DIRAC-CONCENTRATIONS IN AN INTEGRO-PDE MODEL FROM EVOLUTIONARY GAME THEORY

. Nonlocal Lotka-Volterra models have the property that solutions concentrate as Dirac masses in the limit of small diﬀusion. Motivated by the existence of moving Dirac-concentrations in the time-dependent problem, we study the qualitative properties of steady states in the limit of small diﬀusion. Under diﬀerent conditions on the growth rate and interaction kernel as motivated by the framework of adaptive dynamics, we will show that as the diﬀusion rate tends to zero the steady state concentrates (i) at a single location; (ii) at two locations simultaneously; or (iii) at one of two alternative locations. The third result in particular shows that solutions need not be unique. This marks an important diﬀerence of the non-local equation with its local counterpart.


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KING-YEUNG LAM strategies. Then where r j = r(x j ) and k ij = K(x i , x j ), which is the Lotka-Voletrra model of N competing species. The time-dependent problem (1) was considered in [21] in case Ω = R n . Under convexity assumptions on the initial condition and on coefficients of the equation, it was shown that solutions of the time-dependent problem (1) concentrates as a single moving Dirac mass, as ε → 0. Moreover, they showed that the movement of the Dirac mass can be well described by a form of canonical equation, which is connected to the framework of adaptive dynamics [7] underlying the selection process.
Motivated by the work on the time-dependent problem, we will show in this paper that (1) possesses Dirac-concentrated steady states. Furthermore, we shall give three different set of conditions under which the steady state concentrates (i) at a single location; (ii) at two locations simultaneously; or (iii) at two alternative locations. The third result in particular shows that solutions need not be unique. This marks an important difference of the non-local equation (1) with its local counterpart. The steady states in scenarios (i) and (iii) can be considered as evolutionary endpoints corresponding to the single moving Dirac mass found in [21]. The dimorphic steady Dirac mass in scenario (ii) motivates the study of moving Dirac masses supported at two points, which is currently open. We also refer the interested readers to [28] where the existence and structure of positive steady states of a related model is discussed using a bifurcation approach.
Reaction-diffusion equations modeling the evolution of a quantitative trait has a long history (see, e.g. [3,10,15,16,22] for the case when K ≡ 1 is constant). The version studied in this paper, which involves a non-local interaction kernel, was introduced by [26] in the context of competition with neighbors, with See also [1,5,6] for works on the pure selection case. Furthermore, (1) can be rigorously derived from an individual-based, stochastic model in which a finite number of individuals may randomly die or produce an offspring with a rate depending on the competition among conspecifics. Taking the limit of an infinite number of individuals with the correct time scale, (1) can be obtained. We refer the interested reader to [4].
In the model of this paper, we take a simplified point of view where the growth rate r(x) and interaction kernel K(x, y) are prescribed. In general, they may be derived from density-and frequency-dependent interactions among phenotypes, in which case the relative advantage of a trait x against a different trait y depends on the context of their interaction. For instance, in [9,13,17,29] the invasion fitness between phenotypes with different dispersal strategies is obtained in the context of reaction-diffusion equations modeling the two competiting species in a bounded spatial domain. Those results concerning pairwise interactions has implications in the mutation-selection framework [12,11,18,19,24], which concerns populations structured by space and trait.
The remainder of this paper is organized as follows: The mathematical statement of the main results are presented in Section 2. Apriori estimates and the WKB transform are presented in Section 3. In Section 4, Theorems 1 and 2 are proved by the constrained Hamilton-Jacobi equation method pioneered by [8]. In Section 5, the existence of positive steady states and Theorem 3 are proved using a dynamical approach based on persistence theory and the construction of two forward-invariant regions of (1). Finally, the assumptions of our main results and their relation to the framework of adaptive dynamics are discussed in Section 6. 2. Main results. In this paper, we focus on the existence, and multiplicity of steady states of (1) when the trait space is one-dimensional, i.e. Ω = (−1, 1). When there is no ambiguity, we suppress the upper and lower limits in the integral and write, for ρ(y) ∈ L 1 ((−1, 1)), ρ(y) dy = 1 −1 ρ(y) dy. In such case, the steady stateũ ε (x) satisfies In the following we state our three main results.
