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Positive steady state solutions of a Leslie–Gower predator–prey model with Holling type II functional response and density-dependent diffusion Jun Zhou ∗ School of Mathematics and Statistics, Southwest University, Chongqing, 400715, PR China

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Article history: Received 25 September 2012 Accepted 30 December 2012 Communicated by S. Carl MSC: 35J55 35K55 92C15 92C40

abstract In this paper, we consider a Leslie–Gower predator–prey model with Holling type II functional response and density-dependent diffusion under zero Dirichlet boundary condition. Using degree theory, bifurcation theory, energy estimates and asymptotical behavior analysis, the existence and multiplicity of positive steady state solutions were shown under certain conditions on the parameters. © 2013 Elsevier Ltd. All rights reserved.

Keywords: Leslie–Gower predator–prey model with Holling type II functional response Nonlinear diffusion of fraction type Positive steady state solutions Multiplicity Stability

1. Introduction Consider the following prey–predator model with nonlinear diffusions:

c˜1 w ˜ ˜ ˜ ˜ ˜ ˜ − d 1 u = u a − e u − , 1 u˜ + k˜ 1 ˜ ˜ α˜ ˜ − c2 w − ∆ d + w ˜ = w ˜ b , 2 1 + β˜ u˜ u˜ + k˜ 2 u˜ = w ˜ = 0,

x ∈ Ω, x ∈ Ω,

(1.1)

x ∈ ∂Ω,

where Ω ⊂ RN , N ≥ 1, is a bounded open domain with smooth boundary ∂ Ω ; a˜ , b˜ , e˜ , c˜1 , c˜2 , k˜ 1 , k˜ 2 are positive constants; α˜ and β˜ are nonnegative constants. Problem (1.1) models the interactions between a predator, with population density w( ˜ x), and a prey, with population density u˜ (x), inhabiting the region Ω . In reaction terms, a˜ and b˜ are the growth rates of prey u˜ and predator w ˜ , respectively; e˜ measures the strength of competition among individuals of species u˜ ; c˜1 is the maximum value of the per capita reduction rate of u˜ due to w; ˜ k˜ 1 and k˜ 2 measure the extent to which environment provides protection

∗

Tel.: +86 15925367328. E-mail address: [email protected]

0362-546X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2012.12.014

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J. Zhou / Nonlinear Analysis 82 (2013) 47–65

to prey u˜ and to predator w ˜ , respectively; c˜2 has a similar meaning as c˜1 (see [1–3]). In diffusionterms,positive constants

d1 and d2 represent natural dispersive forces of movements of prey and predator, respectively; ∆

α˜ w ˜

1+β˜ u˜

yields a nonlinear

diffusion of fraction type. This nonlinear diffusion describes a prey–predator relationship such that the diffusion of predator is prevented by the density of prey, and β˜ represents the prevention (see [4,5] for a further ecological background). The homogeneous Dirichlet boundary condition ‘‘u˜ = w ˜ = 0 on ∂ Ω ’’ is sometimes said to correspond to a lethal boundary, which can be considered as such a condition under which neither of the two species can exist on the boundary (see [6] or [7, p. 31]). By rescaling as follows u= c1 =

e˜ d1

u˜ ,

c˜1 e˜ 1 d21

w = w, ˜ ,

c2 =

α=

α˜ d2

c˜2 e˜ d1 d2 (1 + α)

,

,

β= k1 =

d1 β˜ e˜ e˜ k˜ 1 d1

,

,

a= k2 =

a˜ d1

,

e˜ k˜ 2 d1

b=

b˜ d2 (1 + α)

,

,

(1.1) is equivalently rewritten as

c1 w −1u = u a − u − , u+k 1 α c2 w − ∆ 1 + w = ( 1 + α)w b − , 1 + βu u + k2 u = w = 0,

x ∈ Ω, x ∈ Ω,

(1.2)

x ∈ ∂Ω,

where α, β are nonnegative constants, and a, b, c1 , c2 , k1 , k2 are positive constants. The system (1.2) is based on a classical predator–prey model of Leslie and Gower [3] with more reasonable Holling type II functional responses [8] in both prey and predator interaction terms (see [9] for more detailed explanation), and the corresponding ODE system is regarded as one of prototypical predator–prey systems in the ecological studies. The kinetic model of (1.2) was proposed based on the biological fact that if the predator w is more capable of switching from its favorite food, say the prey u, to other food options, then it has better ability to survive when the prey population is low. When there is no cross-diffusion (α = 0), problem (1.2) and its related evolution systems were studied in [10–12]. The first reaction–diffusion system with cross-diffusion was proposed by Shigesada et al. in [13] to investigate the habitat segregation phenomena between two species. Since then, a lot of papers have been devoted to this field. In general, many authors have worked on the following reaction–diffusion system with cross-diffusion

−∆[(d1 + α11 u + α12 w)u] = u(a1 − b1 u ± c1 w), −∆[(d2 + α21 u + α22 w)w] = w(a2 ± b2 u − c2 w), u = v = 0,

x ∈ Ω, x ∈ Ω, x ∈ ∂Ω,

(1.3)

where αij (i, j = 1, 2) are nonnegative constants; a1 , bi , ci , di are positive constants; a2 > 0 for competition case (−c1 , −b2 ), a2 ∈ R for prey–predator case (−c1 , +b2 ). The global existence of solutions for the transient version of (1.3) was proved by Chen and Jüngel in [14] for competition case, while the prey–predator case was studied by Li in [15]. The structure of positive solutions of (1.3) for competition case was studied in [16–25], while the prey–predator case was studied in [26–33]. When cross-diffusion of fractional type is concerned, the following model was concerned in recent years

[(1 + αw) u] =u(a− u − c w), −∆ r −∆ 1 + w = w(b + du − w), 1 + βu u = v = 0,

x ∈ Ω, x ∈ Ω,

(1.4)

x ∈ ∂Ω,

where α, β, r are nonnegative constants; a, c , d are positive constants; b ∈ R. When α = 0, (1.4) was studied in [34], in which the author studied the global bifurcation branch of the positive solutions. Furthermore, by analysis for the shadow systems, the nonlinear diffusion effect on the positive solution branch was derived. When α > 0, the model was studied in [4,35]. By using bifurcation theory, the sufficient conditions to ensure the existence of positive solutions were derived in [35], while the limiting behavior of the positive solutions as α → ∞ or β → ∞ was obtained in [4]. Among other things, we are interested in positive solutions of (1.2), which incorporates the cross-diffusion, Holling type II functional response (see the term uu+wk in the first equation of (1.2)), and the modified Leslie–Gower functional response 1

(see the term uw in the second equation of (1.2)). A solution (u, w) of (1.2) is called a positive solution when u > 0 and +k2 w > 0 in Ω . Hence a positive solution corresponds to a coexistence steady-state of prey and predator. The main concern here is the structure of the set of positive solutions of (1.2) under the combined effect of cross-diffusion, Holling type II functional response, and modified Leslie–Gower functional response. It is obvious that the trivial solution of (1.2) is (0, 0) 2

J. Zhou / Nonlinear Analysis 82 (2013) 47–65

a

b=

(a) –

R1

D1

a = g(b)

a

b=

D2

49

(a)

D3

a = g(b) –

D4

R2

b

D5

D6

R3 b

Fig. 1. Illustration of the parameter regions of (a, b) in the main results.

and semi-trivial solutions of (1.2) are (θa , 0) (if a > λ1 ) and (0, k2 θb /c2 ) (if b > λ1 ). Here λ1 is the principal eigenvalue of −∆ with zero Dirichlet boundary, θa and θb are the unique positive solutions of (2.3) with l = a or l = b, respectively. Next, we introduce the main results of this paper. To this end, we define some sets:

λ1 ; 1+α R2 := {(a, b) ∈ R × R : 0 < a < λ1 , 0 < b < λ1 } ; R3 := {(a, b) ∈ R × R : 0 < a < λ1 , b > λ1 } ; λ1 < b < χ (a) − ϵa ; D1 := (a, b) ∈ R × R : a > λ1 , 1+α D2 := {(a, b) ∈ R × R : a > λ1 , χ (a) − ϵa < b < χ (a)} ; D3 := {(a, b) ∈ R × R : a > λ1 , χ (a) < b < λ1 } ; D4 := {(a, b) ∈ R × R : a > g (b), b > λ1 } ; D5 := {(a, b) ∈ R × R : g (b) − ϵb < a < g (b), b > λ1 } ; D6 := {(a, b) ∈ R × R : λ1 − a < g (b) − ϵb , b > λ1 } ,

