EVOLUTIONARILY STABLE AND CONVERGENT STABLE STRATEGIES IN REACTION-DIFFUSION MODELS FOR CONDITIONAL DISPERSAL

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1 EVOLUTIONARILY STABLE AND CONVERGENT STABLE STRATEGIES IN REACTION-DIFFUSION MODELS FOR CONDITIONAL DISPERSAL KING-YEUNG LAM AND YUAN LOU Abstract. We consider a atheatical odel of two copeting species for the evolution of conditional dispersal in a spatially varying but teporally constant environent. Two species are different only in their dispersal strategies, which are a cobination of rando dispersal and biased oveent upward along the resource gradient. In the absence of biased oveent or advection, Hastings showed that the utant can invade when rare if and only if it has saller rando dispersal rate than the resident. When there is a sall aount of biased oveent or advection, we show that there is a positive rando dispersal rate that is both locally evolutionarily stable and convergent stable. Our analysis of the odel suggests that a balanced cobination of rando and biased oveent ight be a better habitat selection strategy for populations. Keywords: Conditional dispersal; Evolutionarily stable strategy; Convergent stable strategy; Reactiondiffusion-advection; Copetition. AMS Classification: 92D35, 92D40, 35K57, 35J Introduction Biological dispersal refers to the oveent of organiss fro one location to another within a habitat. Dispersal of organiss has iportant consequence on population dynaics, disease spread and distribution of species [3, 32, 36, 37]. Unconditional dispersal refers to the oveent of organiss in which the probability of oveent is independent of the local environent, while conditional dispersal refers to those that are contingent on the local environental cues. Understanding the evolution of dispersal is an iportant topic in ecology and evolutionary biology [9, 35]. In the following, we liit our discussion to phenotypic evolution under clonal reproduction, where the iportance of genes and sex are ignored. The evolution of unconditional dispersal is relatively well understood. For instance, Hastings [18] showed that in spatially heterogeneous but teporally constant environent an exotic species can invade when rare if and only if it has a saller dispersal rate than the resident population; See also [26]. It is shown by Dockery et al. [13] that a slower diffusing population can always replace a faster diffusing one, irrespective of their initial distributions; See also [21]. In contrast, faster unconditional dispersal rate can be selected in spatially and teporally varying environents [19, 30]. Conditional dispersal is prevalent in nature. It is well-accepted that conditional dispersal confers a strong advantage in variable environents, as long as ecological cues give an accurate prediction of the habitat [14]. In recent years there has been growing interest in studying the evolution of conditional dispersal strategies in reaction-diffusion odels and ore broadly, advective oveent of populations across the habitat [1, 2, 4, 5, 10, 20, 27, 28, 31, 34, 38]. In this paper we consider a reaction-diffusion-advection odel, which is proposed and studied in [6, 7, 17], for two copeting populations. In this odel, both populations have the sae population dynaics and adopt a cobination of rando dispersal and directed oveent upward along the resource gradient. We Date: February 3,

2 2 KING-YEUNG LAM AND YUAN LOU regard two populations as different phenotypes of the sae species, which are varied in two different traits: rando dispersal rate and directed oveent rate, and inquire into the evolution of these traits in spatially heterogeneous and teporally constant environents. A previous result Theore 7.1, [17]) iplies that if both traits evolve siultaneously, the selection is against the oveent, a prediction siilar to the evolution of unconditional dispersal. As a natural extension of the studies on the evolution of unconditional dispersal rate, in which the directed oveent rate is fixed to be zero, we turn to investigate the evolution of rando dispersal rate with the directed oveent rate, or advection rate, being fixed at a positive value. In this direction, the following results were established see [7, 17]): For any positive advection rate equal for both populations), if one rando dispersal rate is sufficiently sall relative to the advective rate) and the other is sufficiently large, two populations will coexist; if both rando dispersal rates are sufficiently sall, then the population with larger dispersal rate drives the other to extinction. In particular, sall dispersal rate is selected against; if both rando dispersal rates are sufficiently large, then the population with the saller dispersal rate will drive the other to extinction. In particular, large dispersal rate is selected against. These results suggest that interediate rando dispersal rates, relative to the advection rate, are favored. In this paper we focus on the evolution of interediate rando dispersal rates. We closely follow the general approach of Adaptive Dynaics [11, 12, 15, 16]. Roughly speaking, it is an invasion analysis approach across different phenotypes. The first iportant concept is the invasion exponent. The idea is as follows: Suppose that the resident population is at equilibriu and a rare utant population is introduced. The invasion exponent corresponds to the initial exponential) growth rate of the utant population. If the utant has identical trait as the resident, the invasion exponent is necessarily zero. The priary focus of Adaptive Dynaics is to deterine the sign of the invasion exponent which corresponds to success or failure of utant invasions) in the trait space, whose graphical representation is referred to as the Pairwise Invasibility Plot PIP). A central concept in evolutionary gae theory is that of evolutionarily stable strategies [29]. A strategy is called evolutionarily stable if a population using it cannot be invaded by any sall population using a different strategy. A closely related concept is that of convergent stable strategies. We say that a strategy is convergent stable, loosely speaking, if the utant is always able to invade a resident population when the utant strategy is closer to the convergent stable strategy than the resident strategy. We will use the standard abbreviations ESS for evolutionarily stable strategy and CSS for convergent stable strategy. More precise atheatical descriptions of the invasion exponent, ESS and CSS in the reaction-diffusion-advection context will be given in the next section. The goal of this paper is to show that there is selection for interediate rando dispersal rate, relative to directed ovent rate, by rigorously proving the existence of ESS and CSS in the ratio of rando dispersal rate to directed oveent rate. Surprisingly, the ultiplicities of ESS and CSS depend on the spatial heterogeneity in a subtle way. More precisely, if the spatial variation of the environent is less than a threshold value, there exists a unique ESS, which is also a CSS; if the spatial variation of the environent is greater than the threshold value, there ight be ultiple ESS/CSS.