Then, as ε → 0, every positive solutionũ ε (x) of (3) satisfies < 0 for all x, y ∈ [−1, 1], and there existsx ∈ (−1, 1) such that ∂ ∂x Then, as ε → 0, every positive solutionũ ε (x) of (3) satisfies where the positive constants A and B are unqiuely determined by Then, for all ε sufficiently small, (3) has at least two positive solutionsũ ε,+ (x) and u ε,− (x). Moreover, as ε → 0, we havẽ For the ease of exposition, we will postpone the proof for the existence of steady state to Corollary 5.3 in Section 5. r(x) − K(y,y) r(y) as a function of x and y, under the assumptions of Theorems 1, 2 and 3 respectively. Here x and y are the strategy of the invader and resident species respectively. K(x,y) r(x) − K(y,y) r(y) < 0 (resp. > 0) means invasion of resident with strategy "y" by invader with strategy "x" is a success (resp. failure).
There exists C independent of ε > 0 such that sup In particular, the family {ṽ ε } is equicontinuous in the variable x ∈ [−1, 1].
But both cases are impossible, in view of Lemma 3.1.
Proof. By Corollary 3.5, we may pass to a subsequence so that the solution (ṽ ε k ,H ε k ) of (6) converges to some (ṽ, . Assertion (i) follows from Lemma 3.4. Sinceṽ ε is a classical solution of (6), we may apply the stability theorem (see, e.g. [2, Theorem 4.1]) to conclude that the limit functionṽ(x) is a viscosity solution of the Hamilton-Jacobi equation (8). This proves assertion (ii). We next prove (iv). By assumption, x k is a local maximum point ofṽ ε k , so that Since, by the equation (8), we also haveH(x) ≤ 0 for all x, we see that It follows from the uniform convergence ofH ε k →H in [−1, 1] that any limit point This proves (iv). SinceH(x) ≤ 0 and the nodal set of H is nonempty, (iii) is also proved.
Next, we prove a result in the special case whenH(x) has a unique maximum point. 1], and equality holds if and only if x = x .

Proposition 2. Suppose, in addition to the hypotheses of Proposition 1, that for
Proof. We first show a property of the limit functionṽ(x).
Let the maximum ofṽ ε k be attained at x k ∈ [−1, 1]. Then, by Proposition 1(iv), where we used Lemma 3.4 for the last equality. Next, suppose to the contrary that v(x ) = 0 for some x ∈ [−1, 1] \ {x }. The fact thatṽ ≤ 0 implies that x is a local maximum ofṽ. We discuss the two cases separately: is an interior local maximum point ofṽ. Sinceṽ is viscosity solution of (8), we have H(x ) ≥ 0. But this can only happen if x = x , which is a contradiction. In case (ii),ṽ attains a strict local maximum at x = ±1 and there is a sequence x k → x such thatṽ ε k attains a local max at x k . This implies thatH ε k (x k ) ≥ 0. Letting k → ∞, we haveH(x ) ≥ 0 for some x ∈ {1, −1} \ {x }. This again is a contradiction to the assumption onH. Claim 1 is proved.
By Claim 1 and Lemma 3.1, we may pass to a subsequence and assume that (3), and letting ε k → 0, we have This concludes the proof of Proposition 2.

Proof of Theorems 1 and 2.
Proof of Theorem 1. By Proposition 1, we pass to a sequence ε k → 0 so thatṽ By assumption (A) and Lemma 3.1, we may let ε k → 0 to conclude the strict concavity ofH(x)/r(x). This, and Proposition 1(iii), implies the existence of some x ∈ [−1, 1], such thatH(x) ≤ 0 and equality holds iff x = x . (Note that x may depend on the subsequence.) By Proposition 2, we deduce thatũ ε k (x) → r(x ) K(x ,x ) δ 0 (x − x ) in distribution sense. Moreover, The fact thatH(x) is non-positive (Proposition 1(iii)) implies that By (A), we must have x =x. Since the limit point x =x is independent of subsequence ε k → 0, we deduce that in the full limit Proof of Theorem 2. By Proposition 1, we pass to a sequence ε k → 0 so thatṽ . We claim thatH(x)/r(x) is strictly convex. To this end, we compute and observe that the strict convexity ofH(x)/r(x) follows from hypothesis (B) and Claim 2.ṽ(x) < 0 for −1 < x < 1.
To determine the value of the positive constants A and B, we first prove the following estimate.
