TWO-SPECIES COMPETITION WITH HIGH DISPERSAL: THE WINNING STRATEGY. Stephen A. Gourley. (Communicated by Yasuhiro Takeuchi)

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1 MATHEMATICAL BIOSCIENCES AND ENGINEERING Volume, Number, April 5 pp TWO-SPECIES COMPETITION WITH HIGH DISPERSAL: THE WINNING STRATEGY Stephen A. Gourley Department of Mathematics an Statistics, University of Surrey, Guilfor, Surrey, GU 7XH, UK Yang Kuang Department of Mathematics an Statistics, Arizona State University, Tempe, AZ (Communicate by Yasuhiro Takeuchi) Abstract. This paper is motivate by the following simple question: how oes iffusion affect the competition outcomes of two competing species that are ientical in all respects other than their strategies on how they spatially istribute their birth rates. This may provie us with insights into how species learn to compete in a relatively stable setting, which in turn may point out species evolution irections. To this en, we formulate some extremely simple two- species competition moels that have either continuous or iscrete iffusion mechanisms. Our analytical work on these moels collectively an strongly suggests the following in a fast iffusion environment: where ifferent species have the same birth rates on average, those that o well are those that have greater spatial variation in their birth rates. We hypothesize that this may be a possible explanation for the evolution of grouping behavior in many species. Our finings are confirme by extensive numerical simulation work on the moels.. Introuction. Recently, Hutson et al. [9] propose an analyze the reaction iffusion system u t = u + u[α(x) u v], (.) v t = v + v[β(x) u v] on a boune omain, with homogeneous Neumann bounary conitions. Equations (.) were taken as a simple moel of two species that are ientical in all respects, except for their birth rates. It is suppose that species u is a mutation of species v, an thus the ifference between α(x) an β(x) is viewe as small. The limiting resource subject to competition is implicitly assume to be constant (such as habitable space) or has a ynamics much faster than that of the competition mechanism []. In these cases, conventional competition moels such as (.) are plausible an can be employe to stuy issues relate to how iffusion affect the competition outcomes of competing species that are ientical in all respects other than their strategies on how they spatially istribute their birth rates. This may provie us with insights into how species learn to compete in a relatively stable setting, which in turn may point out species evolution irections. Mathematics Subject Classification. 9D5, 35K57. Key wors an phrases. competition, evolution, reaction iffusion, patch population moel, stability, bifurcation. 345

2 346 S. A. GOURLEY AND Y. KUANG Two specific key ecological questions motivate the stuy of system (.) in [9]; () uner what circumstances oes the mutant u invae; an () if it oes invae, oes it rive the original phenotype v to extinction, or will there be coexistence? Some of the results prove in [9] about system (.) are rather striking. For example, when α(x) = β(x) + τg(x), it was shown that for a large class of functions g(x) an small τ, the stability of the two species varies in a complex manner; in particular, stability can change back an forth many times as is increase over (, ). In fact, for any positive integer n, the function g(x) can be chosen (from an open set of possibilities) such that the stability of the semitrivial equilibria (i.e., equilibria with one component zero an the other positive, sometimes known as bounary equilibria) changes at least n times as is increase from zero to infinity. Competition between the species an the mutant thus epens in a particularly elicate way on the balance between the iffusivity an resource utilization as escribe by the form of the reprouction rate β(x). Another important result prove in [9] about system (.) is that there is no optimal form of resource utilization if there is no upper boun on birth rate functions. In other wors, there is no birth rate β(x) for species v that is optimal in the sense that an invaing mutant u with birth rate β(x) + τg(x), subject to a fairness assumption g(x) x =, will necessarily ie out. Sai another way, given a particular value for the iffusivity an a particular spatially epenent birth function β(x), there will always exist a birth rate α(x) for the mutant u that iffers pointwise from β(x) but is the same on average, such that u will invae. The aim of the present paper is twofol. First, we continue the stuy in [9] by proving some further results about system (.). One result we establish concerns the case when β(x) = β, a constant. We prove that if α(x) is nonconstant but has mean value β, an if the iffusivity is sufficiently large, then automatically the mutant u wins, riving v to extinction. This suggests that, for large iffusivities, if v has a constant birth rate then the mutant u has only to vary its birth rate at ifferent points in space while preserving the same mean to win the competition. This is perhaps one possible explanation for the evolution of the aggregation (grouping together) tenency of many animals. Secon, we consier a two-patch moel analogous to (.) an investigate its properties, eluciating in particular how exactly the mutant must vary its birth rate to win the competition an rive the other to extinction. The avantages of using a patch moel are that the analysis can be much more explicit than is possible for a nonautonomous reaction-iffusion system, an also that patch-type moels are particularly amenable to computation. The ynamics of this two-patch moel seems to agree with that of (.). Our analytical work on these moels collectively an strongly suggests the following in a fast iffusion environment: where ifferent species have the same birth rates on average, those that o well are those that have greater spatial variation in their birth rates. We hypothesize that this may be a possible explanation for the evolution of grouping behavior in many species. Our finings are confirme by extensive numerical simulation work on the moels. The paper is organize as follows. We formulate an perform some pointe analysis of an extremely simple two-patch moel for two similar competing species. In section 3, we continue the stuy in [9] by proving some new results about system (.). We conclue with a iscussion section containing implications of our finings, simulation results, an statements of open questions.

