GLOBAL DYNAMICS FOR TWO-SPECIES COMPETITION IN PATCHY ENVIRONMENT

An ODE system modeling the competition between two species in a two-patch environment is studied. Both species move between the patches with the same dispersal rate. It is shown that the species with larger birth rates in both patches drives the other species to extinction, regardless of the dispersal rate. The more interesting case is when both species have the same average birth rate but each species has larger birth rate in one patch. It has previously been conjectured by Gourley and Kuang that the species that can concentrate its birth in a single patch wins if the diffusion rate is large enough, and two species will coexist if the diffusion rate is small. We solve these two conjectures by applying the monotone dynamics theory, incorporated with a complete characterization of the positive equilibrium and a thorough analysis on the stability of the semi-trivial equilibria with respect to the dispersal rate. Our result on the winning strategy for sufficiently large dispersal rate might explain the group breeding behavior that is observed in some animals under certain ecological conditions.

1. Introduction.Gourley and Kuang [3] studied the following two-patch system as a model for two neutrally competing species: where u i (resp., v i ) is the number of species u (resp., v) in patch i, i = 1, 2; the linear birth rates α 1 , α 2 , β 1 , β 2 are positive parameters, and there is a diffusion between the two patches with the same diffusivity (dispersal rate) d for both species.Two species differ only in their birth rates.
These conjectures, if true, suggest that the species that can concentrate its birth in a single patch wins, if the diffusion rate is larger than a critical value.In short, the winning strategy is simply to focus as much birth in a single patch as possible.In this paper, we will establish the following result, which includes Conjectures 1 and 2 as special cases: Theorem 1.4.Suppose that in system (1), Then there exists some positive constant d * so that if d ≥ d * , (ū 1 , ū2 , 0, 0) is globally asymptotically stable among the initial data in R 4  + satisfying u 1 (0) + u 2 (0) > 0; if d < d * , (1) has a unique positive steady state which is globally asymptotically stable among the initial data in R 4  + satisfying u 1 (0) + u 2 (0) > 0 and Our result may explain some grouping behaviors in animal populations which may be advantageous under certain ecological conditions.Theorem 1.4 proves that in fast diffusion scenarios a winning strategy is to concentrate the birth in a single patch.More specifically, for any positive birth rates (β 1 , β 2 ) with β 1 < β 2 , we compare it with the extreme case (0, β 1 + β 2 ); i.e., when σ = β 1 and (α 1 , α 2 ) = (0, β 1 + β 2 ).Theorem 1.4 implies that the birth rates (0, β 1 + β 2 ) provide a winning strategy in the sense that the population adopting it can drive the other population with the birth rates (β 1 , β 2 ) to extinction.This result suggests that for a two-patch habitat it can be more advantageous for the species to have a single breeding site, provided that the dispersal of the species is suitably fast.We do not know whether a similar conclusion can be drawn for multiple-patch models; see also [1].
Theorem 1.4 will be justified in Section 3. The case for larger birth rates for u-species on both patches, namely, β 1 < α 1 , β 2 < α 2 , is discussed in Section 2. The case of interlacing birth rates: We collect in Subsection 3.1 the monotone dynamical system theory to be applied in later sections.The existence of positive equilibrium is studied in Subsection 3.2.The properties and stability of the semi-trivial equilibria are addressed in Subsections 3.3 and 3.4.With the preparation in Subsections 3.1-3.4,we summarize the main results and solve the conjectures in Subsection 3.5.We present the case Two numerical examples illustrating the present theory are given in Section 5.The paper ends with a conclusion section.
The following theorem shows that the coexistence state does not exist for any d > 0.

Interlacing birth rates:
In this section, we consider the following parameters α 1 < β 1 < β 2 < α 2 , i.e., u-species has larger birth rate than v-species in the second patch, while vspecies has larger birth rate than u-species in the first patch.Then we ask how this distribution of birth rates is related to the species persistence or extinction.[3] studied the local stability of semi-trivial equilibria under the following assumption:

Gourley and Kuang
Two conjectures under this parameter condition were posed therein, as mentioned in Section 1. Herein, we consider more general situation: When

condition (H) reduces to condition (A).
We recall the monotone dynamics theory in Subsection 3.1, and discuss the existence of positive equilibrium in Subsection 3.2, properties for the semi-trivial equilibria in Subsection 3.3, the stability of the semi-trivial equilibria in Subsection 3.4, and the coexistence of two species and the extinction of one species in Subsection 3.5.
First, when d = 0, i.e. there is no dispersal, the species with larger birth rate prevails in each patch, as shown in the following result.
The proof of this theorem resembles that of Theorem 2.1, and is omitted.

Monotone dynamics theory.
In this subsection, we recall some monotone dynamical system theories.Denote by R n + = {x ∈ R n : x i ≥ 0, 1 ≤ i ≤ n} the first orthant of R n .Consider the following cones: where nonnegative, and A 2 and A 3 are nonnegative matrices, for some k with 1 ≤ k ≤ n.
Smith [6] showed that the flow φ t (x) generated by the type-K monotone system is type-K monotone with respect to the cone We note that system (1) is a type-K monotone system with respect to since the Jacobian matrix is given by For system (1), let us denote by E 0 := (0, 0, 0, 0) the trivial equilibrium, by E ū := (ū 1 , ū2 , 0, 0), and + , and 0 for t > 0. Define E and E + the sets of all nonnegative equilibria and all positive equilibria for φ t , respectively.Obviously, [E v, E ū] m contains E and E * ∈ (E v, E ū) m for any E * ∈ E + .The following theorem restates Corollary 4.4.3 in [7] for system (1); see also [6,8].Theorem 3.2.If E ū and E v are both linearly unstable, then system (1) is permanent.More precisely, there exist positive equilibria E * and E * * , not necessarily distinct, satisfying In particular, if E * * = E * , then E * attracts all such solutions.Hsu et al. [4] showed that, for two competing-species models, either there is a positive equilibrium representing coexistence of two species, or one species drives the other to extinction.As system (1) satisfies conditions (H1)-(H4) in [4], Theorem B in [4] can be restated as follows.
Theorem 3.3.For system (1), the ω-limit set of every orbit evolved from R 4  + is contained in Γ and exactly one of the following holds: (a) There exists a positive equilibrium A system similar to (1) has been studied in Section 4.4 of [7].Therein, monotone structure was employed to obtain the attracting regions, and global convergence to the semi-trivial equilibrium and the positive equilibrium.However, those results are under conditions on eigenvalues of the Jacobian of the vector field at the semitrivial equilibria.Those quantities depend on the coordinate values of the semitrivial equilibria, and thus the theorem does not provide answers to the conjectures mentioned in Section 1. Indeed, to see the complete scenarios, one needs to elucidate on how the dynamics depend on the parameters and especially, the dispersal rate d.

3.2.
Existence of positive equilibrium.The existence of positive equilibrium for system (1) can be characterized completely, as shown in the following theorem.Theorem 3.4.Under condition (H), there exists a d * > 0 so that system (1) has a unique positive equilibrium Combining each pair of equations ( 2), we obtain Note that a, b are now expressed in terms of parameters, and a 2 > k > b 2 .With (3) we substitute them back to (2) and obtain On the other hand, solving (iii) and (iv), we have The consistency can be verified: from d(a − b) = σ 1 , we see that Hence, the unique positive equilibrium exists for system (1) if and only if or equivalently, First, since b < 1, we have β 2 − bβ 1 > 0, and for all d ≥ 0. Next, we shall find the condition under which the left inequalities of ( 5) and ( 6) hold.These inequalities are equivalent to G(d) > 0 and F (d) > 0 respectively, where Let us study the property for functions F and G.Note that We claim that G (d) < 0, for all d > 0. Indeed, since b = We then compute Next, we show that On the other hand, considering b ≥ β1k β2 and using b < 1, we have and We thus conclude that the system must have a positive equilibrium when and has no positive equilibrium when Since G (d) < 0 for all d > 0, we can deduce that there is a unique point d * with This completes the proof.Corollary 3.5.If σ 1 = σ 2 = σ in condition (H), then d * can be computed exactly as .
By argument similar to above, we have The consistency can be verified as and Hence, the unique positive equilibrium exists for system (1) if and only if or equivalently The left inequalities of ( 7) and ( 8) are equivalent to G(d) > 0 and F (d) > 0, where Notice that , we compute and Since d > 0, we consider Therefore, there are two roots .
Actually, the graph of G(d) only intersects d-axis at one point.Hence, the only solution to equation G(d) = 0 is .
This completes the proof.

Note that (u
Remark 1. From the proof of Theorem 3.4, Corollary 3.5, and (9), we have actually obtained the following results.
Corollary 3.6.Under condition (H), if d < d * , then 3.3.Qualitative properties for semi-trivial equilibria.The following two propositions provide some qualitative properties of the semi-trivial equilibria.
The following proposition can be obtained by arguments similar to those for Proposition 3.7 .
Proof.It can be computed that the two eigenvalues of matrix are negative, under condition (H).The instability of (0, 0, v1 , v2 ) is determined by the sign of the larger eigenvalue (denoted by λ + ) of By a direct calculation, for some d > 0. A direct calculation yields by the equations for vi , i = 1, 2 and Proposition 3.8(ii).But then is a contradiction, as the left-hand side of the inequality is negative, due to v1 < v2 and 0 < σ 1 ≤ σ 2 .Therefore, λ + > 0 for all d > 0.
As for the other eigenvalue similar computation shows that λ − < 0 for any d > 0. This completes the proof.
This completes the proof.4. Other case.Propositions 3.11 and 3.12 actually improve and extend Propositions 1.1, 1.3 in Section 1, reported in [3].Proposition 1.2 can also be generalized in a similar fashion. Consider We need σ 1 < β2−β1 2 to ensure α 1 < α 2 .When σ 1 = σ 2 , condition (H 1 ) reduces to the consideration in [3].The following theorem can be obtained by arguments similar to those in Theorem 3.4 and Corollary 3.5.

Conclusion.
We have characterized the global dynamics for a system modeling competition of two-species living in a two-patchy environment.The coexistence state exists for smaller dispersal rate d < d * where d * can be expressed or estimated by the birth rates.The stability of two semi-trivial equilibria (ū 1 , ū2 , 0, 0) and (0, 0, v1 , v2 ) can also be analyzed completely.For the most interesting case α 1 < β 1 < β 2 < α 2 , where u-species has larger birth rate than v-species in the second patch, while v-species has larger birth rate than u-species in the first patch, (0, 0, v1 , v2 ) is unstable for any d > 0. On the other hand, one of the eigenvalues of the linearized system at (ū 1 , ū2 , 0, 0) changes from positive to negative as d increases from 0 to d * , and (ū 1 , ū2 , 0, 0) becomes globally attractive for d ≥ d * .Thus, for a sufficiently small dispersal rate, the two species can coexist.For a sufficiently large dispersal rate, the two species cannot coexist and the winning strategy is for a species to concentrate its birth on a single patch.This winning strategy might explain the group breeding behavior that is observed in some animals under certain