∗ \ S y∗ such that y − tLy ∈ S y∗ . For an open subset U ⊂ W, let indexW (A, U ) be the Leray–Schauder degree degW (I − A, U, 0), where I is the identity map. The fixed point index of A at y∗ in W is defined by indexW (A, y∗ ) := index(A, U (y∗ ), W), where U (y∗ ) is a small open neighborhood of y∗ in W. Then the following theorem can be obtained from the results of [2,10,18]. Theorem 2.1. Assume that I − L is invertible on W y∗ . If L has property α on W y∗ , then indexW (A, y∗ ) = 0. To begin with, consider the following coupled system which is equivalent to (1.1): 1 −∆u = [u(a − u − c1 v) − αv(−d + c2 u)] 1 + αβ 1 −∆v = [v(−d + c2 u) + βu(a − u − c1 v)] in Ω , 1 + αβ (u, v) = (0, 0) on ∂Ω .
(2.1)
Multiplying the first equation by a positive constant δ which will be determined later, and then subtracting it from the second equation in (2.1), we have 1 [u(a − u − c1 v) − αv(−d + c2 u)] −∆u = 1 + αβ 1 (2.2) −∆(v − δu) = [(1 + αδ)v(−d + c2 u) + (β − δ)u(a − u − c1 v)] in Ω , 1 + αβ (u, v) = (0, 0) on ∂Ω .
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Let w := v − δu; then the following is a system equivalent to (2.2), so (2.1) is obtained: 1 −∆u = [u (a + αdδ − (1 + δ(c1 + αc2 ))u − (c1 + αc2 )w) + αdw] 1 + αβ 1 −∆w = [w (−d(1 + αδ) + (c2 (1 + αδ) − (β − δ)c1 )u) + u(M1 + M2 u)] 1 + αβ (u, v) = (0, 0)
in Ω
(2.3)
on ∂Ω ,
where M1 := −d(1 + αδ)δ + a(β − δ) and M2 := c2 (1 + αδ)δ − (1 + c1 δ)(β − δ). Observe that, if (2.3) has a positive solution (u, w), then (1.1) has a positive solution (u, v) with v = w + δu, and thus it suffices to show that (2.3) has at least one positive solution. Theorem 2.2. Any positive solution (u, w) of (2.3) satisfies u(x) ≤ Q 1 where Q 1 :=
w(x) ≤ Q 2 ,
and
a 1+αδ 1+αδ ( 1+c1 δ
+
α c1 )
1+αδ and Q 2 := αa ( 1+c + 1δ
α c1 ).
Proof. Multiplying the first and second equations in (2.3) by 1 + αδ and α, respectively, and then adding the two equations obtained, we have −∆[(1 + αδ)u + αw] = u(a − (1 + c1 δ)u − c1 w)
in Ω .
Assume that (1 + αδ)u + αw attains its positive maximum at x0 ∈ Ω ; then it easily follows that −∆[(1 + αδ)u(x0 ) + αw(x0 )] = u(x0 )(a − (1 + c1 δ)u(x0 ) − c1 w(x0 )) ≥ 0. This implies a − (1 + c1 δ)u(x0 ) − c1 w(x0 ) ≥ 0, and thus a and w(x0 ) ≤ ca1 . Using these facts, it is easy to see that u(x0 ) ≤ 1+c 1δ max{(1 + αδ)u(x) + αw(x)} = (1 + αδ)u(x0 ) + αw(x0 ) ≤ (1 + αδ) x∈Ω
which yields the desired result.
