Abstract
We study a two-species competition model in a closed advective environment, where individuals are exposed to unidirectional flow (advection) but no individuals are lost through the boundary. The two species have the same growth and advection rates but different random dispersal rates. The linear stability analysis of the semi-trivial steady state suggests that, in contrast to the case without advection, slow dispersal is generally selected against in closed advective environments. We investigate the invasion exponent for various types of resource functions, and our analysis suggests that there might exist some intermediate dispersal rate that will be selected. When the diffusion and advection rates are small and comparable, we determine criteria for the existence and multiplicity of singular strategies and evolutionarily stable strategies. We further show that every singular strategy is convergent stable.
1
Introduction
Hastings [25] showed that in a spatially varying but temporally constant environment, faster dispersal is always selected against if dispersal is completely random; see also Dockery et al. [21] and Altenberg [1] . However, organisms sometimes are forced to move in certain directions, e.g. in advective environments such as rivers, water columns or the gut [4, 26, 30, 40, 43, 44, 45, 46] , or when environmental conditions shift, e.g. movement of temperature isoclines caused by global climate change [6, 37, 41] . Active movement of organisms may also be biased in certain directions, e.g. moving up a resource gradient [5, 8, 18, 23, 42] , or a fitness gradient [10, 12, 17, 19, 27, 28, 29, 35, 39] .
Evolution of biased movement along a resource gradient has received much attention recently [9, 13, 14, 16] . Evolutionarily stable dispersal strategies are found to be those that have comparable dispersal and advection rates [2, 11, 24, 33, 34] : On the one hand, when individuals move upward along the gradient of a resource distribution, and the random dispersal rate is small relative to the advection rate, then individuals can become "too good" at tracking favorable regions. They become overcrowded at only the best quality locations and miss other, reasonably beneficial areas [15, 31, 32, 36] . Higher dispersal rates can help individuals utilize those resources as well. Therefore, slower dispersal is selected against. On the other hand, if random dispersal is large in comparison to resource tracking, the population will spread widely in the habitat so that it only utilizes some average quality of the resource. Therefore, fast dispersal is also selected against. Altogether, intermediate dispersal rates can evolve.
Much less is known about the evolution of dispersal in advective environments, where movement bias is caused by external forces such as river flow, gravity or climate. How should organisms disperse to better persist in advective environments? How should species disperse to avoid the invasion of a mutant with different movement strategies? We have recently attempted to address these questions in [38] in the framework of evolutionary game theory. We considered a two-species competition model in an open advective environment: Individuals are exposed to unidirectional flow, with a net loss of individuals at the downstream end of the domain, e.g. a river. We assumed that two species have the same growth dynamics and advection rates but different random dispersal rates. Among other things, we found that in such context of an open advective environment, unidirectional flow can put slow dispersers at a disadvantage and higher dispersal rates can evolve. In particular, the boundary conditions played an important role in the outcome of evolution. In [38] , we derived several different boundary conditions and explained their biological significance in detail. Two contrasting results from [38] are 1. In a homogeneous (i.e. constant intrinsic growth rate) advective environment with zero Neumann boundary conditions, a population with higher dispersal rate will always displace one with lower dispersal rate. In particular, larger dispersal rates evolve.
2. In a homogeneous advective environment with zero Dirichlet boundary conditions, there seems to evolve a unique, intermediate dispersal rate, which is evolutionarily stable.
In this paper, we consider closed advective environments, where individuals cannot pass through the boundary. We had explicitly excluded this case in [38] . One biological scenario that results in such a closed advective environment is the community of microorganisms in the water column of a lake: Most phytoplankton species have a higher density relative to water, so they tend to sink to the bottom of the water column. Besides that, water turbulence allows phytoplankton to mix randomly within the water column. These two processes of sinking and random mixing can be approximated by advection and diffusion respectively. The upper boundary is the water surface, the lower boundary is the ground. Neither boundary allows individuals to pass through, so that we have a closed environment, which we model by imposing no-flux conditions at both ends. We refer to [22] and references therein.
Following the approach in [38] , we study a system of reaction-diffusion equations for two logistically growing and competing species that differ only in their random diffusion rates:
Here u and v denote the densities of two species (e.g., phytoplankton species) at time t and location x in the bounded interval [0, L]. Function r(x) accounts for the quality of the habitat; a rare single population can grow where r > 0 and will decline where r < 0. Diffusion rates are positive, i.e. d i > 0, and advection points towards larger x, i.e. q > 0. In analogy with rivers, we call x = 0 the upstream end and x = L the downstream end of the habitat. The rest of the paper is organized as follows: In Sect. 2, we consider sufficient conditions for the existence of a single-species steady state and establish various a priori estimates of the steady state. In Sect. 3, we perform a preliminary linear stability analysis of the semi-trivial steady state and find that small dispersal rates should generally be selected against. In Sect. 4, we investigate the selection gradient (see Lemma 4.1 for the definition) for various types of resource functions, and our analysis suggests that for a monotone increasing resource function, there might exist some intermediate dispersal rate that is convergent stable. In Sects. 5-7, we focus on the case when the diffusion and advection rates are small and comparable: Sect. 5 is devoted to further analysis of the selection gradient and to the existence of singular strategies (see Sect. 5 for the definition); in Sect. 6 we give criteria for the existence of evolutionarily stable strategies; in Sect. 7 we show that every singular strategy is convergent stable. Finally, we conclude with some discussions in Sect. 8.
Persistence of a single species
In this section, we consider sufficient conditions for the existence of a single-species steady state (u * (x), 0) of system (1), where u * is a positive solution of the equation
It is known that existence and uniqueness of u * is equivalent to the zero solution of (2) being linearly unstable [7] . Hence, we study conditions for the dominant eigenvalue of
to be positive. In the following, we frequently use the identity
All integrals are definite integrals over the interval [0, L], unless otherwise specified.
x r(x)dx ≥ 0, then the dominant eigenvalue of (3) is positive. In particular, there exists a unique positive solution of (2).
Proof. Using α = q/d 1 and identity (4), we write the eigenvalue problem as
Now, we denote ψ = e −αx ϕ and divide the previous equation by ψ to obtain
Integration by parts yields
where the second equality follows from the no-flux boundary conditions,
Integrating (6) by parts, we arrive at
As r(x) is non-constant, the first term in this equation is always positive (by (7)); as the second term is non-negative, we deduce that λ > 0. 
Proof. Set w = e −αx u * . Then w satisfies
Hence we have w xx (x 0 ) ≤ 0 and w x (x 0 ) = 0. Therefore
) , which implies that
) .
Similarly we can show that
) .
This proves (10).
Lemma 2.4.
Proof. The proof follows directly from Lemma 2.3 and the monotonicity of r(
Proof. Integrating the steady state equation (2) and using the boundary conditions, we find
From this, we obtain
In particular,
Multiplying the steady state equation (2) with g and integrating by parts, we obtain
Dividing by q and letting q → ∞, we see that
3 Local stability of the semi-trivial state (u * , 0)
We perform a preliminary linear stability analysis of the steady state (u * , 0). The linearized system at (u * , 0) decouples, so that it suffices to consider the eigenvalue problem
To determine the stability of the semi-trivial state (u * , 0), we study the sign of the dominant (i.e. the largest) eigenvalue of (14) , denoted by λ. We emphasize that λ = λ(d 1 , d 2 ), as in the following this form is used frequently. Proof. When q = 0, the equation and boundary conditions reduce to the case studied by Hastings [25] . Hence, we know that λ < 0 when d 1 < d 2 and q = 0. By continuous dependence of λ on parameters, we have λ < 0 for small enough values of q.
Lemma 3.1 implies that a faster disperser can not invade when rare by slightly increasing the advection rate of both species. An interesting open problem is to describe the changes of stability of (u * , 0) when d 1 < d 2 are fixed and q varies from small to large. See [3] for related effort.
We are particularly interested in situations where a faster disperser can invade when rare. To this end we first consider the case when q and d 2 are sufficiently large.
Theorem 3.2. Suppose that
Proof. The transformation ϕ = e qx/(2d 2 ) ψ gives
with boundary conditions
Hence, we get the dominant eigenvalue as
Choosing the constant function ξ( (17), we get the lower bound
By Lemma 2.5 and ∫ r > 0 we see that in the limit for large q, λ is positive.
Theorem 3.2 implies that if the advection rate is large, a mutant can invade when rare as long as it disperses sufficiently fast. This is very different from the case of no advection or small advection, where a mutant can not invade when rare if it disperses sufficiently fast.
By Theorem 2 of [16] , as q > 0, we have
The rest of the proof follows from (21) and Lemma 2.4.
Part (i) of Theorem 3.3 implies that if d 1 is suitably small (relative to q) and d 2 is chosen smaller, then the species v can not invade when rare. This means that slow dispersal is always selected against in such advection environments, in contract with the case of no advection where fast dispersal is always selected against. Part (ii) implies that if the resident is a fast disperser (relative to q), a slow dispersing mutant can invade when rare, provided that, e.g., the resource function is monotone increasing. These results suggest that when the resource function is monotone increasing, there might exist some intermediate dispersal rate (comparable to q) will be selected. We will investigate this issue in later sections when both diffusion and advection rates are small.
The selection gradient
Let λ be the dominant eigenvalue of (14) with corresponding eigenfunction ϕ. [20] . If the selection gradient is positive, an invader with d 2 slightly larger than d 1 will invade, and thus evolution will proceed to larger diffusion. If the selection gradient is negative, then the reverse statement holds. The big advantage of calculating the selection gradient is that for d 1 = d 2 , the eigenfunction ϕ in (14) corresponding to the dominant eigenvalue λ = 0 is given by the equilibrium density u * of a single species.
Lemma 4.1. The selection gradient of mutant invader v at the steady-state
The calculations are exactly the same as in Lemma 4.1 of [38] . When r(x) = r = const., we can show that an invader is successful exactly when its diffusion rate is higher than that of the resident.
Lemma 4.2. If r is constant, then the selection gradient is positive.
Proof. Integrating the equation for u * and using the boundary conditions, we find
Therefore, r − u * has to have at least one sign change. We shall show that u * x > 0 in [0, L] and therefore there is exactly one sign change, from positive to negative. First, the following claim follows directly from the defining equation of u * .
In particular, there can be no interior minimum point in {x : u * (x) < r}.
It remains to show that u
Suppose now that u * x = 0 somewhere. Define
Then x 1 < L and u * x (x 1 ) = 0, by our assumption that u * = 0 somewhere. 
. Now we integrate the equation for u * from 0 to x and use the boundary condition again:
The right hand side is zero at x = 0, L. It is decreasing when u * < r and increasing when u * > r. Hence, it is always negative. Therefore, the flux on the left hand side is always negative. This implies that the term (e −αx u * ) x in the selection gradient is negative. Since u * x is positive, this implies (by (22) ) that the selection gradient is positive.
Based on Lemma 4.2, we conjecture that in a homogeneous advective environments with no-flux boundary conditions, populations with higher dispersal rate will always competitively exclude populations with lower dispersal rate.
The main result of this section, which includes the constant resource as a special case, can be stated as follows.
Theorem 4.5. Suppose that r > 0 and r
Theorem 4.5 suggests that when the resource function is monotone increasing, there might exist some intermediate dispersal rate which is convergent stable (but may not be evolutionarily stable, as we will see in Sect. 6). It is interesting to inquire the corresponding scenario for the case of monotone decreasing resource functions.
Before we prove Theorem 4.5, we prove an auxiliary one-dimensional lemma.
where
Corollary 4.7. Let u * be the unique positive solution to For (ii), we observe that
In addition, the assumption
In addition, the assumption 
, which implies by (25) 
Combining with the strict monotonicity of ϕ in [x 1 , x 2 ], we have
which contradicts the monotonicity of c. This shows that
Next, under the assumption that c x > 0 in (0, L), we wish to show that ϕ x > 0 in (0, L). Suppose to the contrary that ϕ x (x 0 ) = 0 for some x 0 ∈ (0, L). Then by the previous claim, x 0 is a local interior minimum of ϕ x in (0, L). Therefore, ϕ xx (x 0 ) = 0 and by (25) , we have c(x 0 ) = ϕ(x 0 ). Together with
Hence ϕ x < 0 in (x 0 , x 0 + δ). This contradicts the fact that ϕ x ≥ 0 in (0, L), and completes the proof.
Proof of Theorem 4.5. The theorem follows directly from Lemma 4.1 and Corollary 4.7.
Singular strategies and the small advection case
When diffusion and advection rates are small, we can use the techniques developed in [33, 34] to obtain a detailed description of the selection gradient and singular strategies. Recall that d is a singular strategy if the selection gradient vanishes atd, i.e.,
We write the model as
From this point on, we adjust notation slightly and denote u * as the positive steady state of the u equation in (26) when v = 0. Here we need some estimates of u * . 
Lemma 5.1 (Convergence of
, which satisfies
Multiplying by −e −qx/d 1 u ′ /ϵ, and integrating by parts, we have
Since
r/2 and the second integral is non-negative for ϵ small. We obtain, by Hölder's inequality that
Since the right hand side is uniformly bounded (by Lemma 5.
. This proves that u ′ ⇀ 0 (weakly) in H 1 . This implies in particular that the right hand side of (28) goes to zero, and we deduce that
Denote by λ the dominant eigenvalue of the linearization of the v equation at (u * , 0). Denote ∂/∂d 2 by ′ and set α = q/d 1 .
Lemma 5.3. The selection gradient satisfies
Proof. The stability of (u * , 0) of (26) is determined by the dominant eigenvalue
We normalize the principal eigenfunction by
We can differentiate (30) with respect to d 2 , and obtain (denoting the derivative with respect to
Set
Multiplying (32) by e −αx u * and integrating by parts, we have
which proves the lemma.
Since the above convergence is again uniform on compact subsets of (0, ∞) in d 1 , the lemma is proved.
Define 
Moreover,d is the unique singular strategy near d 0 .
Example: Consider the linear growth rate r(x) = r 0 +r 1 x. We require r(x) > 0, in particular, r 0 > 0. By rescaling, we may assume L = 1. If r 1 < 0 then there is no positive singular strategy in the limit. If r 1 > 0, then there is a unique positive singular strategy, and it is given by
To see this, replace r(x) = r 0 + r 1 x in the equation
and carry out the calculations. We can find that the integral equality is equivalent to (r 0 + r 1 )e −α = r 0 . This equation is solvable for positive α = q/d if and only if r 1 > 0.
The following two lemmas give sufficient condition for the existence of a unique singular strategy in the limit (i.e. for F (α) to have a unique root).
Lemma 5.7. F (α) has a unique root in either of the two following cases:
(ii) r x (0) > 0 and r xx ≥ 0;
Proof. Without loss we assume L = 1. Write
where all integrals are over [0, 1] . It is easy to see that
Hence, there is at least one root. The derivative of F is
Integrate the last term by parts and get
where the final inequality follows from the assumption (i). Since F ′ is negative, the root is unique.
To prove the claim under the second assumption, we integrate by parts the second integral in the expression (36) . Assuming that α is a root of F , we find
We use this expression to replace the boundary term in (39) and find for any root α
Under the second assumption, this final expression is negative since x < 1. Hence, F ′ (α) < 0 for all roots α, therefore the root is unique. One can immediately conclude the following result regarding the original problem. 
Proof of Lemma 5.8. Let r(x) = e g(x)
. Then
The assumption of the lemma says that g x > 0 and g xx does not change sign. We consider the two cases:
, then the lemma is trivial. We henceforth assume that g x is non-constant and monotone in [0, L]. It is clear then that F (α) changes sign at least once, and any root lies within the interval (min g x , max g x ).
Case (i): g xx ≤ 0. Suppose α 0 is a root, then
where the last strict inequality follows from the fact that h(x) := e −α 0 x+2g g x (α 0 − g x ) changes sign from negative to positive as x varies from 0 to L. More precisely, choose
This proves Case (i). Case (ii): g xx ≥ 0. Then r xx = e g (g xx +g 2 x ) ≥ 0 and the result follows from Lemma 5.7(ii).
Example:
We consider the case of a piecewise linear growth function
where parameters A, B, C are chosen such that r > 0 on the interval Ω = [0, 1]. (We again assume L = 1 without loss of generality.) For 0 < C < 1, this function is not differentiable in Ω. However, one can "smooth" the corner at x = C by altering the function r on an interval of arbitrarily small length around C and with a bounded derivative between -1 and 1. Hence, the integral in the selection gradient can be approximated arbitrarily closely by the piecewise integral
which is well defined. Explicit calculations lead to the expression
By setting y = e −α , we obtain the following equation for y ∈ (0, 1] :
where
The left hand side in (45) is a straight line, the right hand side a root-like function (monotone increasing, concave down). We observe the following.
1. There is exactly one root of F (α) = 0 (no double root) if B < 0. To see this, simply consider the condition that the straight line is below the root function at y = 1, or
Hence, when the resource function has a single hump, then there is precisely one singular strategy.
2. Let B > 0. If F (α) = 0 has a double root, then necessarily C < 1/2. To see this, we notice that the straight line is tangent to the root function at y * = (ii) If C = C * , then F (α) has a double root.
(iii) If C ∈ (C * , 1), then F (α) has no roots.
For existence, we first consider C = 1, then for all B > 0, F (α) = 0 has no root. Now if we decrease C, then the linear function on the LHS is decreasing, while the concave function on the RHS is increasing. Moreover,
and lim C→0 + K 3 y C = K 3 = 2A locally uniformly for y ∈ (0, 1], hence for all B > 0, there are exactly two roots for all C sufficiently small. Finally, C * (B) belongs to (0, 1/2) follows from the previous claim.
4. There exists δ 0 > 0 such that F (α) = 0 has exactly two roots when B, C ∈ (0, δ 0 ).
Criteria for evolutionarily stable strategies
In this section we aim to find criteria for the existence of evolutionarily stable strategies, when the diffusion and advection rates are small. Recall that ′ = ∂ ∂d 2 .
Lemma 6.1. At a singular strategy d, we have λ = λ(d, d) = 0, and
where
where ϕ ′ satisfies (32). Multiplying the above by e −qx/d u * and integrating by parts, we obtain
This proves the lemma.
Proposition 6.2. Suppose for
where ψ satisfies
Proof. We refer to Lemma 4.5 of [33] .
Definition 6.3. Suppose that d 0 is a singular strategy in the limit, i.e.
Our first main result of this section provides an explicit criterion for the existence of a local ESS in the limit.
If r x /r is decreasing and non-constant, then the unique singular strategy d 0 in the limit, given by Lemma 5.8, is also a local ESS in the limit.
As an interesting complement of Theorem 6.4, our next result gives a sufficient condition on the nonexistence of an ESS in the limit. 
(ii) If r x /x is increasing and non-constant, then for all ϵ sufficiently small, the unique singular strategyd is not an ESS. In particular, there are no ESSs for all ϵ sufficiently small.
Remark 6.7. (i) By a different method, it is proved in [2] that when r x /r ≡ c 0 for some positive constant c 0 , then the unique singular strategy is given byd = q/c 0 , which is independent of ϵ > 0. Moreover,d = q/c 0 is an ESS. Our results here explores the stability of this result.
(ii) It is unclear biologically why the shape of the resource function plays such a critical role. It seems interesting to inquire whether the assumption (50) is purely technical or not.
We first prepare by proving a few lemmas.
) then the following conditions are equivalent:
where ψ is given by (49), and y is a solution to
Proof. Let ψ = rz, then by some elementary but tedious computations involving (49), we have
Now, let y = z + ln r d , then (adding an integral constraint for uniqueness)
Or equivalently,
It remains to show the following claim. Recall that ψ, y satisfies (49) and (52) respectively.
First, we show Claim 6.9:
Second, we notice, by multiplying (55) by y and integrating by parts,
Lemma 6.8 follows from Claim 6.9 and (56).
Proof of Theorem 6.5. By Lemma 6.8(ii), it remains to show that
which implies that lim ϵ→0
λ > 0. First, by the fact that ∫ r x (e −qx/d ) x = 0 (singular strategy in limit) and that r x /r increasing, the expression
changes sign exactly once, from negative to positive (since r x /r is assumed to be increasing, non-constant) and ∫ r x (e −qx/d r) x = 0 (singular strategy in the limit). Hence, there exists x 0 ∈ (0, L) such that r x (e −qx/d r) x is non-constant, and
Moreover,
Finally, (58) and (59) implies
which is equivalent to (57), by
Proof of Theorem 6.4. First we show the following calculus lemma.
Lemma 6.10. Suppose that g(t) : [0, L] → (0, ∞) is a decreasing function and it satisfies
Proof. By the monotonicity of LHS with respect to g(x) and of RHS with respect to g(y), it is equivalent to show
where
In particular, for all η ∈ [min g,
This proves (61) and completes the proof of the lemma.
To prove Theorem 6.4, it remains to show that
By (i) r, r x > 0, (ii) r x /r non-constant and decreasing, and (iii) ∫ r x (e −qx/d r) x = 0, we deduce r x (e −qx/d r) x changes sign exactly once, from positive to negative.
Hence, (62) holds provided we can show that
It is enough to show
By (55) and (63),
But then by Lemma 6.10 (here we use (50)), taking g = r x /r, we deduce that for all
This proves (65) and finishes the proof of Theorem 6.4.
Every Singular Strategy is Convergent Stable
In this section we show that every singular strategy is convergent stable, when both diffusion and advection rates are small. The main result can be stated as follows: 
In particular, all assumptions of Theorem 7.1 can be verified under the assumption of Lemmas 5.7 and 5.8. 
Multiplying (66) by e −qx/d 2 ϕ and integrating by parts, we have
Differentiate (67) 
where 
Multiply by −1 ϵ e −qx/d u * and integrate by parts, we have 1 ϵ
The last quantity is of order o(1) as u ′ → 0 in H 1 (Lemma 5.2) and u * is bounded in H 1 (Lemma 5.1).
Lemma 7.5. For each singular strategyd, ifd
Proof. Denote
where the second last inequality follows from Poincaré's inequality and the definition of ϕ ′⊥ in (71). λ 2 is the first strictly negative eigenvalue of (which is (14) with ϵ,
Since one can show (see, e.g. [33, Proposition 4.2]) that λ 2 /ϵ → σ 2 , which denotes the first negative eigenvalue of (assumingd 
Discussion
The question of why individuals disperse has a long and distinguished history in ecology and evolutionary theory. While it is fairly straightforward to understand why dispersal is beneficial in a temporally varying environment, early results in temporally constant but spatially varying environments all point toward the benefit of slower dispersal. Simply put, random dispersal tends to move individuals away from higher-quality regions to lower-quality habitat and therefore slower dispersal should be beneficial. Ever since the work by Hastings [25] , the challenge is to find ecological mechanisms -and mathematical proof -for intermediate or higher dispersal rates to evolve. One such mechanism that has recently received much attention is that random dispersal can -and sometimes needs to -balance individual movement behavior along a resource or fitness gradient. In the absence of strong enough diffusion, intrinsic movement bias towards better environmental conditions may trap individuals and lead to overcrowding. With too strong diffusion, on the other hand, individuals underutilize regions of highest resource quality. Combining these two mechanisms, intermediate dispersal can emerge as an optimal strategy (e.g. [2, 11, 24, 33, 34] ).
Here and in [38] we consider an extrinsic movement bias (advection) as a mechanism that may make higher diffusion rates beneficial. In an open environment, where advection may push individuals out of the domain, some minimal dispersal rate is required for a population to persist in the first place [40] , so that the evolution of slower dispersal is impossible. In fact, when only advection cause boundary loss, then higher diffusion is always beneficial, whereas when diffusion may also cause boundary loss, an intermediate diffusion rate seems most beneficial [38] . In a closed environment that we studied here, advection pushes individuals in one direction but not out of the domain. Therefore, the population concentrates at the downstream boundary and, similarly to the case of strong movement bias to better resources, higher diffusion can be beneficial. The detailed results depend subtly on the actual resource distribution, i.e. the shape of the function r(x).
The example of a piecewise linear resource function could lead to the guess that if the resource decreases in the direction of advection, then higher dispersal rates should evolve as a mechanism for individuals to remain in areas where resources are high. Whereas when resources increase in the direction of advection, then an intermediate optimal dispersal rate might exist. Some diffusion is necessary to get away from the boundary but too much diffusion spreads the population to upstream areas of low resource. However, the more detailed analysis reveals a condition involving the function r x /r for the existence or non-existence of an ESS. Future work will investigate the cases where condition (50) is not satisfied and consider examples when r x /r is not monotone.
The theory developed in this paper can also be extended to other models such as 
We refer to [2, 11] for a detailed derivation of such models and related mathematical results.
It will also be of interest to consider more general ecological models such as             
Note that if r(x) ≡ K(x), system (75) is reduced to system (1). We suspect that the theory of this paper can be extended to the model (75).

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