Global dynamics of a Lotka-Volterra competition patch model
GGlobal dynamics of a Lotka-Volterra competition patch model ∗ Shanshan Chen † Department of Mathematics, Harbin Institute of TechnologyWeihai, Shandong 264209, P. R. China
Junping Shi ‡ Department of Mathematics, William & MaryWilliamsburg, Virginia 23187-8795, USA
Zhisheng Shuai § Department of Mathematics, University of Central FloridaOrlando, Florida 32816, USA
Yixiang Wu ¶ Department of Mathematics, Middle Tennessee State UniversityMurfreesboro, Tennessee 37132, USA
Abstract
The global dynamics of the two-species Lotka-Volterra competition patchmodel with asymmetric dispersal is classified under the assumptions of weakcompetition and the weighted digraph of the connection matrix is strongly con- ∗ S. Chen is supported by National Natural Science Foundation of China (No 11771109), J. Shi issupported by US-NSF grant DMS-1715651 and DMS-1853598, and Z. Shuai is supported by US-NSFgrant DMS-1716445. † Email: [email protected] ‡ Email: [email protected] § Email: [email protected] ¶ Corresponding Author, Email: [email protected] a r X i v : . [ m a t h . C A ] J a n ected and cycle-balanced. It is shown that in the long time, either the compe-tition exclusion holds that one species becomes extinct, or the two species reacha coexistence equilibrium, and the outcome of the competition is determined bythe strength of the inter-specific competition and the dispersal rates. Our maintechniques in the proofs follow the theory of monotone dynamical system and agraph-theoretic approach based on the Tree-Cycle identity. Keywords : Lotka-Volterra competition patch model; global dynamics; weighteddigraph; monotone dynamical system.
MSC 2020 : 92D25, 92D40, 34C12, 34D23, 37C65.
Spatial dispersal of organisms in a heterogeneous environment with uneven resourcedistribution and varying connectivity have long been recognized as key componentsof ecological interactions [3, 4, 11]. Various coupled patch models (or metapopulationmodels) have been proposed to investigate the impact of the environmental hetero-geneity and the connectivity of subregions on the population dynamics. Typically apatch model consist of a system of ordinary differential equations with local dynamicsin each patch coupled with dispersal dynamics between patches.A prominent example of patch models is the Lokta-Volterra competition model ona set of discrete set of habitats [13, 25, 35]. In this paper, we consider the following n -patch two-species Lotka-Volterra competition model: u (cid:48) i = µ u n (cid:88) j =1 ( a ij u j − a ji u i ) + u i ( p i − u i − cv i ) , i = 1 , . . . , n, t > ,v (cid:48) i = µ v n (cid:88) j =1 ( a ij v j − a ji v i ) + v i ( q i − bu i − v i ) , i = 1 , . . . , n, t > ,u (0) = u ≥ ( (cid:54)≡ ) , v (0) = v ≥ ( (cid:54)≡ ) . (1.1)Here u = ( u , . . . , u n ) and v = ( v , . . . , v n ) represent the population densities of twocompeting species in n patches, respectively; n is an integer greater or equal to 2; p i , q i > u i , v i in patch i , respectively;2 , c > µ u , µ v ≥ A = ( a ij ) n × n describes the movement patternbetween patches where a ij ≥ i (cid:54) = j ) is the degree of movement from patch j to patch i . In previous studies of (1.1) [13, 25, 35], it was often assumed that a ij = a ji so thedispersal is symmetric, which is not necessarily assumed here.For our purpose, let L = ( L ij ) n × n denote the connection matrix for the model,where L ij = a ij , i (cid:54) = j, − (cid:88) k (cid:54) = i a ki , i = j. (1.2)Thus (1.1) can be re-written as u (cid:48) i = µ u n (cid:88) j =1 L ij u j + u i ( p i − u i − cv i ) , i = 1 , . . . , n, t > ,v (cid:48) i = µ v n (cid:88) j =1 L ij v j + v i ( q i − bu i − v i ) , i = 1 , . . . , n, t > ,u (0) = u ≥ ( (cid:54)≡ ) , v (0) = v ≥ ( (cid:54)≡ ) . (1.3)The global dynamics of (1.3) when n = 1 is well known, and the regime of bc ≤ < b, c ≤
1, the modelhas a unique positive equilibrium which is globally asymptotically stable. Throughoutout this paper, we assume(A1) b > c >
0, and 0 < bc ≤ p i , q i > i = 1 , , ..., n .(A2) The connection matrix L as defined in (1.2) is irreducible.The assumption (A1) means that the competition between the two species is weakwhile (A2) means that the digraph G associated with A (also L ) is strongly connected.Our results also assume that(A3) The weighted digraph G associated with L is cycle-balanced.3he definition of the weighted digraph G and cycle-balanced will be given in Section2. If A or L is symmetric or n = 2, the assumption (A3) is automatically satisfied. If L is symmetric, the dynamics of (1.3) has been considered in [33] recently.Under the assumptions (A1)-(A3), it is shown that any positive coexistence equi-librium of (1.3) is linearly stable except a special case when bc = 1 (see Theorem 3.1).Together with the theory of monotone dynamical systems [16, 18, 34], the followingalternatives hold for the global dynamics of (1.3): either there exists a unique pos-itive coexistence equilibrium of (1.3) that is globally asymptotically stable; or (1.3)has no coexistence equilibrium and one of the two semitrivial equilibria is globallyasymptotically stable while the other one is unstable (see Theorem 3.2).The results established here resemble the corresponding ones for the reaction-diffusion-advection Lokta-Volterra competition model on a continuous spatial domain: u t = µ u ∆ u − α u ∇ · [ u ∇ Q ( x )] + u ( p ( x ) − u − cv ) , x ∈ Ω , t > ,v t = µ v ∆ v − α v ∇ · [ v ∇ Q ( x )] + v ( q ( x ) − bu − v ) , x ∈ Ω , t > ,µ u ∂u∂n − α u u ∂Q ( x ) ∂n = µ v ∂v∂n − α v v ∂Q ( x ) ∂n = 0 , x ∈ ∂ Ω , t > ,u ( x,
0) = u ≥ , v ( x,
0) = v ≥ , x ∈ ∂ Ω . (1.4)Here u ( x, t ) and v ( x, t ) are the population densities of two competing species at location x ∈ Ω and time t respectively; the habitat Ω is a connected bounded smooth domain in R N for N ≥
1. The combined effect of dispersal rates and environmental heterogeneityon the global dynamics of (1.4) have been studied extensively in recent years in, forexample, [9, 14, 15, 19, 20, 21, 24, 28, 31]. For the diffusive case of α u = α v =0 (corresponding to symmetric L in (1.3)), the global dynamics of (1.4) with weakcompetition bc ≤ α u , α v (cid:54) = 0 (corresponding to asymmetric L in (1.3)) was alsoachieved in [36] under the assumption of µ u /α u = µ v /α v >
0. We refer interestedreaders to the review articles [22, 29] and books [5, 16, 32] for more results for (1.4).Our results here are proved under the similar framework as the ones in [15, 36], but forthe discrete patch model (1.3). We show the effect of dispersal through the connection4atrix L , instead of a diffusion-advection operator as in (1.4), and we utilize the graph-theoretic approach in [26] to bridge the gap between symmetry and asymmetry of L .Many of our results are new and the first of its kind for competition models in a discretespatial landscape (i.e., a patchy environment), even for the case n = 2. Our resultsalso extend the ones in [33] which assumes L to be symmetric.The dynamics of (1.3) with n = 2 (two-patch model) with b = c and µ u = µ v = µ has been studied in [7, 10, 27]. Our results here state that when n = 2 the globaldynamics of the (1.3) is completely determined by the local dynamics of the semitrivialequilibria. Moreover, we generalize the results in [27] to the case n > p i = q i for all i = 1 , , ..., n ; (2) (equal competitiveness and equaldiffusivity) b = c = 1 and µ u = µ v = µ . In this section we provide necessary preliminary results from matrix theory and graphtheory, and also recall the results about the dynamics of a single species patch model.
