The evolutionary limit for models of populations interacting competitively with many resources
aa r X i v : . [ m a t h . A P ] J un The evolutionary limit for models ofpopulations interacting competitively withmany resources
Nicolas Champagnat , Pierre-Emmanuel Jabin , Abstract
We consider a integro-differential nonlinear model that describesthe evolution of a population structured by a quantitative trait. Theinteractions between traits occur from competition for resources whoseconcentrations depend on the current state of the population. Follow-ing the formalism of [15], we study a concentration phenomenon aris-ing in the limit of strong selection and small mutations. We prove thatthe population density converges to a sum of Dirac masses character-ized by the solution ϕ of a Hamilton-Jacobi equation which dependson resource concentrations that we fully characterize in terms of thefunction ϕ . MSC 2000 subject classifications:
Key words and phrases: adaptive dynamics, Hamilton-Jacobi equation withconstraints, Dirac concentration, metastable equilibrium.
We are interested in the dynamics of a population subject to mutation andselection driven by competition for resources. Each individual in the popu-lation is characterized by a quantitative phenotypic trait x ∈ R (for example TOSCA project-team, INRIA Sophia Antipolis – M´editerran´ee, 2004 rte des Lucioles,BP. 93, 06902 Sophia Antipolis Cedex, France,E-mail:
[email protected] Laboratoire J.-A. Dieudonn´e, Universit´e de Nice – Sophia Antipolis, Parc Valrose,06108 Nice Cedex 02, France, E-mail: [email protected] ∂ t u ε ( t, x ) = 1 ε k X i =1 I εi ( t ) η i ( x ) − ! u ε ( t, x ) + M ε ( u ε )( t, x ) , (1.1)where M ε is the mutation kernel M ε ( f )( x ) = 1 ε Z R K ( z ) ( f ( x + εz ) − f ( x )) dz, (1.2)for a K ∈ C ∞ c ( R ) such that R R zK ( z ) dz = 0. Among many other ecolog-ical situations [14], this model is relevant for the evolution of bacteria in achemostat [13, 15]. With this interpretation, u ε ( t, x ) represents the concen-tration of bacteria with trait x at time t , the function η i ( x ) represents thegrowth rate of the population of trait x due to the consumption of a resourcewhose concentration is I εi , and the term − I i , namely I i ( t ) = 11 + R R η i ( x ) u ε ( x ) dx . (1.3)This corresponds to an assumption of fast resources dynamics with respectto the evolutionary dynamics. The resources concentrations are assumed tobe at a (quasi-)equilibrium at each time t , which depends on the currentconcentrations u ε .The limit ε → u ε can be expected when ε →
0. Defining ϕ ε as u ε = e ϕ ε /ε , or ϕ ε = ε log u ε , (1.4)one gets the equation ∂ t ϕ ε = k X i =1 I εi ( t ) η i ( x ) − H ε ( ϕ ε ) , (1.5)2here H ε ( f ) = Z R K ( z ) (cid:0) e ( f ( x + ε z ) − f ( x )) /ε − (cid:1) dz. (1.6)At the limit ε → H ε simply becomes H ( p ) = Z R K ( z ) ( e p z − dz. (1.7)So one expects Eq. (1.5) to lead to ∂ t ϕ = k X i =1 I i ( t ) η i ( x ) − H ( ∂ x ϕ ) , (1.8)for some I i which are unfortunately unknown since one cannot pass to thelimit directly in (1.3). Therefore one needs to find a relation between the ϕ and the I i at the limit. Under quite general assumptions on the parameters(see Lemma 3.1 below), the total population mass R R u ε is uniformly boundedover time. This suggests that max x ∈ R ϕ ( t, x ) = 0 should hold true for all t ≥