a a +α , 1 + c1 δ c1
For simplicity, take δ :=
−(a + d) +
p
(a + d)2 + 4adαβ , 2αd
so that M1 = 0. Then, since β − δ =
d(1+αδ)δ , a
M2 = c2 (1 + αδ)δ − (1 + c1 δ)(β − δ) =
(1 + αδ)c1 dδ a
ac2 − d −δ . c1 d
(2.4)
1 For a positive constant P with P ≥ 1+αβ max{2(1 + c1 + αc2 )Q 1 + (c1 + αc2 )Q 2 , d(1 + αδ) + (β − δ)c1 Q 1 }, define a compact operator A by f 1 (u, w) + Pu −1 A(u, w) := (−∆ + P) , f 2 (u, w) + Pw
where 1 [u (a + αdδ − (1 + δ(c1 + αc2 ))u − (c1 + αc2 )w) + αdw] , 1 + αβ i 1 h f 2 (u, w) := w (−d(1 + αδ) + (c2 (1 + αδ) − (β − δ)c1 )u) + M2 u 2 . 1 + αβ f 1 (u, w) :=
Then, f 1 (u, w) + Pu and f 2 (u, w) + Pw are monotone increasing with respect to u and w, respectively, for all (u, w) ∈ [0, Q 1 ] × [0, Q 2 ]. For the sake of convenience, the following notation is introduced. Notation 2.3. (i) λ1 denotes the principal eigenvalue of −∆ under the homogeneous Dirichlet boundary condition. (ii) E := C D (Ω ) ⊕ C D (Ω ), where C D (Ω ) := {φ ∈ C(Ω ) : φ = 0 on ∂Ω }.
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(iii) N := N Q ⊕ N Q , where N Q := {φ ∈ C D (Ω ) : φ < max{Q 1 , Q 2 } + 1 in Ω }. (iv) W := K ⊕ K , where K := {φ ∈ C D (Ω ) : 0 ≤ φ(x), x ∈ Ω }. (v) N 0 := N ∩ W. Now, to make the operator A positive in N 0 , satisfying the following condition is imposed throughout this section: c2 a − d ≥ β. c1 d
(2.5)
> 0 and (2.4), which implies More precisely, if (2.5) holds, then M2 ≥ 0 follows from the fact that β − δ = d(1+αδ)δ a the positivity of the operator A. In addition, note that (2.3) is equivalent to (u, w) = A(u, w). Therefore, it suffices to prove that A has a positive fixed point in N 0 to show that (2.3) has a positive solution. We point out that (2.3) has no semi-trivial solutions when exactly one of the species is absent. Thus, we only need to calculate indexW (A, N 0 ) and indexW (A, (0, 0)) to investigate the existence of positive steady-state solutions. Lemma 2.4. Assume that (2.5) holds; then indexW (A, N 0 ) = 1. Proof. Define a homotopy Aθ : E → E by −1 θ ( f 1 (u, w) + Pu) Aθ (u, w) = (−∆ + P) θ ( f 2 (u, w) + Pw) for θ ∈ [0, 1]. Then, every fixed point of Aθ is in N 0 but not on ∂ N 0 . Applying the homotopy invariance and normalization properties of the index, we can conclude that indexW (A1 , N 0 ) = indexW (A0 , N 0 ) = 1. √ 2 −(a+d)+ (a+d) +4adαβ In the following lemma, recall that δ = . 2αd Lemma 2.5. Assume that (2.5) holds. If λ1 <
a+αdδ 1+αβ ,
then indexW (A, (0, 0)) = 0.
Proof. By simple calculation, we have W (0,0) = W, S(0,0) = {(0, 0)} and a + αdδ αd + P 1 + αβ 1 + αβ . L := A0 (0, 0) = (−∆ + P)−1 d(1 + αδ) 0 − +P 1 + αβ Assume that L(φ, ψ)T = (φ, ψ)T for (φ, ψ)T ∈ W, that is, a + αdδ αd φ+ ψ −∆φ = 1 + αβ 1 + αβ d(1 + αδ) ψ in Ω , −∆ψ = − 1 + αβ (φ, ψ) = (0, 0) on ∂Ω .
