Positive solutions of some parabolic system with cross-diffusion and nonlocal initial conditions
aa r X i v : . [ m a t h . A P ] N ov Positive solutions of some parabolic system with cross-diffusion and nonlocalinitial conditions
Christoph Walker a a Leibniz Universit¨at Hannover, Institut f¨ur Angewandte Mathematik,Welfengarten 1, D–30167 Hannover, Germany
Abstract
The paper is concerned with a system consisting of two coupled nonlinear parabolic equations with a cross-diffusionterm, where the solutions at positive times define the initial states. The equations arise as steady state equations ofan age-structured predator-prey system with spatial dispersion. Based on unilateral global bifurcation methods forFredholm operators and on maximal regularity for parabolic equations, global bifurcation of positive solutions isderived.
Keywords: cross-diffusion, age structure, global bifurcation, Fredholm operator, maximal regularity
1. Introduction
Consider the situation that an age-structured prey and an age-structured predator population inhabit the samespatial region Ω , that the individuals of both populations undergo spatial fluctuation and that the predator populationexerts a repulsive pressure on the prey population. If u = u ( t, a, x ) ≥ and v = v ( t, a, x ) ≥ denote the densityof the prey and the predator population, respectively, at time t ≥ , age a ∈ [0 , a m ) for a maximal age a m > , andspatial position x ∈ Ω , a simple model reads ∂ t u + ∂ a u − ∆ x (( δ + γv ) u ) = − α u − α uv , t > , a ∈ (0 , a m ) , x ∈ Ω , (1.1) ∂ t v + ∂ a v − δ ∆ x v = − β v + β vu , t > , a ∈ (0 , a m ) , x ∈ Ω . (1.2)These equations are subject to the nonlocal age boundary conditions u ( t, , x ) = Z a m B ( a ) u ( t, a, x ) d a , t > , x ∈ Ω , (1.3) v ( t, , x ) = Z a m B ( a ) v ( t, a, x ) d a , t > , x ∈ Ω , (1.4)the spatial boundary conditions u ( t, a, x ) = 0 , t > , a ∈ (0 , a m ) , x ∈ ∂ Ω , (1.5) v ( t, a, x ) = 0 , t > , a ∈ (0 , a m ) , x ∈ ∂ Ω , (1.6)and are supplemented with time initial conditions. The ∆ x -term in (1.1) describes spatial movement of prey individ-uals. Besides intrinsic dispersion with coefficient δ > , it reflects an increase of the dispersive force on the prey byrepulsive interference with an increase of the predator population. Here, γ ≥ is the predator population pressurecoefficient. We refer to [24] for a derivation of such kind of models (without age-structure). The right hand sides of(1.1) and (1.2) take into account intra- and inter-specific interactions of the two populations with positive coefficients Email address: [email protected] (Christoph Walker)
Preprint submitted to Elsevier September 13, 2018 , α , β , β > . Equations (1.5) and (1.6) describe creation of new individuals with nonnegative birth rates B and B . The reader is referred to [32] and the references therein for further information about linear and nonlinearage-structured population equations with (and without) spatial dispersal. Although such models have a long history,there does not seem to be much literature about equations with nonlinear diffusion. The well-posedness of equations(1.1)-(1.6) might be derived within the general framework presented in [28] though the results therein do not directlyapply due to the cross-diffusion term.We shall remark that the model considered herein is a rather simple biological model and there may be moreaccurate models, e.g. with different maximal ages, nonlocal dependences in the reaction terms etc. The aim of thepresent paper is thus rather to provide a mathematical framework to treat parabolic equations with cross-diffusion andnonlocal initial conditions. Namely, the present paper is dedicated to positive equilibrium (i.e. time-independent)solutions to (1.1)-(1.6) which shall be established based on global bifurcation methods. We write the birth rates inthe form B ( a ) = ηb ( a ) and B ( a ) = ξb ( a ) , where b , b are some fixed birth profiles and the parameters η , ξ measuring the intensities of the fertility shall serve as bifurcation parameters. Aiming at a simple and compactnotation, we let b := b = b and δ = δ = 1 . We emphasize that these simplifications are made merely for thesake of readability and do not impact in any way on the mathematical analysis to follow. To shorten notation further,we shall agree upon the following convention: given a function defined on J := [0 , a m ] and denoted by a lower caseletter, say u or v , we use the corresponding capital letter U or V to denote its age-integral with weight b , i.e., U := Z a m b ( a ) u ( a ) d a , V := Z a m b ( a ) v ( a ) d a . (1.7)Depending on the values of the parameters η and ξ , we are looking for nonnegative and nontrivial functions u = u ( a, x ) and v = v ( a, x ) satisfying the nonlinear parabolic system with cross-diffusion ∂ a u − ∆ x (cid:0) (1 + γv ) u (cid:1) = − α u − α uv , a ∈ (0 , a m ) , x ∈ Ω , (1.8) ∂ a v − ∆ x v = − β v + β vu , a ∈ (0 , a m ) , x ∈ Ω , (1.9)subject to the nonlocal initial conditions u (0 , x ) = ηU , x ∈ Ω , (1.10) v (0 , x ) = ξV , x ∈ Ω , (1.11)and spatial Dirichlet boundary conditions u ( a, x ) = 0 , a ∈ (0 , a m ) , x ∈ ∂ Ω , (1.12) v ( a, x ) = 0 , a ∈ (0 , a m ) , x ∈ ∂ Ω . (1.13)Clearly, of particular interest are coexistence states , that is, solutions ( u, v ) with both components u and v nontrivialand nonnegative.Based on bifurcation techniques, existence of equilibrium solutions was established in [30, 31] for similar systemswith linear diffusion and in [26, 27, 29] for a single equation with nonlinear diffusion. Except for [29], the bifurcationparameter was also chosen to be a measure for the intensity of the fertility as above. Prior to the just cited papers,bifurcation methods were used in [9] to derive positive equilibrium solutions for a single equations with linear dif-fusion. We also refer to [15] where the large time behavior of population dynamics with age-dependence and linearspatial diffusion was analyzed.More attention attracted than the age-structured parabolic equations so far have related elliptic systems of the form − ∆ x [(1 + γv ) u ] = u ( µ − u ± cv ) , − ∆ x v = v ( λ − v ± bu ) in Ω subject to Dirichlet conditions on ∂ Ω with positive constants µ, λ, b, c ; both for the case of linear diffusion γ = 0 (e.g., see [3, 4, 8, 17, 18] and the references therein) and the case with cross-diffusion γ > (e.g., see[10, 11, 13, 14, 23] and the references therein). It is worthwhile to remark that for such elliptic equations with cross-diffusion, the transformation z := (1 + γv ) u leads to a semilinear elliptic system for z and v , for which, when written2n the form ( z, v ) = S ( z, v ) , the corresponding solution operator S enjoys suitable compactness properties. More-over, for such a system, positivity and a-priori bounds of solutions may be derived using the maximum principle, e.g.[10, 13, 23]. For the nonlocal parabolic equations (1.7)-(1.13) under consideration herein, however, a correspond-ing transformation still yields nonlinear second order terms with respect to the spatial variable (or first order timederivatives). Consequently, the underlying solution operator does not enjoy similar compactness properties. As weare interested in global bifurcation of positive solutions, it is thus not clear how to apply directly the unilateral globalbifurcation methods of [16, 21] as in [30], which require compactness of the underlying solution operators. To over-come this deficiency, we shall invoke recent results of Shi & Wang [23] on unilateral global bifurcation for Fredholmoperators, which are based on L´opez-G´omez’s interpretation [16] of the global alternative of Rabinowitz [21] and onthe global bifurcation results of Pejsachowicz & Rabier [19]. These results yield a (global) continuum of positivesolutions. We shall also point out that, due to the cross-diffusion term, no comparison principle in the spirit of [30,Lem.3.2] is available. In some cases, the lack of such a comparison principle prevents us from determining which ofthe possible alternatives the constructed continuum of coexistence states satisfies (see Theorem 2.3 and Theorem 2.4below).
2. Main Results
We suppose throughout this paper that the birth profile b ∈ L + ∞ ((0 , a m )) with b ( a ) > for a near a m (2.1)is normalized such that Z a m b ( a ) e − λ a d a = 1 , (2.2)where λ > denotes the principal eigenvalue of − ∆ x on Ω subject to Dirichlet boundary conditions on ∂ Ω . Beforestating our results on coexistence states in more detail, we first recall some auxiliary results from [30] about semi-trivial states, that is, about solutions ( u, v ) with one vanishing component. Taking for instance v ≡ in (1.8)-(1.9)and using convention (1.7), we obtain the reduced problem ∂ a u − ∆ x u = − α u , u (0 , · ) = ηU , (2.3)subject to Dirichlet boundary conditions. Introducing, for q ∈ ( n + 2 , ∞ ) fixed, the solution space W q := L q ((0 , a m ) , W q,D (Ω)) ∩ W q ((0 , a m ) , L q (Ω)) , where W q,D (Ω) := { u ∈ W q (Ω) ; u = 0 on ∂ Ω } , and letting W + q denote its positive cone, the following result was proved in [30, Thm.2.1, Cor.3.3, Lem.3.6, Cor.3.7]: Theorem 2.1.