3 EVOLUTION OF CONDITIONAL DISPERSAL 3 This paper is a continuation of our recent work [24], where we considered the case when two populations have equal rando dispersal rates but different directed oveent coefficients. 2. Stateent of ain results We shall follow the general approach of Adaptive Dynaics to state and interpret our ain results. Let the habitat be given by a bounded doain Ω in R N with sooth boundary Ω, and denote by n the outward unit noral vector on Ω, and = n. Let x) represent the quality of habitat, or available resource, at location x. Throughout this paper, we assue A1) C 2 Ω), it is positive and non-constant in Ω. Let us recall the reaction-diffusion-advection odel proposed and studied in [6, 7, 17]. u t = d 1 u α 1 u ) + u u v) in Ω 0, ), v 1) t = d 2 v α 2 v ) + v u v) in Ω 0, ), u d 1 α 1u = d 2 v α 2v = 0 on Ω 0, ), ux, 0) = u 0 x) and vx, 0) = v 0 x) in Ω. Here ux, t), vx, t) denote the densities of two populations at location x and tie t in the habitat Ω; d 1, d 2 are their rando dispersal rates, and α 1, α 2 are their rate of directed oveent upward along the resource gradient. The boundary conditions is of no-flux type, i.e. there is no net oveent across the boundary. Now, if we fix the directed oveent rates to be equal, i.e. assue that α 1 = α 2 = α, and set = d 1 /α and ν = d 2 /α. Then 1) can be rewritten as u t = α u u ) + u u v) in Ω 0, ), v 2) t = α ν v v ) + v u v) in Ω 0, ), u u = ν v v = 0 on Ω 0, ), ux, 0) = u 0 x) and vx, 0) = v 0 x) in Ω. It is well known that under assuption A1), any single population persists so long as > 0 and α 0) and reaches an equilibriu distribution given by the unique positive solution to the equation See, e.g. [8]) { α u u ) + u)u = 0 in Ω, 3) u u = 0 on Ω We denote this unique solution of 3) by ũ = ũ, α). Now we are in position to define, atheatically, the invasion exponent of our reaction-diffusion-advection odel. By standard theory, if soe rare population v is introduced into the resident population u at equilibriu i.e. u ũ), then the initial exponential) growth rate of the population of v is given by λ, where λ = λ, ν; α) is the principal eigenvalue of the proble [6]) 4) { α ν ϕ ϕ ) + ϕ ũ) + λϕ = 0 in Ω, ν ϕ ϕ = 0 on Ω, where the positive principal eigenfunction ϕ = ϕ, ν; α) is uniquely deterined by the noralization 5) e / ϕ 2, ν; α) = e / ũ 2, α). Ω In particular, when = ν, we have ϕ, ; α) ũ and λ, ; α) 0 for any α and. This is easy to understand since two species u and v are identical when = ν. Intuitively, 4) gives the exponential growth rate of a population adopting rando dispersal rate ν in the effective environent as given by ũ, after taking the resident population u into account. Ω

4 4 KING-YEUNG LAM AND YUAN LOU Therefore, λ is the invasion exponent in the language of Adaptive Dynaics, and it is the ain object of our investigation. Roughly speaking, a rare utant species v can invade the resident species u at equilibriu if λ < 0 and fails to invade if λ > 0. Since an evolutionarily stable strategy ust be evolutionarily singular, we first study the existence and ultiplicity of evolutionarily singular strategies. The latter refers to the class of strategies where the selection pressure is neutral, i.e. there is neither selection for higher nor for lower dispersal rates. For the convenience of readers we recall the definition of evolutionarily singular strategy. Definition 1. Fix α > 0. We say that > 0 is an evolutionarily singular strategy if λ ν, ; α) = 0. We abbreviate λ ν as λ ν. Our first ain result concerns the existence and ultiplicity of singular strategies. Theore 1. Suppose that ax Ω in Ω Given any δ 0, in{in Ω, ax Ω ) 1 }), for all sall positive α, there is exactly one evolutionarily singular strategy, denoted as ˆ = ˆα), in [δ, δ 1 ]. Moreover, ˆ in Ω, ax Ω ) and ˆ 0 as α 0, where 0 is the unique positive root of the function 6) g 0 ) := e / ), 0 < <. Furtherore, λ ν satisfies, [δ, ˆ); 7) λ ν, ; α) = 0, = ˆ; +, ˆ, δ 1 ]. Ω The ratio ax Ω in Ω is a easureent of the spatial variation of function. Theore 1 says that if the spatial variation of the inhoogeneous environent is suitably sall, then for sall advection rate, there is exactly one evolutionarily singular strategy. Interestingly, this singular dispersal strategy is of the sae order as the advection rate. Moreover, 7) iplies that when the dispersal rate of the resident population is less or greater) than the evolutionarily singular strategy, then a rare utant with a slightly greater resp. saller) dispersal rate ν can invade the resident population successfully. When Ω is an interval and is a onotone function, Theore 1 can be iproved as follows. Theore 2. Suppose that Ω = a, b), x > 0 in [a, b], and b) a) Then for all sall positive α, there is exactly one evolutionarily singular strategy in [0, ). A bit surprisingly, the constant in Theores 1 and 2 is sharp, as shown by the following result. Theore 3. Let Ω = a, b). For any L > 3+2 2, there exists soe function C 2 [a, b]) with, x > 0 and b) a) = L such that for all α > 0 sall, there are three or ore evolutionarily singular strategies. Fro the proof of Theore 3 we know that at least one of the three evolutionarily singular strategies is not an ESS Reark 5.1). It is natural to inquire whether the unique singular strategy in Theores 1 and 2 is evolutionarily stable and/or convergent stable. To address these issues, we first recall the definition of a local ESS. Definition 2. Fix α > 0. A strategy is a local ESS if there exists δ > 0 such that λ, ν; α) > 0 for all ν δ, + δ) \ { }.

5 EVOLUTION OF CONDITIONAL DISPERSAL 5 Under soe further restrictions we are able to show that the unique singular strategy ˆ in Theore 1 is a local ESS. Theore 4. Suppose that Ω is convex with diaeter d and ln L Ω) β 0 /d, where β is the unique positive root of the function t t2 π e 4t 2t 2 e t 1 ). 1 Then for α sufficiently sall, ˆ given in Theore 1 is a local ESS. The assuption of Theore 4 on is ore restrictive than that of Theore 1, as can be seen fro ax Ω in Ω ed ln L e β < We do not know whether an evolutionarily singular strategy, whenever it is unique, is always an ESS. On the other hand, at least one of the singular strategies constructed in Theore 3 is not an ESS, so in general we do not expect every singular strategy to be evolutionarily stable. Our next result shows that for our odel a local ESS is always a local CSS. Again, for the convenience of readers we recall the definition of a local CSS. Definition 3. A strategy is a local convergence stable strategy if there exists soe δ > 0 such that { > 0 if λ, ν; α) = < ν < + δ or δ < ν <, < 0 if ν < < + δ or δ < < ν. Our final result is Theore 5. For any sall δ > 0, there exists soe α 1 > 0 such that if α 0, α 1 ), any local ESS in [δ, δ 1 ] is necessarily a local CSS. In particular, this excludes the possibility of a Garden of Eden type of ESS in the class of rando dispersal rates which are of the sae order as the advection rate. Figure 1a) gives an illustration of Theores 1, 2, 4 and 5, where there is a unique singular strategy which is both locally evolutionarily stable and convergent stable. Figure 1b) illustrates the possibility of ultiple evolutionarily stable strategies, as suggested by Theore 3. Our results, especially Theores 4 and 5, suggest that in spatially varying but teporally constant environent, an evolutionarily stable and convergent stable strategy for rando dispersal is proportional to the directed oveent rate. In the biological context, while tracking the resource gradient ight help a population invade quickly, but if the population overly pursue the resource by excessive advection along the resource gradient, its distribution will only be concentrated near locally ost favorable resources; See [8, 22, 23, 25]. Such strategy ay not be evolutionarily stable since populations using a better ixture of rando oveent and resource tracking can utilize all available resource and thus invade when rare. Hence a balanced cobination of rando and biased oveent ight be a better habitat selection strategy for populations. This paper is a continuation of our recent work [24], where we considered the case when two populations have equal diffusion rates but different advection coefficients. While soe proofs in current paper rely upon those in [24], there exist soe notable differences between these two works: 1) Convergent stability is an iportant aspect in the evolution of dispersal. There were no discussions of CSS in our previous work [24], while in this paper we are able to show that a local ESS is always