0. Together with (1.8), this gives a candidate for the limit dynamicsas a solution to a Hamilton-Jacobi equation with Hamiltonian H and withunknown Lagrange multipliers I i , subject to a maximum constraint. Thelimit population is then composed at time t of Dirac masses located at themaxima of ϕ ( t, · ).This heuristics was justified in [15] in the case of a single resource (and whenthe resources evolve on the same time scale as the population), and the caseof two resources was only partly solved. The mathematical study of theconvergence to the Hamilton-Jacobi equation with maximum constraint andthe study of the Hamilton-Jacobi equation itself have only be done in veryspecific cases [15, 2, 5, 25]. In fact the main problem in this proposed modelis that the number of unknowns (the resources) may easily be larger than thedimension of the constraint (formally equal to the number of points where ϕ = 0).Our goal in this paper is to prove the convergence of ϕ ε to a solution of (1.8),where we give a full characterization of the functions I i . Those are no moreconsidered as Lagrange multipliers for a set of constraints but are given bythe solution ϕ itself. The new resulting model describes the evolution of apopulation as Dirac masses and is formally well posed.The general problem of characterizing evolutionary dynamics as sums ofDirac masses under biologically relevant parameter scalings is a key tool3n adaptive dynamics —a branch of biology studying the interplay betweenecology and evolution [19, 21, 22, 12, 7]. The phenomenon of evolutionarybranching , where evolution drives an (essentially) monotype population tosubdivide into two (or more) distinct coexisting subpopulations, is particu-larly relevant in this framework [22, 17, 18]. When the population state canbe approximated by Dirac masses, this simply corresponds to the transitionfrom a population composed of a single Dirac mass to a population composedof two Dirac masses.Several mathematical approaches have been explored to study this phe-nomenon. One approach consists in studying the stationary behaviour of anevolutionary model involving a scaling parameter for mutations, and then let-ting this parameter converge to 0. The stationary state has been proved to becomposed of one or several Dirac masses for various models (for deterministicPDE models, see [4, 5, 11, 20, 16], for Fokker-Planck PDEs corresponding tostochastic population genetics models, see [3], for stochastic models, see [26],for game-theoretic models, see [10]). Closely related to these works are thenotions of ESS (evolutionarily stable strategies) and CSS (convergence sta-ble strategies) [22, 13], which allow one in some cases to characterize stablestationary states [4, 11, 20, 10].The other main approach consists in studying a simultaneous scaling of mu-tation and selection, in order to obtain a limit dynamics where transitionsfrom a single Dirac mass to two Dirac masses could occur. Here again, de-terministic and stochastic approaches have been explored. The deterministicapproach consists in applying the scaling of (1.1). The first formal resultshave been obtained in [15]. This was followed by several works on other mod-els and on the corresponding Hamilton-Jacobi PDE [5, 25]. For models ofthe type we consider here, rigorous results (especially for the well posednessof the Hamilton-Jacobi eq. at the limit) mainly only exist in the case withjust one resource, see [2] and [1] (one resource but multidimensional traits).The stochastic approach is based on individual-based models, which are re-lated to evolutionary PDE models as those in [11, 20] through a scaling oflarge population [8]. Using a simultaneous scaling of large population andrare mutations, a stochatic limit process was obtained in [6] in the case of amonotype population (i.e. when the limit process can only be composed of asingle Dirac mass), and in [9] when the limit population can be composed offinitely many Dirac masses.Finally note that the total population of individuals is typically very high,for bacteria for example. This is why even stochastic models will usually take4ome limit with infinite populations. Of course, this has some drawbacks.In particular the population of individuals around a precise trait may turnout to be low (even though the total population is large). As in the scalingunder consideration, one has growth or decay of order exp( C/ε ), this is infact quite common. One of the most important open problem would be toderive models that are both able of dealing with very