3 TWO-SPECIES COMPETITION WITH HIGH DISPERSAL 347. A two-patch moel. We propose the following simple two-patch system as a moel for two similar competing species: u = u t (α u v ) + (u u ), u = u t (α u v ) + (u u ), (.) v t = v (β u v ) + (v v ), v t = v (β u v ) + (v v ), in which all parameters are positive, v i is the number of species v in patch i, u i is the same for the mutant u, an there is iffusion between the two patches with iffusivity. Naturally, we assume that v i () an u i (), i =,. It is easy to show that all such solutions exist globally an have nonnegative component values. In fact, if u () + u () >, then one can easily show u i (t) > for t >, i =,. The same is true for v i, i =,. Notice that u t + u (u + u ){max{α, α } t (u + u )}. A similar inequality hols for v +v. Hence, we have the following uniform bouneness result. Lemma.. Solutions of (.) with positive initial values are uniformly boune. In fact lim sup t (u (t)+u (t)) max{α, α }, lim sup(v (t)+v (t)) max{β, β }. t Notice also that u t + v t (u + v ){min{α, β } (u + v )}. A similar inequality hols for u +v. Hence, we have the following patch population persistence result. Lemma.. Assume < min{min{α, β }, min{α, β }} in (.). Then populations in both patches persist. In fact, we have lim inf t (u (t)+v (t)) min{α, β }, lim inf t (u (t)+v (t)) min{α, β }. Except at the en of this section, we assume, without loss of generality, that β > β. The following global stability result for = (which reuces (.) to a ecouple system) is elementary. This result is useful for unerstaning Figure in the iscussion section that epicts a bifurcation iagram of (.). Lemma.3. Assume α < β < β < α an = in (.). Then lim (u (t), u (t), v (t), v (t)) = (, α, β, ). t We stuy the ynamics of system (.) largely through the preictions of linearize analysis of the bounary equilibria (equilibria in which one of u or v is zero), together with numerical simulations to confirm these preictions. Linearize analysis about the bounary equilibria is tractable because the Jacobian matrix at a bounary equilibrium has a block iagonal structure. Unfortunately, it is very ifficult to analytically stuy the linear stability of a coexistence equilibrium.

4 348 S. A. GOURLEY AND Y. KUANG Let us first consier the bounary equilibrium with u = u = (i.e., the mutant is absent). Intuitively, one expects that the v an v components of such an equilibrium woul both be between β an β, an inee it can be shown that the equilibrium equations v (β v ) + (v v ) =, v (β v ) + (v v ) = amit precisely one such solution. After some algebra, one fins that the unique bounary equilibrium (u, u, v, v ) = (,, v, v ) is etermine by v = v (v β ) + v, (.) where v is the largest real root of f(v ) = with f(v ) := v3 + ( ) ( ( ) ) β v β + β v +β +β β β (.3) (small values for actually yiel two roots of f(v ) = between β an β, but only the larger one prouces an amissible value for v ). For the subsequent analysis, we nee some information on the size of v. Now f(β ) = β β <, while f( (β + (β ) + 4β )) = (β + ) β β + + 4β >, since β < β. Hence, v ( (β + (β ) + 4β ), β ). (.4) Linearizing about the equilibrium (,, v, v ) in the usual way, one fins that the eigenvalue equation corresponing to trial solutions proportional to exp(λt) is ( λ (α + α v v )λ + (α v )(α v ) ) ( λ (β + β v v )λ + (β v )(β v ) ) =. (.5) From the structure of the linearization matrix (Jacobian matrix) that le to equation (.5), it is easy to appreciate that the secon quaratic factor is associate with perturbations from (,, v, v ) in which the u i remain zero. We show that the eigenvalues attributable to this secon quaratic factor have negative real parts. To o so, it suffices to show that the coefficient of λ an the constant term are both positive. The coefficient of λ will be positive if v + v > (β + β ), which is obviously true if (β + β ) <, an so it remains to consier the case when (β + β ) >. In this case, graphical consierations in the (v, v ) plane reveal that it is sufficient to check that the intersection of the line v + v = (β + β ) with the curve v = v + v ( β ) is at a value of v less than v. In other wors, we nee to check that the positive root v of g(v ) := v + v ( β ) + ( ) β + β = is less than v. Since we assume β > β, we know that (.4) hols. Therefore, to show that the positive root of g(v ) = is less than v, it is sufficient to show

