ON A NONLOCAL SELECTION-MUTATION MODEL WITH A GRADIENT FLOW STRUCTURE

Transcription

1 ON A NONLOCAL SELECTION-MUTATION MODEL WITH A GRADIENT FLOW STRUCTURE PIERRE-EMMANUEL JABIN AND HAILIANG LIU Abstract. In this paper, we are intereste in an integro-ifferential moel with a nonlinear competition term that escribes the evolution of a population structure with respect to a continuous trait. Uner some assumption, the steay solution is shown unique an strictly positive, an also globally stable. The exponential convergence rate to the steay state is also establishe. 1. Introuction 1.1. The moel an its basic properties. We are intereste in the ynamics of a population of iniviuals with a quantitative trait. The reprouction rate of each iniviual is etermine by its trait an the environment, leaing therefore to selection. On the other han, the influx of mutations into a population over time can counteract the unerlying selection forces. There are ifferent moels featuring balance between two evolutionary forces. In this work, we are concerne with the problem governe by t f(t, x = f(t, x + 1 (1.1a (a(x f(t, x b(x, yf (t, yy, for t > 0, x X, (1.1b f(0, x = f 0 (x 0, x X, (1.1c f = 0, x X, ν where f(t, x enotes the ensity of iniviuals with trait x, X is a subomain of R, ν is the unit outwar normal at a point x on the bounary X. With some effort, the theory presente here coul even be generalize to a complete metric set X enowe with a measure satisfying aequate regularity properties. In the moel, coefficient a(x is the intrinsic growth rate of iniviuals with trait x, an b(x, y > 0 represents the competitive interaction between iniviuals, while the iffusion term plays certain role of mutations in the population ynamics. The trait epenent competition as such appears in many population balance moels of Lotka-Volterra type, see e.g., [3, 11, 13, 14]. In particular, the nonlinear competition effect oes appear in the moel for fish species introuce in [5] in the stuy of the effect of exploitation on these species. Their moel when the number of fish species ten to infinity formally leas to a continuous moel of the form t f(t, x = 1 (a(x f(t, x b(x, y(f(t, y (x, y y. X 1991 Mathematics Subject Classification. Mathematics Subject Classification (000 35B40, 9D15. Key wors an phrases. Aaptive ynamics, Asymptotic analysis, Integro-ifferential equation, convergence rates. 1 X

2 PIERRE-EMMANUEL JABIN AND HAILIANG LIU This equation with = 0 when augmente with a mutation term f is exactly (1.1a. The main ifference between (1.1 an more classical moels is its non-linear competition term b(x, y f (t, y y, compare to the more usual b(x, y f(t, y y. Such a non-linear term significantly changes the effect of competition in particular by increasing the effect of large populations. This may reflect a stricter constraint on resources for instance as in the fish species moel above. Non-linear competitions have selom been stuie from a mathematical point of view an the main goal of the present article is to introuce the require tools an explain how the classical approach for linear competition shoul be moifie. While the analysis is performe for this particular moel, we believe it coul easily be carrie over to other non-linear terms. Note that from a mathematical point of view, an attractive feature of moel (1.1a is its graient flow structure in the sense that (1.1a can be written as (1. t f = 1 δf δf, where the corresponing energy functional is (1.3 F [f] = 1 b(x, yf (t, xf (t, yxy 1 4 a(xf (t, xx + x f(t, x x, so that the energy issipation law F [f] = t t f x 0 hols for all t > 0, at least for classical solutions. Such a graient flow structure is preserve by the finite volume scheme recently propose in [7], in which the authors show that both semi-iscrete an fully iscrete schemes satisfy the two esire properties: positivity of numerical solutions an energy issipation. These ensure that the positive steay state is asymptotically stable. In aition, the numerical solutions of the moel with small mutation are shown to be close to those of the corresponing moel with linear competition (1.5. Numerical schemes with similar methoology have been propose an analyze for the linear selection ynamics governe by (1.5 in [17, 18]. The objective of this paper is to provie a rigorous analysis on global existence of (1.1, an time-asymptotic convergence to the positive steay state, complementing with the numerical results in [7]. Let us remark as well that uner the transformation u = f, the resulting equation from moel (1.1a becomes (1.4 t u(t, x = u u u ( + u(t, x a(x b(x, yu(t, yy. X While the competition term is now linear, the original non-linearity was transfer-e to the iffusion with a new Hamilton-Jacobi like term. Therefore there oes not seem to be any simple way to reuce (1.1a to an alreay stuie case. Of course for rare mutation, the iffusion term may be roppe from moel (1.1a. The resulting equation then becomes the well-known (1.5 t u(t, x = u(t, x ( a(x X b(x, yu(t, yy.