A vector u = ( u , · · · , u n ) (cid:29) ( u ≥ ) means that every entry of u is positive(nonnegative); a vector u > if u ≥ and u (cid:54) = . Let A = ( a ij ) n × n be an n × n matrixand let σ ( A ) be the set of eigenvalues of A . The spectral bound s ( A ) of A is defined as s ( A ) = max { Re λ : λ ∈ σ ( A ) } . A is reducible if we may partition { , , ..., n } into two nonempty subsets E and F such that a ij = 0 for all i ∈ E and j ∈ F . Otherwise A is irreducible .A weighted digraph G = ( V, E ) associated with the matrix A (denoted as G A inshort) consists of a set V = { , , . . . , n } of vertices and a set E of arcs ( i, j ) (i.e.,directed edges from i to j ) with weight a ji , where ( i, j ) ∈ E if and only if a ji > i (cid:54) = j . A digraph is strongly connected if, for any ordered pair of distinct vertices i, j ,there exists a directed path from i to j . A weighted digraph G A is strongly connected ifand only if the weight matrix A is irreducible [2]. A list of distinct vertices i , i , ..., i k with k ≥ directed cycle if ( i m , i m +1 ) ∈ E for all m = 1 , , ..., k − i k , i ) ∈ E .A subdigraph H of G is spanning if H and G have the same vertex set. The weight of a subdigraph H is the product of the weights of all its arcs. A connected subdigraph T of G is a rooted out-tree if it contains no directed cycle, and there is one vertex,called the root, that is not an terminal vertex of any arcs while each of the remainingvertices is a terminal vertex of exactly one arc. A subdigraph Q of G is unicyclic if itis a disjoint union of two or more rooted out-trees whose roots are connected to forma directed cycle. Every vertex of unicyclic digraph Q is a terminal vertex of exactlyone arc, and thus a unicyclic digraph has also been called a contra-functional digraph [12, page 201].A square matrix is called a row (column) Laplacian matrix if all the off-diagonalentries are nonpositive and the sum of each row (column) is zero. For (1.3), we associateit with a row Laplacian matrix: L = (cid:80) k (cid:54) =1 a k − a · · · − a n − a (cid:80) k (cid:54) =2 a k · · · − a n ... ... . . . ... − a n − a n · · · (cid:80) k (cid:54) = n a nk . (2.1)Note that for the connection matrix L defined in (1.2), − L is a column Laplacianmatrix, and the off-diagonal entries of L and −L are the same. Let α i ≥ i -th diagonal element of L . Then( α , α , . . . , α n ) and (1 , , . . . , T (2.2)are the left and right eigenvectors of L corresponding to eigenvalue 0, respectively. If L is irreducible (equivalently, the digraph G associated with A is strongly connected),then L is also irreducible and thus α i > i . We will also call G as the digraphassociated with L , and L as the Laplacian matrix of G throughout this paper.The following Tree-Cycle identity has been established in [26, Theorem 2.2]. Proposition 2.1 (Tree-Cycle Identity) . Let G be a strongly connected weighted digraphand let L be the Laplacian matrix of G as defined in (2.1) . Let α i denote the cofactorof the i -th diagonal element of L . Then the following identity holds for x i , x j ∈ D ⊂ R N , ≤ i, j ≤ n and any family of functions { F ij : D × D → R } ≤ i,j ≤ nn (cid:88) i =1 n (cid:88) j (cid:54) = i,j =1 α i L ij F ij ( x i , x j ) = (cid:88) Q∈ Q w ( Q ) (cid:88) ( s,r ) ∈ E ( C Q ) F sr ( x s , x r ) , (2.3) where Q is the set of all spanning unicyclic digraphs of ( G , A ) , w ( Q ) > is the weightof Q (the product of weights of all directed edges on Q ), and C Q denotes the directedcycle of Q with arc set E ( C Q ) . We recall that a weighted digraph G is said to be cycle-balanced [26, Section 3] iffor any cycle C in G it has a corresponding reversed cycle −C and w ( C ) = w ( −C ). Here −C , the reverse of C , have the same vertices but edges with reserved direction as C .The following result illustrates how the Tree-Cycle Identity can be used to bridgethe gap between symmetry and asymmetry, and will be used later to analyze theeigenvalue problems related to equilibrium stability. Theorem 2.2.
Let G be a strongly connected weighted digraph that is cycle-balanced,and let L be the Laplacian matrix of G as defined in (2.1) . Let α i denote the cofactorof the i -th diagonal element of L . Assume that x i , x j ∈ D ⊂ R N for all ≤ i, j ≤ n and { F ij : D × D → R } ≤ i,j ≤ n be a family of functions satisfying F ij ( x i , x j ) + F ji ( x j , x i ) ≥ , ≤ i, j ≤ n, j (cid:54) = i. (2.4)7 hen the following holds n (cid:88) i =1 n (cid:88) j (cid:54) = i,j =1 α i L ij F ij ( x i , x j ) = n (cid:88) i =1 n (cid:88) j (cid:54) = i,j =1 α i L ij F ji ( x j , x i ) ≥ . (2.5) In addition, the double sum in (2.5) equals 0 if and only if F ij ( x i , x j ) + F ji ( x j , x i ) = 0 for all distinct i, j .Proof. For any unicyclic digraph Q with a directed cycle C Q , reversing the directionsof all directed edges in C Q (keeping the directions of all other directed edges) yieldsanother unicyclic digraph Q (cid:48) . Since G is cycle-balanced, Q (cid:48) is well-defined and w ( Q (cid:48) ) = w ( Q ). Notice that ( s, r ) ∈ E ( C Q ) iff ( r, s ) ∈ E ( C Q (cid:48) ), and ( s, r ) ∈ E ( Q ) − E ( C Q ) iff( s, r ) ∈ E ( Q (cid:48) ) − E ( C Q (cid:48) ). Perform this process to all unicyclic digraph Q on the righthand side of the Tree-Cycle identity (2.3), and we obtain n (cid:88) i =1 n (cid:88) j (cid:54) = i,j =1 α i L ij F ij ( x i , x j )= (cid:88) Q∈ Q w ( Q ) (cid:88) ( s,r ) ∈ E ( C Q ) F sr ( x s , x r )= 12 (cid:104) (cid:88) Q∈ Q w ( Q ) (cid:88) ( s,r ) ∈ E ( C Q ) F sr ( x s , x r ) + (cid:88) Q∈ Q w ( Q (cid:48) ) (cid:88) ( s,r ) ∈ E ( C Q(cid:48) ) F sr ( x s , x r ) (cid:105) = 12 (cid:104) (cid:88) Q∈ Q w ( Q ) (cid:88) ( s,r ) ∈ E ( C Q ) F sr ( x s , x r ) + (cid:88) Q∈ Q w ( Q ) (cid:88) ( r,s ) ∈ E ( C Q ) F