Suppose (2.1) and (2.2) . For each η > there is a unique solution u η ∈ W + q \ { } to equation (2.3) . The mapping ( η u η ) belongs to C ∞ ((1 , ∞ ) , W q ) and satisfies k u η k W q → as η → and k u η k W q → ∞ as η → ∞ . For η > and a ∈ J , the derivative ∂ η u η ( a ) is a (strictly) positive function on Ω . If η > η , then u η ≥ u η . Finally, if η ≤ , then (2.3) has no solution in W + q \ { } . Given ξ > , we let v ξ ∈ W + q \ { } denote the unique solution to the corresponding equation for v when taking u ≡ in (1.8)-(1.9). Though we shall work in the solution space W q , we remark that solutions to (1.7)-(1.13), so inparticular u η and v ξ , are smooth with respect to both a ∈ J and x ∈ Ω (see Lemma 3.3 below). η We first shall keep ξ fixed and regard η as bifurcation parameter. We thus write ( η, u, v ) for solutions to (1.7)-(1.13) with u , v belonging to W + q . Theorem 2.1 entails, for any ξ ≥ , the semi-trivial branch B := { ( η, u η ,
0) ; η > } ⊂ R + × ( W + q \ { } ) × W + q (2.4)3f solutions. For ξ > , an additional semi-trivial branch B := { ( η, , v ξ ) ; η ≥ } ⊂ R + × W + q × ( W + q \ { } ) (2.5)exists. In the latter case, a continuum of coexistence states bifurcates from B : Theorem 2.2.
Suppose (2.1) - (2.2) and let ξ > . There exists η := η ( ξ ) > such that an unbounded continuum C emanates from ( η , , v ξ ) ∈ B , and C \ { ( η , , v ξ ) } ⊂ R + × ( W + q \ { } ) × ( W + q \ { } ) consists of coexistence states ( η, u, v ) to (1.7) - (1.13) . Near the bifurcation point ( η , , v ξ ) , C is a continuous curve.There is no other bifurcation point on B or on B to positive coexistence states. The value of η ( ξ ) corresponding to the bifurcation point is determined in (3.30).We turn to the case ξ < , which is more involved. Recall that B is the only semi-trivial branch of solutions inthis case. Theorem 2.3.
Suppose (2.1) - (2.2) . There is δ ∈ [0 , with the property that, given ξ ∈ ( δ, , there exists η := η ( ξ ) > such that a continuum S emanates from ( η , u η , ∈ B , and S \ { ( η , u η , } ⊂ R + × ( W + q \ { } ) × ( W + q \ { } ) consists of coexistence states ( η, u, v ) of (1.7) - (1.13) . The continuum S (i) is unbounded, or(ii) connects ( η , u η , to a solution ( η ∗ , u, v ) of (1.7) - (1.13) with η ∗ ∈ (0 , and u, v ∈ W + q \ { } .Near the bifurcation point ( η , u η , , S is a continuous curve. There is no other bifurcation point on B to positivecoexistence states. The values of δ and η ( ξ ) are given in (4.5) and (4.6), respectively. Note that alternative (i) above always occursif cross-diffusion is not taken into account, i.e. if γ = 0 , see [31, Thm.1.5]. Alternative (ii) stems from a technicalcondition, and we conjecture that alternative (i) is the generic case also for the present situation where γ > . However,as occurrence of alternative (ii) is not ruled out by our analysis, Theorem 2.3 is rather a local bifurcation result in thisregard. ξ Since the cross-diffusion term involves the predator density v , bifurcation from semi-trivial solution branches ismore intricate when regarding ξ (the measure of the predator fertility) as parameter and keeping η fixed. We shallonly consider the situation η > , the reason being explained in Remark 5.1. We now write ( ξ, u, v ) for solutions to(1.7)-(1.13). Then, there are two semi-trivial branches of solutions T := { ( ξ, , v ξ ) ; ξ > } , T := { ( ξ, u η ,
0) ; ξ ≥ } . Also in this case, bifurcation from the semi-trivial branch T occurs: Theorem 2.4.
Suppose (2.1) - (2.2) and let η > . There exists ξ := ξ ( η ) ∈ (0 , such that a continuum R ofsolutions to (1.7) - (1.13) emanates from ( ξ , u η , ∈ T satisfying the alternatives(i) R \ { ( ξ , u η , } is unbounded in R + × ( W + q \ { } ) × ( W + q \ { } ) , or(ii) R connects T with T .Except for the bifurcation point(s), R consists exclusively of coexistence states. Near the bifurcation point ( ξ , u η , , R is a continuous curve. There is no other bifurcation point on T or on T to positive coexistence states. ξ ( η ) and ξ = ξ ( η ) > , corresponding to the connection point ( ξ , , v ξ ) ∈ T if alternative (ii)occurs, are determined in (5.1) and (5.2), respectively. Given the assumptions of this theorem, if γ = 0 , one can showthat alternative (ii) occurs if η is less than a certain value or if additional assumptions on the birth rates and on α j , β j are imposed, see [30, Thm.2.2].The outline of the paper is as follows: The next section is dedicated to the proof of Theorem 2.2. In Subsection 3.1,we first introduce some notation, and in Subsection 3.2, we provide the necessary auxiliary results needed to performthe actual proof of Theorem 2.2 in Subsection 3.3. Sections 4 and 5 are dedicated to the proofs of Theorem 2.3 andTheorem 2.4, respectively. As these proofs are along the lines of the proof of Theorem 2.2, these sections are keptrather short.
3. Proof of Theorem 2.2
Given two Banach spaces F and F , we shall use the notation L ( F , F ) for the set of bounded linear operatorsand K ( F , F ) for the set of compact linear operators between F and F . We set L ( F ) := L ( F , F ) and similarly K ( F ) := K ( F , F ) . The set of toplinear isomorphisms between F and F is denoted by L is ( F , F ) .Let − ∆ D denote the negative Laplacian on L q := L q (Ω) subject to Dirichlet boundary conditions, that is, − ∆ D := − ∆ x with domain W q,D , where W κq,D := W κq,D (Ω) := { u ∈ W κq ; u = 0 on ∂ Ω } for κ > /q and W κq,D := W κq (Ω) for ≤ κ < /q . As q > n + 2 , we have W − /qq,D ֒ → C ( ¯Ω) by the Sobolevembedding theorem. In particular, int( W − /q, + q,D ) = ∅ , that is, the interior of the positive cone of W − /qq,D is notempty. Let γ denote the trace operator defined by γ u := u (0) for u ∈ W q , which is well-defined owing to theembedding [2, III.Thm.4.10.2] W q ֒ → C (cid:0) [0 , a m ] , W − /qq,D (cid:1) . In fact, we have, due to the interpolation inequality [2, I.Thm.2.11.1], W q ֒ → C − /q − ϑ ([0 , a m ] , W ϑq,D ) , ≤ ϑ ≤ − /q . (3.1)In the following, we let ˙ W + q := W + q \ { } . Recall that an operator A ∈ L ( W q,D , L q ) is said to have maximal L q -regularity provided ( ∂ a + A, γ ) ∈ L is ( W q , L q × W − /qq,D ) , where L q := L q ( J, L q ) . Recall that − ∆ D has maximal L q -regularity. If A : J → L ( W κq,D , L q ) for some κ ≥ ,we write ( Au )( a ) := A ( a ) u ( a ) for a ∈ J and u : J → W κq,D . The following perturbation result is proved in [20,Cor.3.4], which is also valid for R k -valued functions, i.e., if L q is replaced by L q (Ω , R k ) etc.: Lemma 3.1.
Let A ∈ C ( J, L ( W q,D , L q )) be such that A ( a ) has maximal L q -regularity for each a ∈ J and let B ∈ L q ( J, L ( W − /qq,D , L q )) . Then ( ∂ a + A + B, γ ) ∈ L is ( W q , L q × W − /qq,D ) . Given ̺ > and h ∈ C ̺ ( J, C ( ¯Ω)) , we let Π [ h ] ( a, σ ) , ≤ σ ≤ a ≤ a m , denote the unique parabolic evolutionoperator corresponding to − ∆ D + h ∈ C ̺ (cid:0) J, L ( W q,D , L q ) (cid:1) , that is, z ( a ) = Π [ h ] ( a, σ )Φ , a ∈ ( σ, a m ) , defines theunique strong solution to ∂ a z − ∆ D z + hz = 0 , a ∈ [ σ, a m ) , z ( σ ) = Φ , for any given σ ∈ (0 , a m ) and Φ ∈ L q (see [2, II.Cor.4.4.1]). Note that the evolution operator is positive, i.e. Π [ h ] ( a, σ )Φ ∈ L + q , ≤ σ ≤ a ≤ a m , Φ ∈ L + q . − ∆ D has maximal L q -regularity, it follows from Lemma 3.1 that ( ∂ a − ∆ D + h, γ ) ∈ L is ( W q , L q × W − /qq,D ) (3.2)and, in particular, Π [ h ] ( · , ∈ W q for Φ ∈ W − /qq,D . We set H [ h ] := Z a m b ( a ) Π [ h ] ( a,
0) d a .