3 Such a simplifie moel with the usual mutation has been erive from ranom stochastic moels of finite populations (see [8, 9]. This competition moel or its variation arises not only in evolution theory but also in ecology for non-local resources (an x enotes the location there, see e.g. [4, 1, 15]. The moel without mutation is interesting from the point of view of asymptotic behavior; one expects that the population ensity concentrates at large times, see, e.g., [1, 6, 11, 16, 4]. The singular steay-state solutions of the competition moel correspon to highly concentrate population ensities of the form of well separate Dirac masses, which have been shown to happen only asymptotically in moels with mutation [, 10, 19, 0, 1,, 3]. The equation (1.5 amits many generalizations, the most popular of which is (1.6 t u(t, x = ( K(x, yu(t, yy k(xu(t, x + u(t, x a(x b(x, yu(t, yy, X where K(x, y 0 is a mutation kernel satisfying K(x, yy = k(x. Such a competitionmutation moel has been erive from ranom stochastic moels of finite populations (see [8, 9]. The ae term has zero integral, an plays certain role of iffusion, so one also aopts ( (1.7 t u(t, x = u + u(t, x a(x b(x, yu(t, yy, X as a competition-mutation moel. However, the large time solution behavior of this moel remains a challenging issue. Nevertheless, for b(x, y = η(y > 0, the asymptotic solution behavior when mutation tens to vanish has been well stuie, see e.g., [, 1, 3]. Note that the irect competition moel (1.5 is a graient flow t u = grah uner the metric g, h u = g h x, where u H[u] = 1 b(x, yu(xu(yxy a(xu(xx. It is possible to moify the energy to obtain (1.4 irectly as a graient flow. Consier an augmente energy of the form H[u] = 1 b(x, yu(xu(yxy a(xu(xx + 1 x u x. u By a irect calculation we have δh δu = b(x, yu(yy a u u + xu. u By the metric g, h u = g h u δh an, g = grah, g δu u, we have ( a b(x, yu(yy + u xu u. grah = u δh δu = u Hence the graient flow of H is governe by (1.4 which when transforme with f = u leas to (1.1a. In Sect. 1., we present the main results of this article, namely positivity an uniqueness of steay solutions in Theorem 1., convergence to the steay solution in Theorem 1., as well as the exponential convergence rate in Theorem 1.3. In Sect., we prove the results presente in Sect

4 4 PIERRE-EMMANUEL JABIN AND HAILIANG LIU 1.. Main results. The purpose of this paper is to analyze the solution behavior of selection-mutation ynamics (1.1 with the graient flow structure (1. with (1.3. To this en, we make the following assumptions: (1.8a (1.8b (1.8c a L (X, {x; a(x > 0} 0; b L (X X, b m = inf b(x, x,x X x > 0. b(x, y = b(y, x, g L 1 (X\{0}, b(x, yg(xg(yxy > 0. One can check that b efines then a scalar prouct over L 1 (X, g, h b = b(x, yg(xh(yxy with corresponing norm ( g b = b(x, yg(xg(yxy 1/. In what follows we also use the notation R[h] = 1 (1.9 (a h b(x, yh (yy. We work with solutions which are continuous functions of time having values in L (X; enote by C([0, T ]; L (X, an norme by w = sup w(t, L (X. 0 t T Existence of such solutions can be obtaine through a stanar fixe point an extension argument leaing to Theorem 1.1. Let f 0 L (X, an both a an b satisfy the first two assumptions of (1.8. Then (1.1 amits a global weak solution f L (R + ; L (X. Moreover, we have (a sup t>0 f(t, L (X M, (t, x R + X. (b f is stable an epens continuously on f 0 in the following sense: if f is another solution with initial ata f 0, then for every t > 0, f f x e λt f 0 f 0 x, where λ epens only on a, b an f 0. Given that the proof of Theorem 1.1 is rather classical, we only give it in appenix B. The strong competition assumption (1.8c is irectly connecte to the stability of the steay solution. In fact, uner assumption (1.8, the steay solution g is unique an upper-boune.