rs ( x r , x s ) (cid:105) = 12 (cid:104) (cid:88) Q∈ Q w ( Q ) (cid:88) ( s,r ) ∈ E ( C Q ) (cid:16) F sr ( x s , x r ) + F rs ( x r , x s ) (cid:17)(cid:105) ≥ , where the last inequality follows from (2.4).Since L is irreducible, the null space of L is one-dimensional. As a consequence, forany positive left eigenvector ( e , e , . . . , e n ) of L corresponding to eigenvalue 0, thereexists a constant η > e i = ηα i for all i . Therefore, the coefficients α i inTheorem 2.2 can be replaced by the coordinators e i of any positive left eigenvector of L corresponding to eigenvalue 0.Finally we show some necessary and sufficient conditions for a digraph to be cycle-balanced, and enclose the proof in Appendix.8 roposition 2.3. Let G be a strongly connected weighted digraph with n vertices as-sociated with n × n matrix A (assuming a ii = 0 for ≤ i ≤ n ).1. If A is symmetric ( a ij = a ji ), then G is cycle-balanced; and if G is cycle-balanced,then A is sign pattern symmetric, that is, a ij > if and only if a ji > for any j (cid:54) = i .2. If G is cycle-balanced, then G has at least n − arcs.3. If every cycle of G has exactly two vertices, then G is cycle-balanced; and everycycle of G has exactly two vertices if and only if A is sign pattern symmetric and G has exactly n − arcs. In addition, these arcs form two spanning trees in G of the opposite direction: one rooted in-tree and one rooted out-tree. In particular,if n = 2 , then G is cycle-balanced.4. Suppose that G is a complete graph ( a ij > for any i (cid:54) = j ) with at least vertices.Then G is cycle-balanced if and only if each -cycle of G is balanced, that is, forany three distinct vertices i, j, k , we have a ij a jk a ki = a ik a kj a ji . (a) (b) abc def Figure 1: (a) A star migration graph. (b) A digraph that is cycle-balanced if abc = def .The characterization in part 3 of Proposition 2.3 shows that a bi-directional treeis the cycle-balanced network with the minimal number of arcs. One example of such9 2 3 4 5 6 r + s r + s r + s r + s r + sr r r r r Figure 2: Stepping stone modelbi-directional tree is the star graph in Figure 1(a). Because of strong connectivity of G , each row or column of A has at least one non-zero entry. Also for this network, thevalue of the weight a ij does not change the cycle-balanced property of the network,which is not the case for network with longer cycles (like the one in Figure 1(b) orcomplete graphs as in part 4 of Proposition 2.3). We recall results on a single species population model in a heterogeneous environmentof n patches ( n ≥ w (cid:48) i = µ n (cid:88) j =1 L ij w j + w i ( r i − w i ) , i = 1 , . . . , n, t > , (2.6)where w i denotes the population size (or density) in patch i . System (2.6) admitsa trivial equilibrium = (0 , , . . . , Lemma 2.4.
Suppose that the connection matrix L as defined in (1.2) is irreducible,and r i > for all i = 1 , , ..., n . Then the equilibrium is unstable, and (2.6) admitsa unique positive equilibrium w ∗ ( µ, r ) = ( w ∗ , . . . , w ∗ n ) , which is globally asymptoticallystable in R n + − { } . Throughout this paper, we assume that L is irreducible and p i , q i > i =1 , , ..., n . By Lemma 2.4, (1.3) has two semi-trivial equilibria, E = ( w ∗ ( µ u , p ) , ) and E = ( , w ∗ ( µ v , q )), and one trivial equilibria E = ( , ). A positive equilibrium of(1.3), if exists, is denoted by E = ( u, v ) when not causing confusion.An important tool to investigate the global dynamics of the Lotka-Volterra com-petition system (1.3) is the monotone dynamical system theory [16, 18, 23, 34]. Let10 = R n + × R n + equipped with an order ≤ K generated by the cone K = R n + × {− R n + } .That is, for x = (¯ u, ¯ v ) , y = (˜ u , ˜ v ) ∈ X , we say x ≤ K y if ¯ u ≤ ˜ u and ¯ v ≥ ˜ v ; x < K y if x ≤ K y and x (cid:54) = y . The solutions of (1.3) induce a strictly monotone dynamicalsystem in X in the sense that for any two initial data ( u , v ) < K ( u , v ), the cor-responding solutions satisfy ( u ( t ) , v ( t )) < K ( u ( t ) , v ( t )) for all t ≥
0. The notableresults derived from the strictly monotone dynamical system theory are that the globaldynamics of (1.3) is largely determined by the local (linearized) stability of E and E :1. if E is unstable and (1.3) has no positive equilibrium, then E is globally asymp-totically stable; if E is unstable and (1.3) has no positive equilibrium, then E is globally asymptotically stable;2. if E and E are both unstable, then (1.3) has at least one stable positive equi-librium, which is globally asymptotically stable if it is unique;3. if E and E are both locally asymptotically stable, then (1.3) has at least oneunstable positive equilibrium.A key step in the classification results of [15] (and [36]) is show the stability of thepositive equilibrium as they have observed: if every positive equilibrium is stable whenexists, then either E or E is globally asymptotically stable or (4.1) has a positiveglobally stable equilibrium, except for the degenerate case when both E and E arelinearly neutrally stable. In this section we state our main results on the global dynamics of (1.3). Recallthat w ∗ ( µ, r ) is the unique positive steady state of (2.6). Then the following resultstates that every positive equilibrium of (1.3), if exists, is locally asymptotically stableexcept for the degenerate case, when the competition is weak and the network G iscycle-balanced. 11 heorem 3.1. Suppose that (A1)-(A3) hold. A positive equilibrium E = ( u, v ) of (1.3) , if exists, is locally asymptotically stable except for the case bc = 1 and w ∗ ( µ u , p ) = cw ∗ ( µ v , q ) at which case E is linearly neutrally stable.Proof. Let E = ( u, v ) be a positive equilibrium of (1.3). Then it satisfies the followingequations: µ u n (cid:88) j =1 L ij u j + u i ( p i − u i − cv i ) = 0 , i = 1 , . . . , n,µ v n (cid:88) j =1 L ij v j + v i ( q i − bu i − v i ) = 0 , i = 1 , . . . , n. (3.1)Linearizing (1.3) at E , we have the following eigenvalue problem µ u n (cid:88) j =1 L ij φ j + ( p i − u i − cv i ) φ i − u i ( φ i + cψ i ) + λφ i = 0 , i = 1 , . . . , n,µ v n (cid:88) j =1 L ij ψ j + ( q i − bu i − v i ) ψ i − v i ( bφ i + ψ i ) + λψ i = 0 , i = 1 , . . . , n. (3.2)Here ( φ, ψ ) = (( φ , ..., φ n ) , ( ψ , ..., ψ n )) is a principal eigenvector associated with theprincipal eigenvalue λ of (3.2). We normalize ( φ, ψ ) such that φ i > ψ i < i = 1 , ..., n . We will show that λ ≥