6 6 KING-YEUNG LAM AND YUAN LOU Figure 1. Possible pairwise invasibility plots PIP) illustrating Theores 1, 2 Fig. 1a)) and Theore 3 Fig. 1b)). White coloring indicates the utant invades, and black that the utant loses does not invade). On the horizontal axis the resident rando dispersal rate ) is represented, and on the vertical axis the utant rando dispersal rate ν). The evolutionarily singular strategies are given by the intersection of black and white parts of the plane, along the diagonal. In Fig. 1a), the unique singular strategy is an ESS since it lies vertically between black regions). It is also a CSS. In Fig. 1b), there are three singular strategies. The one in the iddle is neither an ESS nor a CSS. a CSS, provided that the advection rate is sall and the rando dispersal rate is of the sae order as the advection rate. 2) When two populations have sae rando dispersal rates d 1 = d 2 := d), we proved in [24] that if the rando dispersal rate d is sall, there is an evolutionarily stable strategy for advection rate. That is, there exists soe ᾱ = ᾱd) such that if α 1 = ᾱ and α 2 is close to but not equal to) ᾱ, then population v can not invade when rare. Moreover, as d 0+, ᾱ/d η, where η is the unique positive root of the function g 1 η) := Ω e η ), 0 < η <. On the other hand, Theore 4 shows that when two populations have sae but sall advection rates α 1 = α 2 := α), there is an evolutionarily stable strategy for rando dispersal rate. That is, there exists soe d = dα) such that if d 1 = d and d 2 is close to but not equal to) d, then population v can not invade when rare. Moreover, as α 0+, d/α ξ, where ξ is the unique positive root of the function g 0 ξ) := Ω e /ξ ), 0 < ξ <. Since ᾱ/d is the optial ratio of advection rate to rando dispersal rate and d/α is the optial ratio of rando dispersal rate to advection rate, our first thought is that ᾱ/d ight be inverse to d/α for sall dispersal rates. But apparently this is not the case since their liits are not inverse to each other; i.e., η 1 ξ. These discussions indicate that understanding the invasion exponent at ũ, 0) is quite subtle, even for sall d i, α i, i = 1, 2. While there are these differences between current paper and our previous work [24], both works suggest that a balanced cobination of rando and biased oveent ight be a better habitat selection strategy

7 EVOLUTION OF CONDITIONAL DISPERSAL 7 for populations, provided that the diffusion and advection rates are sall. It is interesting to inquire whether siilar conclusions can be drawn for large advection rate. This paper is organized as follows. Section 3 is devoted to the study of evolutionarily singular strategies and we establish Theores 1 and 2 there. In Section 4 we deterine whether the evolutionarily singular strategy fro Theore 1 is evolutionarily stable and prove Theore 4. In Section 5 we establish Theore 3. Theore 5 is proved in Section 6. Finally we end with soe biological discussions and interpretations of our results. 3. Evolutionarily Singular Strategies In this section we establish Theores 1 and 2. We first derive the forula of λ ν, ; α) in Subsect Subsect. 3.2 is devoted to various estiates of ũ for sufficiently sall α, which in turn enables us to obtain the liit proble for λ ν, ; α) with sufficiently sall α in Subsect The proofs of Theores 1 and 2 are given in Subsect Materials of this section rely heavily on results fro [24], especially those in Subsects. 3.2 and 3.3. Readers who are interested in learning ore about the technical details ay wish to consult [24] Forula for λ ν. We recall that ũ satisfies { α ũ ũ ) + ũ)ũ = 0 in Ω, 8) n ũ ũ ) = 0 The stability of ũ, 0) is deterined by the sign of the principal eigenvalue λ = λ, ν; α) of 4). Differentiate 4) with respect to ν we get { α ν ϕν ϕ ν ) + ũ)ϕ ν + λϕ ν = α ϕ λ ν ϕ in Ω, 9) n ν ϕ ν ϕ ν ) = ϕ on Ω and e / ϕ ν ϕ = 0. Multiplying 9) by e /ν ϕ and 4) by e /ν ϕ ν, subtracting and integrating the result, we get λ ν 10) e ν ϕ 2 = ϕ e ) ν ϕ. α Setting ν = in 10), then by ϕ, ; α) = ũ and 10) we have the following result: Lea 3.1. For any α > 0 and > 0, the following holds: λ ν, ; α) 11) e / ũ 2 = ũ e / ũ). α 3.2. Estiates. We collect soe known liiting behaviors of ũ as α 0. Lea 3.2 See, e.g. Lea 3.2 of [24]). For any δ > 0, there exists a positive constant C such that for all α and δ, 12) C 1 ũ C in Ω. Moreover, as α 0, ũ L Ω) 0 uniforly for δ 1/δ. Lea 3.3 Lea 3.3 of [24]). For each δ > 0, there exists C > 0 such that for any φ H 1 Ω), ũ 2 φ 2 C ũ L Ω) φ 2 H 1 Ω), where C = C) is independent of α and bounded uniforly for δ.

8 8 KING-YEUNG LAM AND YUAN LOU For later purposes, we state the following corollary, which follows fro Leas 3.2 and 3.3. Corollary 3.4. For each δ > 0, there exists C > 0 such that for all φ i H 1 Ω), i = 1, 2, ũ 2 φ ) 1φ 2 ũ 2 C ũ L Ω) φ 1 2 H 1 Ω) + φ 2 2 H 1 Ω), where C = C) is independent of α and bounded uniforly for δ. Next, we have the following estiate of ũ. Lea 3.5. For each δ > 0, ũ 0 in H1 Ω) as α 0, uniforly for δ 1/δ. Proof. Denote ũ = ũ and differentiate 8) with respect to, we have { α ũ ũ ) + 2ũ)ũ = α ũ in Ω, 13) ũ ũ = ũ Multiply 13) by e / ũ and integrate by parts, we find 14) α e / e / ũ ) 2 + e / 2ũ )e / ũ ) 2 = α ũ e / ũ ). By Hölder s inequality, α e / e / ũ ) 2 + e / 2ũ )e / ũ ) 2 α e / ũ By the fact that 2ũ > 0 uniforly in Ω as α 0 Lea 3.2), we see that uniforly for δ, e /ũ ) 2 = O1) and e / ũ ) 2 = Oα) and hence e / ũ 0 in H 1 Ω). Upon considering 14) again, we conclude that e / ũ 0 in H 1 Ω) as α 0. Next, we show the following lea. Lea 3.6. For each δ > 0, d d as α 0 uniforly, for δ 1/δ. ) ũ e / ũ) 1 2 e / 2 ), Proof. By Lea 3.5, as α 0, ) d ũ e / ũ) d = ũ e / ũ) e / 2 ). ũ e / ũ) + ũ e / ũ ) This copletes the proof. By Leas 3.3 and 3.6, we have the following result. Corollary 3.7. As α 0, 15) ũ e / ũ) g 0 ) := e / ) in C 1 loc0, )). And we see that for α sall, the roots of λ ν, ; α) = 0 and the roots of g 0 are in one-to-one correspondence, provided that the latter roots are non-degenerate.