R1 :=

(a, b) ∈ R × R : a > λ1 , 0 < b <

where b = χ (a) is the inverse function of a = f (b; α, β), and a = f (b; α, β) is defined in Lemma 3.2, g (b) is defined in Lemma 3.3, ϵa and ϵb are defined in Theorems 5.3 and 5.5. Our main results can be summarized as follows (see Fig. 1): 1. the trivial solution (0, 0) is asymptotically stable if (a, b) ∈ int(R2 ), and it is unstable if (a, b) ̸∈ R2 (see Theorem 3.1); 2. if α = 0 or β = 0, the semi-trivial steady state (θa , 0) is locally asymptotically stable if (a, b) ∈ int(R1 ∪ D1 ∪ D2 ∪ D3 ), while it is unstable if (a, b) ∈ int(D4 ∪ D5 ∪ D6 ); if α > 0 and β > 0, the semi-trivial steady state (θa , 0) is locally asymptotically stable if (a, b) ∈ int(R1 ∪ D1 ∪ D2 ), and it is unstable if (a, b) ∈ int(D3 ∪ D4 ∪ D5 ∪ D6 ) (see Corollary 3.4); 3. the semi-trivial solution (0, k2 θb /c2 ) is locally asymptotically stable if (a, b) ∈ int(D5 ∪ D6 ), and it is unstable if (a, b) ∈ int(D4 ) (see Corollary 3.4); 4. problem (1.2) possesses at least one positive solution if (a, b) ∈ int(D3 ∪ D4 ) (see Theorem 4.2); 5. problem (1.2) possesses at least two positive solutions if (a, b) ∈ int D2 and b1 < 0, where b1 is defined in (5.13); or (a, b) ∈ int D5 and a1 < 0, where a1 is defined in (5.4) (see Theorems 5.3 and 5.5); 6. the asymptotic behavior of coexistence region int(D3 ∪ D4 ) is in Fig. 2. The organization of the remaining part of the paper is as follows. In Section 2, we give some preliminaries, which are essential tools in our later study. In Section 3, we consider the stability results about the trivial and semi-trivial solutions. In Section 4, we study the existence of positive solutions by using degree theory. Finally, the multiplicity of positive solutions is established in Section 5. 2. Preliminaries In this section we list some notation, definitions and well-known facts which will be used in the sequel. We use ∥ · ∥X as the norm of Banach space X , ⟨·, ·⟩ as the duality pair of a Banach space X and its dual space X ∗ . For a linear operator L,

50

J. Zhou / Nonlinear Analysis 82 (2013) 47–65

Fig. 2. Asymptotic behavior of the coexistence region, where the left is α → ∞, the middle is β → ∞, and the right is both α → ∞ and β → ∞, where a = f (b; α, β), a = f∗ (b; β), and a = g (b) were defined in Lemmas 3.2 and 3.3.

we use N (L) as the null space of L and R(L) as the range space of L, and we use L[w] to denote the image of w under the linear mapping L. For a multilinear operator L, we use L[w1 , w2 , . . . , wk ] to denote the image of (w1 , w2 , . . . , wk ) under L, and when w1 = w2 = · · · = wk , we use L[w1 ]k instead of L[w1 , w1 , . . . , w1 ]. For a nonlinear operator F , we use Fu as the partial derivative of F with respect to argument u. First we recall some well-known abstract bifurcation theorems. Consider an abstract equation F (λ, u) = 0, where F : R × X → Y is a nonlinear differential mapping, and X , Y are Banach spaces such that X is continuously embedding in Y . The following bifurcation and stability theorems were obtained in [36–38] (see also [39,40]). Theorem 2.1. Let U be a neighborhood of (λ0 , u0 ) in R × X , and let F : U → Y be a twice continuously differentiable mapping. Assume that F (λ, u0 ) = 0 for all (λ, u0 ) ∈ U. At (λ0 , u0 ), F satisfies dim N (Fu (λ0 , u0 )) = codim R(Fu (λ0 , u0 )) = 1 and Fλu (λ0 , u0 )[w0 ] ̸∈ R(Fu (λ0 , u0 )). Here N (Fu (λ0 , u0 )) = span{w0 }. Let Z be the complement of span {w0 } in X . Then the solution set of F (λ, u) = 0 near (λ0 , u0 ) consists precisely of the curves u = u0 and Γ := {(λ(s), u(s)) : s ∈ I = (−ϵ, ϵ)}, where λ : I → R, z : I → Z are C 1 functions such that u(s) = u0 + sw0 + sz (s), λ(0) = λ0 , z (0) = 0, and

λ′ (0) = −

⟨ℓ, Fuu (λ0 , u0 )[w0 , w0 ]⟩ , 2⟨ℓ, Fλu (λ0 , u0 )[w0 ]⟩

where ℓ ∈ Y ∗ satisfies R(Fu (λ0 , u0 )) = {φ ∈ Y : ⟨ℓ, φ⟩ = 0}. Moreover if in addition, Fu (λ, u) is a Fredholm operator for all (λ, u) ∈ U, then the bifurcation curve Γ is contained in Σ , which is a connected component of S, where S := {(λ, u) ∈ U : F (λ, u) = 0, u ̸= u0 }; and either Σ is not compact in U, or Σ contains a point (λ∗ , u0 ) with λ∗ ̸= λ0 . Theorem 2.2. Assume that all assumptions in Theorem 2.1 are satisfied, and let {λ(t ), u(t )} be the solution curve in Theorem 2.1. Then there exists C 2 functions m : (λ0 − ϵ, λ0 + ϵ) → R, z : (λ0 − ϵ, λ0 + ϵ) → X , µ : (−δ, δ) → R, and w : (−δ, δ) → X such that Fu (λ, u0 )z (λ) = m(λ)z (λ),

λ ∈ (λ0 − ϵ, λ0 + ϵ), Fu (λ(t ), u(t ))w(t ) = µ(t )w(t ), t ∈ (−δ, δ),

where m(λ0 ) = µ(0) = 0, z (λ0 ) = w(0) = w0 . Moreover, near t = 0 the functions µ(t ) and −t λ′ (t )m′ (λ0 ) have the same zeros and, whenever µ(t ) ̸= 0, the same sign. More precisely,

−t λ′ (t )m′ (λ0 ) = 1. t →0 µ(t ) lim

Next we recall some well-known facts about linear elliptic equations and diffusive logistic equation. For each q ∈ C (Ω ), let λ1 (q) be the principal eigenvalue of

−1u + q(x)u = λu, u = 0,

x ∈ Ω, x ∈ ∂Ω.

(2.1)

J. Zhou / Nonlinear Analysis 82 (2013) 47–65

51

As is well known, the principal eigenvalue λ1 (q) is given by the following variational characterization:

λ1 (q) =

inf

φ∈H01 (Ω ),∥φ∥L2 (Ω ) =1 Ω

(|∇φ|2 + q(x)φ 2 )dx.

We denote λ1 (0) by λ1 and let φ1 (x) be the positive eigenfunction corresponding to λ1 with ∥φ1 ∥L2 (Ω ) = 1. Furthermore, the principal eigenvalue λ1 (q) has some useful properties as follows (see [16, Proposition A.1] or [41, Proposition 1.1]). Theorem 2.3. (i) If qi ∈ C (Ω ) (i = 1, 2) satisfy q1 ≥ q2 in Ω and q1 ̸≡ q2 , then λ1 (q1 ) > λ1 (q2 ). (ii) For qn ∈ C (Ω ) and q ∈ C (Ω ), let φn ∈ H01 (Ω ) and φ ∈ H01 (Ω ) be the corresponding eigenfunctions of (2.1) satisfying ∥φn ∥L2 (Ω ) = ∥φ∥L2 (Ω ) = 1, where n ∈ N. If limn→∞ ∥qn − q∥L∞ (Ω ) = 0, then limn→∞ λ1 (qn ) = λ1 (q) and limn→∞ φn = φ strongly in H01 (Ω ). (iii) Let (c , d) be an open interval and assume that a mapping β → qβ is continuously differentiable from (c , d) to C (Ω ) with respect to the supremum norm. If φβ ∈ H01 (Ω ) with ∥φβ ∥L2 (Ω ) = 1 is the unique positive eigenfunction corresponding to λ1 (qβ ), then β → λ1 (qβ ) is continuously differentiable from (c , d) to R and d dβ

λ1 (qβ ) =

Ω

∂ qβ 2 φ dx. ∂β β

For q ∈ C (Ω ), let p be a sufficiently large constant such that p − q(x) > 0 for any x ∈ Ω . Define a bounded linear operator T : C (Ω ) → C (Ω ) by u = T v = (−∆ + pI )−1 (p − q(x))v , where u ∈ C (Ω ) is the unique solution of the following problem

−1u + pu = (p − q(x))v, u = 0,

x ∈ Ω, x ∈ ∂Ω.