0, where the equality holds if and only if bc = 1and w ∗ ( µ u , p ) = cw ∗ ( µ v , q ).Multiplying the first equation of (3.2) by u i , the first equation of (3.1) by φ i andtaking the difference, we have µ u n (cid:88) j =1 ,j (cid:54) = i L ij ( φ j u i − φ i u j ) = u i ( φ i + cψ i ) − λu i φ i , i = 1 , . . . , n. (3.3)Let ( α , ..., α n ) be the left eigenvector of L as defined in (2.2). Multiplying both sidesof (3.3) by α i φ i /u i and summing up all the equations, we obtain µ u n (cid:88) i =1 n (cid:88) j =1 ,j (cid:54) = i α i L ij (cid:18) φ i φ j u i − φ i u j u i (cid:19) = n (cid:88) i =1 α i φ i ( φ i + cψ i ) − λ n (cid:88) i =1 α i φ i u i . (3.4)Let { F ij : D × D → R } ≤ i,j ≤ n be a family of functions defined as F ij ( x i , x j ) =12 i φ j u i − φ i u j u i , where D = (0 , ∞ ) × (0 , ∞ ) and x i = ( φ i , u i ) ∈ D . It can be verified that F ij ( x i , x j ) + F ji ( x j , x i ) = (cid:18) φ i φ j u i − φ i u j u i (cid:19) + (cid:18) φ i φ j u j − φ j u i u j (cid:19) = u i u j φ i u i (cid:18) φ j u j − φ i u i (cid:19) + u i u j φ j u j (cid:18) φ i u i − φ j u j (cid:19) = − u i u j (cid:18) φ i u i − φ j u j (cid:19) (cid:18) φ i u i + φ j u j (cid:19) ≤ , and the equal sign holds if and only if φ i /u i = φ j /u j . Since G is cycle-balanced, itfollows from Theorem 2.2 that n (cid:88) i =1 n (cid:88) j =1 ,j (cid:54) = i α i L ij F ij ( x i , x j ) = n (cid:88) i =1 n (cid:88) j =1 ,j (cid:54) = i α i L ij (cid:18) φ i φ j u i − φ i u j u i (cid:19) ≤ , (3.5)and the equal sign holds if and only if φ u = φ u = · · · = φ n u n . (3.6)Similarly, by using (3.1) and (3.2), we obtain µ v n (cid:88) i =1 n (cid:88) j =1 ,j (cid:54) = i α i L ij (cid:18) ψ i ψ j v i − ψ i v j v i (cid:19) = n (cid:88) i =1 α i ψ i ( bφ i + ψ i ) − λ n (cid:88) i =1 α i ψ i v i . (3.7)Similar to (3.5), it follows from Theorem 2.2 that n (cid:88) i =1 n (cid:88) j =1 ,j (cid:54) = i α i L ij (cid:18) ψ i ψ j v i − ψ i v j v i (cid:19) ≥ , (3.8)where the equal sign holds if and only if ψ v = ψ v = · · · = ψ n v n . (3.9)Multiplying (3.7) by c and subtracting it from (3.4), and noticing (3.5) and (3.8),we have − λ n (cid:88) i =1 α i (cid:18) φ i u i − c ψ i v i (cid:19) ≤ − n (cid:88) i =1 α i φ i ( φ i + cψ i ) + n (cid:88) i =1 α i c ψ i ( bcφ i + cψ i ) ≤ − n (cid:88) i =1 α i φ i ( φ i + cψ i ) + n (cid:88) i =1 α i c ψ i ( φ i + cψ i )= − n (cid:88) i =1 α i ( φ i − cψ i )( φ i + cψ i ) ≤ , bc ≤
1. This implies λ ≥ bc = 1, φ i = cψ i for all i , and (3.6) and (3.9) hold.Now we consider the situation when λ = 0. It follows that u = kv for some k > bc = 1. By u = kv and the first equation of (3.1), we have w ∗ ( µ u , p ) = (cid:16) ck (cid:17) u By the second equation of (3.1), we have w ∗ ( µ v , q ) = ( kb + 1) v. Therefore, w ∗ ( µ u , p ) w ∗ ( µ v , q ) = (cid:0) ck (cid:1) u ( kb + 1) v = c. Denote by λ ( µ, h ) the principal eigenvalue of µ n (cid:88) j =1 L ij ψ j + h i ψ i + λψ i = 0 , i = 1 , . . . , n, where h = ( h , . . . , h n ). Then λ ( µ, h ) = − s ( µL + diag ( h i )). Let E = ( w ∗ ( µ u , p ) , )and E = ( , w ∗ ( µ v , q )) be the two semi-trivial equilibria of (1.1). Following [15] wedefine the following parameter subsets of Q = { ( µ u , µ v ) : µ u , µ v > } . S u = { ( µ u , µ v ) : ( w ∗ ( µ u , p ) , ) is linearly stable } = { ( µ u , µ v ) : λ ( µ v , q − bw ∗ ( µ u , p )) > } ,S v = { ( µ u , µ v ) : ( , w ∗ ( µ v , q )) is linearly stable } = { ( µ u , µ v ) : λ ( µ u , p − cw ∗ ( µ v , q )) > } ,S − = { ( µ u , µ v ) : λ ( µ v , q − bw ∗ ( µ u , p )) < , λ ( µ u , p − cw ∗ ( µ v , q )) < } ,S u, = { ( µ u , µ v ) : λ ( µ v , q − bw ∗ ( µ u , p )) = 0 } ,S v, = { ( µ u , µ v ) : λ ( µ u , p − cw ∗ ( µ v , q )) = 0 } ,S , = { ( µ u , µ v ) : λ ( µ v , q − bw ∗ ( µ u , p )) = λ ( µ u , p − cw ∗ ( µ v , q )) = 0 } . We classify the global dynamics of (1.3) according to the diffusion coefficients asfollows. 14 heorem 3.2.
Suppose that (A1)-(A3) hold. Then we have the following mutuallydisjoint decomposition of Q : Q = ( S u ∪ S u, \ S , ) (cid:91) ( S v ∪ S v, \ S , ) (cid:91) S − (cid:91) S , . (3.10) Moreover, the following statements hold for system (1.3) : (i) For any ( µ u , µ v ) ∈ S u ∪ S u, \ S , , E = ( w ∗ ( µ u , p ) , ) is globally asymptoticallystable. (ii) For any ( µ u , µ v ) ∈ S v ∪ S v, \ S , , E = ( , w ∗ ( µ v , q )) is globally asymptoticallystable. (iii) For any ( µ u , µ v ) ∈ S − , (1.3) has a unique positive equilibrium ( u, v ) , which isglobally asymptotically stable. (iv) For any ( µ u , µ v ) ∈ S , , we have bc = 1 , w ∗ ( µ u , p ) ≡ cw ∗ ( µ v , q ) and (1.3) has acompact global attractor consisting of a continuum of equilibria { ( ρw ∗ ( µ u , p ) , (1 − ρ ) w ∗ ( µ u , p ) /c ) : ρ ∈ [0 , } . (3.11) Proof. Step 1