9 EVOLUTION OF CONDITIONAL DISPERSAL Liit Proble for λ ν. By Lea 3.1 and Corollary 3.7, the first step in establishing the existence of evolutionarily singular strategies is to study the roots of g 0 ) = e / ). Since g 0 ) = ) e / 2 1, we observe that 16) g 0 ) < 0 for 0, in Ω ] and g 0 ) > 0 for [ax, ). Ω Proposition 3.8. Let Ω R N. Suppose that ax Ω in Ω Then there exists 0 > 0 such that i) g 0 ) < 0 if [0, 0 ); ii) g 0 ) = 0 if = 0 ; iii) g 0 ) > 0 if > 0. Moreover, g 0 ) > 0. Proof. Let g 0 η) = g 0 1/η), then it suffices to show that g 0 has a unique non-degenerate root in 0, ), which follows fro Proposition 3.8 and Reark 3.11 of [24] Proofs of Theores 1 and 2. Proof of Theore 1. By Proposition 3.8, g 0 ) = e / ) has a unique, non-degenerate root in 0, ). For each δ > 0, ũ e / ũ) e / ) in C 1 [δ, δ 1 ]) as α 0. Hence, we see that for any 0 < δ < in Ω[in{x), x) 1 }], ũ e / ũ) has a unique positive root in [δ, δ 1 ] for all sufficiently sall α. By Lea ũ e /ũ) α λ ν, ; α) = e /ũ. 2 Therefore, given any δ > 0, for sufficiently sall α, λ ν also changes sign exactly once in [δ, δ 1 ]. Moreover, denoting the unique root by ˆ, we see that ˆ 0 as α 0, where 0 is the unique positive root of g 0 ). To show Theore 2, we first prove a useful lea. Lea 3.9. Let φ C 2 [0, 1]) be a solution to { φxx + bx)φ 17) x + gx, cx) φx)) = 0 in 0, 1), φ x 0 at x = 0, 1, where sign gx, s) = sign s and c x > 0 in [0, 1]. Then φ x > 0 in 0, 1). Proof of Lea 3.9. Clai φ x 0 in 0, 1) if c is strictly increasing in [0, 1]. Suppose to the contrary that there exists 0 x 1 < x 2 1 such that φ x < 0 in x 1, x 2 ) and φ x x 1 ) = φ x x 2 ) = 0. Then x 1, x 2 are one-sided local axia of φ x in [x 1, x 2 ]. Hence φ xx x 1 ) 0 φ xx x 2 ), which iplies by 17)), that cx 1 ) φx 1 ) and cx 2 ) φx 2 ). Cobining with the strict onotonicity of φ in [x 1, x 2 ], we have cx 1 ) φx 1 ) > φx 2 ) cx 2 ), which contradicts the strict onotonicity of c. This proves Clai Next, assue further that c x > 0 in [0, 1]. Suppose for contradiction that φ x x 0 ) = 0 for soe x 0 0, 1). Then by the previous clai, x 0 is a local interior iniu of φ x in 0, 1). Therefore, φ xx x 0 ) = 0 and by 17), we have cx 0 ) = φx 0 ). Together with c φ) x x 0 ) = c x x 0 ) > 0,

10 10 KING-YEUNG LAM AND YUAN LOU we derive that c φ > 0 in x 0, x 0 + δ) for soe δ > 0. But then e ) x 0 bs) ds φ x = e x 0 bs) ds gx, c φ) < 0 and φ x x 0 ) = 0. x Hence φ x < 0 in x 0, x 0 + δ). This contradicts Clai 3.10, and copletes the proof. Corollary Suppose Ω = 0, 1),, x > 0 in [0, 1] and let ũ be the unique positive solution of 3). Then i) for all α > 0 and > 0, ũ x > 0 in 0, 1); ii) if > ax [0,1], then e / ũ) x > 0 in 0, 1); iii) if < in [0,1], then e / ũ) x < 0 in 0, 1). Proof of Corollary Firstly, i) follows as a direct consequence of Lea 3.9. For ii), we observe that v := e / ũ satisfies { αvxx + α x v x + e / ve / v) = 0 in 0, 1), v x = 0 at x = 0, 1. In addition, the assuption > ax [0,1] iplies that e / ) x > 0 in [0, 1]. Hence, Lea 3.9 applies and yields e / ũ) x = v x > 0 in 0, 1). For iii), set wx) = v1 x), then w satisfies { αwxx α x w x + e / w e / w) = 0 in 0, 1), w x = 0 at x = 0, 1, where x) = 1 x). In addition, the assuption < in [0,1] iplies that e / ) x > 0 in [0, 1]. Hence, Lea 3.9 applies and yields w x > 0, which is equivalent to e / ũ) x < 0 in [0, 1]. This copletes the proof of the corollary. Proof of Theore 2. Without loss of generality, assue Ω = 0, 1). In view of Lea 3.1 and Theore 1, it suffices to show that λ ν e / ũ 2 = ũ x e / ũ) x 0 for all 0, in [0,1] ) ax [0,1], ), which follows fro Corollary Evolutionarily Stable Strategies In this section we provide a sufficient condition for the evolutionarily singular strategy identified in Theore 1 to be evolutionarily stable. The forula of λ νν, ; α) is given in Subsect We establish various estiates of eigenfunctions for sufficiently sall α in Subsect We apply these results to find the liit of λ νν ˆ, ˆ; α)/α as α 0. The sign of this liit is deterined in Subsect. 4.3, which helps us coplete the proof of Theore Forula for λ νν. Differentiate 9) with respect to ν and denote 2 ϕ ν = ϕ 2 νν and 2 λ ν = λ 2 νν, we have { α ν ϕνν ϕ νν ) + ũ)ϕ νν + λϕ νν = 2α ϕ ν 2λ ν ϕ ν λ νν ϕ in Ω, 18) ϕνν ϕ νν ϕν = 2 Set ν =, we have λ = 0, ϕ, ; α) = ũ and { α ϕνν ϕ νν ) + ũ)ϕ νν = 2α ϕ ν 2λ ν ϕ ν λ νν ũ in Ω, 19) ϕνν ϕ νν ϕν = 2

11 EVOLUTION OF CONDITIONAL DISPERSAL 11 Multiplying 19) by e / ũ, we obtain, via integration by parts, λ νν, ; α) ) 20) e / ũ 2 = 2 ϕ ν e / ũ 2 λ ν, ; α) α α ϕ ν e / ũ where ϕ ν = ϕ ν, ; α) is the unique solution to 9). Hence, we have the following forula for λ νν : Lea 4.1. Suppose that is an evolutionarily singular strategy, i.e., λ ν, ; α) = 0. Then λ νν, ; α) ) 21) e / ũ 2 = 2 ϕ ν e / ũ, α where ϕ ν = ϕ ν, ; α) is the unique solution to { α ϕν ϕ ν ) + ϕ ν ũ) = α ũ in Ω, 22) ϕν ϕ ν = ũ on Ω, e / ϕ ν ũ = Estiates. Let λ k, ϕ k ) be the eigenpairs of 4) counting ultiplicities) with ν = such that λ 1 < λ 2 λ 3. By the transforation φ = e / ϕ and 8), 4) becoes { e 23) / φ) ũ ũ ) ũ e / φ + λ α e/ φ = 0 in Ω, φ = 0 Then λ k /α is the k-th eigenvalue of 23) counting ultiplicities). asyptotic behavior of λ k as α 0. The following result deterines the Proposition 4.2. For each δ > 0, λ k /α σ k as α 0 uniforly for δ 1/δ, where λ k = λ k, ; α) is the k-th eigenvalue of 4) when ν =, and σ k = σ k ) is the k-th eigenvalue of { e 24) / φ) [ ) ] e / φ + σe / φ = 0 in Ω, φ ) φ = 0 Reark 4.1. One can check that all σ k are real, σ 1 = 0, and e / is an eigenfunction corresponding to σ 1. In particular, σ k > 0 for k 2. Proof of Proposition 4.2. It follows fro Proposition 4.2 of [24]. Next, we study the asyptotic behavior of ϕ ν, ; α) as α 0. Recall that ϕ ν, ; α) is the unique solution of 22). We shall assue that = α) is an evolutionarily singular strategy i.e. λ ν, ; α) = 0) and as α 0. Note that if ax Ω in Ω , then = ˆα) is the unique evolutionarily singular strategy as deterined in Theore 1, and = 0 is the unique positive root of g 0 ) = e / ) as deterined in Proposition 3.8. Lea 4.3. Suppose that = α) is an evolutionarily singular strategy and as α 0. By passing to a subsequence, ϕ ν, ; α) ϕ in H 1 Ω) as α 0, where ϕ is the unique solution to { 25) ϕ ϕ ) [ ) ] ϕ = in Ω, ϕ ϕ = on Ω, e / ϕ = 0. Proof. First we estiate ϕ ν L2 Ω) in ters of ϕ ν L2 Ω). To this end, ultiply 22) by e / ϕ ν and integrate by parts, [ e 26) e / e / ϕ ν ) 2 e / e / / ϕ ν ) 2 ] ũ) e / ũ = ũ e / ϕ ν ),