5 TWO-SPECIES COMPETITION WITH HIGH DISPERSAL 349 that the root is less than (β + (β ) + 4β ), an this will follow if g is positive at the latter value. But g( (β + (β ) + 4β )) = ( + β ) + β β + + 4β > as esire. Thus v + v > (β + β ). Checking that (β v )(β v ) > can be one similarly. Our conclusion at this stage is that the bounary equilibrium (,, v, v ) is locally stable to perturbations in which the u i remain zero. It may of course be unstable to perturbations involving the introuction of the mutant u, an this will epen on the relative sizes of the birth rates α i, β i an the iffusivity as we now show. Consierations similar to those alreay escribe lea us to the existence of another bounary equilibrium, (u, u,, ), in which u an u are both between α an α. This equilibrium is linearly stable to perturbations in which the v i remain zero. Next, we prove the following result, which preicts for large iffusivities that if the birth rates for the species v in the two patches are unequal, an if the mutant u increases the isparity between the birth rates (but preserving the same mean), then the mutant will win an rive the original species v to extinction. Proposition.. If β > β an α = β ε, α = β + ε with < ε < β, an is sufficiently large, then (,, v, v ) is unstable an (u, u,, ) is linearly stable. Proof. To show that (,, v, v ) is unstable, it is sufficient to show that the first quaratic factor in the eigenvalue equation (.5) yiels an unstable eigenvalue (one such that Re λ > ). We shall in fact show that a real positive eigenvalue exists, by proving that for sufficiently large, the constant term in the first quaratic factor is negative, i.e., that (α v )(α v ) (α + α v v ) < (.6) (with α = β ε an α = β + ε). This is not immeiately clear since, as, v an v both approach (β + β ) so that the brackete coefficient of in (.6) approaches zero. However, a little asymptotic analysis yiels that v = β + β v = β + β ( ) 4 β (β β ) + O, + ( ) 4 β (β β ) + O ; so that after some algebra, the left han sie of (.6) becomes ( ( ) ( ) β β β β + ε) + + O, which is negative for sufficiently large, since β > β an ε >. The characteristic equation of the linearization about (u, u,, ) is ( λ (β + β u u )λ + (β u )(β u ) ) ( λ (α + α u u )λ + (α u )(α u ) ) = (.7) with α = β ε an α = β + ε, an this time we nee to show for sufficiently large that both quaratic factors prouce only eigenvalues λ with Re λ <. The coefficients of λ in both factors are clearly positive for sufficiently large (u an

6 35 S. A. GOURLEY AND Y. KUANG u epen on but are always between α an α ). Thus, we nee to show that the constant terms are positive, i.e., that an that But for large, (β u )(β u ) (β + β u u ) > (.8) (β ε u )(β + ε u ) (β + β u u ) >. (.9) u = α + α u = α + α 4 ( ) 4 α (α α ) + O, + ( ) 4 α (α α ) + O. Since α = β ε an α = β + ε the left-han sie of (.8) becomes ), ( (β β + ε) (β β ) ) + O ( which is positive for sufficiently large. Similarly, the left-han sie of (.9) is given asymptotically in by (β + β ) + O() >. The proof of Proposition. is complete. In an entirely similar way, we obtain the following proposition: Proposition.. If β > β an α = β + ε, α = β ε with ε > but not too large an is sufficiently large, then (,, v, v ) is linearly stable an (u, u,, ) is unstable. For the case when the iffusivity is large, our preictions thus far for moel (.) mirror the results escribe in [9] regaring the absence of an optimal form of resource utilisation. It follows from Propositions. an. that, in system (.), there is no optimal way for the species v to choose its birth parameters β an β (optimal meaning that the mutant woul ie out whatever the values of its birth parameters α an α, subject to a fairness conition α +α = β +β ). However, Propositions. an. o throw more light on what the mutant s strategy must be for it to win. Essentially, if β an β are unequal an if the mutant wiens the isparity between these birth rates (i.e., aopts a higher birth rate in the patch where the birth rate for v is alreay high, an a lower birth rate than v in the other patch), then the mutant will win (if is large enough). On the other han, if the species v has unequal birth rates an the mutant closes the gap, subject to α + α = β + β, then the mutant will become extinct if is large. It is natural to woner what happens if β = β. The following proposition preicts that in this case if the iffusivity is large then the mutant can win, riving v to extinction, simply by introucing some isparity in its birth rate between the two patches. Proposition.3. Let β = β = β > an α = β ε, α = β + ε with ε of either sign an ε < β. Then, if is sufficiently large, (,, v, v ) is unstable an (u, u,, ) is linearly stable. Proof. The proof is similar to that of Proposition.. Note that when β = β, v = v = β.