We first show the mutually disjoint decomposition of Q . For simplic-ity of notations, we denote u ∗ = w ∗ ( µ u , p ) and v ∗ = w ∗ ( µ v , q ). Denote by ψ =( ψ , . . . , ψ n ) > (respectively, φ = ( φ , . . . , φ n ) > ) the principal eigenvector withrespect to λ ( µ v , q − bu ∗ ) (respectively, λ ( µ u , p − cv ∗ )). Then we have µ u n (cid:88) j =1 L ij φ j + ( p i − cv ∗ i ) φ i + λ ( µ u , p − cv ∗ ) φ i = 0 , i = 1 , . . . , n,µ v n (cid:88) j =1 L ij ψ j + ( q i − bu ∗ i ) ψ i + λ ( µ v , q − bu ∗ ) ψ i = 0 , i = 1 , . . . , n. (3.12)Note that u ∗ = ( u ∗ , · · · , u ∗ n ) and v ∗ = ( v ∗ , · · · , v ∗ n ) satisfy µ u n (cid:88) j =1 L ij u ∗ j + u i ( p i − u ∗ i ) = 0 , i = 1 , . . . , n,µ v n (cid:88) j =1 L ij v ∗ j + v i ( q i − v ∗ i ) = 0 , i = 1 , . . . , n. (3.13)15ultiplying the first equation of (3.12) by u ∗ i , the first equation of (3.13) by φ i andtaking the difference, we have µ u n (cid:88) j =1 ,j (cid:54) = i L ij ( φ j u ∗ i − φ i u ∗ j ) = ( cv ∗ i − u ∗ i ) u ∗ i φ i − λ ( µ u , p − cv ∗ ) φ i u ∗ i , i = 1 , . . . , n (3.14)Let ( α , ..., α n ) be the left eigenvector of L as defined in (2.2). Multiplying both sidesof (3.14) by α i u ∗ i /φ i , and taking the sum, we obtain µ u n (cid:88) i =1 n (cid:88) j =1 ,j (cid:54) = i α i L ij (cid:18) φ j ( u ∗ i ) φ i − u ∗ i u ∗ j (cid:19) = n (cid:88) i =1 α i ( cv ∗ i − u ∗ i )( u ∗ i ) − λ ( µ u , p − cv ∗ ) n (cid:88) i =1 α i ( u ∗ i ) . (3.15)Let F ij = F ij ( φ i , φ j ) = φ j ( u ∗ i ) φ i − u ∗ i u ∗ j . Since F ij + F ji = φ j ( u ∗ i ) φ i − u ∗ i u ∗ j + φ i ( u ∗ j ) φ j − u ∗ j u ∗ i = (cid:32) (cid:112) φ j √ φ i u ∗ i − √ φ i (cid:112) φ j u ∗ j (cid:33) ≥ , and the equality holds if and only if φ i /u ∗ i = φ j /u ∗ j , it follows from Theorem 2.2 that n (cid:88) i =1 n (cid:88) j =1 ,j (cid:54) = i α i L ij (cid:18) φ j ( u ∗ i ) φ i − u ∗ i u ∗ j (cid:19) ≥ , (3.16)and the equal sign holds if and only if φ u ∗ = φ u ∗ = · · · = φ n u ∗ n . (3.17)Similarly, by using (3.12) and (3.13), we have µ v n (cid:88) i =1 n (cid:88) j =1 ,j (cid:54) = i α i L ij (cid:18) ψ j ( v ∗ i ) ψ i − v ∗ i v ∗ j (cid:19) = n (cid:88) i =1 α i ( bu ∗ i − v ∗ i )( v ∗ i ) − λ ( µ v , q − bu ∗ ) n (cid:88) i =1 α i ( v ∗ i ) , (3.18)and n (cid:88) i =1 n (cid:88) j =1 ,j (cid:54) = i α i L ij (cid:18) ψ j ( v ∗ i ) v i − v ∗ i v ∗ j (cid:19) ≥ , (3.19)16here the equal sign holds if and only if ψ v ∗ = ψ v ∗ = · · · = ψ n v ∗ n . (3.20)Multiplying (3.18) by c and subtracting it from (3.15), and noticing (3.16) and(3.15), we have λ ( µ u , p − cv ∗ ) n (cid:88) i =1 α i ( u ∗ i ) + λ ( µ v , q − bu ∗ ) c n (cid:88) i =1 α i ( v ∗ i ) ≤ n (cid:88) i =1 α i ( cv ∗ i − u ∗ i )( u ∗ i ) + n (cid:88) i =1 α i ( bcu ∗ i − cv ∗ i )( cv ∗ i ) ≤ − n (cid:88) i =1 α i ( cv ∗ i − u ∗ i ) ( cv ∗ i + u ∗ i ) ≤ , (3.21)where the last two inequalities are equalities if and only if cv ∗ = u ∗ and bc = 1. Thisimplies that ( S u ∪ S u, \ S , ) ∩ ( S v ∪ S v, \ S , ) = ∅ , (3.22)which proves (3.10). Step 2
As in [15], we rule out the possibility of coexistence equilibrium in the followingtwo cases:(A) ( µ u , µ v ) ∈ S u, \ S , ;(B) ( µ u , µ v ) ∈ S v, \ S , .It suffices to consider case (A). To the contrary, we assume that there exists a coexis-tence equilibrium ( U ∗ , V ∗ ) for some ( µ u , µ v ) ∈ S u, \ S , and ( b, c ) = ( b , c ) satisfying17 c ≤
1. So, λ ( µ v , q − b u ∗ ) = 0 and λ ( µ u , p − c v ∗ ) <
0. We define G ( b, c, u, v ) = µ u n (cid:88) j =1 L j u j + u ( p − u − cv )... µ u n (cid:88) j =1 L nj u j + u n ( p n − u n − cv n ) µ v n (cid:88) j =1 L j v j + v ( q − bu − v )... µ v n (cid:88) j =1 L nj v j + v n ( q n − bu n − v n ) . Then we compute the Jacobian matrix of G evaluated at ( b, c, u, v ) = ( b , c , U ∗ , V ∗ ): DG ( u,v ) ( b , c , U ∗ , V ∗ ) = µ u L + diag( p − U ∗ − c V ∗ ) − diag( c U ∗ ) − diag( b V ∗ ) µ v L + diag( q − b U ∗ − V ∗ ) . By Theorem 3.1, the principal eigenvalue of DG ( u,v ) ( b , c , U ∗ , V ∗ ) is negative andso all of its eigenvalues are on the right half plane of the complex plane. There-fore, DG ( u,v ) ( b , c , U ∗ , V ∗ ) is invertible. By G ( b , c , U ∗ , V ∗ ) = 0 and the implicitfunction theorem, there exist positive solutions ( u ( b, c ) , v ( b, c )) of G ( b, c, u, v ) = 0 for( b, c ) close to ( b , c ), where ( u ( b, c ) , v ( b, c )) is continuously differentiable in ( b, c ) with( u ( b , c ) , v ( b , c )) = ( U ∗ , V ∗ ). By the definition of G , ( u ( b, c ) , v ( b, c )) is a positiveequilibrium of (1.3). Noticing λ ( µ u , p − c v ∗ ) <
0, we may choose (ˇ b, ˇ c ) close to( b , c ) with 0 < ˇ c < c , ˇ b > b , and ˇ b ˇ c ≤ u (ˇ b, ˇ c ) , v (ˇ b, ˇ c )) with λ ( µ u , p − ˇ cv ∗ ) <
0. Since ˇ b > b , we have λ ( µ v , q − ˇ bu ∗ ) > λ ( µ v , q − b u ∗ ) = 0 . Then (ˇ b, ˇ c ) ∈ S u , which means ( u ∗ ,
0) is globally asymptotically stable. This contradictsthat (1.3) has a positive equilibrium ( u (ˇ b, ˇ c ) , v (ˇ b, ˇ c )). Step 3
By Theorem 3.1 and Steps 2-3, the claims (i)-(iv) follow from the strictly mono-tone dynamical system theory [16, 18, 23, 34].18 tep 4
Finally, we show (iv). By (3.21), if λ ( µ u , p − cv ∗ ) = λ ( µ v , q − bu ∗ ) = 0 then u ∗ = cv ∗ and bc = 1, i.e. S , ⊂ { ( µ u , µ v ) : u ∗ = cv ∗ and bc = 1 } . On the other hand, if u ∗ = cv ∗ and bc = 1, from µ u n (cid:88) j =1 L ij u ∗ j + u ∗ i ( p i − u ∗ i ) = 0, wehave µ u n (cid:88) j =1 L ij v ∗ j + v ∗ i ( p i − cv ∗ i ) = 0 , which implies λ ( µ u , p − cv ∗ ) = 0. Similarly, λ ( µ v , q − bu ∗ ) = 0. So we have { ( µ u , µ v ) : u ∗ = cv ∗ and bc = 1 } ⊂ S , . Hence, S , = { ( µ u , µ v ) : u ∗ = cv ∗ and bc = 1 } . (3.23)It is easy to check that (1.3) has a continuum of equilibria (3.11) when ( µ u , µ v ) ∈ S , ,which is a global attractor by [17, Theorem 3] in the sense that every solution of (1.3)converges to an equilibrium in (3.11). The result in [17, Theorem 3] is for reaction-diffusion competition systems, which also holds for patch models. In this section, we apply the results obtained in Section 3 to two special situations.The following result is needed for the applications (see [1, 6]).
Lemma 4.1.
Suppose that n × n matrix A is irreducible and quasi-positive (i.e. off-diagonal entries are nonnegative) with s ( A ) = 0 . Let η = ( η , η , ..., η n ) T be a positiveright eigenvector of A corresponding to eigenvalue s ( A ) = 0 and D = diag ( d j ) be adiagonal matrix. Then, dda s ( aA + D ) ≤ , for all a > , where the equality holds if and only if D is a multiple of the identity matrix I . Moreover,the following limits hold: lim a → s ( aA + D ) = max ≤ i ≤ n { d i } and lim a →∞ s ( aA + D ) = n (cid:88) i =1 η i d i / n (cid:88) i =1 η i . .1 Example (A) Firstly, we consider a situation that two species compete for the common resource (i.e. p = q = r ): u (cid:48) i = µ u n (cid:88) j =1 L ij u j + u i ( r i − u i − cv i ) , i = 1 , . . . , n, t > ,v (cid:48) i = µ v n (cid:88) j =1 L ij v j + v i ( r i − bu i − v i ) , i = 1 , . . . , n, t > ,u (0) = u ≥ ( (cid:54)≡ ) , v (0) = v ≥ ( (cid:54)≡ ) , (4.1)where r > . We remark that the continuous space version of model (4.1) has beeninvestigated in [36]. Then we have the following results. Theorem 4.2.
Suppose that < bc ≤ and ( A − ( A holds. Let θ = ( θ , θ , ..., θ n ) be a positive right eigenvector of L corresponding to s ( L ) = 0 with (cid:80) ni =1 θ i = 1 . Thenthe following statements hold for (4.1) : (i) If r = δθ for some δ > , then we have: (i ) If ( b, c ) = (1 , , there exists a compact global attractor consisting of a con-tinuum of equilibria { ( ρr, (1 − ρ ) r ) : ρ ∈ [0 , } ;(i ) If b ≥ and c < , E = ( w ∗ ( µ u , r ) , ) is globally asymptotically stable; (i ) If b < and c ≥ , E = ( , w ∗ ( µ v , r )) is globally asymptotically stable; (i ) If b < and c < , there exists a unique positive equilibrium E = (cid:18) − c − bc r, − b − bc r (cid:19) , which is globally asymptotically stable. (ii) If r (cid:54) = δθ for any δ > , then we have: (ii ) Suppose µ u < µ v . Then there exist b ∗ < and c ∗ > with b ∗ c ∗ > such thatif b < b ∗ and c < c ∗ (4.1) has a unique positive equilibrium which is globallyasymptotically stable; if b ≥ b ∗ , E = ( w ∗ ( µ u , r ) , ) is globally asymptoticallystable; if c ≥ c ∗ , E = (0 , w ∗ ( µ v , r )) is globally asymptotically stable; ) Suppose µ u > µ v . Then there exist b ∗ > and c ∗ < with b ∗ c ∗ > such thatif b < b ∗ and c < c ∗ (4.1) has a unique positive equilibrium which is globallyasymptotically stable; if b ≥ b ∗ , E = ( w ∗ ( µ u , r ) , ) is globally asymptoticallystable; if c ≥ c ∗ , E = (0 , w ∗ ( µ v , r )) is globally asymptotically stable; (ii ) Suppose µ u = µ v . Then if b < and c < , (4.1) has a unique positiveequilibrium which is globally asymptotically stable; if b ≥ and c < , E = ( w ∗ ( µ u , r ) , ) is globally asymptotically stable; if c ≥ and b < , E = ( , w ∗ ( µ v , r )) is globally asymptotically stable; if ( b, c ) = (1 , , thereexists a compact global attractor consisting of a continuum of steady states { ( ρw ∗ ( µ u , r ) , (1 − ρ ) w ∗ ( µ v , r )) : ρ ∈ [0 , } . Proof. (i) If r = δθ for some δ >
0, we have w ∗ ( µ u , r ) = w ∗ ( µ v , r ) = r. For the case b = c = 1, a direct computation yields λ ( µ v , r − bw ∗ ( µ u , r )) = λ ( µ v , ) = 0 ,λ ( µ u , r − cw ∗ ( µ v , r )) = λ ( µ u , ) = 0 . Then it follows from Theorem 3.2 (iv) that (i ) holds.For the case b ≥ c <
1, we have λ ( µ u , r − bw ∗ ( µ v , r )) = λ ( µ u , (1 − b ) r ) ≥ ,λ ( µ u , r − cw ∗ ( µ v , r )) = λ ( µ u , (1 − c ) r ) < . Then it follows from Theorem 3.2 (i) that E = ( w ∗ ( µ u , r ) , ) is globally asymptoticallystable, which proves (i ). Similarly, we can prove (i ).For the case b < c <
1, we have λ ( µ v , r − bw ∗ ( µ u , r )) = λ ( µ v , (1 − b ) r ) < ,λ ( µ u , r − cw ∗ ( µ v , r )) = λ ( µ u , (1 − c ) r ) < . Then it follows from Theorem 3.2 (iii) that statement (i ) holds.21ii) Suppose r (cid:54) = δθ for any δ >
0. Then r − w ∗ ( µ u , r ) (cid:54) = γ (1 , . . . ,
1) for any γ ∈ R .If this is not true, then there exists γ ∈ R such that r − w ∗ ( µ u , r ) = γ (1 , . . . , w ∗ := w ∗ ( µ u , r ) satisfies µ u n (cid:88) j =1 L ij w ∗ j + w ∗ i ( r i − w ∗ i ) = 0 , we conclude that − γ /µ u is the principal eigenvalue of L with w ∗ ( µ u , r ) being a positiveeigenvector. Then γ = 0, and r = w ∗ ( µ u , r ) = δ θ for some δ >
0, which is acontradiction. Similarly, we obtain that r − w ∗ ( µ v , r ) (cid:54) = γ (1 , . . . ,
1) for any γ ∈ R .Then it follows from Lemma 4.1 that λ ( µ, r − w ∗ ( µ u , r )) and λ ( µ, r − w ∗ ( µ v , r )) isstrictly increasing for µ ∈ (0 , ∞ ).Note that λ ( µ u , r − w ∗ ( µ u , r )) = 0 and λ ( µ v , r − w ∗ ( µ v , r )) = 0 . (ii ) If µ u < µ v , we have λ ( µ v , r − bw ∗ ( µ u , r )) | b =1 > λ ( µ u , r − cw ∗ ( µ v , r )) | c =1 < . Since λ ( µ v , r − bw ∗ ( µ u , r )) | b =0 < λ ( µ v , r − bw ∗ ( µ u , r )) is strictly increasing in b , there exists b ∗ ∈ (0 ,
1) such that λ ( µ v , r − bw ∗ ( µ u , r )) < , b < b ∗ , = 0 , b = b ∗ > , b > b ∗ . (4.2)Since λ ( µ u , r − cw ∗ ( µ v , r )) is strictly increasing in c and lim c →∞ λ ( µ u , r − cw ∗ ( µ v , r )) = ∞ , there exists c ∗ > λ ( µ u , r − cw ∗ ( µ v , r )) < , c < c ∗ , = 0 , c = c ∗ > , c > c ∗ . (4.3)We claim b ∗ c ∗ >
1. To see it, we first note that b ∗ c ∗ < b, c ) such that b < b ∗ and c < c ∗ with bc <
1. For such ( b, c ), we have22 ( µ v , r − bw ∗ ( µ u , r )) < λ ( µ u , r − cw ∗ ( µ v , r )) <
0, i.e. both E and E arestable, which is impossible by Theorem 3.2.Suppose to the contrary that b ∗ c ∗ = 1. Since λ ( µ v , r − b ∗ w ∗ ( µ u , r )) = λ ( µ u , r − c ∗ w ∗ ( µ v , r )) = 0, we have w ∗ ( µ u , r ) = c ∗ w ∗ ( µ v , r ) by Theorem 3.2 (iv). Putting thisinto µ u n (cid:88) j =1 L ij w ∗ j ( µ u , r ) + w ∗ i ( µ u , r )( r i − w ∗ i ( µ u , r )) = 0 ,µ v n (cid:88) j =1 L ij w ∗ j ( µ v , r ) + w ∗ i ( µ v , r )( r i − w ∗ i ( µ v , r )) = 0 , we obtain w ∗ i ( µ v , r ) = µ v − µ u c ∗ µ v − µ u r i < r i for all i = 1 , ..., n. Therefore, noticing (cid:80) ni =1 L ij = 0 for all j = 1 , ..., n , we have0 = n (cid:88) i =1 ( µ v n (cid:88) j =1 L ij w ∗ j ( µ v , r )+ w ∗ i ( µ v , r )( r i − w ∗ i ( µ v , r ))) = n (cid:88) i =1 w ∗ i ( µ v , r )( r i − w ∗ i ( µ v , r )) > , which is a contradiction. This proves the claim.It follows from (4.2)-(4.3), b ∗ c ∗ > ) holds.Using similar arguments for (ii ), we can prove (ii ). For the case (ii ), we have b ∗ = c ∗ = 1 and its proof is similar to (ii ). Remark . We have a complete classification of the global stability of Example A byTheorem 4.2. In Figure 3, we have plotted a picture to illustrate our results for thecase (ii ) (i.e. r (cid:54) = δθ and µ u < µ v ). In this subsection, we consider a special case: u (cid:48) i = µ n (cid:88) j =1 L ij u j + u i ( p i − u i − v i ) , i = 1 , . . . , n, t > ,v (cid:48) i = µ n (cid:88) j =1 L ij v j + v i ( q i − u i − v i ) , i = 1 , . . . , n, t > ,u (0) = u ≥ ( (cid:54)≡ ) , v (0) = v ≥ ( (cid:54)≡ ) . (4.4)23 * * c (1, 1) bc=1I IIIII Figure 3: Illustration of Theorem 4.2 (ii ) ( r (cid:54) = δθ and µ u < µ v ) for Example A.Here, b ∗ < c ∗ >