12 12 KING-YEUNG LAM AND YUAN LOU which can be rewritten as e / e / ϕ ν ) 2 + e / e / ũ) 2 e / ϕ e 27) / ũ) 2 ν ) 2 = 2 e / e / ũ) e / ϕ ν) e / ϕ e / ũ ν 1 ũ e / ϕ ν ). By Hölder s inequality, this yields 1 e / e / ϕ ν ) e / e / ũ) 2 e / ϕ e / ũ) 2 ν ) e / ũ 2 ) C ũ 2 e / ϕ ν ) 2 + e / ϕ ν ) [ C 2 + ũ ] e / ϕ ν ) 2 + C ) C e / ϕ ν ) 2 + ũ L Ω) e / ϕ ν 2 H 1 Ω) + 1. The second and last inequalities follow fro Leas 3.2 and 3.3 respectively. Hence for each δ > 0, there is soe constant C independent of α sall and δ such that ) 28) ϕ ν 2 C ϕ ν ) ), Next, we show that ϕ ν H 1 Ω) is bounded uniforly as α 0. By applying the Poincaré s inequality and λ 2 α e / ϕ ν ) 2 = e / e / ϕ ν ) 2 ũ e / ϕ ν ) C ϕ ν 2 H 1 Ω) + 1). e e / e / / ϕ ν ) 2 ) ũ) ũ By Proposition 4.2, λ2 α σ 2 as α 0. Observe that σ 1 = 0 and is siple, one can deduce that σ 2 > 0 Reark 4.1). So cobining with 28), we deduce that ϕ ν 2 H 1 Ω) C ϕ ν H1 Ω) + 1). Hence ϕ ν H 1 Ω) is bounded independent of α sall and ϕ ν converges weakly in H 1 Ω) to soe ϕ 0 H 1 Ω) satisfying e / ϕ 0 = 0. Passing to the liit using the weak forulation of 22), we see that ϕ 0 = ϕ by uniqueness. This proves the lea. The following result is a direct consequence of Leas 4.1 and 4.3. Corollary 4.4. Suppose that = α) is an evolutionarily singular strategy and as α 0. Then λ νν, ; α) 2 29) li = ϕ e / ), α 0 α e / 2 where ϕ is the unique solution of 25).

13 EVOLUTION OF CONDITIONAL DISPERSAL Liit proble for λ νν. In this subsection we study, for sufficiently sall α, the sign of λ νν ˆ, ˆ; α) where ˆ is the unique evolutionarily singular strategy deterined in Theore 1. By Corollary 4.4, it suffices to study the sign of ϕ e /0 ), where ϕ is the unique solution of 25) with = 0, and 0 is a positive root of g 0 ). A sufficient condition for the uniqueness of 0 is given in Proposition 3.8. The ain result of this subsection is Proposition 4.5. Suppose that Ω is convex with diaeter d. If is non-constant and d ln L Ω) β 0, then λ νν ˆ, ˆ; α) 2 li = α 0 α e / 0 2 ϕ e /0 ) > 0, where ϕ is the unique solution to 25) with = 0 ), β is the unique positive root of t t 2 π e 4t 2t 2 e t 1 ), 1 and 0 is given in Proposition 3.8. That β 0 is well-defined follows fro the following lea. Lea 4.6. Let f 4 t) = ) ) t 2 π e 4t 2t e t 1 1, then f 4 has a unique root β in 0, 1). In fact, we have, t 0, β 0 ); f 4 t) = 0, t = β 0 ; +, t β 0, ). Proof of Lea 4.6. One can show that i) li t 0 f 4 t) = 1, ii) f 4 is strictly increasing in [0, 0.8] and that iii) f 4 t) > 0 for all t > 0.8. ) 2 t f 4 t) = e 4t π 1 e t 2te t 1 + e t ) d dt f 4t) = 2t π 2 + e 4t 4 3e t ) 1 e t ) 2 2te t 1 + e t ) + e 4t = 2t π 2 + > 2t π 2 + e 4t 1 e t [ 4 3e t ) 1 e t e t 2t 1) ) ] t 2 1 e t e t 1 + e t 2t 1) e 4t [ 4 3e t 1 e t ) 2e t 1 ) + e t 2t 1) ] = 2t π 2 + 2e 6t 1 e t [ t)e t + 2e 2t]. t 1 e t > 1 Since the first ter is positive and t > t 0 := 0, f 4 is strictly increasing if t t 0, t 1 ], where t 1 := ln 5+t0)+ 5+t 0) Therefore we ay assue t > t 1. Siilarly, f 4 is strictly increasing if t t 1, t 2 ], where t 2 := ln 5+t1)+ 5+t 1) Thus defining an increasing sequence {t i } t=0 by induction t i+1 = ln 5 + t i) t i ) 2 24, 4 then we see that f 4 is strictly increasing for t 0, li i t i ), where li i t i > 0.8. Also, for all t 0.8 by t e t 1 < 1), This copletes the proof. ) 2 ) t 2t e 4t π e t 1 1 > ) 2 t e 4t > 0. π )

14 14 KING-YEUNG LAM AND YUAN LOU Let w denote the unique solution to 30) { w w ) [ ) ] w = in Ω, w w = on Ω, e / w = 0. It is clear that Proposition 4.5 follows fro the following result: Proposition 4.7. Suppose that Ω is convex with diaeter d. If is non-constant, > 0 satisfies g 0 ) = 0 and d ln L Ω) β 0, then 31) Proof. First, consider the transforation w = satisfies w e / ) > 0. z ln ). After soe tedious but direct calculations, z 32) { 2 e / z) = e / ) in Ω, z = 0 Integrating over Ω, we have 33) 0 = 2 e / z) = e / ) = e / 2 1 ). Multiplying by z and integrate by parts, we have 34) 2 e / z 2 = z e / ) = ze / 2 1 ). Lea 4.8. w e / ) = 1 ) 2 ln 1 1 e / 2 1 ) 2 2 e / z 2. Proof of Lea 4.8. w e / ) [ = z ln )] e / ) ln = z) e / ) e / ) = e / 1 ) z + z ln ) e / ) = 2 e / ln ) z + z ln ) e / ) = ln ) 2 e / z) + z ln ) e / ) [ 1 = 2 ln + ) ] + z e / )