7 TWO-SPECIES COMPETITION WITH HIGH DISPERSAL The reaction-iffusion moel. Motivate by the preictions in the analysis for the two-patch moel in the previous section, the purpose of this section is to establish some results for the reaction-iffusion system (.), which complement the results prove in [9]. Proposition.3 in particular leas us to woner whether some analogous result might hol for (.). Let us consier the system u t = u + u[β + εg(x) u v], v t = v + v[β u v] (3.) subject to u/ n = v/ n = on, where n is the outwar-pointing unit normal on. In this section we will assume that β >, ε, g(x) is nonconstant an g(x) x =. (3.) Clearly, system (3.) has a bounary equilibrium (u, v) = (, β) corresponing to the mutant being absent, an another bounary equilibrium (ũ(x), ) (original species v is absent), in which ũ(x) is the solution of ũ + ũ[β + εg(x) ũ] =, ũ/ n = on. (3.3) Assumptions (3.) assure us of the existence of a unique positive solution ũ of (3.3). We will prove the following result: Proposition 3.. Let (3.) hol. Then, if is sufficiently large, the equilibrium (u, v) = (, β) is unstable as a solution of (3.), an the equilibrium (ũ(x), ) is linearly stable. Proof. Linearizing (3.) about (, β) furnishes an eigenvalue problem from which one equation ecouples. To show that (, β) is unstable, it is sufficient to show the existence of a positive eigenvalue λ to the eigenvalue problem λφ = φ + εg(x)φ, φ/ n = on (3.4) that results from trial solutions with temporal epenence of the form exp(λt). Let λ be the principal eigenvalue of (3.4) (the eigenvalue of greatest real part), an φ > be the corresponing eigenfunction (φ > follows from Theorem. in [5]). Division by φ an integration over yiels λ = φ x + ε g(x) x φ }{{} = = φ φ x >, since φ is nonconstant (if φ were constant then g(x) woul have to be constant). Thus, λ > an so (, β) is unstable. Next, we emonstrate that (ũ(x), ) is linearly stable for sufficiently large values of. Linear stability of this equilibrium is etermine by the eigenvalue problem λφ = φ + (β + εg(x) ũ(x))φ ũ(x)ψ, λψ = ψ + (β ũ(x))ψ, φ/ n = ψ/ n = on. (3.5) Certain facts concerning this eigenvalue problem follow from remarks on p464 of [9]. The eigenvalues can be examine using a suitable positive operator (see [6]), an it can be establishe that (3.5) has a principal eigenvalue (a simple real eigenvalue that

8 35 S. A. GOURLEY AND Y. KUANG is larger than the real part of any other eigenvalue). Also, an very importantly, the principal eigenvalue for (3.5) coincies with the principal eigenvalue of the scalar problem λψ = ψ + (β ũ(x))ψ, ψ/ n = on (3.6) (see [8]). To emphasize epenence on the large parameter, let λ an ψ > be the principal eigenvalue an corresponing eigenfunction of (3.6), with ψ normalize such that ψ x =. We aim to show that λ <. Multiplying λ ψ = ψ + (β ũ(x))ψ by ψ an integrating over yiels λ = ψ ψ x + (β ũ(x))ψ x = ψ x + (β ũ(x))ψ x. (3.7) But ũ(x) satisfies (3.3) an it is known from [7] that, as, ũ(x) (β + εg(x)) x = β uniformly for x. (3.8) Also, there is a well-known comparison theorem for eigenvalue problems of the form λu = u + a(x)u on homogeneous Neumann or Dirichlet bounary conitions, which states that if a(x) ã(x) for all x, then the principal eigenvalue of the problem with a(x) excees that of the corresponing problem in which a(x) is replace by ã(x) ([5], Thm.6). An if a(x) equals a constant a, then the principal eigenvalue for the homogeneous Neumann problem is simply a itself. From these facts, it follows that the principal eigenvalue λ of (3.6) is between sup x β ũ(x) an sup x β ũ(x), an so, by (3.8), λ as. It is therefore easily seen from (3.7) that ψ x as. So, for large, ψ approximates to a constant, ψ / (by the normalization conition), an it remains to show that λ <. But, from (3.7), λ (β ũ(x))ψ x (β ũ(x)) x. Finally, iviing (3.3) by ũ an integrating yiels ũ (β ũ(x)) x = x = ũ ũ ũ x <, since ũ(x) is nonconstant. Hence λ <. Our next result, concerning the reaction-iffusion system (.), mirrors Proposition. on the two-patch moel. The result preicts that if the birth rate for v is nonconstant an the mutant has a birth rate with higher variability but the