(2.6)
Then, the strong maximum principle obviously allows us to conclude that ψ ≡ 0 in Ω . Let ϕ > 0 be the principal eigenfunction corresponding to λ1 . Multiplying ϕ by the first equation after substituting ψ = 0 in (2.6), and then integrating it on Ω , we have Z Z Z a + αdδ a + αdδ a + αdδ 0= ϕ ∆φ + φ dx = φ ∆ϕ + ϕ dx = ϕφ −λ1 + dx. 1 + αβ 1 + αβ 1 + αβ Ω Ω Ω Since λ1 <
a+αdδ 1+αβ ,
φ ∈ K and ϕ > 0 in Ω , it follows that φ ≡ 0 in Ω . This shows that I − L is invertible on W.
Furthermore, it is easy to see that L has property α. More precisely, for t := (λ1 + P)/( a+αdδ 1+αβ + P), it is easy to check that t ∈ (0, 1), (φ, 0) ∈ W (0,0) \ S(0,0) , and (φ, 0)T − tL(φ, 0)T ∈ S(0,0) . Therefore, Theorem 2.1 leads to the conclusion that indexW (A, (0, 0)) = 0. By Lemmas 2.4 and 2.5, since indexW (A, N 0 ) 6= indexW (A, (0, 0)), we have the following theorem which provides sufficient conditions for the existence of positive solutions to (1.1).
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Theorem 2.6. Assume that c2ca−d ≥ β. Then (1.1) has at least one positive solution provided that λ1 < 1d √ 2 −(a+d)+ (a+d) +4adαβ . δ= 2αd
a+αdδ 1+αβ
for
1 Corollary 2.7. (i) If β ≤ min{ c2ca−d , a−λ αλ1 }, then (1.1) has at least one positive solution. 1d
(ii) If
a−λ1 αλ1
< β < min{ c2ca−d , 1d
(λ1 +d)(a−λ1 ) }, αλ21
then (1.1) has at least one positive solution.
a+αdδ 1 Proof. (i) Since the condition β ≤ a−λ αλ1 is equivalent to (1+αβ)λ1 ≤ a, it is easy to see that λ1 < 1+αβ is satisfied, and thus the desired result follows from Theorem 2.6. 1 (ii) Since a−λ d, so that 2λ1 (1 + αβ) − a + d > 0. In addition, since αλ1 < β, we have 2λ1 (1 + αβ) > 2a > a − p 1 λ1 (1 + αβ) − a − αdδ = 2 (2λ1 (1 + αβ) − a + d − (a + d)2 + 4adαβ) and [2λ1 (1 + αβ) − a + d]2 − (a + 1) d)2 − 4adαβ = 4(αβ + 1)(λ21 αβ − (a − λ1 )(λ1 + d)), it follows from another given assumption β < (λ1 +d)(a−λ 2
that λ1 <
a+αdδ 1+αβ .
αλ1
Therefore, Theorem 2.6 leads to the conclusion that (1.1) has at least one positive solution.
ˆ d, α, ci , λ1 ) such Remark 2.8. In view of Corollary 2.7(i), we conclude that there exists a positive constant βˆ := β(a, ˆ that (1.1) has a positive solution provided that β ≤ β. Biologically, this implies that the prey and predator species may coexist when the prey can survive alone without its predator (i.e., a > λ1 ) and its corresponding intrinsic growth rate is greater than some level (i.e., a > cd2 ), provided that the cross-diffusion β which is induced on the prey by its predator is sufficiently small. 3. The non-existence of positive steady-state solutions In this section, some sufficient conditions for the non-existence of positive solutions of (1.1) are provided. Theorem 3.1. (i) If λ1 ≥ max{a, c2 a(1 +
α c1 ) − d},
then (1.1) has no positive solution.