(2.2)

Let r (T ) be the spectral radius of T . Then the relationship between λ1 (q) and r (T ) can be given as follows (see [42, Proposition 1] or [43, Lemmas 2.1 and 2.3]). Theorem 2.4. Let q ∈ C (Ω ) and let p be a sufficiently large number such that p > q(x) for any x ∈ Ω . Then we have (i) λ1 (q) > 0 if and only if r ((−∆ + pI )−1 (p − q(x))) < 1; (ii) λ1 (q) < 0 if and only if r ((−∆ + pI )−1 (p − q(x))) > 1; (iii) λ1 (q) = 0 if and only if r ((−∆ + pI )−1 (p − q(x))) = 1. Consider the following steady state problem for the logistic equation with linear diffusion

−1u = u(l − u), u = 0,

x ∈ Ω, x ∈ ∂Ω,

(2.3)

where l is a positive constant. Then the following results are well known (see [42, Lemma 1] and [44, Propositions 6.1–6.4]). Theorem 2.5. (i) If l ≤ λ1 , then (2.3) has no nontrivial solutions. (ii) If l > λ1 , then there exists a unique positive solution θl (x) of (2.3) satisfying 0 < θl (x) < l for all x ∈ Ω . (iii) liml→λ+ θl (x) = 0 uniformly in Ω . More precisely, 1

θl =

Ω

−1 φ13 dx (l − λ1 )φ1 + o(l − λ1 ) as l → λ+ 1 .

(iv) liml→∞ θl (x) = ∞ and liml→∞ θl (x)/l = 1 uniformly in K , where K is any compact subset of Ω . (v) The mapping l → θl is C 1 from (λ1 , ∞) to C (Ω ) and θl (x) is strictly increasing with respect to l. More precisely,

∂θl = (−∆ + (2θl − l)I )−1 θl , ∂l where (−∆ + (2θl − l)I )−1 is the inverse operator of −∆ + (2θl − l)I with zero Dirichlet boundary condition. Finally, we introduce some concepts of fixed point index theory in a cone [45]. Let E be a Banach space and W ⊂ E be a closed convex set. W is called a total wedge in E if γ W ⊂ W for all γ ≥ 0 and W − W = E. For y ∈ W , define Wy = {x ∈ E : y + γ x ∈ W for some γ > 0} and Sy = {x ∈ W y : −x ∈ W y }. Then W y is a wedge containing W , y, −y, while Sy is a closed subset of E containing y. Let T be a compact linear operator on E which satisfies T (W y ) ⊂ W y . We say that T has property a on W y if there exist t ∈ (0, 1) and ω ∈ W y \ Sy such that (I − tT )ω ∈ Sy . Let A : W → W be a compact operator with a fixed point y ∈ W , and let D be a relatively open subset of W such that A has no fixed point on the boundary of D. We denote by degW (I − A, D) the degree of I − A in D relative to W , and by indexW (A, y) the fixed point index of A at y relative to W . The following result is well-known (see [45], [43, Theorem D] or [46, Lemma 4.1]).

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J. Zhou / Nonlinear Analysis 82 (2013) 47–65

Theorem 2.6. Assume that W is a total wedge, and let A : W → W be a compact operator with a fixed point y ∈ W and it is Fréchet differentiable at y. Let L = A′ (y) be the Fréchet derivative of A at y. Then L maps W y into itself. Moreover, if I − L is invertible on W y , then the following results hold. (i) If L has property a on W y , then indexW (A, y) = 0. (ii) If L does not have property a on W y , then indexW (A, y) = (−1)σ , where σ is the sum of multiplicities of all eigenvalues of L which is greater than 1. 3. Stability analysis of the trivial and semi-trivial solutions of (1.2) In this section, we analyze the trivial and semi-trivial solutions of (1.2). It is obvious that the trivial solution of (1.2) is

(0, 0) and semi-trivial solutions of (1.2) are (θa , 0) (if a > λ1 ) and (0, k2 θb /c2 ) (if b > λ1 ). Here θa and θb are the unique positive solutions of (2.3) with l = a or l = b, respectively. The main result of this section is the following theorem. Theorem 3.1. Consider the system (1.2). (i) The trivial steady state (0, 0) is locally asymptotically stable if a < λ1 and b < λ1 , while it is unstable if a > λ1 or b > λ1 .

(ii) Assume that a > λ1 . Then the semi-trivial steady state (θa , 0) is locally asymptotically stable if λ1 −

(1+α)b(1+βθa ) 1+α+βθa

> 0,

(1+α)b(1+βθ ) while it is unstable if λ1 − 1+α+βθ a < 0. a (iii) Assume that b > λ1 . Then the semi-trivial steady state (0, k2 θb /c2 ) is locally asymptotically stable if a < λ1 it is unstable if a > λ1

c1 k 2 θ b c2 k1

c1 k2 θb c2 k 1

, while

.

Proof. We only prove case (ii) since the proofs of the other two cases are similar. From the linearization principle, the stability of (θa , 0) is determined by studying the following eigenvalue problem

c1 θa ψ = λφ, −1φ + (2θa − a)φ + θa + k1 α ψ − (1 + α)bψ = λψ, −∆ 1 + 1 + βθa φ = ψ = 0,

x ∈ Ω, x ∈ Ω,

(3.1)

x ∈ ∂Ω.

Since (3.1) is not completely coupled, we only need to consider the following two eigenvalue problems

−1φ + (2θa − a)φ = λφ, φ = 0,

x ∈ Ω, x ∈ ∂Ω,

(3.2)

and

−∆ 1 +

α 1 + βθa

ψ − (1 + α)bψ = λψ,

x ∈ Ω,

ψ = 0,

(3.3)

x ∈ ∂Ω.

Then it follows from [47, p. 76] that the eigenvalues of (3.1) are the union of the eigenvalues of (3.2) and (3.3). Denote the principal eigenvalue of (3.2) and (3.3) by λ∗ and λ∗ , respectively. Then

λ∗ = λ1 (2θa − a) > λ1 (θa − a) = 0. In order to determine the sign of λ∗ , letting ϕ =

−1ϕ −

ϕ = 0,

1+α+βθa 1+βθa

ψ , (3.3) is equivalent to

(1 + α)b(1 + βθa ) 1 + βθa ϕ=λ ϕ, 1 + α + βθa 1 + α + βθa

x ∈ Ω, x ∈ ∂Ω.

By the variational characterization of principal eigenvalue, we have

λ = ∗

inf

ϕ∈H01 (Ω ), ϕ̸≡0

Ω

b(1+βθa ) 2 |∇ϕ|2 dx − Ω (1+α) ϕ dx 1+α+βθa . 1+βθa ϕ 2 dx Ω 1+α+βθa

J. Zhou / Nonlinear Analysis 82 (2013) 47–65

53

a Since 0 < 1+α+βθ < 1 for x ∈ Ω , a

1+βθ

(1 + α)b(1 + βθa ) > 0, 1 + α + βθa (1 + α)b(1 + βθa ) if λ1 − < 0. 1 + α + βθa (1+α)b(1+βθ ) Combining the above results, one can see that if λ1 − 1+α+βθ a > 0, then all eigenvalues of (3.1) are positive, and thus a b(1+βθa ) (θa , 0) is locally asymptotically stable. On the other hand, if λ1 − (1+α) < 0, then (3.1) has a negative eigenvalue, 1+α+βθa which implies the instability of (θa , 0). (1 + α)b(1 + βθa ) > λ − , 1 1 + α + βθa ∗ λ (1 + α)b(1 + βθa ) < λ1 − , 1 + α + βθa

if λ1

−

Next we make some explanations to Theorem 3.1. It is obvious that

(1+α)b(1+βθa ) 1+α+βθa

= λ1 − b if α = 0 or β = 0;

b(1+βθa ) − (1+α) 1+α+βθa

< 0 if b > λ1 ;

(1+α)b(1+βθa ) 1+α+βθa

> 0 if b <

1. λ1 − 2. λ1

3. λ1 −

λ1 1+α

.

Thus, 1. when α = 0 or β = 0, (θa , 0) is locally asymptotically stable if b < λ1 , while it is unstable if b > λ1 ; 2. when α > 0 and β > 0, (θa , 0) is locally asymptotically stable if b < λ1 /(1 + α), while it is unstable if b > λ1 . In order to analyze the case of λ1 /(1 + α) ≤ b ≤ λ1 when α > 0 and β > 0, we define a curve C1 in the (b, a)-plane by

(1 + α)b(1 + βθa ) 2 = 0, a ≥ λ1 , λ1 /(1 + α) ≤ b ≤ λ1 , C1 = (b, a) ∈ R : λ1 − 1 + α + βθa

(3.4)

where θa is the unique positive solution of (2.3) with l = a if a > λ1 and θa = 0 if a = λ1 . Then we have the following lemma, which describes the profile of C1 . Lemma 3.2. Assume α > 0 and β > 0 and let C1 be the curve defined by (3.4). (i) Then curve C1 can be expressed as

C1 = {(b, a) ∈ R2 : a = f (b; α, β), λ1 /(1 + α) < b ≤ λ1 }. Here f (b; α, β) is a strictly decreasing C 1 function. Furthermore, it satisfies the following properties: lim

b→λ1 /(1+α)

f (b; α, β) = ∞,

f (λ1 ; α, β) = λ1 ,

f ′ (λ1 ; α, β) = −

1+α

λ1 αβ

.