Then H [ h ] ∈ K ( W − /qq,D ) owing to standard regularizing effects of the parabolic evolution operator Π [ h ] and thecompact embedding W q,D ֒ → W − /qq,D . Moreover, as Π [ h ] ( a, for a ∈ (0 , a m ) is strongly positive on W − /qq,D by[6, Cor.13.6], the same holds true for H [ h ] due to (2.1), that is, H [ h ] Φ ∈ int( W − /q, + q,D ) , Φ ∈ W − /q, + q,D \ { } . (3.3)The corresponding spectral radius r ( H [ h ] ) can thus be characterized according to the Krein-Rutman theorem [1,Thm.3.2] (see [30, Lem.3.1]): Lemma 3.2.
For h ∈ C ̺ ( J, C ( ¯Ω)) with ̺ > , the spectral radius r ( H [ h ] ) > is a simple eigenvalue of H [ h ] with a corresponding eigenfunction belonging to int( W − /q, + q,D ) . It is the only eigenvalue of H [ h ] with a positiveeigenfunction. Moreover, if h and g both belong to C ̺ ( J, C ( ¯Ω)) with g ≥ h but g h , then r ( H [ g ] ) < r ( H [ h ] ) . In particular, the normalization (2.2) implies r ( H [0] ) = 1 (3.4)since any positive eigenfunction of − ∆ D is an eigenfunction of H [ h ] as well. Moreover, writing the solution to (2.3)in the form u = Π [ α u ] ( · , u (0) , Theorem 2.1 together with Lemma 3.2 imply η r ( H [ α u η ] ) = ξ r ( H [ β v ξ ] ) = 1 , η, ξ > , (3.5)since u η (0) , v ξ (0) ∈ ( W − /q, + q,D ) \ { } . The aim is to apply the global bifurcation results of [23, Thm.4.3,Thm.4.4] in order to establish Theorem 2.2. Wefirst provide the necessary tools.Let ξ > be fixed and let v ξ ∈ ˙ W + q denote the solution to (1.7)-(1.13) with u ≡ provided by Theorem 2.1.Throughout we use the convention (1.7). Notice that for each a ∈ J , the operator A ( a ) , given by A ( a ) u := − div x ((1 + γv ξ ( a )) ∇ x u ) , u ∈ W q,D , has maximal L q -regularity due to its divergence form, the positivity of v ξ ∈ W q , (3.1), and e.g. [2, I.Cor.1.3.2,III.Ex.4.7.3, III.Thm.4.10.7]. Moreover, A ∈ C ( J, L ( W q,D , L q )) by (3.1). Noticing also that B ( a ) u := − γ div x (cid:0) u ∇ x v ξ ( a ) (cid:1) + α v ξ ( a ) u , a ∈ J , u ∈ W − /qq,D , defines an operator B ∈ L q ( J, L ( W − /qq,D , L q )) , it follows from Lemma 3.1 that T := ( ∂ a + A + B , γ ) − ∈ L ( L q × W − /qq,D , W q ) (3.6)is well-defined. Observe that A ξ u := ( A + B ) u = − ∆ D (cid:0) (1 + γv ξ ) u (cid:1) + α v ξ u , u ∈ W q,D . T := ( ∂ a − ∆ D + 2 β v ξ , γ ) − ∈ L ( L q × W − /qq,D , W q ) . To derive a bifurcation from a point ( η , , v ξ ) ∈ B for a suitable η = η ( ξ ) , we write ( η, u, v ) = ( η, u, v ξ + w ) for a solution to problem (1.7)-(1.13), which is then equivalent to ∂ a u − ∆ D (cid:0) (1 + γv ξ ) u (cid:1) = ∆ D (cid:0) γwu (cid:1) − α u − α u ( v ξ + w ) , a ∈ (0 , a m ) , x ∈ Ω , (3.7) ∂ a w − ∆ D w = − β w − β v ξ w + β ( v ξ + w ) u , a ∈ (0 , a m ) , x ∈ Ω , (3.8)subject to u (0 , x ) = ηU ( x ) , x ∈ Ω , (3.9) w (0 , x ) = ξW ( x ) , x ∈ Ω . (3.10)The solutions ( η, u, w ) to (3.7)-(3.10), in turn, are the zeros of the map F : R × W q × ˆ W q → W q × W q , defined by F ( η, u, w ) := (cid:18) u − T (cid:0) ∆ D ( γwu ) − α u − α uw , ηU (cid:1) w − T (cid:0) − β w + β ( v ξ + w ) u , ξW (cid:1) (cid:19) , (3.11)where ˆ W q := { w ∈ W q ; w ( a, x ) > − / γ , a ∈ J , x ∈ ¯Ω } is an open subset of W q owing to (3.1). Clearly, F is smooth and F ( η, ,
0) = 0 for η ∈ R . As for the Frech´etderivatives at ( η, u, w ) we compute F ( u,w ) ( η, u, w )[ φ, ψ ] = (cid:18) φ − T (cid:0) ∆ D ( γψu ) + ∆ D ( γwφ ) − α uφ − α wφ − α uψ , η Φ (cid:1) ψ − T (cid:0) − β wψ + β ψu + β ( v ξ + w ) φ , ξ Ψ (cid:1) (cid:19) (3.12)and F η, ( u,w ) ( η, u, w )[ φ, ψ ] = (cid:18) − T (0 , Φ)0 (cid:19) (3.13)for ( φ, ψ ) ∈ W q × W q . The choice of W q as solution space is to have a suitable functional setting to work with inthe framework of maximal regularity. However, as it is needed later on, we note that solutions to (1.7)-(1.13), i.e. to(3.7)-(3.10), are smooth. The proof is a bootstrapping argument which we provide for the reader’s ease. Lemma 3.3. If ( η j , u j , v j ) is a bounded sequence in R × W q × ˆ W q of solutions to (1.7) - (1.13) , then ( u j ) and ( v j ) are bounded in C ε ( J, C ε ( ¯Ω)) ∩ C ε ( J, C ε ( ¯Ω)) for some ε > .Proof. To stick with the notation of [2], let ( E, A ) := ( L q , − ∆ D ) and let [( E α , A α ); α ≥ be the correspondinginterpolation scale induced by the real interpolation functors ( · , · ) α,q . Putting F := E − /q . = W − /qq,D , F := E − /q , it follows from [2, V.Thm.2.1.3] that the F -realization of ∆ D , again denoted by ∆ D , has domain F and is thegenerator of an analytic semigroup { e a ∆ D ; a ≥ } on F . Thus, k e a ∆ D k L ( F µ ,F ν ) ≤ c a µ − ν , a ∈ J \ { } , µ, ν ∈ (0 , , (3.14)where F µ := ( F , F ) µ,q for µ ∈ (0 , . Note that the almost reiteration property [2, V.Thm.1.5.3] ensures F θ + ֒ → E θ − /q ֒ → F θ − , < θ − < θ < θ + < . (3.15)Let now ( η j , u j , v j ) be a sequence of solutions to (1.7)-(1.13) in R × W q × ˆ W q with | η j | + k ( u j , v j ) k X ≤ B , j ∈ N ,for some B > . Writing ∂ a v j − ∆ D v j = − β v j + β v j u j =: f j , v j (0) = ξV j =: v j , (3.16)7t follows from the continuity of pointwise multiplication W q × W q → W q (owing to q > n + 2 and Sobolev’sembedding) and (3.1) that k f j k C ( J,F ) ≤ c ( B ) , j ∈ N , (3.17)while (1.7), (2.1), and the embedding E ֒ → F /q − ε with ε > sufficiently small entail k v j k F /q − ε ≤ c ( B ) , j ∈ N , (3.18)for some constant c ( B ) > . Thus, from (3.14), (3.16), (3.17), and (3.18), for j ∈ N , k v j k L ( J,F − ε ) ≤ Z a m k e a ∆ D k L ( F /q − ε ,F − ε ) k v j k F /q − ε d a + Z a m Z a k e ( a − σ )∆ D k L ( F ,F − ε ) k f j ( σ ) k F d σ d a ≤ c ( B ) . Therefore, from (2.1) we conclude that ( v j ) is bounded in F − ε . Since ( f j ) is bounded in C ( J, F ) , we deduce from(3.16) and [2, II.Thm.5.3.1] that ( v j ) is bounded in C ε ( J, F − ε ) for some ε > sufficiently small. Now, taking [25,Thm.5.3.4,Thm.5.4.1] into account which guarantee E − /q . = (cid:0) D (∆ D ) , D (∆ D ) (cid:1) − /q,q ֒ → W − /q ) q,D , with D (∆ kD ) denoting the domain of the k -th power of ∆ D equipped with its graph norm, we obtain F − ε = ( E − /q , E − /q ) − ε,q ֒ → (cid:0) W − /q ) q,D , W − /q ) q,D (cid:1) − ε,q . = W − /q − εq,D ֒ → C ε ( ¯Ω) for ε > sufficiently small by Sobolev’s embedding theorem since q > n + 2 . Consequently, ( v j ) is bounded in C ε ( J, C ε ( ¯Ω)) ∩ ˆ W q . (3.19)But then, since ∂ a u j − ∆ D (cid:0) (1 + γv j ) u j (cid:1) = − α u j − α u j v j , u j (0) = η j U j , (3.20)we similarly conclude that ( u j ) is bounded in C ε ( J, C ε ( ¯Ω)) , where the analogue of (3.14) holds due to (3.19)and [2, II. § (∆ D v j ) is bounded in C ε ( J, C ε ( ¯Ω)) while ( − β v j − β v j u j ) is bounded in C ε ( J, C ε ( ¯Ω)) . From (3.16) we derive that ( ∂ a v j ) is bounded in C ε ( J, C ε ( ¯Ω)) and similarly we derive this for ( ∂ a u j ) .Noticing that C ε ( J, C ε ( ¯Ω)) embeds compactly in C ¯ ε ( J, C ε ( ¯Ω)) and C ε ( J, C ε ( ¯Ω)) in C ε ( J, C ¯ ε ( ¯Ω)) for ¯ ε ∈ (0 , ε ) , we deduce: Corollary 3.4.