15 EVOLUTION OF CONDITIONAL DISPERSAL 15 where we used 33) in the second equality, integrated by parts in the second last equality and used 32) in the last line. By 33) and 34), 1 w e / ) = 2ln + ) 1 e / ) + z e / ) = 1 2ln + ) 1 e / 2 1 ) 2 e / z 2 = 1 ) 2 ln 1 1 e / 2 1 ) 2 2 e / z 2. This copletes the proof of Lea 4.8. By Lea 4.8, 31) is equivalent to 35) 2 e / z 2 < 1 ) 2 ln 1 1 e / 2 1 ) 2. In particular we ay assue without loss of generality by replacing z by z + C) that z satisfies 32) and 36) e / 2 z = 0. Set L = ax Ω in Ω. Since L 1 < enough to show the following: < L by 16)) and that ft) = ln t t 1) is strictly decreasing for t > 0, it is 37) 2 e / z 2 1 ) 2 ln L L 1 1 e / 2 1 ) 2. To this end, we prove the following Poincaré-type inequality. Lea e / z 2 1 ˆγ 2 e / 2 1 ) 2, where ˆγ 2 is the second or first positive) eigenvalue of 38) [ ) ] 2 e / φ φ = 0 + ˆγe / 2 φ = 0 in Ω, Proof of Lea 4.9. Let ˆγ k, φ k ) be the eigenpairs of 38), canceling 1 2 on both sides, we have 39) { 2 e / φ) + ˆγe / 2 φ = 0 in Ω, φ = 0

16 16 KING-YEUNG LAM AND YUAN LOU Observe that γ 1 = 0 and φ 1 = 1. Now set z = k=2 a kφ k a 1 = 0 by 36)) and 1 = k=1 b kφ k, then one can deduce that a k = b k ˆγ k and b 1 = 0. Now, 2 e / z 2 = = [ ] z 2 e / z) z 2 e / 1 ) b k = φ k ˆγ k k=2 = 1 2 e / 1 ˆγ 2 ) ) 2 e / b k φ k k=2 b 2 k ˆγ k φ 2 k 2 e / 1 ) 2. k=2 And the lea follows. Next, we derive a lower estiate of ˆγ 2 in ters of L. Recall that L = ax Ω in Ω and set s = ln L Ω), d = diaω). Then by ean value theore, we have Suppose ds β 0, then ln L = ln ax Ω in Ω d ln L Ω). 40) L e ds e β0 < The following clai follows fro 0 < ds β 0 and Lea 4.6. Clai ) 2 [ ] ds 2ds e 4ds π e ds 1 1. Next, we show the following clai. Clai Suppose that Ω is convex with diaeter d, then By 40), Clai 4.11 is proved if we can show that [ is strictly decreasing in t ) 2 [ ] ds 2 ln L L 4 π L 1 1. ) 2 ln t f 3 t) = t 4 t 1 1 = t 4 2 ln t t + 1) t 1 1, ]. Now, d dt f 3t) = t 5 4 5t) t 1) 2 2 ln t t + 1) + t 5 2 t)t 1). t 1) 2

17 EVOLUTION OF CONDITIONAL DISPERSAL 17 For t 1, ln t = ln t 1 ) > t 1 1) + t 1 1) 2 2. Hence, d dt f 3t) < t 5 { [ 4 5t) 2t 1 t 1) 2 1) + t 1 1) 2 t + 1 ] + 2 t)t 1) } [ Therefore f 3 t) is strictly decreasing in = t 7 { [ 4 5t) 2t1 t) + 1 t) 2 t 1) 2 t 2 t 1) ] + t 2 2 t)t 1) } = t 7 { 4 5t) t 2 + 3t 1) + 2t 2 t 3} t 1 = t 7 4t 2 13t + 4) = 4t 7 t 13 ) 105 t 13 + ) , ]. This proves Clai Recalling that ˆγ 2 is the second or first positive) eigenvalue of 38), we clai the following: Clai ˆγ 2 > [ ] 1 2 ln L L 1 1. We prove Clai Let f 2 t) := t 2 e t. If L [1, ], then ft) fl 1 ) for all t [L 1, L]. Hence by L 1 < < L and eigenvalue coparison, ˆγ 2 > N 2 Ω) N 2 Ω) ax Ω ax Ω e / e / ) 2 1 in Ω f 2 )) ) 2) 1 f 2 2 L 1 ) N 2 Ω) e 1/L ln 2 L Ω) L2) 1 f2 L 1 ) = N 2 Ω) ln 2 π ) 2 s 2 L 4 d [ ] 1 2 ln L L 1 1 L Ω) L 4 where the last inequality follows fro Clai 4.11, whereas N 2 Ω) is the second Neuann eigenvalue of the Laplacian of Ω and the second last inequality follows fro the following optial estiate of N 2 Ω) for convex doains due to Payne and Weinberger. Theore 6 [33]). Suppose that Ω is a convex doain in R N with diaeter d, then N 2 Ω) π2 d 2. This proves Clai Finally, by Lea 4.9 and Clai 4.12, we have 2 e / z 2 < 1 ) 2 ln L L 1 1 e / 2 1 ) 2. This proves 37) and hence Proposition 4.7.

18 18 KING-YEUNG LAM AND YUAN LOU 4.4. ESS: Theore 4. Proof of Theore 4. By the assuptions, ) ln ax Ω d ln L in Ω Ω) β 0 < ln ). Hence by Theore 1), for all sufficiently sall, there exists a unique evolutionarily singular strategy, denoted by ˆ = ˆα) in Ω, ax Ω ). Moreover, ˆ 0 as 0, where 0 is the unique positive root of g 0 ) guaranteed by Proposition 3.8. Then, by Corollary 4.4 and Proposition 4.5, we have λ νν ˆ, ˆ; α) 41) li = 2 ϕ e /0 ) > 0. α 0 α e / 0 2 As λˆ, ˆ; α) = λ ν ˆ, ˆ; α) = 0 for all α sall, there exists δ 1 = δ 1 α) > 0 such that λˆ, ν; α) > 0 if ν ˆ δ 1, ˆ + δ 1 ) \ {ˆ}. Thus, the strategy = ˆα) is a local ESS. The following result will be needed in the next section. Lea Let w denote the unique solution of 30). If g 0) < 0, then w e / ) < 0. Proof. Suppose 42) 0 > g ) = 1 2 e / 2 2 ). ) Define as before z by w = z ln such that z satisfies 32). Then z can be characterized as a global iniizer of the functional J z) = e / 2 z 2 + ze / 2 1 ). 2 By 34), we have Jz) = 2 e / 2 z 2. Hence, ) ln Jz) J e / 2 z 2 1 e / 2 / ln + e = 1 e / [ ln 1 )] 2 33) = 1 e / [ ln ) < 1 [ e / = 1 [ ] 2 ln ) e / ) ) 2 ln 1 )] ) ln 1 ) ] By Lea 4.8, this is equivalent to w e / ) < 0.