(ii) Let K1 be any compact subset of (λ1 /(1 + α), λ1 ], K2 be any compact subset of (λ1 , ∞), and K3 be any compact subset of (0, ∞). Then, 1. limα→∞ f (b; α, β) = f∗ (b; β) uniformly for b ∈ K1 and β ∈ K3 ; λ1 uniformly for a ∈ K2 and α ∈ K3 ; 2. limβ→∞ f −1 (a; α, β) = 1+α 3. limα→∞ limβ→∞ f −1 (a; α, β) = 0 uniformly for a ∈ K2 . Here b = f −1 (a; α, β), a ≥ λ1 , is the inverse function of a = f (b; α, β) and f∗ (b; β) is a strictly decreasing C 1 function. Furthermore, it satisfies the following properties: lim f∗ (b; β) = ∞,

b→0

f∗ (λ1 ; β) = λ1 ,

f∗′ (λ1 ; β) = −

1

λ1 β

.

Proof. We first prove (i). Set S (a, b) = λ1 (ϕ(1 + βθa , b)),

(a, b) ∈ [λ1 , ∞) × (λ1 /(1 + α), λ1 ],

(3.5)

where

ϕ(z , b) = −

(1 + α)bz . z+α

By Theorem 2.5, 1 + βθa is a continuous and strictly increasing function with respect to a such that lima→λ1 (1 + βθa (x)) = 1 uniformly in Ω and lima→∞ (1 + βθa (x)) = ∞ uniformly in any compact subsets of Ω . Since ϕ(z , b) is strictly decreasing with respect to both z and b, it follows from Theorem 2.3 that S (a, b) is strictly decreasing with respect to both a and b.

54

J. Zhou / Nonlinear Analysis 82 (2013) 47–65

Since lima→λ1 ϕ(1 + βθa , b) = −b uniformly in Ω , it follows from Theorem 2.3(ii) that S (λ1 , b) = lim λ1 (ϕ(1 + βθa , b)) = λ1 − b.

(3.6)

a→λ1

By the variational characterization of principal eigenvalue, S (a, b) =

inf

φ∈H01 (Ω ),∥φ∥L2 (Ω ) =1

|∇φ|2 dx +

Ω

Ω

ϕ(1 + βθa , b)φ 2 dx .

Recall φ1 is the positive eigenfunction of λ1 with ∥φ1 ∥L2 (Ω ) = 1. Then S (a, b) ≤

|∇φ1 |2 dx +

Ω

Ω

ϕ(1 + βθa , b)φ12 dx.

Since lima→∞ ϕ(1 + βθa , b) = −(1 + α)b for any x ∈ Ω , by Lebesgue’s dominate convergence theorem, lim S (a, b) ≤ lim

a→∞

a→∞

Ω

|∇φ1 |2 dx +

ϕ(1 + βθa , b)φ12 dx

= ∥∇φ1 ∥2L2 (Ω ) − (1 + α)b

Ω

Ω

φ12 dx = λ1 − (1 + α)b.

(3.7)

On the other hand, S (a, b) = ∥∇φa,b ∥2L2 (Ω ) +

Ω

ϕ(1 + βθa , b)φa2,b dx,

(3.8)

where φa,b (x) is the positive eigenfunction corresponding to λ1 (ϕ(1 + βθa , b)) with ∥φa,b ∥L2 (Ω ) = 1. Then

∥∇φa,b ∥2L2 (Ω ) = S (a, b) −

Ω

ϕ(1 + βθa , b)φa2,b dx ≤ λ1 + (1 + α)b.

By the reflexive property of H01 (Ω ), there exists a function φ∞,b ∈ H01 (Ω ) with ∥φ∞,b ∥L2 (Ω ) = 1 and a subsequence of {φa,b }a , denoted by {φa,b }a again, such that 1. φa,b ⇀ φ∞,b weakly in H01 (Ω ) as a → ∞, 2. φa,b → φ∞,b strongly in L2 (Ω ) as a → ∞. Then, by (3.8), lim S (a, b) ≥ ∥∇φ∞,b ∥2L2 (Ω ) − (1 + α)b

a→∞

Ω

2 φ∞, b dx ≥ λ1 − (1 + α)b.

(3.9)

Thus, by (3.7) and (3.9), lim S (a, b) = λ1 − (1 + α)b.

a→∞

(3.10)

It follows from (3.6) and (3.10) that for each b ∈ (λ1 /(1 + α), λ1 ], there exists a unique ab ∈ [λ1 , ∞) such that S (ab , b) = 0. Define a function f (b; α, β) by f (b; α, β) = ab ,

b ∈ (λ1 /(1 + α), λ1 ].

Clearly f (b; α, β) is a continuous function for b ∈ (λ1 /(1 + α), λ1 ], and it satisfies f (λ1 ; α, β) = λ1 by the fact that S (λ1 , λ1 ) = 0. By Theorems 2.3 and 2.5, S (a, b) satisfies that

∂θa 2 φ ∂ϕ(1 + βθa , b) 2 ∂ a a,b φa,b dx = −(1 + α)bαβ dx < 0, ∂ a ( 1 + α + βθa )2 Ω Ω ∂S ∂ϕ(1 + βθa , b) 2 1 + βθa ( a, b ) = φa,b dx = −(1 + α) φ 2 dx < 0. ∂b ∂ b 1 + α + βθa a,b Ω Ω ∂S ( a, b ) = ∂a

By the implicit function theorem, f (b; α, β) is a C 1 -function for b ∈ (λ1 /(1 + α), λ1 ) and ∂S (f (b; α, β), b) f ′ (b; α, β) = − ∂∂ Sb < 0. (f (b; α, β), b) ∂a

By Theorems 2.5 and 2.3, the following results hold true (see [41, p. 432]):

(3.11) (3.12)

J. Zhou / Nonlinear Analysis 82 (2013) 47–65

1. lima→λ1 2. limb→λ1 3. lima→λ1

55

θa (x) = 0 uniformly in Ω ; φf (b;α,β),b = φ1 strongly in H01 (Ω ); −1 ∂θa = Ω φ13 dx φ1 uniformly in Ω . ∂a

From (3.11) and (3.12), it follows that

λ1 αβ ∂S (λ1 , λ1 ) = − , ∂a 1+α

∂S (λ1 , λ1 ) = −1, ∂b

and thus ∂S

(λ1 , λ1 ) 1+α f ′ (λ1 ; α, β) = − ∂∂ Sb =− . λ1 αβ (λ1 , λ1 ) ∂a Next we prove limb→λ1 /(1+α) f (b; α, β) = ∞. Assume on the contrary that limb→λ1 /(1+α) f (b; α, β) = a∗ < ∞. Since f (b; α, β) is strictly decreasing, it follows from Theorem 2.3(i) and the monotone property of ϕ that 0 = S (f (b; α, β), b) > S (a∗ , b) = λ1 (ϕ(1 + βθa∗ , b)). Since θa∗ (x) < a∗ for all x ∈ Ω , then

ϕ(1 + βθa∗ , b) > −

(1 + α)b(1 + β a∗ ) . 1 + α + β a∗

Consequently

(1 + β a∗ ) (1 + α)b(1 + β a∗ ) → λ1 × 1 − >0 0 > λ1 (ϕ(1 + βθa∗ , b)) > λ1 − 1 + α + β a∗ 1 + α + β a∗ as b → λ1 /(1 + α), which is a contradiction. So, (i) is true. By Lebesgue’s dominate convergence theorem,

1. limα→∞ λ1 −

(1+α)b(1+βθa ) 1+α+βθa

= λ1 (−b(1 + βθa )) uniformly for b ∈ K1 and β ∈ K3 ; (1+α)b(1+βθa ) 2. limβ→∞ λ1 − 1+α+βθ = λ1 − (1 + α)b uniformly for a ∈ K2 and α ∈ K3 . a

Then the conclusions of (ii) follow by similar analysis as (i).

Similarly, we have the following lemma. Lemma 3.3. Define a curve C2 in the (b, a)-plane by

C2 = (b, a) ∈ R : a = λ1 2

c1 k2 θb c2 k1

, b ≥ λ1 ,

where θb is the unique positive solution of (2.3) with l = b if b > λ1 and θb = 0 if b = λ1 . Then the curve C2 can be expressed as

C2 = {(b, a) ∈ R2 : a = g (b), b ≥ λ1 }. Here g (b) is a strictly increasing C 1 function. Furthermore, it satisfies the following properties: g (λ1 ) = λ1 ,

g ′ (λ1 ) =

c1 k2 c2 k1

,

lim g (b) = ∞.

b→∞

By virtue of the above analysis and Theorem 3.1, the stability results for the semi-trivial steady states (θa , 0) and

(0, k2 θb /c2 ) read as follows (see Fig. 1, where b = χ (a) is the inverse function of a = f (b; α, β)).