5 If ax 0, the steay state is strictly positive. The case ax < 0 is less obvious. Recalling [5, Theorem 3.13(b] which claims that there exists a unique positive λ 1 an the positive function ψ D(L 1 such that aψ x > 0 an (1.10 λ 1 = x ψ x aψ x = inf { x v x av x : v D(L 1 an } av x > 0, where D(L 1 = {u H (X : n u X = 0} is the omain of the Laplace operator L 1 u = u, we can show the steay state is still strictly positive if λ 1 < 1/. More precisely, we have the following result. Theorem 1.. There exists g 0 solution in the sense of istribution to (1.11 g + R[g] = 0, x X ν g = 0, on X. Moreover, (i If ax 0 or ax < 0 with λ 1 < 1/, then there exists a unique positive solution such that 0 < g min g g max < in X. (ii If ax < 0 with λ 1 1/, there is no positive steay solution. Thanks to this result we can show the convergence of f(t, towars g: Theorem 1.3. Assume both a an b satisfy (1.8. Consier any non-negative f 0 L 1 (X L (X. Then the corresponing solution f(t, of (1.1 is such that (1.1 F [f(t, ] < 0 as long as f is not a steay solution. t As a consequence (1.13 lim t f(t, g( L (X = 0. An moreover, there exists C epening on initial ata f 0 an g 0 such that f(t, x g(x x Ce rt t > 0, for ax 0 or ax < 0 with λ 1 1, where of course g = 0 if λ 1 > 1/. For ax < 0 an λ 1 = 1, f(t, x x C t > t The proofs of theorems 1. an 1.3 rely on a careful use of the competition assumption an is given in the next section.. Proofs of the results.1. Proof of Theorem 1.. The existence of a non-negative steay state follows a rather classical variational argument which, for the sake of completeness, is given in appenix A. Consier a non-negative an non 0 steay state g. Multiplying (1.11 by g an then integrating over the omain X, we have g (a b g x = g gx = g x 0. 5

6 6 PIERRE-EMMANUEL JABIN AND HAILIANG LIU This implies that leaing to the upper boun a g L b m g 4 L 0, g L a b m <. This irectly implies that g L. By Sobolev embeing, g L p for some p > an we can iterate to fin g L p. Iterating a finite number of times, this finally gives g L since a L which is the optimal smoothness on g. We hence have the elliptic problem g + c(xg = 0 with boune c(x an g. By the stanar Harnack inequality we have sup g C inf g in any small ball within X. Hence g > 0 an boune from above unless it is ientical zero. We next prove the uniqueness. Let g 1 an g be two positive solutions of (1.11, then using the positivity of b, thir assumption in (1.8 0 (g1 g(xb(x, y(g1 g(yyx = (g 1 g/g 1 g 1 (x b(x, yg1(yyx (g1/g g g (x b(x, yg(yyx = (g 1 g/g 1 ( g 1 (x + a(xg 1 (xx + (g g1/g ( g (x + a(xg (xx = (g 1 g/g 1 g 1 (x + (g g1/g g (x, by using the equation (1.11. Hence by integrating by part ( 0 x g 1 g 1g x g g x g 1 g1 x g 1 x ( x g g 1g x g 1 g1 x g x g x ( = x g 1 g 1 x g g As a conclusion g 1 = g, leaing to g 1 = g. g + xg g x g 1 x 0. g 1.. Proof of Theorem 1.3: Convergence. Let us first prove (1.1. A irect calculation shows that t F = t f x 0. Thus it only remains to show that for some t 0 0, if t f(t 0, x 0 for all x X, then t f(t, x 0 for all x X an t 0. i For t > t 0. Since t f(t 0, x 0 for all x X, then f(t 0, x is a steay solution. By the uniqueness implie by (b in Theorem 1.1, we have f(t, x = f(t 0, x, x X