Any bounded and closed subset of { ( η, u, w ) ∈ R × W q × ˆ W q ; F ( η, u, w ) = 0 } is compact. Let now ( η, u, w ) ∈ R × W q × ˆ W q be fixed. We shall show that L := F ( u,w ) ( η, u, w ) ∈ L ( W q × W q ) is a Fredholm operator. To this end, we introduce, for a ∈ J , the operators A ij ( a ) ∈ L ( W q,D , L q ) by A ( a ) φ : = − ∆ D (cid:0) (1 + γv ξ ( a ) + γw ( a )) φ (cid:1) + α ( v ξ ( a ) + w ( a )) φ + 2 α u ( a ) φ ,A ( a ) ψ : = − ∆ D (cid:0) γu ( a ) ψ (cid:1) + α u ( a ) ψ ,A ( a ) φ : = − β ( v ξ ( a ) + w ( a )) φ ,A ( a ) ψ : = − ∆ D ψ + 2 β ( v ξ ( a ) + w ( a )) ψ − β ψu ( a ) , a ∈ J , φ, ψ ∈ W q,D and set A ( a ) := (cid:20) A ( a ) A ( a ) A ( a ) A ( a ) (cid:21) , a ∈ J .
Moreover, we define D ( a ) h := (cid:18) − ∆ D (cid:0) (1 + γv ξ ( a )) h (cid:1) + α v ξ ( a ) h − ∆ D h + 2 β v ξ ( a ) h (cid:19) , h = ( h , h ) ∈ W q,D × W q,D , a ∈ J , so that D ∈ L ( W q,D × W q,D , L q × L q ) for a ∈ J , and we also define ℓ [ η ] ∈ L ( W q × W q , W q,D × W q,D ) by ℓ [ η ] z := (cid:18) η Φ ξ Ψ (cid:19) , z = ( φ, ψ ) ∈ W q × W q . It then readily follows from (3.12) that, given z = ( φ, ψ ) and h = ( h , h ) in W q × W q , the equation Lz = h isequivalent to ∂ a z + A ( a ) z = ∂ a h + D ( a ) h , a ∈ J , z (0) = ℓ [ η ] z + h (0) . (3.21)In the sequel, we use the notation X := L q × L q , X := W q × W q , X θ := W θq,D × W θq,D , θ ∈ [0 , . Let us first observe that
Remark 3.5.
The space X can be equipped with an equivalent norm, which is continuously differentiable at allpoints except zero.Proof. According to [22], since X = W q × W q is separable, the statement is equivalent to say that the dual space X ′ = W ′ q × W ′ q of X is separable. But, since W q is dense in L q , the separable space L ′ q = L q ′ is dense in W ′ q ,where /q + 1 /q ′ = 1 . So X ′ is separable.Investigation of (3.21) requires the following information on the involved operators: Lemma 3.6.
The above defined operators ( ∂ a + A , γ ) and ( ∂ a + D , γ ) both belong to L is ( X , X × X − /q ) , and ℓ [ η ] belongs to K ( X , X − /q ) .Proof. Writing A ( a ) ψ = − ∆ D (cid:0) γu ( a ) ψ (cid:1) + α u ( a ) ψ = − div x (cid:0) γu ( a ) ∇ x ψ (cid:1) + (cid:8) α u ( a ) ψ − div x (cid:0) ψγ ∇ x u ( a ) (cid:1)(cid:9) and using (3.1), it is readily seen that A can be written in the form A := A + A := (cid:20) A ˜ A A (cid:21) + (cid:20) A A (cid:21) with A ∈ C ( J, L ( X , X )) , A ∈ L q ( J, L ( X − /q , X )) . (3.22)Recalling γ ( v ξ ( a, x ) + w ( a, x )) ≥ / , ( a, x ) ∈ J × ¯Ω , due to the positivity of v ξ and w ∈ ˆ W q , it follows as in (3.6) that A ( a ) and A ( a ) have maximal L q -regularityfor each fixed a ∈ J . Consequently, the problem ∂ a z + A ( a ) z + ˜ A ( a ) z = f ( a ) , a ∈ J , z (0) = z ,∂ a z + A ( a ) z = f ( a ) , a ∈ J , z (0) = z , f = ( f , f ) ∈ X and z = ( z , z ) ∈ X − /q a unique solution z = ( z , z ) ∈ X given by z = (cid:0) ∂ a + A ( a ) , γ (cid:1) − ( f − ˜ A ( a ) z , z ) ,z = (cid:0) ∂ a + A ( a ) , γ (cid:1) − ( f , z ) , and there is some constant c independent of f and z such that k z k X ≤ c (cid:0) k f k X + k z k X − /q ) . Therefore, ( ∂ a + A ( a ) , γ ) ∈ L is ( X , X × X − /q ) for each a ∈ J , whence ( ∂ a + A , γ ) ∈ L is ( X , X × X − /q ) by (3.22) and Lemma 3.1. Analogously we deduce the statement on ( ∂ a + D , γ ) . Since W q,D embeds compactly in W − /qq,D , the assertion on ℓ [ η ] ∈ L ( X , X ) is immediate.Based on Lemma 3.6, we have Σ := ( ∂ a + A , γ ) − ∈ L ( X × X − /q , X ) and Q := (cid:2) w ℓ [ η ] (cid:0) Σ(0 , w ) (cid:1)(cid:3) ∈ K ( X − /q ) . We now show that L is indeed a Fredholm operator. The proof is along the lines of [26, Lem.2.1]. Proposition 3.7.
Let ( η, u, w ) ∈ R × W q × ˆ W q and L = F ( u,w ) ( η, u, w ) ∈ L ( X ) . Then L is a Fredholm operatorof index zero. More precisely, rg( L ) = (cid:8) h ∈ X ; h (0) + ℓ [ η ](Σ( ∂ a h + D h, ∈ rg(1 − Q ) (cid:9) (3.23) is closed in X and ker( L ) = (cid:8) Σ(0 , w ) ; w ∈ ker(1 − Q ) (cid:9) with dim(ker( L )) = codim(rg( L )) = dim(ker(1 − Q )) < ∞ . Proof.