Corollary 3.4. Let a = f (b; α, β) and a = f∗ (b; β) be the functions defined in Lemma 3.2, and let a = g (b) be the function defined in Lemma 3.3. Then, we have the following. (i) Assume a > λ1 . 1. If α = 0 or β = 0, the semi-trivial steady state (θa , 0) is locally asymptotically stable if b < λ1 , while it is unstable if b > λ1 . 2. For fixed α > 0 and β > 0, the semi-trivial steady state (θa , 0) is locally asymptotically stable if 0 < b ≤ λ1 /(1 + α) or a < f (b; α, β), while it is unstable if a > f (b; α, β) or b ≥ λ1 . 3. For fixed β > 0, the stable region of (θa , 0) shrinks to {(b, a) : 0 < b < λ1 , λ1 < a < f∗ (b; β)} and the unstable region of (θa , 0) expands to {(b, a) : 0 < b < λ1 , a > f∗ (b; β)} ∪ {(b, a) : b ≥ λ1 , a > λ1 } as α → ∞.

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J. Zhou / Nonlinear Analysis 82 (2013) 47–65

4. For fixed α > 0, the stable region of (θa , 0) shrinks to {(b, a) : 0 < b < λ1 /(1 + α), a > λ1 } and the unstable region of (θa , 0) expands to {(b, a) : b > λ1 /(1 + α), a > λ1 } as β → ∞. 5. The stable region of (θa , 0) disappears and the unstable region of (θa , 0) expands to {(b, a) : b > 0, a > λ1 } as both α and β tend to ∞. (ii) Assume b > λ1 . Then the semi-trivial steady state (0, k2 θb /c2 ) is locally asymptotically stable if a < g (b), while it is unstable if a > g (b). 4. Existence of positive solutions By using the transformation

v = 1+

α w, 1 + βu

(1.2) can be rewritten as follows:

−1u = F (u, w(u, v)) := uρ(u, v), −1v = G(u, w(u, v)) := vϱ(u, v), u = v = 0,

x ∈ Ω, x ∈ Ω, x ∈ ∂Ω,

(4.1)

where c1 w

F (u, w) = u a − u − G(u, w) = (1 + α)w

w(u, v) =

u + k1

b−

(1 + β u)v 1 + α + βu

,

c2 w

u + k2

,

, c1 (1 + β u)v

ρ(u, v) = a − u −

, (u + k1 )(1 + α + β u) (1 + α)(1 + β u) c2 (1 + β u)v ϱ(u, v) = b− . 1 + α + βu (u + k2 )(1 + α + β u)

It is obvious that (1.2) has a positive solution if and only if (4.1) has a positive solution. So, we will study the existence of positive solutions of (1.2) through (4.1). Problem (4.1) has no positive solutions if a ≤ λ1 or b ≤ λ1 /(1 + α). Indeed let (u, v) be a positive solution of (4.1), then

−1u < au, −1v < b(1 + α)v, u = v = 0,

x ∈ Ω, x ∈ Ω, x ∈ ∂Ω,

and it follows from the property of the principal eigenvalue that a > λ1 and b > λ1 /(1 + α). Since we are interested in positive solutions, throughout this section, we assume that a > λ1 and b > λ1 /(1 + α) hold. By maximum principle, one can get that all nonnegative solutions (u, v) of (4.1) satisfy u(x) ≤ a,

v(x) ≤ M (a, b) :=

b(a + k2 )(1 + α + aβ) c2

,

x ∈ Ω.

(4.2)

Now we introduce the following notations:

• E = C0 (Ω ) × C0 (Ω ), where C0 (Ω ) = {u ∈ C (Ω ) : u(x) = 0, x ∈ ∂ Ω }. It is obvious that E is a Banach space with the norm

∥(u, v)∥E = max |u(x)| + max |v(x)|. x∈Ω

x∈Ω

• W = K × K , where K = {u ∈ C (Ω ) : u(x) ≥ 0 for x ∈ Ω }. • D = {(u, v) ∈ W : u(x) < a + 1, v(x) < M (a, b) + 1 for x ∈ Ω }, where M (a, b) are defined in (4.2). From (4.2), nonnegative solutions of (4.1) must be in D. Define a positive and compact operator A : D → E by A(u, v) := (−∆ + pI )

−1

F (u, w(u, v)) + pu G(u, w(u, v)) + pv

,

where p is a sufficiently large number such that p + ρ(u, v) > 0

and

p + ϱ(u, v) > 0

for (u, v) ∈ D.

(4.3)

J. Zhou / Nonlinear Analysis 82 (2013) 47–65

57

Note that (4.1) is equivalent to (u, v) = A(u, v) by the regularity of elliptic equations, and therefore it suffices to prove that A has a nontrivial fixed point in D in order to show the existence of positive solutions of (4.1). To this end we need to compute the fixed point index of the trivial and semi-trivial solutions of (4.1). It is easy to see that (4.1) has a trivial solution (u, v) = (0, 0) and two semi-trivial solutions (θa , 0) for a > λ1 and (0, k2 (1 + α)θb /c2 ) for b > λ1 . Moreover the following lemma holds. Lemma 4.1. The following conclusions hold true. (i) degW (I − A, D) = 1. (ii) indexW (A, (0, 0)) = 0 if a > λ1 and b ̸= λ1 . (iii) Assume a > λ1 . Then, (1+α)b(1+βθ ) 1. indexW (A, (θa , 0)) = 1 if λ1 − 1+α+βθ a > 0;

a (1+α)b(1+βθa ) < 0. 2. indexW (A, (θa , 0)) = 0 if λ1 − 1+α+βθ a (iv) Assume b > λ1 . Then, c k θ 1. indexW (A, (0, k2 (1 + α)θb /c2 )) = 1 if a < λ1 1c 2k b ; 21 c k θ 2. indexW (A, (0, k2 (1 + α)θb /c2 )) = 0 if a > λ1 1c 2k b . 2 1

Proof. (i) For each t ∈ [0, 1], we define a positive and compact operator At : D → E by At (u, v) = (−∆ + pI )−1

tF (u, w(u, v)) + pu tG(u, w(u, v)) + pv

.

Then A1 = A, At has no fixed point on ∂ D, and At (D) ⊂ W . Thus degW (I − At , D) is well defined for all t ∈ [0, 1]. By the homotopy invariance of Leray–Schauder degree and (0, 0) is the only fixed point of A0 in D, we obtain that degW (I − A, D) = degW (I − A0 , D) = indexW (A0 , (0, 0)). Set L0 = A0(u,v) (0, 0) = (−∆ + pI )

−1

p 0

0 . p

It is easy to see that I − L0 is invertible on W (0,0) = K × K and r (L0 ) < 1 by Theorem 2.4(i). Since r (L0 ) < 1, then L0 does not have property a on W (0,0) . Thus indexW (A0 , (0, 0)) = 1 by Theorem 2.6(ii). (ii) Let L = A(u,v) (0, 0), then L = (−∆ + pI )−1

p + Fu + Fw wu Gu + Gw wu

Fw wv p + Gw wv

(u,v)=(0,0)

= (−∆ + pI )−1

p+a 0

0 p+b

.

Assume that L(ξ , η) = (ξ , η) for some (ξ , η) ∈ W (0,0) = K × K . Then it is easy to verify that ξ = η ≡ 0 since a ̸= λ1 and b ̸= λ1 . Thus I − L is invertible on W (0,0) . Since a > λ1 , by Theorem 2.4 (ii), we see that ra := r ((−∆ + pI )−1 (p + a)) > 1 and ra is the principal eigenvalue of the operator (−∆ + pI )−1 (p + a) with a corresponding eigenfunction φa (x) > 0 in Ω and φa |∂ Ω = 0. Set ta = 1/ra ∈ (0, 1), then (φa , 0) ̸∈ S(0,0) = {(0, 0)}, but (I − ta L)(φa , 0) = (0, 0) ∈ S(0,0) . This shows that L has property a, and thus indexW (A, (0, 0)) = 0 by Theorem 2.6(i). (iii) Let L = A(u,v) (θa , 0), then L = (−∆ + pI )

−1

p + Fu + Fw wu Gu + Gw wu

Fw wv p + Gw wv

p − (2θa − a)

= (−∆ + pI )−1

0

−

(u,v)=(θa ,0)

c1 θa (1 + βθa )

(θa + k1 )(1 + α + βθa ) . (1 + α)b(1 + βθa ) p+ 1 + α + βθa

Assume that L(ξ , η) = (ξ , η) for some (ξ , η) ∈ W (θa ,0) = C (Ω ) × K , i.e. (ξ , η) satisfies

c1 θa (1 + βθa ) −1ξ + (2θa − a)ξ = − η, (θ + k1 )(1 + α + βθa ) a (1 + α)b(1 + βθa ) −1η = η, 1 + α + βθa ξ = η = 0,

x ∈ Ω, x ∈ Ω, x ∈ ∂Ω.