9 TWO-SPECIES COMPETITION WITH HIGH DISPERSAL 353 same mean, then the mutant will win an rive v to extinction. For the purposes of establishing this result, it will be convenient to write (.) as u t = u + u[β + δ g(x) u v], v t = v + v[β + δ g(x) u v], u(x, ) = u (x), v(x, ) = v (x), u/ n = v/ n = on (3.9) in which β, δ, δ >, g(x) is nonconstant, an g(x) x =. (3.) Cantrell an Cosner [] establishe some results that have a similar flavor to those of this section in that they involve hypotheses on the means of spatially varying coefficients (see, in particular, Proposition 3.9 in [], which eals with large iffusivities). However, their theorems o not inclue system (3.9), because the birth rates in (3.9) have exactly the same mean. System (3.9) has a unique bounary equilibrium of the form (ũ(x), ), where ũ(x) > satisfies ũ + ũ[β + δ g(x) ũ] =, ũ/ n = on (3.) an a unique equilibrium of the form (, ṽ(x)), with ṽ(x) > satisfying ṽ + ṽ[β + δ g(x) ṽ] =, ṽ/ n = on. (3.) We prove Proposition 3. below. Before oing so, we point out that Proposition 3., which effectively aresses the case δ =, is not a particular case of Proposition 3.. The proof of the latter leans heavily on the assumption δ > ; it is this fact that assures us that ṽ(x) is nonconstant, which is essential for the proof. Proposition 3.. Assume (3.) hols an that δ > δ. Then, if is sufficiently large, the equilibrium (, ṽ(x)) is unstable as a solution of (3.9), an (ũ(x), ) is linearly stable. Proof. We first show that (, ṽ(x)) is unstable. The linearization about this equilibrium leas to the following eigenvalue problem, corresponing to trial solutions with temporal epenence exp(λt): λφ = φ + (β + δ g(x) ṽ(x))φ, λψ = ψ ṽ(x)φ + (β + δ g(x) ṽ(x))ψ, φ/ n = ψ/ n = on. (3.3) The principal eigenvalue of (3.3) coincies with the principal eigenvalue of the problem λφ = φ + (β + δ g(x) ṽ(x))φ, φ/ n = on. (3.4) Let λ an φ > be the principal eigenvalue an corresponing eigenfunction. We wish to show that λ >. Let Θ = φ /ṽ; then Θ satisfies λ ṽ Θ = (ṽ Θ) + [(δ δ )g(x)ṽ ]Θ, Θ/ n = on. (3.5) As in [9], λ is given by the variational characterization ṽ Θ x + (δ δ )g(x)ṽ Θ x λ = sup {Θ W, (): Θ } ṽ Θ x.

10 354 S. A. GOURLEY AND Y. KUANG (Note that [9] use a ifferent notational convention; their λ is our λ.) The choice Θ = yiels (δ δ ) ṽ g(x) x λ, ṽ x an therefore it suffices to show that ṽ g(x) x >. This can be shown using the equation for ṽ, equation (3.). Multiplying (3.) by ṽ an integrating yiels δ ṽ g(x) x = ṽ x + ṽ (ṽ β) x. We know from [7] that as, ṽ(x) (β + δ g(x)) x = β uniformly for x. (3.6) Using also that ṽ is nonconstant, it follows that for sufficiently large, δ ṽ g(x) x > β (ṽ β) x = β ṽ x ṽ > since ṽ is nonconstant. Thus ṽ g(x) x >, so λ >. The above argument cannot be reverse to conclue that (, ṽ(x)) is stable when δ > δ (otherwise we coul have inferre the stability properties of the other bounary equilibrium (ũ(x), ) without further effort). Therefore, we must stuy the linear stability of (ũ(x), ) separately, an a somewhat ifferent strategy is require to establish its linear stability uner the conition δ > δ, for sufficiently large. The eigenvalue problem resulting from the linearization about (ũ(x), ) is λφ = φ + (β + δ g(x) ũ(x))φ ũ(x)ψ, λψ = ψ + (β + δ g(x) ũ(x))ψ, φ/ n = ψ/ n = on, (3.7) an as before, it suffices to consier the principal eigenvalue of the scalar problem λψ = ψ + (β + δ g(x) ũ(x))ψ, ψ/ n = on. (3.8) Let λ an ψ > be the principal eigenvalue an eigenfunction of (3.8). We want to show that λ <. Let Φ = ψ /ũ; then Φ satisfies Define λ ũ Φ = (ũ Φ) + [(δ δ )g(x)ũ ]Φ, Φ/ n = on. (3.9) C() = sup {φ W, (): gũ φ <} ũ φ x. (3.) g(x)ũ φ x It can be shown that, for sufficiently large, ũ g(x) x >. This means that φ = const. is not an amissible function in (3.), so, for all sufficiently large, C() <.