Owing to (3.21) and Lemma 3.6, for z, h ∈ X , the equation Lz = h is equivalent to z = Σ( ∂ a h + D h,
0) + Σ(0 , z (0)) , (3.24) (1 − Q ) z (0) = ℓ [ η ] (cid:0) Σ( ∂ a h + D h, (cid:1) + h (0) . (3.25)If belongs to the resolvent set of Q ∈ K ( X − /q ) , then (3.24), (3.25) entail a trivial kernel ker( L ) . Moreover, inthis case, for an arbitrary h ∈ X , there is a unique z (0) ∈ X − /q solving (3.25), thus the corresponding z ∈ X given by (3.24) is the unique solution to Lz = h . This easily gives the assertion in this case.Otherwise, if is an eigenvalue of Q ∈ K ( X − /q ) , then (3.24), (3.25) yield the characterization of ker( L ) and rg( L ) as stated. In particular, since Σ is an isomorphism, we deduce dim(ker( L )) = dim(ker(1 − Q )) whichis a finite number because 1 is an eigenvalue of the compact operator Q . Moreover, rg( L ) is closed in X since M := rg(1 − Q ) is closed by the compactness of Q and due to Lemma 3.6 and (3.1). To compute codim(rg( L )) ,note that codim( M ) = dim(ker(1 − Q )) < ∞ , hence M is complemented in X − /q leading to a direct sum decomposition X − /q = M ⊕ N . Denoting by P M ∈ L ( X − /q ) a projection onto M along N , we set P h := Λ (cid:0) ∂ a h + D h , P M h (0) − (1 − P M ) ℓ [ η ](Σ( ∂ a h + D h, (cid:1) , h ∈ X , (3.26)where Λ := ( ∂ a + D , γ ) − ∈ L ( X × X − /q , X ) , and obtain P ∈ L ( X ) from Lemma 3.6. Since (cid:0) ∂ a + D (cid:1) ( P h ) = ∂ a h + D h , γ ( P h ) = P M h (0) − (1 − P M ) ℓ [ η ](Σ( ∂ a h + D h, , the characterization (3.23) actually implies that P maps X into rg( L ) . Furthermore, if h ∈ rg( L ) , then (3.23) alsoensures P h = Λ( ∂ a h + D h, h (0)) = h , P (rg( L )) = rg( L ) . Thus P = P with rg( P ) = rg( L ) is a projection and X = rg( L ) ⊕ ker( P ) . Since Λ is anisomorphism, we obtain ker( P ) = { h ∈ X ; ∂ a h + D h = 0 , h (0) ∈ N } , from which we deduce the equality of the dimension of N and ker( P ) and thus the statement. Corollary 3.8.
For k ∈ (0 , and ( η, u, w ) ∈ R × W q × ˆ W q , (1 − k ) F ( u,w ) ( η, ,
0) + kF ( u,w ) ( η, u, w ) ∈ L ( X ) is a Fredholm operator of index zero.Proof. Since, by (3.12), F ( u,w ) ( η, , φ, ψ ] = (cid:18) φ − T (cid:0) , η Φ (cid:1) ψ − T (cid:0) β v ξ φ , ξ Ψ (cid:1)(cid:19) , (3.27)the operator (1 − k ) F ( u,w ) ( η, ,
0) + kF ( u,w ) ( η, u, w ) has the same structure as F ( u,w ) ( η, u, w ) .It follows from Lemma 3.3 that the operator A ξ , given by A ξ ( a ) φ := − ∆ D (cid:0) (1 + γv ξ ( a )) φ (cid:1) + α v ξ ( a ) φ , a ∈ J , φ ∈ W q,D , (3.28)belongs to C ε ( J, L ( W q,D , L q )) , while the positivity of v ξ ensures that − A ξ ( a ) is for each a ∈ J the generator of ananalytic semigroup on L q . Consequently, it generates a parabolic evolution operator Π A ξ ( a, σ ) , ≤ σ ≤ a ≤ a m , inview of [2, II.Cor.4.4.2]. Note that Π A ξ ( a, for a > is strongly positive on W − /qq,D , see e.g. [6, Cor.13.6]. Wethen set G ξ := Z a m b ( a ) Π A ξ ( a,
0) d a (3.29)and obtain from (2.1) and the compact embedding of W q,D in W − /qq,D that G ξ ∈ K ( W − /qq,D ) is strongly positive.Thus, by the Krein-Rutman theorem, r ( G ξ ) > is a simple eigenvalue of G ξ with an eigenvector in the interior ofthe positive cone W − /q, + q,D . Let then η := η ( ξ ) := 1 r ( G ξ ) > , ker(1 − η G ξ ) = span { Φ } , Φ ∈ int( W − /q, + q,D ) . (3.30)We define φ ∗ ( a ) := Π A ξ ( a, , a ∈ J , Φ ∗ := Z a m b ( a ) φ ∗ ( a ) d a , (3.31)and, using the notation of Subsection 3.1, ψ ∗ ( a ) := Π [2 β v ξ ] ( a, + Z a Π [2 β v ξ ] ( a, σ ) (cid:0) β v ξ ( σ ) φ ∗ ( σ ) (cid:1) d σ , a ∈ J , (3.32)where Ψ := ξ (cid:0) − ξH [2 β v ξ ] (cid:1) − (cid:18)Z a m b ( a ) Z a Π [2 β v ξ ] ( a, σ ) (cid:0) β v ξ ( σ ) φ ∗ ( σ ) (cid:1) d σ d a (cid:19) . Note that Ψ is well-defined since − ξH [2 β v ξ ] is invertible owing to Lemma 3.2, (3.5), and v ξ ∈ ˙ W + q which ensure r ( ξH [2 β v ξ ] ) < . Also note, from (3.2) and (3.6), that φ ∗ and ψ ∗ both belong to ˙ W + q . Lemma 3.9.
The kernel of F ( u,w ) ( η , , is spanned by ( φ ∗ , ψ ∗ ) , and F η, ( u,w ) ( η , , φ ∗ , ψ ∗ ] does not belong tothe range of F ( u,w ) ( η , , . Moreover, Φ = η Φ ∗ . roof. Observe that ( φ, ψ ) ∈ X belonging to the kernel of F ( u,w ) ( η , , is equivalent to ∂ a φ − ∆ D (cid:0) (1 + γv ξ ) φ (cid:1) + α v ξ φ = 0 , φ (0) = η Φ ,∂ a ψ − ∆ D ψ + 2 β v ξ ψ = β v ξ φ , ψ (0) = ξ Ψ , according to (3.27) and the definitions of T and T . Now, the first assertion follows from (3.28)-(3.32) by solving for φ and ψ . Next, suppose F η, ( u,w ) ( η , , φ ∗ , ψ ∗ ] belongs to the range of F ( u,w ) ( η , , . Then, in view of (3.13),(3.27), and the definition of T , there is φ ∈ W q with ∂ a φ + A ξ φ = 0 , φ (0) = η Φ − Φ ∗ , so φ ( a ) = Π A ξ ( a, φ (0) , a ∈ J and whence (1 − η G ξ ) φ (0) = − Φ ∗ . Since Φ = η Φ ∗ by definition of φ ∗ and(3.30), we conclude Φ ∈ ker(1 − η G ξ ) ∩ rg(1 − η G ξ ) what is impossible since η G ξ is compact with simple eigenvalue . Having established the necessary auxiliary results in the previous subsection, we are now in a position to proveTheorem 2.2 by applying [23, Thm.4.3,Thm.4.4]. Recall that, writing ( η, u, v ) = ( η, u, v ξ + w ) , the solutions ( η, u, v ) to (1.7)-(1.13) are obtained as the zeros ( η, u, w ) of the smooth function F defined in (3.11). Also recall that η = η ( ξ ) is given in (3.30).As in the second part of the proof of Lemma 3.9, ker( F ( u,w ) ( η , , ∩ rg( F ( u,w ) ( η , , { } , whence X = span { ( φ ∗ , ψ ∗ ) } ⊕ rg( F ( u,w ) ( η , , by [5, Lem.2.7.9] and Lemma 3.9. In view of Proposition 3.7 and Lemma 3.9 we may apply [23, Thm.4.3]. Therefore,there are ε > and continuous functions η : ( − ε, ε ) → R , ( θ , θ ) : ( − ε, ε ) → rg( F ( u,w ) ( η , , with η (0) = η and ( θ , θ )(0) = (0 , such that the solutions to (1.7)-(1.13) near ( η , , v ξ ) are exactly the semi-trivial ones (˜ η, , v ξ ) , ˜ η ≥ , and the ones lying on the curve Γ := Γ + ∪ Γ − ∪ { ( η , , v ξ ) } , where Γ ± := (cid:8)(cid:0) η ( s ) , sφ ∗ + sθ ( s ) , v ξ + sψ ∗ + sθ ( s ) (cid:1) ; 0 < ± s < ε (cid:9) . Moreover, Γ is contained in C ∗ , which is a connected component of the closure of S := { ( η, u, v ξ + w ) ; F ( η, u, w ) = 0 , ( u, w ) = (0 , } . Being merely interested in positive solutions, we first note:
Lemma 3.10.