19 EVOLUTION OF CONDITIONAL DISPERSAL Counterexaple for ax Ω in Ω > To study the ultiplicity of evolutionarily singular strategies when α is sall, by 16) it suffices to consider the nuber of roots of g 0 ) for in Ω, ax Ω ). Equivalently, let η = 1, it suffices to consider the nuber of roots of 43) g 1 η) = e η 2 1 η) for η 1 ax Ω, 1 in Ω Suppose that Ω = 0, 1) and is non-decreasing i.e. 0), then by aking the substitution s = ηx), 44) η 2 g 1 η) = where hs) = e s 1 s). 1 0 η 2 η)1 η)e η dx = η1) η0) ). x)hs)ds, Proposition 5.1. If Ω = 0, 1) and x) = a + b a)x for soe 0 < a < b, then g 1 η) has exactly one root in [0, ). Proof. By 16), it suffices to consider η [ b 1, a 1]. Now x) = b a. By 44), ) d η 2 ) dη b a g 1η) = d ηb hs)ds dη ηa = hηb)b hηa)a = [ b1 ηb)e ηb] + [ a1 ηa)e ηa] < 0 for all η [b 1, a 1 ], as both ters on the last line are negative. Hence η 2 g 1 η) is strictly decreasing in [b 1, a 1 ] and has exactly one root. Proposition 5.2. Let Ω = 0, 1), then for each L > 3+2 2, there exists C 2 [0, 1]) satisfying, > 0 such that g 0 ) has at least three roots in in Ω, ax Ω ). Proof. Consider again g 1 η) = g 0 1/η) defined in 43). Fix L > Set in Ω = a = 1 and ax Ω = b = L. Choose η 1/L, 1) such that η a < 2 2 and η b > Such η exists as L > Note that 45) sh s) + 2hs) > 0 if and only if s < 2 2 or s > Hence, by our choice of η, η a)h η a) + 2hη a) > 0 and η b)h η b) + 2hη b) > 0. Furtherore, as η a < 1 < η b, we also have 46) hη a) > 0 > hη b). Let x) be the piecewise linear function given by 0) = a, 1) = b, and [ ] ɛ L 1 x 0, [ L 1 ] x) = L 2 x 1 ɛ L 2, 1 ) L 3 := b ɛ) a+ɛ) ɛ 1 x ɛ L 1, 1 ɛ L 2, L ɛ 1 L 2

20 20 KING-YEUNG LAM AND YUAN LOU where ɛ > 0 is to be chosen sall, and L 1, L 2 are to be chosen positive and large. Note that L 3 b a reains uniforly bounded ) for sall ɛ and L 1, L 2 1. By the choices of and η, we see that η satisfies η. Then 1 ax Ω, 1 in Ω ηa+ɛ) ηb η 2 g 1 η) = L 1 hs) ds + L 2 ηa := L 1 I 1 + L 2 I 2 + L 3 I 3. ηb ɛ) Both I 3 and d dη I 3 are uniforly bounded for ɛ > 0 sall: ηb ɛ) hs) ds + L 3 hs) ds ηa+ɛ) I 3 = Hηb ɛ)) Hηa + ɛ)), Hs) = e s s, d dη I 3 = hηb ɛ))b ɛ) hηa + ɛ))a + ɛ). It is easy to see that for sufficiently sall ɛ, I 1 = I 1 η, ɛ) ɛhη a) > 0 and I 2 = I 2 η, ɛ) ɛhη b) < 0. Therefore, for each sall but fixed ɛ, there exists L 1, L 2 arbitrarily large such that 47) 1 ax Ω, 1 in Ω L 1 hη b) L 2 hη a) and g 1 η ) = 0. It reains to fix sufficiently sall ɛ > 0 such that for L 1, L 2 arbitrarily large, we have d 48) L 1 dη I d 1 + L 2 η=η dη I 2 +. η=η ) ) 1 1 Since then g 1 ax Ω > 0, g 1 in Ω < 0 and g 1η ) > 0 and therefore g 1 has at least 3 roots in ). Firstly we copute d dη I 1. d dη I 1 = hηa + ɛ))a + ɛ) hηa))a = a[hηa + ɛ)) hηa)] + ɛhηa + ɛ)) = ɛ[ηah θ 1 ) + hηa + ɛ))] where θ 1 ηa, ηa + ɛ)), and li ɛ 0 [ηah θ 1 ) + hηa + ɛ))] = ηah ηa) + hηa). Siilarly, d dη I 2 = ɛ[ηbh θ 2 ) + hηb ɛ))], where θ 2 ηb ɛ), ηb) and li ɛ 0 [ηbh θ 2 ) + hηb ɛ))] = ηbh ηb) + hηb). In view of 47), a sufficient condition for 48) is which is equivalent to hη b)[η a)h η a) + hη a)] + hη a)[η b)h η b) + hη b)] > 0 49) hη b)[η a)h η a) + 2hη a)] + hη a)[η b)h η b) + 2hη b)] > 0, which holds by 45) and 46). We have proved the proposition. In conclusion, we have found a piecewise C 1 function and soe η > 0 such that the stateent of the proposition holds. Although the constructed is only piecewise C 1, one can approxiate it by C 2 Ω) functions such that ax Ω in Ω = ax Ω in Ω and in W 1, Ω).

21 EVOLUTION OF CONDITIONAL DISPERSAL 21 Reark 5.1. For the singular strategy = α) such that := 1/η, by Lea 4.1 we have ) λ νν, ; α) li e / 2 = 2 ϕ e / ). α 0 α Since g 0 ) = 1 ) g 1η ) < 0, the last ter above is negative by Lea Hence λ 2 νν < 0 for α sufficiently sall. That is, the evolutionarily singular strategy = α) is not an ESS for all α sufficiently sall. 6. Convergence Stable Strategy This section is devoted to the proof of Theore 5. In fact, we will prove a stronger result. Theore 7. For each δ > 0, there exists α 1 > 0 such that for all α 0, α 1 ), if ˆ is an evolutionarily singular strategy satisfying ˆ [δ, δ 1 ] and d 50) dt λ νt, t; α) = λ ν + λ νν )ˆ, ˆ; α) > 0, t=ˆ then ˆ is a local CSS. Corollary 6.1. For all α sufficiently sall, the unique evolutionarily singular strategy ˆ fro Theore 1 is a local CSS. In particular, the local ESS fro Theore 4 is also a local CSS. Proof. By Theore 1, ˆ = ˆα) 0, the latter being the unique positive root of g 0 ). By Lea 3.1, Corollary 3.7 and Proposition 3.8, as α 0 we have d dt λ νt, t; α) g 0 0 ) > 0. t=ˆ Hence the desired result follows fro Theore 7. For the sake of brevity we abbreviate the invasion exponent as λ, ν); i.e., oitting its dependence on α. Firstly, 51) λ, ) = 0 for all. Secondly, for any evolutionarily singular strategies = ˆ 52) λ ˆ, ˆ) = λ ν ˆ, ˆ) = 0 and hence any evolutionarily singular strategy ˆ, ˆ) is a critical point of λ on the -ν plane. Therefore, the sign of λ, ν) in a neighborhood of ˆ, ˆ) is deterined by the Hessian of λ, which satisfies by 51) again) 53) λ ˆ, ˆ) + 2λ ν ˆ, ˆ) + λ νν ˆ, ˆ) = 0. By the proof of Theore 1, λ ν ˆ, ˆ) = 0 and d dt λ νt, t) t=ˆ 0, hence λ changes sign in every neighborhood of ˆ, ˆ).