11 TWO-SPECIES COMPETITION WITH HIGH DISPERSAL 355 Letting φ be the actual extremizing function in (3.), a stanar argument of variational calculus yiels that φ must satisfy (ũ φ ) + C()gũ φ =. (3.) With the notation being use in this paper, λ, as a function of δ, is convex []. When δ = δ, it follows from (3.9) that λ = (with Φ = constant) an when δ = δ + C() (< δ ), λ is again zero by (3.). By convexity, λ must be negative (so that (ũ(x), ) is linearly stable) for values of δ such that δ + C() < δ < δ. (3.) The right-han inequality in (3.) hols by hypothesis. We show that the left han one hols for sufficiently large, by emonstrating that C() as. (3.3) But we know that ũ β as, uniformly in x. Therefore, as, φ x C() sup {φ W, ():, (3.4) gφ <} g(x)φ x which is strictly negative (φ = const. is not amissible since g(x) x = ). Thus (3.3) hols. The proof of Proposition 3. is complete. In a similar way, we have the following proposition: Proposition 3.3. Assume (3.) hols an that δ < δ. Then, if is sufficiently large, the equilibrium (, ṽ(x)) is linearly stable as a solution of (3.9) an (ũ(x), ) is unstable. Next, we emonstrate that, if is sufficiently large, system (3.9) cannot possess a coexistence equilibrium. Doing so enables us to make statements on the global ynamics of (3.9) by employing the powerful theory of monotone ynamical systems [4]. Proposition 3.4. Let (3.) hol, with δ δ. Then, if is sufficiently large, system (3.9) has no coexistence equilibrium. Proof. For a contraiction, suppose there exist sequences { i }, {ũ i }, {ṽ i } with i, ũ i (x) >, ṽ i (x) >, an By the maximum principle, i ũ i + ũ i [β + δ g(x) ũ i ṽ i ] =, i ṽ i + ṽ i [β + δ g(x) ũ i ṽ i ] =, ũ i / n = ṽ i / n = on. ũ i β + δ g, ṽ i β + δ g (3.5) so that ũ i an ṽ i are boune inepenently of i. Now set φ i = ũ i / ũ i. Then φ i satisfies ( ) β + δ g(x) ũ i ṽ i φ i + φ i =, φ i / n = on. i By the regularity properties of solutions of elliptic equations in their epenence on the equation coefficients [5], it follows that as i, φ i φ in C (), where φ satisfies φ =, φ/ n = on an φ =. The solution of the latter problem is simply φ =. Thus φ i, uniformly in x.

12 356 S. A. GOURLEY AND Y. KUANG Also, if we ivie the equation for ũ i in (3.5) by i an recall that ũ i is boune inepenently of i, we conclue (again by elliptic regularity) that ũ i (x) must approach, as i, a limit function ũ(x) satisfying ũ =, ũ/ n = on. But solutions of this problems are constants. Thus, as i, ũ i (x) µ for some constant µ. Similarly, ṽ i (x) ν for some constant ν. If we now integrate the equation for ũ i in (3.5) an then ivie by ũ i, we obtain φ i (x)(β + δ g(x) ũ i (x) ṽ i (x)) x =. Taking the limit as i then yiels (β + δ g(x) µ ν) x =. Since g(x) x =, it follows that µ + ν = β. Next, we erive some inequalities that will be neee later. Note that, for all i, (ũ i (x) + ṽ i (x) β) x = i ũ i x >. This follows by iviing the equation for ũ i by ũ i an then integrating. Furthermore, for large but finite i, ũ i (x)g(x) x > an ṽ i (x)g(x) x >. (3.6) To see the first of these inequalities (the secon is erive similarly), integrate the equation for ũ i in (3.5), an let i be large but finite (so that i is large but finite) to obtain δ ũ i (x)g(x) x = ũ i (x)(ũ i (x) + ṽ i (x) β) x µ (ũ i (x) + ṽ i (x) β) x Define u i = ũ i /ṽ i. Then u i satisfies >. ũ i i (ṽ i u i ) + u i ṽ i (δ δ )g(x) =. Multiplying this by u i, integrating, an then replacing u i by ũ i /ṽ i yiels i ṽi (ũ i /ṽ i ) x = ũ i (δ δ )g(x) x. (3.7) Similarly, i ũ i (ṽ i /ũ i ) x = ṽi (δ δ )g(x) x. (3.8)