7 7 for all t > t 0. ii For 0 t t 0, we enote w(t, x = f(t, x f(t 0, x for x X an 0 < t < t 0, so that w is a solution to where t w = w + S(t, x, in (t, x (0, t 0 X, ν w = 0, x X, w(t 0, x = 0, x X, A irect estimate using (B.1 gives S(t, x = R[f(t, ] R[f(t 0, ]. (.1 S(t, λ w(t, for λ = 1 ( a + 3 b M. Suppose for some t 1 (0, t 0, w(t 1, x 0 in X, then Λ(t = X xw x X w x is well efine in a neighborhoo of t 1. By a irect calculation we have t Λ(t = x w t x wx w t wx ( w x w x = [ w x = [ w x w t wx + x w w x ] ( w + Λw( w + Sx, x w x ] w t wx since t w = w + S. This gives t Λ(t = [ w x w x ( w + Λw + ( 14 S x ] S( w + Λwx S x w x 1 λ. Hence, Λ(t must be boune in [t 1, t 0, where t 0 t 0 is the first instant in (t 1, t 0 ] at which w(t, 0 in Ω.

8 8 PIERRE-EMMANUEL JABIN AND HAILIANG LIU On the other han, t ( log 1 = w x w x = w x w t wx w( w + Sx = Λ(t Λ(t + λ. Swx w x Thus w x 0 as long as Λ(t is boune, contraicting to the assumption that w( t 0, 0 in Ω. Hence w 0 in Ω [0, t 0 ], an our proof of (1.1 is now complete. Define, for any initial ata f 0, the usual ω-limit set as ω(f 0 = s>0 {f(t,, t s}. The previous analysis shows that if f ω(f 0, then f must be a steay solution. Results in Theorem 1. an (1.1 then imply that the ω-limit set contains only g as the unique point, hence ( Proof of Theorem 1.3: Exponential convergence. In the case g > 0, we introuce the auxiliary functional [ ( ] f g f G = g log x, g which is boune from below [ f g G ( ] f g g 1 x = 1 (f g x. A irect calculation gives t G = (f g f t f x [ f = (f g f g g 1 ] b(x, y(f (y g (yy x ( = x f f g xg + xg g f xf x 1 (f g (xb(x, y(f g (yyx where D(f, g = D(f, g, g x ( f g x + 1 (f g (xb(x, y(f g (yyx. We claim that there exists µ > 0 such that (. D(f, g µ f/g 1 L.

9 Assuming that this is correct for the time being, this gives ( f t G µ g 1 x µ G. gmax By Gronwall lemma, we have Hence ( G(t G(0 exp µ t. gmax f(t, g( L G(t ( G(0 µ We are thus left to prove (.. Using the lower boun an the usual Poincaré s inequality we have ( f g x x g f g minc X inf c g c x, where C X is a constant epening only on X. As usual the minimum is achieve at ( c = 1 f x. X X g As a consequence it suffices to fin µ inepenent of c 0 such that C X gmin f g c x + 1 f g b µ f g 1 L. If c = 1, the inequality is obvious for µ µ 1 = C X g min. For c 1, we first estimate f g b = f c g + (c 1g b g max = (c 1 g b + (c 1 f c g, g b + f c g b ɛ(c 1 g b ɛ 1 ɛ f c g b, for any 0 < ɛ < 1 by using Young s inequality. Note that we have f c g b b f cg L f + cg L b g max( f + c g f/g c L b g max M (c + 1 f/g c L ; t. M = max{m, g }. Hence f g b ɛ(c 1 g b ɛ 1 ɛ b gmax M (c + 1 f/g c L. Therefore C X gmin f g c x + 1 f g b ( C X gmin ɛ 1 ɛ b gmax M (c + 1 f/g c + ɛ (c 1 g b 1 C X g min f/g c + ɛ (c 1 g b, 9

10 10 PIERRE-EMMANUEL JABIN AND HAILIANG LIU C by taking ɛ = X gmin C X gmin + b g M. max (c+1 On the other han still using the Young inequality that for any 0 < η < 1 a + b η a η 1 η b, one has that f g c f x = g c f x η g 1 This finally gives C X gmin f g c x + 1 f g b η ( C X gmin f g 1 x + c 1 Therefore it is enough to take η such that giving us C X gmin f g c ɛ (c + 1 g b η X 1 η 0, x η X 1 η c 1. ɛ (c + 1 g b η X 1 η x + 1 f g f b µ g 1 for µ = η C X gmin. Finally we investigate the case when 0 is the only non-negative steay solution, which is the case when λ 1 1 an ax < 0. In such case, the convergence rate can also be establishe, by introucing instea G = 1 f x. A irect calculation by integration by parts gives t G = ff t x = x f x + 1 x, af x 1 f b. If λ 1 > 1, then from [5, Theorem 3.11] it follows that there exists ν > 0 such that x v x 1 (.3 av x ν v for any v D(L 1. Hence we have t G ν f 1 f b νg. This leas to G G(0e νt, hence f(t, f 0 e rt with r = ν. If λ 1 = 1, we have t G = x f x + λ 1 af x 1 f b 1 f b,.

11 where we have use the efinition for λ 1 when af > 0, an the inequality remains vali when af 0. Hence t G b m f 4 = b m G. This upon integration over [0, t] using f = G gives f(t, f 0 (1 + b m f 0 t 1, t > 0. This completes the proof of Theorem 1.3. Appenix A. Existence of a stationary solution There are several ways to construct a non-negative solution of (1.11. We give here an example of a variational construction. Let g H 1 (X be a weak solution to (1.11 in the sense that (A.1 x g x wx + R[g]wx = 0 w H 1 (X an g 0, g 0. We claim that this weak solution is equivalent to the nonzero critical point of the functional [ 1 F [w] = 4 (b w w 1 ] aw + + x w x, where w + = max(w, 0. Inee, a weak solution of (A.1 is obviously a critical point of F [w]. Conversely, if g H 1 (X is a critical point of F [w], then 0 = F [g], g = [ x g + 1 g (b g x = 0, where g = min(g, 0. We see that g = 0, which implies g 0. Hence g is a weak solution of (A.1. We next prove the existence of a minimizer for the variational problem F. Proposition A.1. There exists g A := {g H 1 (X, g 0}, such that F (g = inf F [w]. w H 1 (X Moreover, (i If ax 0 or ax < 0 with λ 1 < 1/, then g is not ientically 0; (ii If ax < 0 with λ 1 1/, g 0. Proof. By Young s inequality, we have F [w] = 1 (b w w x 1 a w+ x + x w x b m w 4 L 1 a w L + x w x x w x a, 4b m 11

12 1 PIERRE-EMMANUEL JABIN AND HAILIANG LIU so that F is boune from below. In fact set m = inf w H 1 F [w], then a 4bm m <. Select a minimizing sequence {g k } k=1 so that lim F [g k] = m. k Then we have sup k x g k L <, an enoting C = sup k F (g k 1 4 b m g k 4 L 1 a g k L + C, which implies that g k L a b m + 1 b m a + b m C <. Hence {g k } is a boune sequence in H 1 (X. There exists g H 1 (X an a subsequence of {g k } (still enote g k such that g k g weakly in H 1 (X, g k g strongly in L (X. On the other han, note that w + w. Therefore F (g + F (g, clearly one may always replace g k by its positive part an as a consequence we may assume that g k 0. Hence g 0 as well by strong convergence in L. A irect calculation shows that xg i k x xg i x xg i xg i k x = xg i ( xg i k xgx i 0, i = 0, 1 as k. Note also that b g k g k x b g k g g k g x = b ( g k g k g ( g k g k g x + b ( g k g k g g k g x b g g x as k. These together with lim k a gk x = a g x, ensure that But by g A, it follows that m = lim k F [g k ] F [g]. F [g] = m = min F [w]. w A This proves the existence of a non negative minimizer. To prove that g is not ientically 0, when ax 0 or ax < 0 yet λ 1 < 1/, we iscuss case by case, keeping in min that F (0 = 0. (i If ax > 0, one consiers the function ientically equal to ɛ with ɛ small F (ɛ = ɛ4 4 X b ɛ a < 0, for ɛ small enough;

13 (ii If ax = 0, one takes instea w = ɛ(1 + δv with v satisfying avx > 0 an n v = 0 on X, so that [ F (w = ɛ ɛ (b (1 + δv (1 + δv x ] +δ ( x v x av x δ avx < 0 for ɛ, δ suitable small. (iii If ax < 0 an λ 1 < 1, we take w = τψ with τ > 0 so that F [τψ] = τ 4 ( (b ψ ψ x + τ λ 1 1 aψ x 4 τ ( ( 1 aψ 4 ψ b τ 4 λ x 1 < 0 ψ b for τ sufficient small. Hence, in all these three cases the minimizer cannot be 0. Finally we show 0 is the only minimizer if ax < 0 an λ 1 1. Note that for any v satisfying av x 0 we have F [v] = 1 (b v v x 1 av x + x v x v b + x v ; 13 otherwise if av x > 0, then F [v] 1 4 v b + ( λ 1 1 av x 1 4 v b. That is we have F [v] 0 = F [0], in such case 0 is the only minimizer. Finally note that if a an b are not constant then the Euler-Lagrange equation for the variational problem, which is (1.11, oes not amit constants as solution. Thus in that case the minimizer cannot be constant either. Appenix B. Proof of Theorem 1.1 We begin with a formal a priori estimate, then we inicate how to make the argument rigorous. (a The existence theory is base on the following a priori estimate for solutions to (1.1. It is obtaine after multiplying the equation by f. t f x + f x = a b m f (a f x b(x, yf (yy f x b m ( ( γ f x f x.

14 14 PIERRE-EMMANUEL JABIN AND HAILIANG LIU Here b m = inf x,y X b(x, y > 0 an γ := a b m. A irect Gronwall lemma then gives { } f γ (t, xx 1 + (γ f 0 max f L 1e a t 0 x, γ =: M. In aition, t f xτ 1 f0 x + a t, t > b m (b In orer to prove the existence of a solution with the above properties, we consier a stanar fixe point an extension argument. For T > 0, let Γ := {f C([0, T ]; L (X, f M, 0 t T }, where we recall that f = sup t [0, T ] f(t,. L (X. The set Γ is not empty (since 0 Γ, an Γ is close. If f Γ, then f M. If f 1, f Γ, then It follows that (B.1 R[f 1 ] R[f ] = 1 a (f 1 f 1 f 1 b f f b f = 1 a (f 1 f 1 (f 1 f b f f b (f f 1. R[f 1 ] R[f ] 1 a f 1 f + 1 f 1 f b f b f f 1 f f 1 + f λ f 1 f, for λ = 1( a + 3 b M. Let T = 1 an observe that T epens only on f λ 0, a an b. Define a mapping Φ on C([0, T ]; L (X onto itself by w = Φ[f], where w is the unique solution in C([0, T ]; L (X to t w = w + R[f], ν w X = 0, where obviously R[f] C([0, T ]; L (X. Observe that a fixe point of Φ is a solution of (1.1. We claim that Φ maps Γ onto itself. To see this, let f Γ, then Therefore t w x + R[f] λ f. w x = wr[f]x λ w f. Hence w T λ f T λm = 1 M which proves our claim. Next, we show that Φ is a contraction on Γ. If f 1, f Γ, we have w i = Φ[f i ], an v = w 1 w solves t v = v + R[f 1 ] R[f ].

15 15 Hence t v x + which thanks to (B.1 leas to (B. which upon integration gives x v = v(r[f 1 ] R[f ]x v R[f 1 ] R[f ], t v λ f 1 f. v λt f 1 f 1 f 1 f, an so Φ is a contraction on Γ. We may apply Banach s fixe point theorem to conclue that Φ has a unique fixe point in Γ. Note that we have also prove that if f 0 L (X, then for any solution f we have f(t, L (X M. This allows us to conclue that the solution of the problem exists for all time. To see this, we apply the above local existence result repeately on T + nσ t T + (n + 1σ, where σ = σ( f(t, L (X, an eventually we obtain a solution for all time. Moreover, we have from (B. that if f 1 an f are two solutions (f 1 f (t, e λt (f 1 f (0,, t > 0. Acknowlegments H. Liu is partially supporte by the National Science Founation uner Grant DMS an by NSF Grant RNMS (Ki-Net P.E. Jabin is partially supporte by NSF Grant DMS an an by NSF Grant RNMS (Ki-Net References [1] G. Barabás, G. Meszzéna, When the exception becomes the rule: the isappearance of limiting similarity in the Lotka-Volterra moel, J Math Biol 58(1 ( [] G. Barles, S. Mirrahimi, B. Perthame, Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result, Methos an Applications of Analysis 16(3 ( [3] H. Berestycki, G. Nain, B. Perthame, L. Ryzhik, The non-local Fisher KPP equation: travelling waves an steay states, Nonlinearity (1 ( [4] B. Bolker, W. Pacala, Using moment equations to unerstan stochastically riven spatial pattern formation in ecological systems, Theoretical Population Biology 5 ( [5] K. J. Brown, S. S. Lin, On the existence of positive eigenfunctions for an eigenvalue problem with inefinite weight function, J. Math. Anal. Appl. 75 ( [6] R. Bürger, The mathematical theory of selection, recombination an mutation. Wiley, New York (000. [7] W.-L. Cai, H. Liu, A finite volume metho for nonlocal competition-mutation equations with a graient flow structure, Mathematical Moeling an Numerical Analysis (MAN. In press (017. [8] N. Champagnat, R. Ferrière, S. Méléar, Unifying evolutionary ynamics: from iniviual stochastic processes to macroscopic moels, Theor Popul Biol 69 ( [9] N. Champagnat, R. Ferrière, S. Méléar, From iniviual stochastic processes to macroscopic moels in aaptive evolution, Stoch Moels 4(1 ( [10] Y. Cohen, G. Galiano, Evolutionary istributions an competition by way of reaction-iffusion an by way of convolution, Bulletin of Mathematical Biology 75(1 (

16 16 PIERRE-EMMANUEL JABIN AND HAILIANG LIU [11] L. Desvillettes, P.E. Jabin, S. Mischler, G. Raoul, On selection ynamics for continuous structure populations, Commun Math Sci 6(3 ( [1] N. Fournier, S. Méléar, A microscopic probabilistic escription of a locally regulate population an macroscopic approximations, Ann Appl Probab 14(4 ( [13] S. Génieys, B. Perthame, Dynamics of nonlocal Fisher concentration points: a nonlinear analysis of Turing patterns, Mathematical Moeling in Natural Phenomenon (4 ( [14] S. Génieys, V. Volpert, P. Auger, Aaptive ynamics: moeling Darwin s ivergence principle, C. R. Aca. Sc. Paris, Biologies 39(11 ( [15] S. S. Gourley, Travelling front of a nonlocal Fisher equation, J Math Biol 41 ( [16] P.E. Jabin, G. Raoul, On selection ynamics for competitive interactions, J Math Biol 63(3 ( [17] H. Liu, W.-L. Cai, N. Su, Entropy satisfying schemes for computing selection ynamics in competitive interactions, SIAM J. Numer. Anal. 53(3 ( [18] W.-L. Cai, P. Jabin, H. Liu, Time-asymptotic convergence rates towars the iscrete evolutionary stable istribution, Mathematical Moels an Methos in Applie Sciences (M3AS, 5(8 ( [19] A. Lorz, S. Mirrahimi, B. Perthame, Dirac mass ynamics in multiimensional nonlocal parabolic equations, Communications in Partial Differential Equations, 36 (6 ( [0] S. Mirrahimi, B. Perthame, E. Bouin, P. Millien, Population formulation of aaptive meso-evolution: theory an numerics, Mathematics an Biosciences in Interaction ( [1] S. Mirrahimi, B. Perthame, J. Y. Wakano, Direct competition results from strong competition for limite resource, J. Math. Biol. 68 (4 (014, [] B. Perthame, G. Barles, Dirac concentrations in Lotka-Volterra parabolic PDEs. Iniana Univ Math J 57(7 ( [3] G. Raoul, Long time evolution of populations uner selection an vanishing mutations, Acta Appl Math 114(1- ( [4] G. Raoul, Local stability of evolutionary attractors for continuous structure populations, Monatsh Math 165(1 ( [5] K. Shirakihara, S. Tanaka, Two fish species competition moel with nonlinear interactions an equilibrium catches, Res. Popul. Ecol. 0 ( University of Marylan, Department of Mathematics, College Park, MD aress: pjabin@cscamm.um.eu Iowa State University, Mathematics Department, Ames, IA aress: hliu@iastate.eu