1, and b ∗ c ∗ >
1. In I, there exists a globally asymptotically stablepositive equilibrium; in II, E is globally asymptotically stable; in III, E is globallyasymptotically stable.Here the two species have the same intraspecific competition coefficients and diffusionrates, but different resources availability. The case n = 2 (two-patch model) of (4.4)has been investigated by [7, 10, 27]. In [10], it was conjectured that if q − σ = p µ ∗ the u -only semitrivial equilibrium is globallyasymptotically stable. In [7], it was shown that such a threshold result no longer holdswhen the inequalities on the birth rates p i , q i are relaxed. We consider the general casewith n ≥ Theorem 4.4.
Suppose that (A1)-(A3) hold. If p > q , then E = ( w ∗ ( µ, p ) , ) is lobally asymptotically stable for (4.4) ; if p < q , then E = ( , w ∗ ( µ, q )) is globallyasymptotically stable for (4.4) .Proof. We only show the case of p > q . For simplicity of notations, we denote u ∗ = w ∗ ( µ, p ) and v ∗ = w ∗ ( µ, q ). By Theorem 3.2, it suffices to consider the signs of λ ( µ, q − u ∗ ) and λ ( µ, p − v ∗ ), where u ∗ and v ∗ also depend on µ . Let φ be a positive eigenvectorcorresponding to λ ( µ, q − u ∗ ). Then µ n (cid:88) j =1 L ij φ j + ( p i − u ∗ i ) φ i + ( q i − p i ) φ i + λ ( µ, q − u ∗ ) φ i = 0 . (4.5)Denote A = µL + diag( p j − u ∗ j ) and D = diag( q j − p j ). Clearly, u ∗ is a positive righteigenvector of A corresponding with eigenvalue 0. So s ( A ) = 0. Therefore, by Lemma4.1, s ( aA + D ) is strictly decreasing in a andlim a → s ( aA + D ) = max ≤ j ≤ n { q j − p j } ≤ a →∞ s ( aA + D ) = n (cid:88) i =1 η i ( q i − p i ) < , where η = ( η , . . . , η n ) is the positive right eigenvector of A corresponding to s ( A ) with (cid:80) ni =1 η i = 1. This yields λ ( µ, q − u ∗ ) = − s ( A + D ) > . (4.6)Similarly, we can prove that λ ( µ, p − v ∗ ) < . This, combined with (4.6), impliesthat ( µ, µ ) ∈ S u ∪ S u, \ S , , and consequently, E is globally asymptotically stable byTheorem 3.2.Next we consider the case when the resources of the two competing species are notcomparable. Theorem 4.5.
Suppose that (A1)-(A3) hold, and p (cid:54)≥ q and q (cid:54)≥ p . Let θ = ( θ , θ , ..., θ n ) be a positive right eigenvector of L corresponding to s ( L ) = 0 satisfying (cid:80) ni =1 θ i = 1 .Then the following statements hold for (4.4) :1. There exists µ > such that (4.4) has a unique positive equilibrium which isglobally asymptotically stable for < µ < µ . . If n (cid:88) j =1 θ j ( p j − q j ) > , then there exists µ > µ such that E is globally asymp-totically stable for µ > µ ; on the other hand if n (cid:88) j =1 θ j ( p j − q j ) < , then thereexists µ > µ such that E is globally asymptotically stable for µ > µ .Proof. We claim that there exist m, M > mθ ≤ u ∗ ≤ M θ for all µ >
0. Tosee that, we may choose m, M > u = mθ is a lower solution and u = M θ is an upper solution of u (cid:48) i = µ n (cid:88) j =1 L ij u j + u i ( p i − u i ) , i = 1 , . . . , n. (4.7)Since u ∗ is the unique globally asymptotically stable positive equilibrium of (4.7), wehave mθ ≤ u ∗ ≤ M θ for all µ > u ∗ is the unique positive equilibrium of (4.7), we have u ∗ i → p i as µ → µ → u ∗ ≥ mθ . By Lemma 4.1and lim µ → u ∗ = p , we have lim µ → λ ( µ, q − u ∗ ) = − max ≤ i ≤ n { q i − p i } <
0. Similarly,lim µ → λ ( µ, p − v ∗ ) = − max ≤ i ≤ n { p i − q i } <
0. Therefore, there exists µ > λ ( µ, q − u ∗ ) , λ ( µ, p − v ∗ ) < < µ < µ . Therefore by Theorem 3.2, (4.4) hasa unique positive equilibrium which is globally asymptotically stable for 0 < µ < µ .2. We first claim: lim µ →∞ u ∗ = (cid:80) ni =1 θ i p i (cid:80) ni =1 θ i θ. (4.8)To see that, for any µ k → ∞ , there exists a subsequence, still denoting by itself,such that the corresponding positive solution u ∗ k = ( u k , ..., u nk ) of (4.7) satisfies u ∗ k → u ∞ ≥ k → ∞ . Dividing both sides of (4.7) by µ k and taking k → ∞ , weobtain that n (cid:88) j =1 L ij u ∞ j = 0, which means that u ∞ = lθ for some l ≥
0. Summingup all the n equations in (4.7) and noticing n (cid:88) i =1 L ij = 0 for all 1 ≤ j ≤ n , we have n (cid:88) i =1 u ik ( p i − u ik ) = 0. Taking k → ∞ , we have n (cid:88) i =1 lθ i ( p i − lθ i ) = 0. Since l > u ∗ ≥ mθ , we have l = (cid:80) ni =1 θ i p i / (cid:80) ni =1 θ i . This proves (4.8).26hen by Lemma 4.1 and (4.8), we have lim µ →∞ λ ( µ, q − u ∗ ) = n (cid:88) i =1 θ i ( p i − q i ) . Similarly,lim µ →∞ λ ( µ, p − v ∗ ) = n (cid:88) i =1 θ i ( q i − p i ) . If n (cid:88) j =1 θ j ( p j − q j ) >
0, there exists µ > λ ( µ, q − u ∗ ) > λ ( µ, p − v ∗ ) < µ > µ . Therefore, by Theorem 3.2, E isglobally asymptotically stable for µ > µ . The case n (cid:88) j =1 θ j ( p j − q j ) < Remark . By Theorem 4.5, one may expect that there exists a critical value µ ∗ such that if µ < µ ∗ , (4.4) has a unique globally stable positive equilibrium while if µ > µ ∗ either E or E is globally asymptotically stable for (4.4). However, this resultdoes not hold in general. Indeed, in [7], it has shown that for n = 2 there mightexist 0 < µ ∗ < µ ∗ < µ ∗ such that (4.4) has a globally asymptotically stable positiveequilibrium exactly when µ ∈ (0 , µ ∗ ) ∪ ( µ ∗ , µ ∗ ).The following result characterizes the asymptotic limit of the unique positive equi-librium of (4.4) when the diffusion rate µ approaches zero. Theorem 4.7.