The curve Γ + lies in R + × ˙ W + q × ˙ W + q .Proof. Let u s := sφ ∗ + sθ ( s ) and v s := v ξ + sψ ∗ + sθ ( s ) . Then u s (0) = s Φ + o ( s ) and v s (0) = ξV ξ + s Ψ + o ( s ) in W − /qq,D as s → + by (3.1). Thus, it follows from Φ , V ξ ∈ int( W − /q, + q,D ) that u s (0) , v s (0) ∈ int( W − /q, + q,D ) for s ∈ (0 , ε ) with ε > small enough, whence u s , v s ∈ ˙ W + q for s ∈ (0 , ε ) due to the parabolic maximum principle[6, Thm.13.5] and (1.7)-(1.13).Now, invoking Corollary 3.4, Remark 3.5, and Corollary 3.8 we obtain from [23, Thm.4.4] (see also [23, Rem.4.2.1])further information about the global character of the continuum. More precisely, if C + denotes the connected compo-nent of C ∗ \ Γ − containing Γ + , then C + satisfies the alternatives:12i) C + intersects with the boundary of R × W q × ˆ W q , or(ii) C + is unbounded in R × W q × ˆ W q , or(iii) C + contains a point ( η, , v ξ ) with η = η , or(iv) C + contains a point ( η, u, v ξ + w ) with ( u, w ) = (0 , and ( u, w ) ∈ rg( F ( u,w ) ( η , , .Due to Lemma 3.10, the continuum C := C + ∩ ( R + × W + q × W + q ) of solutions to (1.7)-(1.13) contains thecurve Γ + . Furthermore, we have: Lemma 3.11. C \ { ( η , , v ξ ) } ⊂ R + × ˙ W + q × ˙ W + q is unbounded.Proof. We first show that C + does not reach the boundary of R + × ˙ W + q × ˙ W + q at some point ( η, u, v ) = ( η , , v ξ ) .Suppose otherwise, i.e. let there be a sequence ( η j , u j , v j ) in C ∩ ( R + × ˙ W + q × ˙ W + q ) converging toward some point ( η, u, v ) = ( η , , v ξ ) not belonging to R + × ˙ W + q × ˙ W + q . Since u j and v j are nonnegative, the limits u and v are aswell. So u ≡ or v ≡ because ( η, u, v ) / ∈ R + × ˙ W + q × ˙ W + q . We claim that neither is possible. Suppose first thatboth u and v identically vanish. As v j ∈ ˙ W + q , ψ j := v j / k v j k W q is well-defined in ˙ W + q , has norm 1, and ∂ a ψ j − ∆ D ψ j = − β v j ψ j + β ψ j u j , ψ j (0) = ξ Ψ j . The proof of Lemma 3.3 shows that ( ψ j ) is bounded in C ε ( J, C ε ( ¯Ω)) ∩ C ε ( J, C ε ( ¯Ω)) for some ε > and sowe may assume without loss of generality that ( ψ j ) converges in ˙ W + q to some ψ satisfying ∂ a ψ − ∆ D ψ = 0 , ψ (0) = ξ Ψ . Thus ψ ( a ) = e a ∆ D ψ (0) , a ∈ J , and ψ (0) = ξH [0] ψ (0) implying ξr ( H [0] ) = 1 by the Krein-Rutman theorem.However, this contradicts (3.4) and ξ > . Next, assume u vanishes identically but v . Then ( η, u, v ) = ( η, , v ) and the uniqueness statement of Theorem 2.1 implies v = v ξ . Thus, ( η, , v ξ ) ∈ B is a bifurcation point to positivecoexistence states. By Lemma 3.3, we may assume ( v j ) converges to v ξ in C ε ( J, C ε ( ¯Ω)) ∩ C ε ( J, C ε ( ¯Ω)) forsome ε > . Moreover, as above we may assume that ( φ j ) , defined by φ j := u j / k u j k W q , converges in ˙ W + q to some φ satisfying ∂ a φ − ∆ D (cid:0) (1 + γv ξ ) φ (cid:1) = − α φv ξ , φ (0) = η Φ . (3.33)Therefore, φ ( a ) = Π A ξ ( a, φ (0) , a ∈ J , and φ (0) = ηG ξ φ (0) . Thus η = η by the Krein-Rutman theorem and(3.30). This yields the contradiction ( η, u, v ) = ( η , , v ξ ) . Finally, suppose v ≡ but u . Then we have ( η, u, v ) = ( η, u, what gives u = u η with η > by Theorem 2.1 since u ∈ ˙ W + q , and so ( η, u, v ) = ( η, u η , ∈ B is a bifurcation point to positive coexistence states. As above we may assume that ( ψ j ) , given by ψ j := v j / k v j k W q ,converges to some ψ ∈ ˙ W + q satisfying ∂ a ψ − ∆ D ψ = β ψu η , ψ (0) = ξ Ψ . This readily implies ξr ( H [ − β u η ] ) what is impossible since ξ > and r ( H [0] ) < r ( H [ − β u η ] ) according to(3.4) and Lemma 3.2.Consequently, C + intersects with the boundary of R + × ˙ W + q × ˙ W + q only at ( η , , v ξ ) , whence C = C + . So neitheralternative (i) nor (iii) above is possible. Suppose (iv) occurs, and let ( φ, ψ ) ∈ X and ( η, u, v ξ + w ) ∈ C + be with ( u, w ) = F ( u,w ) ( η , , φ, ψ ] . Then φ − u = T (0 , η Φ) by (3.27). Recall, from the definition of φ ∗ and Lemma 3.9, that φ ∗ = T (0 , η Φ ∗ ) with Φ ∗ ∈ int( W − /q, + q,D ) . The latter implies κη Φ ∗ + φ (0) − u (0) ∈ int( W − /q, + q,D ) for some κ > . Defining p := κφ ∗ + φ − u ∈ W q , we obtain p = T (0 , η ( κ Φ ∗ + Φ)) , that is, ∂ a p + A ξ p = 0 with p (0) = η P + η U . Hence (1 − η G ξ ) p (0) = η U . Since u ∈ ˙ W + q by assumption and thus U ∈ ( W − /q, + q,D ) \ { } , this last equation does notadmit a positive solution p (0) according to (3.30) and [1, Thm.3.2] in contradiction to p (0) ∈ int( W − /q, + q,D ) by thechoice of κ . So (iv) is impossible as well, and we conclude that C \{ ( η , , v ξ ) } ⊂ R + × ˙ W + q × ˙ W + q is unbounded.13o finish off the proof of Theorem 2.2 we merely have to remark that ( η , , v ξ ) is the only bifurcation point. Lemma 3.12.
There is no other bifurcation point on B or on B to positive coexistence states.Proof. Exactly the same arguments as in the first step of the proof of Lemma 3.11 show that there is neither abifurcation point ( η, u η , ∈ B nor ( η, , v ξ ) ∈ B to positive coexistence states.
4. Proof of Theorem 2.3
As the proof of Theorem 2.3 is similar to the one of Theorem 2.2, we merely sketch it and point out the necessarymodifications.We shall derive a bifurcation from the branch B by linearizing around a point ( η, u η , with a suitable η = η tobe determined. First note that the smooth branch U := { ( η, u η ); η > } in (1 , ∞ ) × ˙ W + q of solutions to (2.3) providedby Theorem 2.1 extends to a smooth branch U ∗ := { ( η, u η ); η > η ∗ } in ( η ∗ , ∞ ) × W q passing through ( η, u ) = (1 , ,where η ∗ ∈ (0 , and − u η ∈ ˙ W + q for η ∈ ( η ∗ , . Indeed, application of [26, Thm.2.4], [27, Prop.2.5] (with ε < in[27, Eq.(2.17)], see the proof of [30, Prop.3.4]) shows that the branch U of positive solutions extends smoothly witha branch { ( η ( ε ) , u η ( ε ) ); − ε < ε ≤ } , where − u η ( ε ) ∈ ˙ W + q for ε ∈ ( − ε , . Thus, fixing ε ∈ ( − ε , , it followsthat w := − u η ( ε ) ∈ ˙ W + q satisfies ∂ a w − ∆ D w = α w , w (0) = η ( ε ) W , whence w (0) = η ( ε ) H [ − α w ] w (0) and thus η ( ε ) r ( H [ − α w ] ) = 1 by the Krein-Rutman theorem. Due to Lemma 3.2and (3.4), we have r ( H [ − α w ] ) > r ( H [0] ) = 1 and so η ( ε ) < . We thus get the desired smooth extension U ∗ of U bychoosing η ∗ sufficiently close to . Consequently, the solutions ( η, u, v ) = ( η, u η − w, v ) to problem (1.7)-(1.13) canbe obtained as the zeros ( η, w, v ) of the smooth map F : ( η ∗ , ∞ ) × W q × ˆ W q → W q × W q , defined by F ( η, w, v ) := (cid:18) w − T (cid:0) − ∆ D ( γv ( u η − w )) − α u η w + α w + α ( u η − w ) v , ηW (cid:1) v − T (cid:0) − β v + β v ( u η − w ) , ξV (cid:1) (cid:19) , (4.1)where the set ˆ W q is as in Section 3 and T := ( ∂ a − ∆ D , γ ) − ∈ L ( L q × W − /qq,D , W q ) . Clearly, F ( η, ,
0) = 0 for η ∈ ( η ∗ , ∞ ) and the Frech´et derivatives at ( η, w, v ) are given by F ( w,v ) ( η, w, v )[ φ, ψ ] = φ − T (cid:0) − ∆ D ( γψ ( u η − w )) + ∆ D ( γvφ ) − α ( u η − w ) φ + α ψ ( u η − w ) − α vφ , η Φ (cid:1) ψ − T (cid:0) − β vψ − β vφ + β ( u η − w ) ψ , ξ Ψ (cid:1) (4.2)and F η, ( w,v ) ( η, u, w )[ φ, ψ ] = (cid:18) − T (cid:0) − ∆ D ( γψu ′ η ) − α u ′ η φ + α ψu ′ η , Φ (cid:1) − T (cid:0) β u ′ η ψ , (cid:1) (cid:19) (4.3)for ( φ, ψ ) ∈ W q × W q with dashes referring to derivatives with respect to η . It is then straightforward to modify theproofs of Lemma 3.6 and Proposition 3.7 in order to derive the analogue of Corollary 3.8: Lemma 4.1.