22 22 KING-YEUNG LAM AND YUAN LOU 6.1. Forulae for λ and λ. Denote by the subscript derivatives with respect to. Differentiate 4), the equation of ϕ with respect to, we have { α ν ϕ ϕ ) + ũ)ϕ + λϕ = ũ ϕ λ ϕ in Ω, 54) ν ϕ ϕ = 0 Suppose = ˆ is an evolutionarily singular strategy, and set ν = = ˆ, then by 52), 54) becoes { α ϕ ϕ ) + ũ)ϕ = ũ ϕ in Ω, 55) ϕ ϕ = 0 Multiplying 54) by e /ν ϕ and integrate by parts, we have 56) λ e /ν ϕ 2 = e /ν ũ ϕ 2. Differentiate 56) and set ν = = ˆ and divide by α, we have λ 57) e /ˆ ũ 2 = 1 e /ˆ ũ ũ e /ˆ ũ ũϕ α α α where ϕ satisfies 55). when = ν = ˆ, 6.2. Asyptotic liit of λ. Proposition 6.2. Suppose that = α) is an evolutionarily singular strategy and as α 0, then necessarily li sup α 0 1 α λ, ) < 0. We prove the proposition with three leas. Lea 6.3. Set ν =, then as α 0, a subsequence ϕ ϕ in H 1 Ω), where ϕ satisfies { 58) ϕ ϕ ) [ ) ] ϕ = in Ω, ϕ ϕ = In particular, if we define ϕ = ϕ e / ϕ ũ e / ũ ũ 2 then ϕ ϕ in H 1 Ω), where ϕ is the unique solution to { 59) ϕ ϕ ) [ ) ] ϕ = in Ω, ϕ ϕ = on Ω and e / ϕ = 0. Proof. It follows fro the observation that ϕ + ϕ ν = ũ + Cũ for soe constant C, where ϕ ν is the unique solution to 22) and ũ 0 in H 1 Ω) Lea 3.5). 1 Lea 6.4. li sup e /ˆ ũ ũ 2 0. α 0 α Proof. First, we derive the equations for ũ and ũ. { α ũ ũ ) + 2ũ)ũ = α ũ in Ω, 60) ũ ũ = ũ 61) { α ũ ũ ) + 2ũ)ũ = 2α ũ + 2ũ ) 2 in Ω, ũ ũ ũ = 2

23 EVOLUTION OF CONDITIONAL DISPERSAL 23 Set = ˆ in 61), ultiply by 1 e /ˆ α ũ and integrate by parts, we have 1 e /ˆ ũ 2 ũ = 2 ũ e /ˆ ũ) 2 e /ˆ ũũ ) 2 α α 2 ũ e /ˆ ũ) 0. Here the last quantity is of order o1) as ũ H1 Ω) 0 as α 0 by Lea Lea 6.5. li sup e / ϕ ũ ũ < 0. α 0 α Proof. Multiply 55) by e / ϕ, then 1 e / ϕ ũ ũ α [ ] = e / e / ϕ ) 2 ũ)e / ϕ 2 λ 2 α σ 2 e / ϕ ) 2 e / ϕ ) 2 where the second last inequality follows fro Poincaré s inequality and the last ter is negative by the spectru estiate in Proposition 4.2, Reark 4.1 and Lea 6.3. Proposition 6.2 follows fro 57), Lea 6.4 and ϕ 0 Lea 6.5). Proof of Theore 7. For any evolutionarily singular strategy ˆ, we have λ ˆ, ˆ) = λ ν ˆ, ˆ) = 0 = λˆ, ˆ). Therefore the sign of λ depends on the Hessian of λ. We first clai Clai 6.6. Given δ > 0, for all α sall, λ ˆ, ˆ) < 0 for any evolutionarily singular strategy ˆ [δ, δ 1 ]. Suppose to the contrary that for soe α = α j 0 and evolutionarily singular strategies ˆ = ˆ j, 62) li sup α 0 λ α 0. Now let α = α j 0 in 57), then by Leas 6.4 and 6.5, we have ) λ li sup e / 2 < 0. α 0 α And we obtained a contradiction to 62). This proves Clai 6.6. Denote a = λ ˆ, ˆ) < 0, b = λ ν ˆ, ˆ) and c = λ νν ˆ, ˆ). We assue in addition that b + c > 0. Then by 53) we can write By Taylor s theore, b = a b + c) and c = a + 2b + c). λ, ν) = a 2 ˆ)2 + b ˆ)ν ˆ) + c 2 ν ˆ)2 + o ˆ + ν ˆν ) ν) = a 2 a + 2b + c) ˆ)2 [a + b + c)] ˆ)ν ˆ) + ν ˆ) 2 + o ˆ + ν ˆν ) ν) [ 2 a ] = ν) 2 ν) b + c)ν ˆ) + o ˆ + ν ˆν ) < 0

24 24 KING-YEUNG LAM AND YUAN LOU for all ˆ + ν ˆ sall and either > ν ˆ or < ν ˆ, by considering the cases i) ˆ ν ˆ O1) and ii) ˆ ν ˆ separately. This copletes the proof of Theore 7. Proof of Theore 5. Suppose that ˆ = ˆα) is a sequence of local ESS such that ˆ > 0. By Clai 6.6, λ ˆ, ˆ) < 0 for all α sufficiently sall. Suppose ˆ is a local ESS, i.e. λ νν ˆ, ˆ; α) 0, then by 53), λ ν + λ νν )ˆ, ˆ; α) = 1 2 λ νν λ )ˆ, ˆ; α) > 0. And Theore 5 follows fro Theore Discussions We considered a atheatical odel of two copeting species for the evolution of conditional dispersal in a spatially heterogeneous but teporally constant environent. The two species have the sae population dynaics but different dispersal strategies, which are a cobination of rando dispersal and biased oveent upward along the resource gradient. If there is no biased oveent for both species, Hastings [18] showed that the utant can invade when rare if only if it has saller rando dispersal rate than the resident. In contrast, we show that positive rando dispersal rate can evolve when both species adopt soe biased oveent. More precisely, if both species have sae but sall advection rate, a positive rando dispersal rate is locally evolutionarily stable and convergent stable. We further deterine the asyptotic behavior of this evolutionarily stable dispersal rate for sufficiently sall advection rate. A recurring thee in the study of evolution of dispersal is resource atching. For unconditional dispersal in spatially heterogeneous but teporally constant environents, selection is against dispersal since slower dispersal helps individuals better atch resources. However, if the population is able to tract local resouce gradients, thus exhibiting a for of conditional dispersal, then interediate dispersal rate ight be selected. On one hand, sall rando dispersal is selected against as it will cause the individuals to concentrate at the locally ost favorable locations and severely underatch other available resources so that the utant can invade when rare. On the other hand, large rando dispersal is selected against as well since the population is only using average resource and is therefore also underatching resources in high quality patches. Our results suggest that a balanced cobination of rando and directed oveent can help the population better atch resources and could be a better habitat selection strategy for populations. Furtherore, when the spatial variation of the environent is not large, there ay exist a unique rando dispersal rate which is evolutionarily and convergent stable. However, ultiple evolutionarily stable strategies ay appear if the spatial variation of the environent is large. Acknowledgeent. This research was partially supported by the NSF grant DMS and has been supported in part by the Matheatical Biosciences Institute and the National Science Foundation under grant DMS The author would like to acknowledge helpful discussions with Frithjof Lutscher and the hospitality of Center for Partial Differential Equations of East China Noral University, Shanghai, China, where part of this work is done. References [1] F. Belgace and C. Cosner, The effects of dispersal along environental gradients on the dynaics of populations in heterogeneous environent, Canadian Appl. Math. Quart, )

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