Suppose that (A1)-(A3) hold. Let Ω u = { i : 1 ≤ i ≤ n, p i > q i } and Ω v = { i : 1 ≤ i ≤ n, p i < q i } . Suppose that Ω u and Ω v are not empty with Ω u ∪ Ω v = { , , ..., n } . Let u = ( u , u , ..., u n ) and v = ( v , v , ..., v n ) , where u i = p i , if i ∈ Ω u , , if i ∈ Ω u , and v i = if i ∈ Ω u ,q i if i ∈ Ω u . Let ( u, v ) be the unique positive equilibrium of (4.4) when µ is small, then lim µ → ( u, v ) =( u , v ) .Proof. By Theorem 4.5, there exists µ > u, v ) that is globally asymptotically stable. By the definition, when µ = 0,( u , v ) is a solution of the following system µ n (cid:88) j =1 L ij u j + u i ( p i − u i − v i ) = 0 , i = 1 , . . . , n,µ n (cid:88) j =1 L ij v j + v i ( q i − u i − v i ) = 0 , i = 1 , . . . , n. (4.9)27e show that (4.9) has a continuum of solutions emanating ( µ, ( u, v )) = (0 , ( u , v )).To see this, we define F ( µ, ( u, v )) = µ n (cid:88) j =1 L j u j + u ( p − u − v )... µ n (cid:88) j =1 L nj u j + u n ( p n − u n − v n ) µ n (cid:88) j =1 L j v j + v ( q − u − v )... µ n (cid:88) j =1 L nj v j + v n ( q n − u n − v n ) . Then we compute the Jacobian matrix of F evaluated at ( µ, ( u, v )) = (0 , ( u , v )): DF ( u,v ) (0 , ( u , v )) = diag( p − u − v ) − diag( u ) − diag( v ) diag( q − u − v ) . By the assumption p i (cid:54) = q i for all i and the definition of u and v , we can see that DF ( u,v ) (0 , ( u , v )) is invertible. Therefore, by the implicit function theorem, thereexists µ ∗ > u ( µ ) , v ( µ )) for each 0 ≤ µ < µ ∗ , where( u ( µ ) , v ( µ )) is continuous in µ . By the definition of ( u , v ), we may choose µ ∗ smallsuch that u i ( µ ) > i ∈ Ω u and v i ( µ ) > i ∈ Ω v for all 0 ≤ µ ≤ µ ∗ .We show that u i ( µ ) > i ∈ Ω v and v i ( µ ) > i ∈ Ω u for µ close to zero. To see that, fix i ∈ Ω v . Then, p i < q i , u i = 0 and v i = q i .Differentiating µ n (cid:88) j =1 L i j u j + u i ( p i − u i − v i ) = 0 with respect to µ and evaluatingat ( µ, ( u, v )) = (0 , ( u , v )), we obtain u (cid:48) i (0) = (cid:80) nj =1 L i j u j q i − p i = (cid:80) j ∈ Ω u L i j u j q i − p i . By the assumption, (cid:88) j ∈ Ω u L i j >
0. So u (cid:48) i (0) >
0. Therefore, u i ( µ ) ≈ u i (0) + u (cid:48) i (0) µ > µ close to zero. Since i ∈ Ω v was arbitrary, u i ( µ ) > i ∈ Ω v when µ isclose to zero. Similarly, v i ( µ ) > i ∈ Ω u when µ is close to zero.28e can find µ ∗ < µ ∗ such that the solution ( u ( µ ) , v ( µ )) of (4.9) is positive for0 < µ < µ ∗ . Then the conclusion follows from the uniqueness of the positive solutionof (4.9) and the continuity of ( u ( µ ) , v ( µ )) in µ . Appendix
Proof of Proposition 2.3.
1. If A is symmetric, then G is clearly cycle-balanced fromthe definition. Now we assume that G is cycle-balanced. Suppose a ij >
0; that is,there is an arc ( j, i ) from vertex j to vertex i in G . Since G is strongly connected, thereexists a path from i to j . Therefore, the arc ( j, i ) belongs to some cycle C . If G iscycle-balanced, then for any cycle C , its reverse −C is also a cycle in G . This implies a ji >
0. Hence A must be sign pattern symmetric.2. Since G has n vertices and it is strongly connected, any of its spanning tree has n − G , thus G has at least 2( n − A be a sign pattern symmetric n × n matrix with exactly 2( n −
1) positiveentries, and assume A is irreducible. Let A + = ( a + ij ) n × n be defined by a + ij = a ij when i > j and a + ij = 0 when i ≤ j , then the subdigraph G + associated with A + is a tree.Similarly let A − = ( a − ij ) n × n be defined by a − ij = a ij when i < j and a − ij = 0 when i ≥ j ,then the subdigraph G − associated with A − is also a tree. The digraph G is the unionof two disjoint trees G + and G − . It is easy to see every cycle of G has exactly twovertices, and G is cycle-balanced as every 2-cycle is naturally balanced. This provessuch a bi-directional tree is cycle-balanced. If n = 2 then every cycle of G has twovertices. Hence it must be cycle-balanced.Now assume G is strongly connected, every cycle of G has exactly two vertices and G is cycle-balanced. From part 1, A is sign pattern symmetric. So we only need toprove G has exactly 2( n −
1) arcs. Let T be a spanning tree of G , then T has n − T yields −T , which is also a subdigraphof G . This implies that G has at least 2( n −
1) arcs, n − T and n − −T . If in addition to the 2( n −
1) arcs in the spanning tree T and its reverse −T ,29here exists at least one more arc, say ( i a , i b ), which is not in T ∪ ( −T ). But thereis a path P from i b to i a in T ∪ ( −T ) because of the property of spanning tree. Thelength of P is at least 2 as ( i a , i b ) (cid:54)∈ T ∪ ( −T ), so the union of P and ( i a , i b ) is a cyclewith length at least 3, which contradicts with the assumption that every cycle of G hasexactly two vertices. Therefore G has exactly 2( n −
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