This yields Claim 4. We conclude the proof by determining A and B. To this end we integrate (3) over −1 < x < 0, then Using (10) and using Claim 4, we may let k → ∞ to obtain Similarly, we may repeat integrate (3) over 0 < x < 1 and repeat the above arguments to obtain .
5. Proof of Theorem 3. Consider now the time-dependent problem (1) in case In this section, let u ε (x, t) be a solution of (13).
Lemma 5.2. There exists C > 0, such that for any t 0 > 1, Proof. For each y and t, extend u ε (x, t), r(x) and K(x, y) on the boundary x = ±1 by reflection, we may assume that u ε satisfies the same equation in (−3, 3) × [0, ∞). Hence, we have by application of the local maximum principle [20,Theorem 7.36].
The following proposition from persistence theory, which is a special case of [27,Theorem 6.2], is the key to proving Theorem 3. (ii) X is forward-invariant with respect to the semiflow generated by (13) in C([0, 1]; [0, ∞)). (iii) X is not the singleton set of the trivial function. Then (3) has a positive solutionũ ε (x) lying in X.
To see the claim, we apply the Harnack inequality (for parabolic equations on bounded domain with Neuman boundary condtiions), due to J. Huska [14,Theorem 2.5], to obtain Claim 5 thus follows upon taking t → ∞, and using Lemma 5.1.
Claim 6. The semiflow Φ t , restricted to the forward-invariant set X, has a compact attractor A of neighborhood of compact sets. i.e. every compact subsets K 0 ⊂⊂ X has a neighborhood N such that We use [27,Theorem 2.30] to show the claim. It suffices to show that the semiflow Φ t is (i) point-dissipative; (ii) asymptotically smooth; and (iii) eventually bounded on every compact subset K 0 of X. Here we refer the readers to [27,Definition 2.25] for the definitions of (i) -(iii). Point-dissipativity is a direct consequence of Lemmas 5.1 and 5.2.
Next, we prove asymptotic smoothness. First, we combine the parabolic Krylov-Safanov estimate [25] (see also [20,Corollary 7.36]) and the local maximum principle (Lemma 5.2) to obtain, for each ε > 0, 0 < γ < 1 and 0 < δ < T , the existence of a constant C > 0 such that for any t 0 ≥ 0, Now, let X 1 be a forward-invariant, bounded, closed subset of X, let t i → ∞ and p i ∈ X 1 , then then again there exists M > 0 such that sup u0∈K0 u 0 (y) dy ≤ M , and Lemma 5.1 implies that By (17), i.e. the family {Φ ti (p i )} i is uniformly bounded in C γ ([−1, 1]) and hence has a convergent subsequence in C ([0, 1]). This demonstrates that Φ t is asymptotically smooth. Finally, let K 0 be a compact susbet of X, then again there exists M > 0 such that sup u0∈K0 u 0 (y) dy ≤ M and (18) holds. Hence, Lemma 5.2 says that if u 0 ∈ K 0 , then i.e. the semiflow Φ t is eventually bounded on every compact subset K 0 of X. This proves Claim 6.
Claim 7. For each t ∈ (0, 1], Φ t : X → X is compact. Fix t ∈ (0, 1] and a bounded subset B of X, then by (17), there exists C = C(t) such that where u ε (x, t) is the solution of (13) with initial condition u 0 . By Lemma 5.1, the last term can be estimated by C max{ u 0 L 1 ((−1,1)) , r * /K * }. Hence we may take supremum over u 0 ∈ B, so that Φ t (B) is a bounded subset of C γ ([−1, 1]) and is precompact in X. This proves Claim 7.
This is a direct consequence of the strong maximum principle [20,Theorem 2.7]. Finally, by the above setup, and Claims 5, 6, 7 and 8, we may apply [27, Theorem 6.2] to conclude the existence of at least one positive solutionũ ε (x) of (3) in X.
(iv) g − (x) = x + 1 in some neighborhood of −1, and that Proof. We will first construct g − (x). By assumption (C), we have Hence, By subtracting a small positive constant from h − (x) and modifying in a small neighborhood of −1, one may obtain a smooth function g 0 such that g 0 (x) = x + 1 in a small neighborhood of −1 and g 0 (x) < h − (x) in [−1, 1] and x 0 g 0 (y) dy > 0 for x ∈ (−1, 1). Finally, further subtract from g 0 a positive function supported in a neighborhood of 1, we obtain g − (x) with all desired properties (i) to (iv).
In view of (iii'), the last property is equivalent to (ii").