For k ∈ (0 , and ( η, w, v ) ∈ ( η ∗ , ∞ ) × W q × ˆ W q , (1 − k ) F ( w,v ) ( η, ,
0) + k F ( w,v ) ( η, w, v ) ∈ L ( X ) is a Fredholm operator of index zero.
14o determine the bifurcation point, let us observe that r ( H [ − β u η ] ) > is a strictly increasing function of η on (1 , ∞ ) according to Theorem 2.1 and Lemma 3.2. Since u η depends continuously on η in the topology of W q byTheorem 2.1, we obtain from [2, II.Lem.5.1.4] that the evolution operator Π [ − β u η ] ( a, and hence H [ − β u η ] dependcontinuously on η with respect to the corresponding operator topologies. Together with the fact that the spectral radiusconsidered as a function K ( W − /qq,D ) → R + is continuous (see [7, Thm.2.1]), we conclude that (cid:0) η r ( H [ − β u η ] ) (cid:1) ∈ C (cid:0) (1 , ∞ ) , (1 , ∞ ) (cid:1) is strictly increasing (4.4)with lim η → r ( H [ − β u η ] ) = 1 . Defining δ ∈ [0 , by δ := 1lim η →∞ r ( H [ − β u η ] ) , (4.5)it follows that for any ξ ∈ ( δ, fixed we find a unique η := η ( ξ ) > with ξ = 1 r ( H [ − β u η ] ) . (4.6)We may then choose Ψ ∈ int( W − /q, + q,D ) spanning ker (cid:0) − ξH [ − β u η ] (cid:1) . Define ( φ ⋆ , ψ ⋆ ) ∈ W q × ˙ W + q by ψ ⋆ := Π [ − β u η ] ( · ,
0) Ψ and φ ⋆ := Π [2 α u η ] ( · , + N ψ ⋆ , Φ := η (cid:0) − η H [2 α u η ] (cid:1) − (cid:18)Z a m b ( a )( N ψ ⋆ )( a ) d a (cid:19) , with ( N ψ ⋆ )( a ) := Z a Π [2 α u η ] ( a, σ ) (cid:0) − ∆ D ( γu η ( σ ) ψ ⋆ ( σ )) + α u η ( σ ) ψ ⋆ ( σ ) (cid:1) d σ , a ∈ J , where the invertibility of − η H [2 α u η ] is due to (3.5). The analogue of Lemma 3.9 then reads: Lemma 4.2.
The kernel of F ( w,v ) ( η , , is spanned by ( φ ⋆ , ψ ⋆ ) and F η, ( w,v ) ( η , , φ ⋆ , ψ ⋆ ] does not belong tothe range of F ( w,v ) ( η , , .Proof. That ker( F ( w,v ) ( η , , { ( φ ⋆ , ψ ⋆ ) } follows as in the proof of Lemma 3.9. To check the transver-sality condition, suppose F η, ( w,v ) ( η , , φ ⋆ , ψ ⋆ ] belongs to the range of F ( w,v ) ( η , , . Recall (4.2), (4.3) and let v ∈ W q be such that v − T ( β u η v, ξV ) = − T ( β u ′ η ψ ⋆ , . Choose τ > with τ Ψ − v (0) ∈ int( W − /q, + q,D ) .Since ψ ⋆ = T ( β u η ψ ⋆ , ξ Ψ ⋆ ) , it follows that p := τ ψ ⋆ − v satisfies ∂ a p − ∆ D p − β u η p = β u ′ η ψ ⋆ , p (0) = ξP , from which we deduce (cid:0) − ξH [ − β u η ] (cid:1) p (0) = ξβ Z a m b ( a ) Z a Π [ − β u η ] ( a, σ ) (cid:0) u ′ η ( σ ) ψ ⋆ ( σ ) (cid:1) d σ d a . However, invoking [1, Thm.3.2] and (4.6), this is impossible since p (0) ∈ int( W − /q, + q,D ) by the choice of τ and sincethe right hand side is positive and nonzero due to (2.1) and the positivity of ψ ⋆ and of u ′ η stated in Theorem 2.1.As Corollary 3.4 holds also for F , we may proceed as in Subsection 3.3 to derive from [23, Thm.4.3,Thm.4.4]that a continuum S + in ( η ∗ , ∞ ) × W q × W q of solutions to (1.7)-(1.13) bifurcates from ( η , u η , satisfying thealternatives:(i) S + intersects with the boundary of ( η ∗ , ∞ ) × W q × ˆ W q , or(ii) S + is unbounded in ( η ∗ , ∞ ) × W q × ˆ W q , or 15iii) S + contains a point ( η, u η , with η = η , or(iv) S + contains a point ( η, u η − w, v ) with ( w, v ) = (0 , and ( w, v ) ∈ rg( F ( w,v ) ( η , , .Moreover, near the bifurcation point, S + is a continuous curve Γ + = (cid:8) ( η ( s ) , u η ( s ) − sφ ⋆ − sθ ( s ) , sψ ⋆ + sθ ( s )) ; 0 < s < ε (cid:9) for a continuous real-valued function η and some continuous W q -valued functions θ j with η (0) = 0 and θ j (0) = 0 .Since u η ( s ) (0) and ψ ⋆ = Ψ both belong to int( W − /q, + q,D ) , it follows as in Lemma 3.10 that Γ + is a subset of (1 , ∞ ) × ˙ W + q × ˙ W + q for ε > sufficiently small. For the continuum S , given by S := S + ∩ (cid:0) [ η ∗ , ∞ ) × W + q × W + q (cid:1) , we have: Lemma 4.3. S \ { ( η , u η , } is a subset of [ η ∗ , ∞ ) × ˙ W + q × ˙ W + q consisting of coexistence states ( η, u, v ) to (1.7) - (1.13) . The continuum S is unbounded or it connects ( η , u η , to a solution ( η ∗ , u, v ) of (1.7) - (1.13) with u, v ∈ ˙ W + q .Proof. First suppose S + \ { ( η , u η , } does not reach the boundary of ( η ∗ , ∞ ) × ˙ W + q × ˙ W + q , so S = S + . Thenneither (i) nor (iii) above is possible. Suppose (iv) occurs. Then there are ( η, u η − w, v ) ∈ S and ( φ, ψ ) ∈ W q × W q such that ( w, v ) = F ( w,v ) ( η , , φ, ψ ] . Hence p := κψ ⋆ + ψ − v ∈ W q , with κ > chosen such that p (0) belongsto ∈ int( W − /q, + q,D ) , satisfies ∂ a p − ∆ D p − β u η p = β u η v , p (0) = ξP + ξV , so that (cid:0) − ξH [ − β u η ] (cid:1) p (0) = ξV + ξβ Z a m b ( a ) Z a Π [ − β u η ] ( a, σ ) (cid:0) u η ( σ ) v ( σ ) (cid:1) d σ d a . Since v ∈ ˙ W + q by assumption, this last equation does not admit a positive solution p (0) according to [1, Thm.3.2] inview of (4.6). However, this contradicts p (0) ∈ int( W − /q, + q,D ) . So (iv) is impossible as well, and we conclude thatif S + \ { ( η , , v ξ ) } does not reach the boundary of ( η ∗ , ∞ ) × ˙ W + q × ˙ W + q , then S = S + is unbounded. Otherwise,suppose S \ { ( η , , v ξ ) } reaches the boundary of ( η ∗ , ∞ ) × ˙ W + q × ˙ W + q at a point ( η, u, v ) = ( η , u η , and choosea sequence ( η j , u j , v j ) in S ∩ (( η ∗ , ∞ ) × ˙ W + q × ˙ W + q ) converging toward ( η, u, v ) . Since u j and v j are nonnegativeand η j > η ∗ , the limits u and v are nonnegative as well and η ≥ η ∗ . So u ≡ or v ≡ or η = η ∗ . We claim thatnecessarily η = η ∗ and u, v ∈ ˙ W + q . We proceed as in Lemma 3.11. If both u ≡ and v ≡ , then the limit ψ ∈ ˙ W + q of ψ j := v/ k v j k W q satisfies ∂ a ψ − ∆ D ψ = 0 with ψ (0) = ξ Ψ leading to the contradiction ξr ( H [0] ) = ξ dueto (3.4). If v ≡ but u , then u ∈ ˙ W + q satisfies ∂ a u − ∆ D u = − α u and u (0) = ηU and thus u = u η withnecessarily η > by the uniqueness statement of Theorem 2.1. Hence ψ ∈ ˙ W + q satisfies ∂ a ψ − ∆ D ψ = β u η ψ with ψ (0) = ξ Ψ giving the contradiction η = η by (4.6). If u ≡ but v , then v ∈ ˙ W + q satisfies ∂ a v − ∆ D v = − β v and v (0) = ξV what is impossible according to Theorem 2.1 since ξ < . Therefore, neither u ≡ nor v ≡ and weconclude η = η ∗ . This proves the claim. Lemma 4.4.