58

J. Zhou / Nonlinear Analysis 82 (2013) 47–65

(1+α)b(1+βθ ) (1+α)b(1+βθ ) Since λ1 − 1+α+βθ a ̸= 0, then η ≡ 0, and then ξ ≡ 0 follows by λ1 (2θa − a) > 0. If λ1 − 1+α+βθ a > 0, one can a a (1+α)b(1+βθ ) show indexW (A, (θa , 0)) = 1 by the similar arguments as (i). If λ1 − 1+α+βθ a < 0, one can show indexW (A, (θa , 0)) = a 0 by the similar arguments as (ii). For the case (iv), the proof is similar to (iii).

The following existence theorem is a consequence of Lemma 4.1. Theorem 4.2. Assume that a > λ1 . Then (1.2) (or equivalently (4.1)) admits a positive steady state if one of the following conditions holds.

b(1+βθa ) < b ≤ λ1 and λ1 − (1+α) < 0; 1+α+βθa c1 k2 θb . (ii) b > λ1 and a > λ1 c k 2 1 (i)

λ1 1+α

Proof. First we consider the case λ1 /(1 + α) < b < λ1 and λ1 −

(1+α)b(1+βθa ) 1+α+βθa

< 0. Assume on the contrary that (4.1) has

no positive solution. Since a > λ1 and b < λ1 , (4.1) has only a trivial solution (0, 0) and a semi-trivial solution (θa , 0). Then it follows from the definition of degree (see, e.g., [48]) that degW (I − A, D) = indexW (A, (0, 0)) + indexW (A, (θa , 0)).

(4.4)

By Lemma 4.1, the left-hand side of (4.4) is equal to one, while the right-hand side of (4.4) is equal to zero. This is a (1+α)b(1+βθa ) contradiction, and thus (4.1) has a positive solution if λ1 /(1 + α) < b < λ1 and λ1 − 1+α+βθ < 0. a

Now we prove that (4.1) has a positive solution if b = λ1 and λ1 − a sequence {(bn , un , vn )}∞ n=1 such that

λ1 < bn < λ1 , 1+α

λ1

(1 + α)bn (1 + βθa ) − 1 + α + βθa

< 0,

(1+α)λ1 (1+βθa ) 1+α+βθa

< 0. For a fixed a (>λ1 ), there exists

lim bn = λ1

n→∞

and (un , vn ) is a positive solution of (4.1) with b = bn . By (4.2), there exists a constant C independent of n such that ∥un ∥∞ +∥vn ∥∞ ≤ C . It follows from the regularity of elliptic equations that (un , vn ) converges to (u0 , v0 ) in C 2 (Ω )× C 2 (Ω ), and obviously (u0 , v0 ) is a nonnegative solution of (4.1) with b = λ1 . Assume on the contrary that (4.1) has no positive solution when b = λ1 . Then u0 ≡ 0 or v0 ≡ 0. First we assume that v0 ≡ 0. Let φn := vn /∥vn ∥L∞ (Ω ) , where n ∈ N. By the second equation of (4.1) with b = bn and (u, v) = (Un , vn ), φn satisfies −1φn = φn ϱ(un , vn ), x ∈ Ω , (4.5) φn = 0, x ∈ ∂Ω. Since ∥φn ϱ(un , vn )∥L∞ (Ω ) ≤ C for some constant C independent of n, then the regularity results of elliptic equations imply that there exists a nonnegative function φ0 ∈ C 2 (Ω ) with ∥φ0 ∥L∞ (Ω ) = 1 such that limn→∞ φn = φ0 in C 2 (Ω ), and it satisfies

−1φ = φ ϱ(u , 0) = (1 + α)λ1 (1 + β u0 ) φ , 0 0 0 0 1 + α + β u0 φ0 = 0,

x ∈ Ω,

(4.6)

x ∈ ∂Ω.

Since φ0 is nonnegative and nontrivial, by strong maximum principle, φ0 > 0 in Ω , so 0 = λ1 −

(1+α)λ1 (1+β u0 ) 1+α+β u0

, which

implies that u0 ≡ 0. Letting ψn := un /∥un ∥L∞ (Ω ) , by similar argument as above, we can get a = λ1 from the first equation of (4.1), which contradicts the fact that a > λ1 . Thus (4.1) has a positive solution if λ1 /(1 + α) < b ≤ λ1 and λ1 −

(1+α)b(1+βθa ) 1+α+βθa

< 0.

Finally we prove the existence of a positive solution of (4.1) if b > λ1

and a > λ1

c1 k2 θb c2 k1

.

Assume on the contrary that (4.1) has no positive solution. Since a > λ1 and b > λ1 , (4.1) has a trivial solution (0, 0) and two semi-trivial solutions (θa , 0) and (0, k2 (1 + α)θb /c2 ). Then, degW (I − A, D) = indexW (A, (0, 0)) + indexW (A, (θa , 0)) + indexW (A, (0, k2 (1 + α)θb /c2 )), thus we can easily proceed a contradiction by Lemma 4.1, thus (4.1) has a positive solution.

Similar to Corollary 3.4, the coexistence result (Theorem 4.2) reads as follows (see Figs. 1 and 2).

J. Zhou / Nonlinear Analysis 82 (2013) 47–65

59

Corollary 4.3. Let a = f (b; α, β) and a = f∗ (b; β) be the functions defined in Lemma 3.2, and let a = g (b) be the function defined in Lemma 3.3. Then, we have the following. (i) If α = 0 or β = 0, then (4.1) has a positive solution if b > λ1 and a > g (b). (ii) For fixed α > 0 and β > 0, (4.1) has a positive solution if (a, b) ∈ CE , where

CE = {(b, a) : λ1 /(1 + α) < b ≤ λ1 , a > f (b; α, β)} ∪ {(b, a) : b > λ1 , a > g (b)}. (iii) For fixed β > 0, the coexistence region CE expands to CE′ as α → ∞, where

CE′ = {(b, a) : 0 < b ≤ λ1 , a > f∗ (b; α, β)} ∪ {(b, a) : b > λ1 , a > g (b)}. (iv) For fixed α > 0, the coexistence region CE expands to CE′′ as β → ∞, where

CE′′ = {(b, a) : λ1 /(1 + α) < b ≤ λ1 , a > λ1 } ∪ {(b, a) : b > λ1 , a > g (b)}. (v) The coexistence region CE expands to CE′′′ as both α → ∞ and β → ∞, where

CE′′′ = {(b, a) : 0 < b ≤ λ1 , a > λ1 } ∪ {(b, a) : b > λ1 , a > g (b)}. 5. Multiplicity of positive solutions In this section, we will study the multiplicity results about the positive solutions of (4.1) by Theorem 2.1. First, we consider the bifurcation from the semi-trivial solution (u, v) = (0, k2 (1 + α)θb /c2 ) for fixed b > λ1 with bifurcation parameter a. By linearizing (4.1) at (0, k2 (1 + α)θb /c2 ), we obtain the following eigenvalue problem

c1 k2 1φ + a − θb φ = λφ, c2 k1 1 + α − 2αβ k2 2 αβ bk2 1 ψ + θ + θ φ + (b − 2θb )ψ = λψ, b b c2 c2 φ = ψ = 0,

x ∈ Ω, (5.1)

x ∈ Ω, x ∈ ∂Ω.

A necessary condition for bifurcation is that the principal eigenvalue of (5.1) is zero, which occurs if a = λ1

c1 k2 c2 k 1

θb = g (b)

by Lemma 3.3. Let Φ be the positive eigenfunction corresponding to a = g (b). We assume that Φ is normalized so that ∥Φ ∥L2 (Ω ) = 1. Since λ1 (2θb − b) > 0, −∆ + 2θb − b is invertible. Define

Ψ = (−∆ + 2θb − b)

−1

1 + α − 2αβ k2 c2

θ + 2 b

αβ bk2 c2

θb Φ .

(5.2)

With the functions defined above, we have the following result regarding the bifurcation of positive solutions of (4.1) from (0, k2 (1 + α)θb /c2 ) at a = g (b). Theorem 5.1. Let b > λ1 be fixed. Then a = g (b) is a bifurcation value of (4.1) where positive solutions bifurcate from the line of semi-trivial solutions Γ0 = {(a, 0, k2 (1 + α)θb /c2 ) : a > 0}; near (g (b), 0, k2 (1 + α)θb /c2 ), there exists δ > 0 such that all the positive solutions of (4.1) lie on a smooth curve Γ1 = (a(s), u(s), v(s)) : 0 < s < δ and

a(s) = g (b) + sa1 + sa2 (s), u(s) = sΦ + su (s, x), 1 k ( 1 + α) 2 v(s) = θb + sΨ + sv1 (s, x).

(5.3)

c2

Here s → (a2 (s), u1 (s, x), v1 (s, x)) is a smooth function from (0, δ) to R × X × X for X = C02+σ (Ω ) with σ ∈ (0, 1) such that a2 (0) = 0, u1 (0, x) = v1 (0, x) = 0 and

a1 =

1+ Ω

c1 k2 αβ c1 k2 − (1 + α)c2 k1 c2 k21

θb Φ 3 dx +

c1

(1 + α)k1

Ω

Φ 2 Ψ dx.

(5.4)

Moreover a = g (b) is the unique bifurcation value for which positive solutions bifurcate from Γ0 . Proof. Let X = C02+σ (Ω ) and Y = C σ (Ω ) with σ ∈ (0, 1). Define a nonlinear mapping F : R × X × X → Y by

F (a, u, v) =

1u + F (a, u, w(u, v)) , 1v + G(u, w(u, v))

where F , G and w are defined in (4.1).

(5.5)

60

J. Zhou / Nonlinear Analysis 82 (2013) 47–65

We consider the bifurcation at (a, u, v) = (g (b), 0, k2 (1 + α)θb /c2 ). From straightforward calculation,

1ξ + (Fu + Fw wu )ξ + Fw wv η , 1η + (Gu + Gw wu )ξ + Gw wv η u ξ Fa (a, u, v) = , Fa(u,v) (a, u, v)[ξ , η] = , 0 0 2 A1 ξ + 2B1 ξ η + C1 η2 , F(u,v)(u,v) (a, u, v)[ξ , η]2 = A2 ξ 2 + 2B2 ξ η + C2 η2

F(u,v) (a, u, v)[ξ , η] =

where A1 = Fuu + 2Fuw wu + Fww wu2 + Fw wuu , B1 = Fw wuv + Fuw wv + Fww wu wv , C1 = Fw wvv + Fww wv2 A2 = Guu + 2Guw wu + Gww wu2 + Gw wuu , B2 = Gw wuv + Guw wv + Gww wu wv , C2 = Gw wvv + Gww wv2 . At (a, u, v) = (g (b), 0, k2 (1 + α)θb /c2 ), it is easy to see that the kernel

N

F(u,v) g (b), 0,

k2 (1 + α) c2

θb

= span{(Φ , Ψ )}

and the range space

R F(u,v) g (b), 0,

k2 (1 + α) c2

θb

= (h¯ 1 , h¯ 2 ) ∈ Y × Y : h¯ 1 (x)Φ (x) = 0 . Ω

Then

Fa(u,v) g (b), 0,

k2 (1 + α) c2

k2 (1 + α) θb [Φ , Ψ ] = (Φ , 0) ̸∈ R F(u,v) g (b), 0, θb c2

Φ dx ̸= 0. Thus we can apply Theorem 2.1 to conclude that the set of positive solutions to (4.1) near (g (b), 0, k2 (1 + Ω α)θb /c2 ) is a smooth curve

since

2

Γ1 = {(a(s), u(s), v(s)) : s ∈ (0, δ)}

(5.6)

such that a(0) = g (b), u(s) = sΦ + o(s), v(s) = k2 (1 + α)θb /c2 + sΨ + o(s). Moreover, by Theorem 2.1,

⟨ℓ, F(u,u)(u,u) (g (b), 0, k2 (1 + α)θb /c2 )[Φ , Ψ ]2 ⟩ , a1 = a′ (0) = − 2⟨ℓ, Fa,(u,v) (g (b), 0, k2 (1 + α)θb /c2 )[Φ , Ψ ]⟩ where ℓ is a linear functional on Y × Y defined as ⟨ℓ, (h¯ 1 , h¯ 2 )⟩ =

h (x)Φ (x)dx. Thus, Ω ¯1 c k αβ c k 2c1 2 Ω −1 + 1 22 − (1+α)1c 2k θb Φ 3 dx − (1+α) Φ 2 Ψ dx k1 Ω c2 k1 2 1 a1 = − 2 Ω Φ 2 dx αβ c1 k2 c1 k2 c1 3 = 1+ − θ Φ dx + Φ 2 Ψ dx. b 2 ( 1 + α) c k ( 1 + α) k c k 2 1 1 Ω 2 1 Ω

Finally we prove that a = g (b) is the unique bifurcation point where positive solutions bifurcate from (0, k2 (1 +α)θb /c2 ). Suppose that there is a sequence {(an , un , vn )}n≥1 of positive solutions of (4.1) such that lim (an , un , vn ) =

n→∞

a, 0 ,

k2 (1 + α) c2

θb .

Let φn = un /∥un ∥L∞ (Ω ) . From the first equation of (4.1) with a = an ,

−1φ = φ a − u − c1 wn , n n n n un + k1 φn = 0,

x ∈ Ω, x ∈ ∂Ω,

(5.7)

J. Zhou / Nonlinear Analysis 82 (2013) 47–65

where wn =

(1+β un )vn . 1+α+β un

61

By (4.2) and the regularity theory for elliptic equations, there exists a subsequence of {φn }n≥1 such

that it converges uniformly in X to some nonnegative function φ ∈ X with ∥φ∥L∞ (Ω ) = 1. It follows from (5.7) that (a, φ) satisfies

−1φ = φ a − c1 k2 θ , b c2 k1 φ = 0,

x ∈ Ω,

(5.8)

x ∈ ∂Ω.

By the Krein–Rutman Theorem, a = λ1

c1 k2 c2 k1

θb = g (b). This completes the proof.

Theorem 5.2. Let b > λ1 be fixed and let a1 be defined as in (5.4). If a1 ̸= 0, then there exists δ˜ ∈ (0, δ] such that for ˜ , the positive solution (a(s), u(s), v(s)) bifurcating from (g (b), 0, k2 (1 + α)θb /c2 ) is not degenerate, where δ is the s ∈ (0, δ) constant in Theorem 5.1. Moreover, (u(s), v(s)) is unstable if a1 < 0, and it is stable if a1 > 0. Proof. In order to study the stability of the bifurcation positive solution (u(s), v(s)) of (4.1), we consider the following eigenvalue problem

L(s)[ξ (s), η(s)] = µ(s)

ξ (s) = η(s) = 0,

ξ ( s) , η(s)

x ∈ Ω, x ∈ ∂Ω,

where L(s) := −F(u,v) (a(s), u(s), v(s)) =

L11 L21

L12 L22

,

L11 = −∆ − Fu (a(s), u(s), w(u(s), v(s))) − Fw (a(s), u(s), w(u(s), v(s)))wu (u(s), v(s)), L12 = −Fw (a(s), u(s), w(u(s), v(s)))wv (u(s), v(s)), L21 = −Gu ((u(s), w(u(s), v(s)))) − Gw (u(s), w(u(s), v(s)))wu (u(s), v(s)), L22 = −∆ − Gw ((u(s), w(u(s), v(s))))wv (u(s), v(s)). Furthermore,

lim L(s) = s→0+

−

−∆ +

c1 k2

c2 k1 1 + α − 2αβ k2 c2

θb − g (b) θb2 −

αβ bk2 c2

0

θb

−∆ + 2θb − b

.

θb − g (b) = 0 and λ1 (2θb − b) > λ1 (θb − b) = 0, 0 is the first eigenvalue of L0 with corresponding eigenfunction (Φ , Ψ ). Moreover all the real parts of other eigenvalues of L0 are apart from 0. By the perturbation theory of linear operators [49], we know that for s > 0 small, L(s) has a unique eigenvalue µ(s) such that lims→0+ µ(s) = 0 and all other eigenvalues of L(s) have positive real parts and are apart from 0. Now we determine the sign of µ(s) for s > 0 by Theorem 2.1. Consider the following eigenvalue problem k2 (1 + α) φ(a) θb [φ(a), ψ(a)] = γ (a) , x ∈ Ω, −F(u,v) a, 0, ψ(a) c2 φ(a) = ψ(a) = 0, x ∈ ∂Ω. Since λ1

c1 k2 c2 k1

Then φ(a) satisfies

−1φ(a) + c1 k2 θ − a φ(a) = γ (a)φ(a), b c2 k1 φ(a) = 0,

x ∈ Ω,

(5.9)

x ∈ ∂Ω.

Since γ (g (b)) = 0 and φ (g (b)) = Φ , differentiating (5.9) with respect a at a = g (b),

−1ϕ − Φ + c1 k2 θ − g (b) ϕ = γ ′ (g (b)) Φ , b c2 k1 ϕ = 0,

x ∈ Ω,

(5.10)

x ∈ ∂Ω,

where ϕ = φ ′ (g (b)). Multiplying both sides of (5.10) with Φ and integrating the result over Ω , by (5.1) and the fact that Φ is the positive eigenfunction corresponding to a = g (b),

γ ′ (g (b))

Ω

Φ 2 dx = −

Ω

Φ 2 dx,

62

J. Zhou / Nonlinear Analysis 82 (2013) 47–65

Fig. 3. Possible bifurcation diagram of u when a1 < 0.

and

γ ′ (g (b)) = −1.

(5.11)

Since a1 ̸= 0, it follows from Theorem 2.1 and (5.11) that µ(s) ̸= 0 for s > 0 small and lim

s→0+

µ(s) s

= −γ ′ (g (b)) a′ (0) = a1 .

(5.12)

Since all the other eigenvalues of L have positive real parts and are apart from 0, then the conclusions follow from (5.12).

Based on the above preparations, we give the multiplicity result on positive solutions of (4.1) as follows. Theorem 5.3 (See D5 in Fig. 1). Assume that the conditions of Theorem 5.1 are satisfied, and a1 be defined as in (5.4). If a1 < 0, then there exists a positive constant ϵb ∈ (0, g (b) − λ1 ) such that problem (4.1) has at least two positive solutions if g (b) − ϵb < a < g (b), and it has at least one positive solution if a ≥ g (b) − ϵb . Proof. From Theorem 5.1, (4.1) has a curve {(a(s), u(s), v(s)) : s ∈ (0, δ)} of positive solutions near (g (b), 0, k2 (1+α)θb /c2 ). Since a1 < 0, a(s) < g (b) for s > 0 small. Assume on the contrary that (4.1) has a unique positive solution (ˆu, vˆ ) when a < g (b) but near g (b). Then it is obvious that (ˆu, vˆ ) must be the positive function bifurcating from (g (b), 0, k2 (1 +α)θb /c2 ), which was obtained from Theorem 5.1. That is (ˆu, vˆ ) = (u(s), v(s)), which is not degenerate by Theorem 5.2. Thus I − A(u,v) (ˆu, vˆ ) : W (ˆu,ˆv ) → W (ˆu,ˆv ) is invertible. Recall that A is the operator defined in (4.3). Since W (ˆu,ˆv ) − S(ˆu,ˆv ) = ∅, A(u,v) (ˆu, vˆ ) does not have property a. Consequently indexW (A, (ˆu, vˆ )) = ±1.

Notice that λ1 < a < g (b) for s > 0 small and λ1 −

(1+α)b(1+βθa ) 1+α+βθa

< 0 if b > λ1 . It follows from Lemma 4.1 that

1 = degW (I − A, D)

= indexW (A, (0, 0)) + indexW (A, (θa , 0)) + indexW (A, (0, k2 (1 + α)θb /c2 )) + indexW (A, (ˆu, vˆ )) = 0 + 0 + 1 ± 1, which is a contradiction. Thus if a < λ1 (c1 k2 θb /(c2 k1 )) and near λ1 (c1 k2 θb /(c2 k1 )), there exist at least two positive solutions of (4.1). By Theorem 2.1, the curve Γ1 of the bifurcating positive solution is contained in a connected component Σ of the set of positive solutions of (4.1). Moreover either the closure of Σ contains another semi-trivial solution on {(a, 0, k2 (1 +α)θb /c2 ) : a > 0} or the closure of Σ contains semi-trivial solution {(a, θa , 0) : a > λ1 } or Σ is unbounded. By Theorem 5.1, a = g (b) is the unique bifurcation value for positive solutions of(4.1) from the line of semi-trivial solution {(a, 0, k2 (1 + α)θb /c2 ) : a > 0}, so the first alternative is not possible. Since λ1 −

(1+α)b(1+βθa ) 1+α+βθa

< 0 if b > λ1 , it follows from Theorem 3.1(ii) that (a, θa , 0) cannot be a bifurcating point for positive solutions for all a > λ1 , so the second alternative is not possible. Thus Σ must be unbounded. Furthermore, by (4.2) and regularity theory of elliptic equations, for each a > 0 there exists C (a) > 0 such that 0 < ∥(u)∥X + ∥v∥X ≤ C (a), and there is no positive solutions when a ≤ λ1 . Thus there exists ϵb ∈ (0, g (b) − λ1 ) such that the projection of Σ on the a-axis contains an interval [g (b) − ϵb , ∞). In particular (4.1) has at least two positive solutions if g (b) − ϵb < a < g (b), and it has at least one positive solutions if a ≥ g (b) − ϵb (see Fig. 3).

J. Zhou / Nonlinear Analysis 82 (2013) 47–65

63

Remark 5.4. 1. Since limb→λ1 g (b) = λ1 , ϵb = 0, when b = λ1 . 2. a1 < 0 can be achieved by fixing α, β, c2 , k2 > 0, b > λ1 and letting c1 = k1 = ε > 0 in (5.4). Then Φ and Ψ are all independent of ε , while

lim a1 = lim

ε→0+

ε→0+

1+ Ω

k2 αβ k2 1 θb Φ 3 dx − θb Φ 3 dx + Φ 2 Ψ dx = −∞. (1 + α)c2 c2 ε Ω 1+α Ω

Thus a1 < 0 if ε > 0 small enough. Next, we consider the bifurcation from the semi-trivial solution (u, v) = (θa , 0) for a > λ1 with bifurcation parameter b ∈ (λ1 /(1 + α), λ1 ). Recall the results of Lemma 3.2 that a = f (b; α, β) is the solution

λ1

(1 + α)b(1 + βθa ) − 1 + α + βθa

= 0.

For fixed α, β > 0, let b = χ (a), a ∈ [λ1 , ∞) be the inverse function of a = f (b; α, β). It follows from Lemma 3.2 that 1. b = χ (a) is a strictly decreasing function with respect to a; 2. χ(λ1 ) = λ1 and lima→∞ χ (a) = b/(1 + α). By linearizing (4.1) at (θa , 0), we obtain the following eigenvalue problem

c1 θa (1 + βθa ) ψ = λφ, 1φ + (a − 2θa )φ − (θa + k1 )(1 + α + βθa ) (1 + α)b(1 + βθa ) 1ψ + ψ = λψ, 1 + α + βθa φ = ψ = 0,

x ∈ Ω, x ∈ Ω, x ∈ ∂Ω.

A necessary condition for bifurcation is that the principal eigenvalue of (5.13) is zero, which occurs if b = χ (a). Let Ψ˜ be the positive eigenfunction corresponding to b = χ (a). We assume that Ψ˜ is normalized so that ∥Ψ˜ ∥L2 (Ω ) = 1. Since λ1 (2θa − a) > 0, −∆ + 2θa − a is invertible. Define

˜ = −(−∆ + 2θa − a)−1 Φ

c1 θa (1 + βθa )

(θa + k1 )(1 + α + βθa )

Ψ˜ ,

and

b1 =

c2 (1+α) Ω θa +k2

2

1+α)χ(a) ˜ ˜ 2 Ψ˜ 3 dx − Ω αβ( Φ Ψ dx (1+α+βθa )2 . 1+βθa (1 + α) Ω 1+α+βθa Ψ˜ 2 dx

1+βθa

1+α+βθa

(5.13)

Then similar to the proof of Theorem 5.3, we get the following result. Theorem 5.5 (See D2 in Fig. 1). Let a > λ1 , α > 0, β > 0 be fixed, b ∈ (λ1 /(1 + α), λ1 ), and b1 be defined as in (5.13). If b1 < 0, then there exists a positive constant ϵa ∈ (0, χ (a) − λ1 /(1 + α)) such that problem (4.1) has at least two positive solutions if χ (a) − ϵa < b < χ (a), and it has at least one positive solution if b ≥ χ (a) − ϵa . Remark 5.6. 1. Since (4.1) has no positive solution when a = λ1 and lima→∞ χ (a) = λ1 /(1 + α), ϵa = 0 if a = λ1 or a tends to ∞. ˜ and Ψ˜ are both independent of c2 and k2 , 2. Since Φ αβ(1+α)χ(a) ˜ ˜ 2 Φ Ψ dx Ω (1+α+βθa )2 1+βθa + α) Ω 1+α+βθa Ψ˜ 2 dx

lim b1 = lim b1 = −

c2 → 0

k2 →∞

(1

< 0.

Thus b1 < 0 if c2 is small enough or k2 is large enough. 3. It is obvious that b1 > 0 if α = 0 or β = 0, so the conditions in Theorem 5.5 do not hold without cross-diffusion. Furthermore, if we assume c2 is small enough or k2 is large enough, then combined with 2, Theorem 5.5 and [9, Theorem 3.1], we can conclude that problem (4.1) has a unique positive solution if α = β = 0, and (4.1) has at least two positive solutions if α > 0 and β > 0. Acknowledgments This work was done when the author visited College of William and Mary in 2012. The author would like to thank Department of Mathematics, College of William and Mary for warm hospitality, and thank Prof. Junping Shi for constant encouragement and helpful discussions.

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