13 TWO-SPECIES COMPETITION WITH HIGH DISPERSAL 357 We arrive at a contraiction by consiering (3.7) an (3.8) for large but finite i so that ũ i µ an ṽi ν. Then ũ i (β ṽ i), an so (3.7) becomes i ṽi (ũ i /ṽ i ) x = (β ṽ i ) (δ δ )g(x) x = ṽi (δ δ )g(x) x β(δ δ ) ṽ i g(x) x = i ũ i (ṽ i /ũ i ) x β(δ δ ) ṽ i g(x) x so that i ṽ i (ũ i /ṽ i ) x + i ũ i (ṽ i /ũ i ) x = β(δ δ ) ṽ i g(x) x, } {{ } > which prouces a contraiction if δ > δ. It can be shown in a similar way that, for large but finite i, i ṽi (ũ i /ṽ i ) x + i ũ i (ṽ i /ũ i ) x = β(δ δ ) ũ i g(x) x, }{{} > which prouces a contraiction in the case when δ > δ. The proof of Proposition 3.4 is complete. The nonexistence of a coexistence state for large values of the iffusivity now enables us to make stronger statements on the outcome of the competition between u an v, by using results in [4] (Chap. 7). Before oing so, note that system (3.9) is transforme by the introuction of the new variables u = u, v = v into u t = u + u [β + δ g(x) u + v ], vt = v + v [β + δ g(x) u + v ], u (x, ) = u (x), v (x, ) = v (x), u / n = v / n = on. (3.9) Solutions of (3.9) remain in the fourth quarant, since solutions of (3.9) remain positive by the maximum principle. Accoringly, system (3.9) is a cooperative system, in the sense that the reaction part of the first equation is non-ecreasing with respect to v, while that of the secon equation is non-ecreasing with respect to u (note that v ). In abstract notation, the semiflow Φ efine by Φ t (φ) = (u (t, φ), v (t, φ)), where (u (t, φ), v (t, φ)) is the solution of (3.9) satisfying (u (, φ), v (, φ)) = φ, is strongly monotone ([4], p3). It follows that Φ is strongly orer preserving ([4], p3). The trichotomy given in ([4], p7) then applies an states that, for the original system (3.9), either a coexistence state exists, or solutions of (3.9) approach one of the bounary equilibria. The nonexistence of a coexistence state leaves us with just the latter alternative. In other wors, knowlege of the equilibria an their local stability, together with the powerful results in [4], enables us to make statements about the global ynamics of (3.9). We thus have the following theorem on the outcome of the competition between u an v for large values of. Theorem 3.. Assume (3.) hols, an let be sufficiently large. Then,. if δ > δ an u (x), the solution of (3.9) satisfies (u(x, t), v(x, t)) (ũ(x), ) as t ;

14 358 S. A. GOURLEY AND Y. KUANG 3.5 v v.5 u u u, u.5 v, v Figure. A bifurcation iagram of (.) with α =, α = 3, β =.5, β =.5, an initial conition (.5,.5,.5,.5).. if δ < δ an v (x), the solution of (3.9) satisfies (u(x, t), v(x, t)) (, ṽ(x)) as t. 4. Discussion. This paper has been inspire by the work of Hutson et al. [9] an is motivate by the simple question of how iffusion affects the competition outcomes of two competing species that are ientical in all respects other than their strategies on how they spatially istribute their birth rates. This may provie us with insights into how species learn to compete in a relatively stable setting an this in turn may point out species evolution irections. To this en, we formulate some simple, though artificial, two-species competition moels that incorporate either continuous or iscrete iffusion mechanisms. Our analytical work on these moels collectively an strongly suggests that, in a fast iffusion environment, a species will have a higher chance of success if it tries to aopt greater spatial variation in its birth rate than another competing species with a birth rate that is the same on average. This suggests that, subject to species having the same overall average birth rate over the omain, those species that aopt the greatest spatial variation in their birth rates have the greatest chance of success, which may in turn provie an explanation for the evolution of grouping behavior in animal populations in high-iffusion situations. Our finings are confirme by extensive numerical simulation work on the moels, an a main purpose of this section is to report these informative numerical finings. Specifically, in this section, we selectively present some numerical simulation results that not only confirm but also complement the preictions of the analytical results. We also attempt to state some biological implications of our analytical an numerical finings. In aition, we mention a few open mathematical questions for future work. The numerical simulations are carrie out using the routines oe3s an pepe that are part of MATLAB. Clearly, Figure confirms the results of Lemmas.. an Proposition.. It also strongly suggests the following two conjectures.

15 TWO-SPECIES COMPETITION WITH HIGH DISPERSAL u u u 3 v v v u, u, u v, v, v Figure. A bifurcation iagram of (4.) with α =, α = 3, α 3 = 5, β =.5, β = 3, β 3 = 4.5, an initial conition (.5,.5,.5,.5,.5,.5). Conjecture. Assume in (.), β > β an α = β ε, α = β + ε with < ε < β, an is sufficiently large. If u ()+u () >, then lim(u, u, v, v ) = (u, u,, ). Conjecture. Assume in (.), β > β an α = β ε, α = β + ε with < ε < β. Assume is small enough so that (.) has a positive steay state E. If u () + u () >, v () + v () >, then lim(u, u, v, v ) = E. These conjectures, if true, suggest that the species that can concentrate its birth in a single patch wins, if the iffusion rate is large enough (in Fig. one nees only >.3, far less than the maximum birth rates of either species). In short, the winning strategy is simply to focus as much birth in a single patch as possible. Inee, this is also numerically confirme by a similar bifurcation iagram (Fig. ) for the following three-patch moel of two similar species competition. u = u t (α u v ) + (u + u 3 u )/, u = u t (α u v ) + (u + u 3 u )/, u 3 = u t 3 (α 3 u 3 v 3 ) + (u + u u 3 )/, (4.) v t = v (β u v ) + (v + v 3 v )/, v t = v (β u v ) + (v + v 3 v )/, v 3 t = v 3(β 3 u 3 v 3 ) + (v + v v 3 )/. Our work in the previous section strongly suggests that a similar winning strategy hols for the continuous iffusion moels (3.) an (3.9). Specifically, the winning strategy here is to concentrate birth in as small an area as possible. The two simulation figures (Figs. 3 an 4) not only confirm the analytical results but also provie glimpses of how the two species evolve into the two limiting scenarios: extinction of one species an coexistence.