(ii) There exists a positive constant β := β(a, d, α, λ1 ) such that (1.1) has no positive solution provided that β ≥ β. (iii) There exists a positive constant α := α(a, d, β, λ1 ) such that (1.1) has no positive solution provided that α ≥ α. Proof. (i) Let (u, v) be a positive solution of (1.1). Multiplying the first and second equations in (1.1) by u and v, respectively, and then integrating these equations on Ω , we have Z Z Z α ∇u∇vdx = u 2 (a − u − c1 v)dx − |∇u|2 dx, ΩZ ΩZ Ω Z (3.1) −β ∇u∇vdx = v 2 (−d + c2 u)dx − |∇v|2 dx. Ω
Ω
Ω
Since (1.1) is exactly equivalent to (2.3) when δ = 0, it follows from Theorem 2.2 that the positive solution (u, v) satisfies α α a u ≤ 1+ a and v ≤ 1 + . c1 c1 α Applying the Poincar´e inequality to the first equation in (3.1), we have Z Z Z 2 α ∇u∇vdx ≤ u (a − u − c1 v)dx − λ1 u 2 dx Ω Ω Ω Z Z = (a − λ1 )u 2 dx − u 2 (u + c1 v)dx < 0. Ω
Ω
The last inequality follows from the given assumption a ≤ λ1 . Therefore, using the Poincar´e inequality again and another given assumption c2 a(1 + cα1 ) − d ≤ λ1 , the following contradiction is derived from the second equation in (3.1):
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0 < −β
Z
Z ∇u∇vdx =
ZΩ
Ω
v (−d + c2 u)dx − 2
Z
|∇v|2 dx Ω
v 2 (−d + c2 u − λ1 )dx Z α − λ1 dx ≤ 0. v 2 −d + c2 a 1 + ≤ c1 Ω ≤
Ω
(ii) Suppose, by contradiction, that (1.1) has a positive solution (u, v). Multiplying the first and second equations in (1.1) by v and u, respectively, and then integrating these equations over Ω , we have Z Z Z ∇u∇vdx = |∇v|2 dx, uv(a − u − c1 v)dx − α Ω Ω Ω Z Z Z ∇u∇vdx = |∇u|2 dx. uv(−d + c2 u)dx + β Ω
Ω
Ω
The above two identities yield Z Z Z Z 2 2 α |∇v| dx + β |∇u| dx − (a + d) uvdx = − uv ((1 + c2 )u + c1 v) dx. Ω
Ω
Ω
(3.2)
Ω
Applying the Poincar´e inequality to the left-hand side of (3.2), we have Z Z Z 2 2 α |∇v| dx + β |∇u| dx − (a + d) uvdx Ω Ω ΩZ Z Z ≥ αλ1 v 2 dx + βλ1 u 2 dx − (a + d) uvdx Ω Ω Ω Z Z Z 2 v 2 u + ≥ αλ1 v 2 dx + βλ1 u 2 dx − (a + d) dx 2 Ω Ω Ω 2 Z Z (a + d) a+d 2 = αλ1 − v dx + βλ1 − u 2 dx 2 2 Ω Ω
(3.3)
1 for an arbitrary positive constant . Now, fix a positive constant 0 with 0 ≤ 2αλ a+d ; then it follows obviously that a+d . This is a contradiction to the fact that the right-hand side of (3.2) is (3.3) is non-negative for β ≥ β := 2 0 λ1 negative. (iii) This can be shown like in the proof of (ii). Thus, the proof is omitted.
Remark 3.2. In view of Theorem 3.1, we may conclude that if the cross-diffusion rate of the prey or its predator is sufficiently large, then the prey and predator species cannot coexist. In other words, the large cross-diffusion coefficients α and β tend to mean no positive coexistence. Acknowledgements The authors thank the anonymous referee for his/her valuable comments and suggestions for improving the content of this article. The first author (W. Ko) would like to thank Professor Y. Du for his warm hospitality, mathematical advice, and encouragement in Australia. The first author’s work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2006-214-C00006). References [1] [2] [3] [4] [5] [6] [7]
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