There is no other bifurcation point to positive coexistence states on B .Proof. The assumption ( η, u η , ∈ B being a bifurcation point to positive coexistence states corresponds to thecase η > , u , and v ≡ in the proof of Lemma 4.3 and analogously implies η = η .This completes the proof of Theorem 2.3. 16 . Proof of Theorem 2.4 Again, the main part of the proof of Theorem 2.4 is a straightforward modification of Section 3, and we thusomit details. Let η > be fixed. Linearization around ( ξ, u η , ∈ T entails the existence of a continuum R + in R × W q × W q of solutions to (1.7)-(1.13) bifurcating from ( ξ , u η , , where ξ := ξ ( η ) ∈ (0 , is given by ξ := 1 r ( H [ − β u η ] ) . (5.1)Near the bifurcation point ( ξ , u η , , R + is a continuous curve in R + × ˙ W + q × ˙ W + q and it can be shown exactly as inLemma 3.11 or Lemma 4.3 that R := R + ∩ ( R + × W + q × W + q ) satisfies the alternatives:(a) R \ { ( ξ , u η , } is unbounded in R + × ˙ W + q × ˙ W + q , or(b) R reaches the boundary of R + × ˙ W + q × ˙ W + q at some point ( ξ, u, v ) = ( ξ , u η , with u ≡ or v ≡ .If (b) occurs, choose a sequence ( ξ j , u j , v j ) in R ∩ ( R + × ˙ W + q × ˙ W + q ) converging toward ( ξ, u, v ) = ( ξ , u η , .Putting φ j := u j / k u j k W q and ψ j := v j / k v j k W q , we may assume that φ j → φ and ψ j → ψ in W q . If both u ≡ and v ≡ , then φ ∈ ˙ W + q satisfies ∂ a φ − ∆ D φ = 0 with φ (0) = η Φ and so ηr ( H [0] ) contradicting (3.4) and η > . If u but v ≡ , then u = u η according to Theorem 2.1, and ψ ∈ ˙ W + q satisfies ∂ a ψ − ∆ D ψ = β u η ψ with ψ (0) = ξ Ψ . Hence ξ = ξ by (5.1) what is impossible since ( ξ, u, v ) = ( ξ , u η , . Therefore, the only possibilityis u ≡ but v . In this case necessarily ξ > and v = v ξ in view of Theorem 2.1. So R joins up with T at ( ξ, , v ξ ) . We remark that then the relation ηr ( G ξ ) = 1 (5.2)with G ξ given in (3.29) must hold, since φ ∈ ˙ W + q satisfies (3.33). That no other bifurcation point(s) on T or T exist(s) is immediate by the previous observations. This yields Theorem 2.4. Remark 5.1.
Since the operator A ξ in (3.28) does not yield a suitable maximum principle, there is no analogue ofLemma 3.2 for the spectral radius of G ξ and the only information we have on r ( G ξ ) is that it is positive for each ξ > as observed in Section 3.2. Consequently, given η > , we cannot decide a priori whether (5.2) holds forsome ξ > . However, for η > , if R joins up with T , then (5.2) must occur and the connection point ( ξ, , v ξ ) on T is determined by this relation. Then again, the relation (5.2) is also a sufficient condition for the existence of acontinuous curve of positive coexistence solutions bifurcating from T .The same difficulty arises when considering bifurcation from T with respect to ξ when η < is fixed. In this case, (5.2) is again a necessary and sufficient condition for the existence of a bifurcation point on T to a curve of positivecoexistence states, which then extends to an unbounded continuum. References [1] H. Amann.
Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces.
SIAM Rev. (1976), no. 4, 620-709.[2] H. Amann. Linear and Quasilinear Parabolic Problems, Volume I: Abstract Linear Theory.
Birkh¨auser 1995.[3] J. Blat, K.J. Brown.
Bifurcation of steady-state solutions in predator-prey and competition systems.
Proc. Royal Soc. Edinburgh (1984),21-34.[4] J. Blat, K.J. Brown.
Global bifurcation of positive solutions in some systems of elliptic equations.
SIAM J. Math. Anal. (1986), 1339-1353.[5] B. Buffoni, J. Toland. Analytic Theory of Global Bifurcation.
Princeton Series in Applied Mathematics, Princeton University Press, 2003.[6] D. Daners, P. Koch-Medina.
Abstract Evolution Equations, Periodic Problems, and Applications.
Pitman Res. Notes Math. Ser., , Long-man, Harlow 1992.[7] G. Degla.
An overview of semi-continuity results on the spectral radius and positivity.
J. Math. Anal. Appl. (2008), 101-110.[8] M. Delgado, J.L´opez-G´omez, A. Su´arez.
On the symbiotic Lotka-Volterra model with diffusion and transport effects.
J. Differential Equations (2000) 175-262.[9] M. Delgado, M. Molina-Becerra, A. Su´arez.
Nonlinear age-dependent diffusive equations: A bifurcation approach.
J. Differential Equations (2008), 2133-2155.[10] M. Delgado, M. Montenegro, A. Su´arez.
A Lotka-Volterra symbiotic model with cross-diffusion.
J. Differential Equations (2009), no. 5,2131-2149.[11] D. Horstmann.
Remarks on some Lotka-Volterra type cross-diffusion models.
Nonlinear Anal. Real World Appl. (2007) 90-117.
12] P. Korman, A. Leung.
On the existence and uniqueness of positive steady-states in the Volterra-Lotka ecological models with diffusion.
Appl.Anal. (1987) 145-160.[13] K. Kuto. Stability of steady-state solutions to a prey-predator system with cross-diffusion.
J. Differential Equations (2004) 293-314.[14] K. Kuto, Y. Yamada.
Multiple coexistence states for a prey-predator system with cross-diffusion.
J. Differential Equations (2004) 315-348.[15] M. Langlais.
Large time behavior in a nonlinear age-dependent population dynamics problem with spatial diffusion.
J. Math. Biol. (1988)319-346.[16] J. L´opez-G´omez. Spectral Theory and Nonlinear Functional Analysis.
Research Notes in Mathematics. Chapman & Hall 2001.[17] Y. Lou.
Necessary and sufficient condition for the existence of positive solutions of certain cooperative system.
Nonlinear Anal. (1996)1079-1095.[18] C.V. Pao. Nonlinear Parabolic and Elliptic Equations.
Plenum Press, New York, 1992.[19] J. Pejsachowicz, P. Rabier.
Degree theory for C Fredholm mappings of index . J. Anal. Math. (1998), 289-319.[20] J. Pr¨uß. Maximal regularity for evolution equations in L p -spaces. Conf. Semin. Mat. Univ. Bari No. (2002), 139 (2003).[21] P.H. Rabinowitz.
Some global results for nonlinear eigenvalue problems.
J. Functional Analysis (1971), 487-513.[22] G. Restrepo. Differentiable norms in Banach spaces.
Bull. Amer. Math. Soc. (1964), 413-414.[23] J. Shi, X. Wang. On global bifurcation for quasilinear elliptic systems on bounded domains.
J. Differential Equations (2009), no. 7,2788-2812.[24] N. Shigesada, K. Kawasaki, E. Teramoto.
Spatial segregation of interacting species.
J. Theoretical Biology, (1979), 83-89.[25] H. Triebel. Interpolation Theory, Function Spaces, Differential Operators.
Second edition. Johann Ambrosius Barth. Heidelberg, Leipzig1995.[26] Ch. Walker.
Positive equilibrium solutions for age and spatially structured population models.
SIAM J. Math. Anal. (2009), 1366-1387.[27] Ch. Walker. Global bifurcation of positive equilibria in nonlinear population models.
J. Differential Equations (2010), 1756-1776.[28] Ch. Walker.
Age-dependent equations with non-linear diffusion.
Discrete Contin. Dyn. Syst. (2010), no. 2, 691-712.[29] Ch. Walker. Bifurcation of positive equilibria in nonlinear structured population models with varying mortality rates.
To appear in: Ann. Mat.Pura Appl. (arXiv:1002.1788[math.AP])[30] Ch. Walker.
On positive solutions of some system of reaction-diffusion equations with nonlocal initial conditions.
To appear in: J. ReineAngew. Math. (arXiv:1003.4698[math.AP]).[31] Ch. Walker.
On nonlocal parabolic steady-state equations of cooperative or competing systems.
Preprint (2010) (arXiv:1008.3125[math.AP]).[32] G.F. Webb.
Population models structured by age, size, and spatial position.
In: P. Magal, S. Ruan (eds.)
Structured Population Models inBiology and Epidemiology.