16 36 S. A. GOURLEY AND Y. KUANG u(x,t) v(x,t) Distance x.8 Time t Distance x.8 Time t Figure 3. Simulation of system (3.9) on the omain x [, ]. Parameter values were β =, δ =.3, δ =.4, an =., an we took g(x) = sin πx. For initial ata, small numbers of the mutant u were introuce throughout the omain initially inhabite mainly by v. The outcome: u wins an v goes extinct. Figure 3 shows the result of a simulation of (3.9) in a situation in which δ > δ. The simulation confirms the preictions of Proposition 3. an Theorem 3. in that the mutant u wins in these circumstances with v going extinct. The simulation also shows that the iffusivity oes not, in fact, have to be particularly large for the mutant to win. Other numerical simulations, results of which are not inclue here, support the theoretical preictions for the case when δ < δ. Figure 4 shows the effect of lowering the value of the iffusivity. Parameter values this time were the same as in Figure 3, except the iffusivity was lowere to =. an the initial conitions for u an v were ientical (to be certain of not biasing the outcome). The result is that u an v can coexist at this value of, but they become spatially segregate. In conclusion, we woul like to emphasize that extensive work exists on patch population ynamics (e.g., [6], [3], []) an on population moels involving reaction iffusion-equations (e.g., [], [3], [4]), many of which eal with both competition an preator-prey interactions. Acknowlegments. The research of Yang Kuang is supporte in part by two NSF grants (DMS-7779 an DMS-34388). Corresponence shoul be irecte to Stephen Gourley. REFERENCES [] Britton, N. F. (986). Reaction-Diffusion Equations an Their Applications to Biology. Lonon: Acaemic Press. [] Cantrell, R. S. an C. Cosner. (998). On the effects of spatial heterogeneity on the persistence of interacting species. J. Math. Biol., 37, 3-45.

17 TWO-SPECIES COMPETITION WITH HIGH DISPERSAL 36 u(x,t) v(x,t) Time t Time t Distance x..4 Distance x.6.8 Figure 4. Simulation of system (3.9) on the omain x [, ]. Parameter values were β =, δ =.3, δ =.4, an =., an we took g(x) = sin πx. The initial ata was taken as u(x, ) = v(x, ) =. for all x [, ]. The outcome is that the species can coexist. [3] Cantrell, R. S. an C. Cosner. (3). Spatial Ecology via Reaction-Diffusion Equations. New York: John Wiley an Sons. [4] Cosner, C. an Y. Lou. (3). Does movement towar better environments always benefit a population? J. Math. Anal. Appl., 77, [5] Gilbarg, D. an N. S. Truinger. (983). Elliptic Partial Differential Equations of Secon Orer. n eition. Berlin: Springer-Verlag. [6] Hess, P. (99). Perioic-Parabolic Bounary Value Problems an Positivity. Pitman Research Notes in Mathematics Series, 47. Harlow, UK: Longman Scientific an Technical. [7] Hutson, V., J. Lopez-Gomez, K. Mischaikow an G. Vickers. (995). Limit behaviour for a competing species problem with iffusion, in Dynamical Systems an Applications, Worl Sci. Ser. Appl. Anal., 4, River Ege, NJ: Worl Scientific. [8] Hutson, V., K. Mischaikow, an P. Polácik. (). The evolution of ispersal rates in a heterogeneous time-perioic environment. J. Math. Biol., 43, [9] Hutson, V., Y. Lou, K. Mischaikow an P. Polácik. (3). Competing species near a egenerate limit. SIAM. J. Math. Anal. 35, [] Kato, T. (98). Superconvexity of the spectral raius, an convexity of the spectral boun an the type. Math. Z. 8, [] Kuang, Y. (). Basic properties of mathematical population moels, J. Biomath., 7, 9-4. [] Kuang, Y., W. Fagan an I. Lolaze. (3). Bioiversity, habitat area, resource growth rate an interference competition, Bull. of Math. Biol., 65, [3] Kuang, Y. an Y. Takeuchi. (994). Preator-prey ynamics in moels of prey ispersal in two patch environments, Math. Biosci.,, [4] Smith, H. L. (995). Monotone Dynamical Systems: An Introuction to the Theory of Competitive an Cooperative Systems. Provience: American Mathematical Society. [5] Smoller, J. (993). Shock Waves an Reaction-Diffusion Equations. New York: Springer- Verlag.

18 36 S. A. GOURLEY AND Y. KUANG [6] Takeuchi, Y. (996). Global Dynamical Properties of Lotka-Volterra Systems. River Ege, NJ: Worl Scientific. Receive on January 7, 5. Revise on march 7, 5. aress: aress: