INVADING THE IDEAL FREE DISTRIBUTION

. Recently, the ideal free dispersal strategy has been proven to be evolutionarily stable in the spatially discrete as well as continuous setting. That is, at equilibrium a species adopting the strategy is immune against invasion by any species carrying a diﬀerent dispersal strategy, other conditions being held equal. In this paper, we consider a two-species competition model where one of the species adopts an ideal free dispersal strategy, but is penalized by a weak Allee eﬀect. We will show rigorously in this case that the ideal free disperser is invasible by a range of non-ideal free strategies, illustrating the trade-oﬀ between the advantage of being an ideal free disperser and the setback caused by the weak Allee eﬀect. Moreover, an integral criterion is given to determine the stability/instability of one of the semi-trivial steady states, which is always linearly neutrally stable due to the degeneracy caused by the weak Allee eﬀect.

1. Introduction. Habitat selection plays a pivotal role in a species' life history, i.e. its endeavor to survive and reproduce. While accounting for all factors in this complicated process is a formidable task, ecologists attempt to identify elements which capture the essential mechanisms of habitat choice. One such perspective is known as ideal free distribution (IFD) theory, proposed by Fretwell and Lucas in their study of birds [9]. This framework operates from two assumptions: (i) each individual has complete knowledge of its environment to determine the most favorable locations for growth and (ii) each individual is able to freely move to these "best" spots. Provided the assumptions hold, the theory predicts that the resulting distribution of the species will be proportional to the amount of available resources at each location in the habitat [9].
Although the presuppositions of IFD theory are regarded by some in the empirical research community as being overly simplistic [14], there are a number of experimental studies confirming its predictions (see for example [8,11,21]). Moreover, this simple principle provides a striking, quantitative link between habitat choice and evolution of dispersal that allows for rigorous theoretical investigation.
For instance, a type of movement known as "balanced dispersal" arises from the evolution of dispersal in patchy environments. Balanced dispersal, as described in [2], "implies that at equilibrium individuals in any patch will have the same fitness, namely zero, because all populations are at carrying capacity". This means it is capable of producing an ideal free distribution. Since individuals at all locations have the same fitness, there is no advantage for anyone to move to another location to increase fitness. Furthermore, if the resident species adopts the balanced dispersal strategy, then it cannot be invaded by another rare species playing a different strategy. In the game-theoretic setting, the balanced dispersal strategy, regarded as a strategy played by the resident population, is an evolutionarily stable strategy (ESS).
The spatially continuous case is taken up by Cantrell et. al. [4] where a reactiondiffusion-advection model of two competing species was proposed. (See system (1) below.) The species disperse via a combination of diffusion and advection, the latter component capturing the conditional, or biased, aspect of their movement. Among all of the advection modes (admissible under their formulation), only one allows for the possibility for the resident to equilibrate to a form of ideal free distribution. Sufficient conditions were obtained for the ideal free dispersal strategy to be defended against any invader, showing that IFD is a local ESS. Later it was further shown in [1] that this IFD strategy is a global ESS. (See Theorem 2.1.) We note that a similar result was recently proved in [15] from a slightly different modeling perspective.
All of the above lines of research show that in the context of an environment that is spatially variable but temporally constant, the species adopting IFD is able to resist invasion from any other dispersal strategy (with other properties being held equal). But how robust is this conclusion? For instance, can the ideal free disperser, vulnerable to some fitness cost or constraint, still resist invasion by a rare species? The subtlety of this question lies in the fact that at equilibrium, an ideal free dispersing resident leaves absolutely no resource for an invader to draw upon for growth.
We narrow our inquiry into this multi-faceted question by assuming that the ideal free disperser is subjected to a weak Allee effect, which causes a reduction in its fitness at low density. This question was first posed in [23], where numerical evidence presented therein suggested that the ideal free disperser can indeed be invaded by a certain range of non-IFD strategies. Our present paper presents analytical justification for this phenomenon, and offers conditions and intuition as to how the ideal free disperser may give way to an invading species, and in some cases even be driven to extinction.

Model and main results.
Let Ω be a bounded domain in R N for N ≥ 2, with smooth boundary ∂Ω. Consider the following reaction-diffusion-advection model of Lotka-Volterra type: in Ω × (0, ∞), where u(x, t) and v(x, t) represent species' densities at location x ∈ Ω and time t; d 1 and d 2 are their respective diffusion rates;P (x),Q(x) ∈ C 2 (Ω) specify the advective direction and the corresponding speeds; m(x) describes the quality of the habitat at location x. Throughout this paper, we assume that (M1): m ∈ C 2 (Ω) is positive and non-constant. Finally, ∂ n is the outward normal derivative on ∂Ω and reflecting boundary conditions are imposed, i.e. there is no net movement across any point on ∂Ω.
Note that the species u and v in (1) have identical population dynamics but different movement strategies. Setting P (x) =P (x) d1 and Q(x) =Q (x) d2 , we reformulate our system to match the model in [4]: As pointed out in [4], if we set P = ln m, then (m, 0) is a steady state of (2). That is, in the absense of competitor, the species adopting dispersal strategy P = ln m exactly matches the local carrying capacity at equilibriumũ ≡ m. (Note also that the net flux is zero as well: ∇ũ −ũ∇ ln m ≡ 0.) Hence P = ln m is a form of IFD strategy. The significance here is that the strategy yielding IFD is a globally evolutionarily stable strategy (Theorem 1.2 in [1]). Mathematically, the result can be stated as follows: is non-constant and positive. Given any d 1 , d 2 > 0, if P = ln m, and Q − ln m is not constant, then (m, 0) is the globally asymptotically stable steady state of (2) among any nonnegative and not identically zero initial data.
Theorem 2.1 holds on the premise that both species have identical population dynamics. To explore the robustness of IFD, we subject the ideal free disperser to a weak Allee effect by replacing the reaction term for species u in (2) by u 2 (m − u − v) [23]. Our question then becomes, is the ideal free disperser subject to a weak Allee effect invadable by a species playing a different strategy? Suppose that P ≡ ln m and Q = β ln m (0 ≤ β < ∞), we obtain in Ω. ( Mathematically, our goal is to find β ∈ [0, ∞) such that (m, 0) is unstable in system (3). Before stating our main results, we make a few preliminary remarks. System (3) has two semitrivial steady states (m, 0) and (0, v * ), where v * is the unique positive solution of In this paper, we are interested in the local and global stabilities of the steady states of (3). The mathematical subtlety lies in the fact that, due to the degeneracy caused by the IFD and the weak Allee effect, both of the semi-trivial steady states are linearly neutrally stable. i.e. the principal eigenvalues of the linearized system at (m, 0) or (0, v * ) are identically zero. Therefore, we need to argue for nonlinear stability/instability directly. Our staring point is the following integral characterization of the local stability of (0, v * ), proved in Section 3.
Theorem 2.2. Let v * be the unique positive solution of (4). ( Using the criterion in Theorem 2.2, we can determine the local behavior of system (3) near (0, v * ) for β close to 1. Confirming the conjecture in [23], we demonstrate that (0, v * ) does indeed change stability as β crosses the threshold value of 1.
is locally asymptotically stable. If we assume in addition that (M2): m ∈ C 3 (Ω) has a unique critical point x 0 ∈ Ω, which is a non-degenerate local (hence global) maximum, and ∂ n m| ∂Ω ≤ 0. then (0, v * ) is unstable for all β sufficiently large.
Biologically, Proposition 2.3 (ii) indicates that there is a range of strategies which prevent invasion by an ideal free disperser. This is interesting because we can mathematically justify that the cost of the weak Allee effect is enough to offset the invasion success enjoyed by a resource matching strategy.
While the mathematics are compelling in their own right, the biological implications of the instability of (m, 0) are quite strong: a rare species v can invade an ideal free disperser with significant resident population and drive it to extinction. This result seems counter intuitive as species u should have large enough density to minimize the cost of the weak Allee effect (which expresses itself mostly when the species is rare) and therefore dominate resource acquisition throughout the habitat. However, because v has slightly stronger advection, numerical simulations suggest that v is able to quickly establish itself at resource maxima. Focusing on the most abundant resource sites, species v eventually overtakes u, forcing it towards less favorable locations and eventually to extinction.
The paper is organized as follows: In Section 3 we determine conditions for the local stability of (0, v * ). In Section 4, we discuss and prove the global dynamics of system (3), making use of the non-existence results of positive steady states contained in the appendix sections. Finally, we present our conclusions and discuss the intuition behind Theorem 2.2 in Section 5.
3. Local stability of (0, v * ). In this section we discuss the local asymptotic stability of (0, v * ). First, we recall several definitions from dynamical systems.
Next, we state and prove the integral characterization of local stability of (0, v * ).

Theorem 3.2.
Let v * be the unique positive solution of (4).
Remark 1. In [18], a PDE model in population genetics was considered where a degeneracy analogous to Allee effect is present. Among other things, the change in stability of the trivial solution was demonstrated via variational and degreetheoretical methods and a similar integral condition was obtained (See Theorem 1.1 therein). In our local stability analysis of (0, v * ), a non-variational proof using the upper and lower solution method is presented. See also [25] for a Crandall-Rabinowitz type bifurcation analysis of a single species model with weak Allee effect which is "non-degenerate".
Proof of Theorem 3.2. Define

KING-YEUNG LAM AND DANIEL MUNTHER
First we prove (i). Suppose C β = Ω m 2 (m − v * ) < 0. We will construct, for all > 0 sufficiently small, a pair of super/subsolution of the form and z → z ∞ uniformly as → 0, where z ∞ is the unique solution to Now we begin the construction. First, let w be the unique solution to Note that the existence of w follows from the fact that Secondly, for z , we have the following lemma.
Lemma 3.3. There exists 0 > 0, and a small neighborhood U of z ∞ , such that for all ∈ (− 0 , 0 ), the problem (10) has a unique solution z in U. Moreover, z → z ∞ uniformly as → 0. Here z ∞ is the unique solution to (8).
Then F(z ∞ , 0) = 0 and First, from the equation of v * , we observe that zero is the principal eigenvalue of Hence, the principal eigenvalue σ 1 of Therefore σ 1 > 0 and zero is not an eigenvalue of D z F and the claim is proved. Lemma 3.3 follows from an application of the Implicit Function Theorem.
Now, defineū and v as in (6). Then the second equation of (7) is equivalent to (10), which holds by definition of z . It remains to show the first inequality in (7): where w and z satisfy (9) and (10) respectively. Now, one can verify that, similar as in Case (i), with z → z ∞ as → 0 (z ∞ being the unqiue solution to (8)). This proves (ii).

Remark 2.
By exploiting the super/subsolution method, as in Theorem 3.2, one can in fact show that when 3.1. Local stability of (0, v * ) when β ≈ 1.
Proposition 3.5. There exists 0 > 0 such that Note that (i) is Lemma 3.4 and Lemma 4.1 from [23]. We need some preparation for the proof of (ii).
Proof. Let w 1 = ϕ − ln m, then ϕ satisfies To see the claim, multiply (15) by me −ϕ and integrate the resulting equation in Ω, This proves Claim 3.7. Finally, where ϕ is non-constant as m is non-constant.
This also gives an alternative proof of (i) for β less than and close to 1.
Proof. First, we state the following result which follows from Theorem B.1 in Appendix B.

Remark 4.
The assumption (M2) can be relaxed here. In fact, by the proof of Theorem 3.5 of [3] (with modification as in Lemma 2.2(ii) of [16]), the conclusion of Lemma 3.9 and hence Proposition 3.8 hold provided the set of critical points of m is of measure zero.
Because of the degeneracy of the linearization of (3) at (m, 0), we will determine the local stability of (m, 0) indirectly through the global dynamics, which is the subject of the following section.
4. Global dynamics. We now discuss the global dynamics of system (3) for β ∈ [0, ∞). We begin with a result from [23]. Since (3) gives rise to a strongly monotone dynamical system, the proof of Theorem 4.1 in [23] follows a standard procedure: establish the stability of one of the two semi-trivial steady states and then show that the system does not admit any strictly positive steady states. We already know that (0, v * ) is unstable for all β ∈ (0, 1) (Lemma 4.1 in [23]). In order to ensure the non-existence of positive steady states, we prescribe the following restriction on m and Ω. Proof. See Section A.
With non-existence of positive steady states for β sufficiently close to 1, we can infer the global dynamics of the system from the local stability results of (0, v * ) established in the previous section. Proof. By Lemma 4.1 in [23] we have that (0, v * ) is (linearly) unstable for all β ∈ (0, 1). Proposition 4.2 says that for all β ∈ (1 − 1 , 1), (3) has no positive steady states. Hence by monotone dynamical system theory [12,26], the result is established.
This result provides analytic justification to the prediction made in [23] and demonstrates that the ideal free disperser not only may be driven to extinction, but it may not be able to even invade.
However, as β grows larger, the stability of (0, v * ) changes to be unstable again (see Theorem 2.3 in [23]), indicating that the ideal free disperser is not significantly affected by weak Allee effect. In order to capture the global picture, we want to demonstrate that (3) has no positive steady states for large β. This immediately leads to the global stability result: Theorem 4.6. Suppose (M1) holds. Then for some β > 1, (m, 0) is globally asymptotically stable for β > β .

5.
Discussion. This study builds on the work in [1], where it was proved that the IFD strategy adopted by species u is evolutionarily stable. In other words, everything else being held equal, an established population of u is immune against invasion by any rare competitor species adopting a different dispersal strategy. In this paper, by establishing the global asymptotic stability of (0, v * ) for β close to but greater that 1, we have shown that the ideal free disperser u is sometimes invadable by a rare competitor species v with a dispersal strategy that is not ideal free, provided that the fitness of u has a weak Allee effect. This illustrates the trade-off, in the dynamics of competing species in a spatially variable environment, between the advantage of an IFD strategy on the one hand, and the setback of a weak Allee effect on the other.
Not only is this biologically interesting, but the mathematics on which this notion rests are curious. First, the instability of the semi-trivial steady state (m, 0) is established indirectly, through the local asymptotic stability and the non-existence of positive steady states. This indirect method is adopted due to the highly degenerate nature of (m, 0) as a steady state of (3).
Second, we discuss the local stability of (0, v * ), or equivalently, the invasibility of an established species v by species u. As shown above (see Section 3), the success of u depends on the sign of the following integral condition: To give an intuitive connection between (16) and the growth/decay of a rare ideal free disperser u, we first linearize system (3) at (0, v * ), setting u(x, t) = ψ(x)e −λt + O( 2 ) and v(x, t) = v * + φ(x)e −λt + O( 2 ). Substituting these expressions into (3) and letting → 0, we can show that λ = 0 is the principal eigenvalue and ψ satisfies the following: By the maximum principle and after suitable rescaling, ψ m ≡ 1 in Ω, and therefore, ψ = m in Ω. This means that apart from the principal eigenfunction m, all other modes of invasion for species u decay exponentially to zero. Therefore, we can write u(x, t) = m + O( 2 , e −λ2t ), where λ 2 > 0 is the second eigenvalue of system (3) linearized at (0, v * ).
If we integrate the equation for u in (3) over Ω we have: Thus, we see that the growth or decay (on average) of u depends on the sign of C β . What is remarkable is that C β changes from positive to negative as β increases to surpass the critical value 1. Locally, this means that (0, v * ) changes from being unstable (for β < 1) to asymptotically stable for β > 1. Using steady state profiles, we further illustrate this connection: Note the resource m(x) = 3 sin(πx)+0.5 is black and the dotted blue curve represents v * in each subfigure.
• For β ∈ (β 2 , 1), v * under-matches m near the global maximum x 0 and overmatches away from this maximum (see Figure 1(i)) In this case, (16) is positive, indicating that u can invade. Here u, when rare, will have a positive growth rate for all time near the maximum point x 0 , which is enough for it to overcome the weak Allee effect. • For β ∈ (1, β 3 ), we see that v * over matches near x 0 and under matches elsewhere (see Figure 1(ii)). While u can have positive growth rates away from x 0 , the integral condition specifies that this is not enough to overcome the Allee effect and compete with resident species v * . • For larger β, while v * still dominates resource acquisition near x 0 , its concentration near x 0 implies that C β is negative again (see Figure 1(iii)) and, by making use of resources away from x 0 , u is able to invade. To discuss the global dynamics, we broaden our scope and consider the following slightly more general system: in Ω.
Note that (18) contains (3) as a special case. Being a monotone dynamical system (under appropriate changes in variable), the global dynamics of (18) are determined by the structure of positive steady states: If, for some β = 1, (18) does not have any positive steady states, then one of the semitrivial steady states (m, 0) or (0, v * ) must be globally asymptotically stable. For this reason, let us discuss, for each p > 1, the global bifurcation diagram of positive steady states of (18), by treating β ≥ 0 as the bifurcation parameter.
β* Figure 2. Bifurcation diagram illustrating the ideal free strategy tradeoff with the weak Allee effect, i.e. for system (18) with p > 1.
By compactness, the bifurcation diagram changes continuously as p 1. In this case the branch Γ either goes to β = ∞, or remains bounded and merges with Γ 0 . The first alternative is impossible, since β 4 depends only only on the smallness of v * and is independent of p ∈ [1, 2). Hence, as p 1, dist(Γ, Γ 0 ) → 0. For this reason, we end with the following conjecture. Conjecture 1. There exists δ 0 > 0 such that for all p ∈ (1, 1 + δ 0 ), the set of positive steady states of (18) consists of two disjoint branches: Γ 0 as defined above and another branch Γ. Moreover, dist(Γ, Γ 0 ) → 0 as p 1.
It is easy to see the claim when t = 0, 1. For t ∈ (0, 1), L t is an irreducible, cooperative operator with zero as an eigenvalue corresponding a positive eigenfunction (1, 1). It follows that zero must be the principal eigenvalue, and is necessarily simple. This proves the claim for t ∈ [0, 1], and the rest follows by continuity.
whereψ is given by the unique solution to Proof of Lemma A.8. The existence of (φ,ψ) and the limit lim →0ψ = 1/d 1 can be proven in a similar manner as Lemma A.1 and is omitted. For the remaining part, we assume (M3) and write the equation ofψ in radial coordinate: where the last inequality follows from the fact thatψ → 1/d 1 as → 0. Hence ∇ψ ∞ = ψ r ∞ = O(δ) andψ r < 0 for sufficiently small. The rest of the proof is similar to the proof of Lemma A.2 and is omitted. Now, similar to the procedure in Case (i), let (φ,ψ) be given by Lemma A.8. Multiply the first equation of (32) byφ and the second byψ. Upon integrating by parts and adding the equations, we have where in the last line, we have used the fact that ∇ψ ∞ ≤ O(δ) (Lemma A.8). By Lemma A.8, the last term is non-zero for sufficiently small , δ. This is a contradiction.
Case ( where v * = v * ( ) is the positive steady state of (4) when β = 1 + . By Lemma A.5, s = o(1). Moreover, First, observe that u u ∞ → m m ∞ in C 1 (by dividing the equation of u by u ∞ , and using compactness). We deduce from u = u ∞ m m ∞ + o(1) that Multiply (37) by m and integrate, we have and hence u ∞ = m ∞ s + o(s). Therefore by (37), we deduce And the desired conclusion follows upon canceling sm from both sides.
First, observe that u 4 satisfies where the estimate follows from (36). Hence by Poincaré's inequality, Simplifying, By the uniform invertibility of the corresponding linear problem (see Claim 3.4), we have v 4 C 1 ≤ o( ). This proves Claim A.10.
Next, by integrating the equation of u over Ω and dividing by s 2 , we have by Claims A.9 and A.10, and the fact that Recalling that ∂ ∂β m 2 (m − v * )| β=1 < 0 (see Remark 3), the last line is nonzero for all s, sufficiently small. This contradiction completes the proof of Proposition 4.2.
Appendix B. Non-existence of positive steady states when β 1.
Proof of Theorem B.1. This is a special case of Theorem 1.6 of [16]. For completeness' sake, we shall give a short proof here, which follows from the methods developed in [5]. First, we fix, by the non-degeneracy of x 0 , two positive constants κ, K > 0 such that Then there exists β 1 = β 1 (d) ≥ 2/ such that for all β ≥ β 1 and d 2 ≥ d.
It remains to show that M = v * L ∞ (Ω) is bounded independent of β large. Now by (44), On the other hand, by fixing R 1 > R 0 suitably large, we have (independent of β large) in Ω \B R1/ √ β (x 0 ). Therefore, the maximum of v * is attained in B R1/ Next, we change coordinates x = x 0 + y √ β , then V (y) = v * (x 0 + y √ β ) is a positive solution to (46) and x m(x 0 )y uniformly for y ∈ B 2R1 (0), all the coefficients of (47) are bounded. We may apply the Harnack inequality to V in B 2R1 (0), which implies that for some c depending on dimension and m(x), but independent of β large, v * ≥ c v * ∞ for x ∈ B R1/ √ β (x 0 ) (or y ∈ B R1 (0)). Hence Combining (45)  Hence M = v * L ∞ (Ω) is bounded independent of β large. This proves the theorem.
Next, we show the nonexistence of positive steady states when β is sufficiently large.
Proof of Proposition 4.5. Suppose to the contrary that for some fixed d 1 , d 2 > 0, there exists a sequence β k → ∞ and positive solution (u k , v k ) of (3) corresponding to β = β k .
By a comparison argument, u k < m inΩ for each k.
To see the claim, notice that 0 < v k < v * (by comparison) and v * → 0 in L p (Ω) for all p ∈ [1, ∞) (by Theorem B.1). Hence the claim follows from elliptic L p estimates applied to the equation of u.
where N is defined in (42). Hence by comparing v k andv in Ω\B R0/ √ β k (x 0 ) (details similar to proof of Theorem B.1) And Lemma B.4 follows from Bounded Convergence Theorem.
Rewriting the equation of u k , we have Divide by m − u k L ∞ (Ω) , letting w = (m − u k )/(m m − u k L ∞ (Ω) ), where mu 2 k → m 3 in L ∞ (Ω) and the right hand side goes to zero in L p (Ω). Hence by L p estimate, w → 0 weakly in W 2,p (Ω) and strongly in C 1,α (Ω). i.e. m − u k m − u k L ∞ (Ω) → 0 uniformly in Ω as β k → ∞.
This is a contradiction. Therefore, (3) has no positive solutions for β large.