large populations andstill treat correctly the small subpopulations (keeping the stochastic effectsor at least truncating the population with less than 1 individual).There are already some attempts in this direction, mainly proposing modelswith truncation, see [24] and very recently [23]. For the moment however thetruncation is of the same order as the maximal subpopulation.In order to state our main result, we need some regularity and decay assump-tions on the η i , namely η i > , ∃ ¯ η ∈ C ( R ) , ∀ x, k X i =1 (cid:16) | η i ( x ) | + | η ′ i ( x ) | + | η ′′ i ( x ) | (cid:17) ≤ ¯ η ( x ) , (1.9)where C ( R ) is the set of continuous function, tending to 0 as x → ±∞ .In order to characterize the resources I i ( t ) involved in (1.8), we introduce asort of metastable equilibrium. To this aim, we need an assumption on thenumber of possible roots of the reproduction rate ∃ ¯ k ≤ k, ∀ I . . . I k ∈ [0 , , the function k X i =1 I i η i ( x ) − k roots . (1.10)We also require an invertibility condition on the matrix η i ( x j ) ∀ x . . . x ¯ k distinct, the ¯ k vectors of coordinates ( η i ( x j )) i =1 ... ¯ k are free(1.11)Then we may uniquely define the metastable measure associated with a set ω by Proposition 1.1
For any closed ω ⊂ R , there exists a unique finite nonneg-ative measure µ ( ω ) satisfyingi) supp µ ⊂ ω ii) denoting ¯ I i ( µ ) = 1 / (1 + R η i ( x ) dµ ( x )) , k X i =1 ¯ I i ( µ ) η i ( x ) − ≤ in ω, k X i =1 ¯ I i ( µ ) η i ( x ) − on supp µ. I i are directly obtained by I i ( t ) = ¯ I i ( µ ( { ϕ ( t, . ) = 0 } )) . (1.12)We prove Theorem 1.1
Assume K ∈ C ∞ c ( R ) , R R zK ( z ) dz = 0 , (1.9) , (1.10) , (1.11) ,that the initial data u ε ( t = 0) > or ϕ ε ( t = 0) are C , satisfy sup ε Z R u ε ( t = 0 , x ) dx < ∞ , sup ε k ∂ x ϕ ε ( t = 0 , · ) k L ∞ ( R ) < ∞ , (1.13)inf ε inf x ∈ R ∂ xx ϕ ( t = 0 , x ) > −∞ , (1.14) and that ϕ ε ( t = 0 , · ) converges to a function ϕ for the norm k · k W , ∞ ( R ) .Then up to an extraction in ε , ϕ ε converges to some ϕ uniformly on anycompact subset of [0 , T ] × R and in W ,p ([0 , T ] × K ) for any T > , p < ∞ andany compact K . In particular, ϕ is continuous. The function I εi converges to I i in L p ([0 , T ]) for any T > , p < ∞ , where I i is defined from ϕ as in (1.12),and I i is approximately right-continuous for all t ≥ . The function ϕ is asolution to (1.8) almost everywhere in t, x with initial condition ϕ ( t = 0 , · ) = ϕ . Moreover if one defines ψ = ϕ − P ki =1 R t I i ( s ) ds η i ( x ) , then ψ is aviscosity solution to ∂ t ψ ( t, x ) = H ∂ x ψ + k X i =1 Z t I i ( s ) ds η i ( x ) ! . (1.15)We recall that a function f on [0 , + ∞ ) is approxiamtely right-continuous at t ≥ t is a point of Lebesgue right-continuity of f , i.e.lim s → s Z t + st | f ( θ ) − f ( t ) | dθ = 0 . Notice that, under the assumptions of Theorem 1.1, ϕ ε ( t = 0 , x ) → −∞ when x → ±∞ since R R u ε < ∞ and ϕ ε is uniformly Lipschitz. Be alsocareful that we assume k ϕ ε ( t = 0 , · ) − ϕ k W , ∞ ( R ) → ϕ ε ( t = 0)(and thus ϕ ) is not bounded.In the proofs below, C denotes a numerical constant which may change fromline to line. 6 Proof of Prop. 1.1
Assume that two measures µ and µ satisfy both points of Prop. 1.1. Wefirst prove that they induce the same ressources ¯ I i and then conclude thatthey are equal. ¯ I i . The argument here is essentially an adaptationof [20]. First note that Z R k X i =1 ¯ I i ( µ ) η i ( x ) − ! dµ + Z R k X i =1 ¯ I i ( µ ) η i ( x ) − ! dµ ≤ , (2.1)since µ and µ are non negative and by the point ii , P ki =1 ¯ I i ( µ j ) η i ( x ) − ω for j = 1 , P ki =1 ¯ I i ( µ j ) η i − µ j , onehas for instance Z R k X i =1 ¯ I i ( µ ) η i − ! dµ = Z R k X i =1 ( ¯ I i ( µ ) − ¯ I i ( µ )) η i ! dµ = k X i =1 ( ¯ I i ( µ ) − ¯ I i ( µ )) Z R η i dµ = k X i =1 ( ¯ I i ( µ ) − ¯ I i ( µ )) (1 / ¯ I i ( µ ) − , by the definition of ¯ I i ( µ ).Since one has k X i =1 ( ¯ I i ( µ ) − ¯ I i ( µ )) (1 / ¯ I i ( µ ) − / ¯ I i ( µ )) = k X i =1 ( ¯ I i ( µ ) − ¯ I i ( µ )) ¯ I i ( µ ) ¯ I i ( µ ) ≥ . one deduces from (2.1) that¯ I i ( µ ) = ¯ I i ( µ ) , i = 1 . . . k. (2.2) µ . It is not possible to deduce that µ = µ directlyfrom (2.2). This degeneracy (the possibility of having several equilibrium7easures, all corresponding to the same environment) is the reason why werequire additional assumptions on the η i .First of all by Assumption (1.10), point i and thanks to (2.2), we know that µ and µ are supported on at most ¯ k points { x , . . . , x ¯ k } , which are theroots of P i ¯ I i ( µ ) η i ( x ) − P i ¯ I i ( µ ) η i ( x ) −
1. Therefore one may write µ j = ¯ k X l =1 α jl δ x l . Now (2.2) tells that R η i dµ = R η i dµ which means that ¯ k X l =1 α l η i ( x l ) = ¯ k X l =1 α l η i ( x l ) , ∀ i = 1 . . . k. To conclude it remains to use condition (1.11) and get that α l = α l . The basic idea to get existence is quite simple: Solve the equation ∂ t ν = k X i =1 ¯ I i ( ν ) η i ( x ) − ! ν, (2.3)and obtain the equilibrium measure µ as the limit of ν ( t ) as t → + ∞ .This is done by considering the entropy L ( ν ) = k X i =1 log ¯ I i ( ν ) + Z dν = − k X i =1 log (cid:16) Z η i dν (cid:17) + Z dν. (2.4)As − log is convex and η i ≥
0, then L itself is a convex function of ν .Moreover if ν ( t ) solves (2.3), one has ddt L ( ν ( t )) = − Z k X i =1 ¯ I i ( ν ) η i ( x ) − ! dν. (2.5) Existence and uniqueness are trivial for (2.3), for example by Cauchy-Lipschitz theo-rem in the set of finite positive measures equipped with the total variation norm. ν and the equilibrium measure we arelooking for to be the minimum of L .Since the η i are bounded, one finds L ( ν ) ≥ − C + c Z dν, for two numerical constants C and c . Consequently L is bounded from belowon M ( ω ) the set of nonnegative Radon measures on ω . It also gets itsinfimum on a bounded part of M ( ω ). As any ball of M ( ω ) is compact forthe weak-* topology (dual of continuous functions with compact support), L attains its infimum, or M = { ν ∈ M ( ω ) , L ( ν ) ≤ L ( ν ′ ) ∀ ν ′ ∈ M ( ω ) } 6 = ∅ . Now take any µ ∈ M then take ν the solution to (2.3) with ν ( t = 0) = µ . L ( ν ) is non increasing and since it is already at a minimum initially, it isnecessarily constant. By (2.5), this means that k X i =1 ¯ I i ( µ ) η i − µ j . Hence µ is in fact a stationary solution to (2.3) and it satisfies point i andthe second part of point ii of Prop. 1.1. Note by the way that the uniquenessargument in fact tells that there is a unique element in M .It only remains to check the first part of point ii . By contradiction assumethat there exists a point x ∈ ω s.t. k X i =1 ¯ I i ( µ ) η i ( x ) − > . Let α > ν α = µ + αδ x ∈ M ( ω ). Now compute L ( ν α ) = Z dµ + α − k X i =1 log (cid:18) Z η i dµ + αη i ( x ) (cid:19) = Z dµ + α − k X i =1 (cid:18) log (cid:18) Z η i dµ (cid:19) + α η i ( x )1 + R η i dµ + O ( α ) (cid:19) = L ( µ ) − α k X i =1 ¯ I i ( µ ) η i ( x ) − ! + O ( α ) . L ( ν α ) < L ( µ ) for α small enough which is impossible as µ is an absoluteminimum of L .Consequently the first part of ii is satisfied and the proof of Prop. 1.1 com-plete. (1.5) We denote by BV loc ( R ) the set of functions on R with bounded variation onany compact subset of R , by M ( ω ) the set of signed Radon measures on thesubset ω of R equipped with the total variation norm.We show the following estimates on the solution to (1.5) Lemma 3.1
Let ϕ ε be a solution to (1.5) with initial data ϕ ε such that R R e ϕ ε ( x ) /ε dx < ∞ , ∂ x ϕ ε ∈ L ∞ ( R ) and ∂ xx ϕ ε uniformly lower bounded. Thenfor any T > k ∂ t ϕ ε k L ∞ ([0 ,T ] × R ) + k ∂ x ϕ ε k L ∞ ([0 ,T ] ,BV loc ( R ) ∩ L ∞ ( R )) + k ∂ tx ϕ ε k L ∞ ([0 ,T ] ,M ) ≤ C T , ∀ t ≤ T, x ∈ R , ∂ xx ϕ ε ( t, x ) ≥ − C T , H ε ( ϕ ε ) ≥ − C T ε, ∀ t ≤ T, Z R u ε ( t, x ) dx ≤ C T , ϕ ε ( t, x ) ≤ C T ε log 1 /ε. where C T only depends on the time T , R u ε ( t = 0) dx , k ∂ x ϕ ε k L ∞ ( R ) and inf x ∂ xx ϕ ε ( x ) . Proof.
We start with the easy bound on the total mass.
Step 0: Bound on the total mass.
First notice that because of (1.9), thereexists
R > ∀| x | > R, k X i =1 η i ( x ) ≤ / . Let ψ be a regular test function with support in | x | > R , taking values in100 ,
1] and equal to 1 on | x | > R +1. Using the fact that I εi ( t ) ≤
1, we compute ddt Z R ψ ( x ) u ε ( t, x ) dx ≤ − ε Z R ψ ( x ) u ε ( t, x ) dx + 1 ε Z R K ( z )( ψ ( x − εz ) − ψ ( x )) u ε ( t, x ) dz dx ≤ − ε Z R ψ ( x ) u ε ( t, x ) dx + C Z R u ε ( t, x ) dx. On the other hand as each η i >
0, one has for some constant CI εi ( t ) = 11 + R η i u ε dx ≤ C R (1 − ψ ) u ε dx . Therefore with the same kind of estimate ddt Z R (1 − ψ ( x )) u ε ( t, x ) dx ≤ C Z R u ε ( t, x ) dx + 1 ε C R (1 − ψ ) u ε dx − ! Z R (1 − ψ ( x )) u ε ( t, x ) dx. Summing the two ddt Z R u ε ( t, x ) dx ≤ ε (cid:18) C R (1 − ψ ) u ε dx − (cid:19) Z R (1 − ψ ( x )) u ε ( t, x ) dx − ε Z R ψ ( x ) u ε ( t, x ) dx + C Z R u ε ( t, x ) dx. Since the sum of the first two terms of the r.h.s. is negative if R u ε is largerthan a constant independent of ε , this shows that R u ε ( t, x ) dx remains uni-formly bounded on any finite time interval. Step 1: Bound on ∂ x ϕ ε . This is a classical bound for solutions to someHamilton-Jacobi equations. Here we still have to check that it remains trueuniformly at the ε level. Compute ∂ t ∂ x ϕ ε = k X i =1 I εi ( t ) η ′ i ( x )+ Z K ( z ) e ϕε ( t,x + εz ) − ϕε ( t,x ) ε ∂ x ϕ ε ( t, x + εz ) − ∂ x ϕ ε ( t, x ) ε dz.
11e first observe that, as I εi ∈ [0 ,
1] and P i | η ′ i ( x ) | ≤ ¯ η ( x ) | ∂ t ∂ x ϕ ε | ≤ ¯ η ( x ) + 2 ε Z R K ( z ) e | z | k ∂ x ϕ ε ( t, · ) k L ∞ ( R ) k ∂ x ϕ ε ( t, · ) k L ∞ ( R ) dz ≤ Cε e C k ∂ x ϕ ε ( t, · ) k L ∞ ( ρ ) , since K has compact support. This entails k ∂ x ϕ ε ( t, · ) k L ∞ ( R ) ≤ Cε Z t e C k ∂ x ϕ ε ( s, · ) k L ∞ ( R ) ds, from which easily follows that ∂ x ϕ ε ∈ L ∞ ([0 , t ε ] , R ) for some t ε >
0, whichmay (for the moment) depend on ε .Now we use the classical maximum principle. Fix t ∈ [0 , T ] such that C ε,t := k ∂ x ϕ ε ( t, · ) k L ∞ ( R ) < ∞ . For any x ∈ R such that ∂ x ϕ ε ( t, x ) > sup y ∂ x ϕ ε ( t, y ) − α , where the constant α > ∂ t ∂ x ϕ ε ( t, x ) ≤ ¯ η ( x ) + Z R K ( z ) e | z | C t,ε αε dz ≤ C (cid:16) αε e C C t,ε (cid:17) . Therefore, choosing α = εe − C C t,ε , we obtain ddt sup x ∂ x ϕ ε ( t, x ) ≤ C, for a constant C independent of ε . Using a similar argument for the mini-mum, we deduce that t ε > T and that ∂ x ϕ ε is bounded on [0 , T ] × R by aconstant depending only on T and k ∂ x ϕ ε k L ∞ ( R ) . Step 2: First bound on H ε ( ϕ ε ) and bounds on ∂ t ϕ ε and ϕ ε . Simply note that − ≤ H ε ( ϕ ε ( t ))( x ) = Z R K ( z ) e ϕε ( t,x + εz ) − ϕε ( t,x ) ε dz − Z R K ( z ) dz ≤ Z K ( z ) e | z | k ∂ x ϕ ε k L ∞ ([0 ,T ] , R ) dz ≤ C. Consequently, directly from Eq. (1.5), | ∂ t ϕ ε | ≤ ¯ η ( x ) + C, hence concluding the full Lipschitz bound on ϕ ε .12o get the upper bound on ϕ ε , simply note that because of the uniformLipschitz bound on ϕ ε ϕ ε ( t, y ) ≥ ϕ ε ( t, x ) − C T | y − x | , so Z R u ε ( t, y ) dy ≥ Z R e ϕ ε ( t,x ) /ε e − C T | y − x | /ε dy ≥ C T ε e ϕ ε ( t,x ) /ε . Hence the bound on the total mass yields that ϕ ε ≤ C T ε log 1 /ε . Step 3: BV bound on ∂ x ϕ ε . As for ∂ x ϕ ε , we begin with a maximum (actually,minimum) principle. First from (1.5) ∂ t ∂ xx ϕ ε ≥ − ¯ η ( x ) + Z R K ( z ) e ϕε ( t,x + εz ) − ϕε ( t,x ) ε ∂ xx ϕ ε ( t, x + εz ) − ∂ xx ϕ ε ( t, x ) ε dz + Z R K ( z ) e ϕε ( t,x + εz ) − ϕε ( t,x ) ε ( ∂ x ϕ ε ( t, x + εz ) − ∂ x ϕ ε ( t, x )) ε dz. The last term is of course non negative and so with the same argument asbefore, we get ddt inf x ∂ xx ϕ ε ( t, x ) ≥ − C, where C does not depend on ε . This proves the uniform lower bound on ∂ xx ϕ ε . On the other hand, for any measurable subset A of [ x , x ], Z x x ( I x ∈ A − I x A ) ∂ xx ϕ ε ( t, x ) dx = Z x x ∂ xx ϕ ε ( t, x ) dx − Z x x ∂ xx ϕ ε ( t, x ) I x A dx ≤ ∂ x ϕ ε ( t, x ) − ∂ x ϕ ε ( t, x ) + C | x − x | ≤ k ∂ x ϕ ε k L ∞ ([0 ,T ] , R ) + C | x − x | . This indeed shows that ∂ xx ϕ ε ( t, · ) belongs to M ([ x , x ]) with total variationnorm less than 2 k ∂ x ϕ ε k L ∞ ([0 ,T ] , R ) + C | x − x | . Thus, ∂ xx ϕ ε belongs to thespace L ∞ ([0 , T ] , M ( R )), which entails ∂ x ϕ ε ∈ L ∞ ([0 , T ] , BV loc ( R )).Finally, differentiating (1.5) once in x , one has | ∂ tx ϕ ε ( t, x ) | ≤ ¯ η ( x ) + Z K ( z ) e | z | k ∂ x ϕ ε k L ∞ | ∂ x ϕ ε ( t, x + εz ) − ∂ x ϕ ε ( t, x ) | ε dz ≤ ¯ η ( x ) + C Z K ( z ) Z z | ∂ xx ϕ ε ( t, x + εθ ) | dθdz. Z x x | ∂ tx ϕ ε | dx ≤ Z x x ¯ η ( x ) dx + C Z x + ερx − ερ | ∂ xx ϕ ε | dx, where ρ is such that the support of K is included in the ball centered at 0 ofradius ρ . This ends the proof of all the bounds on the derivatives of ϕ ε . Conclusion.
It only remains to show the sharp lower bound on H ε ( ϕ ε ). Letus write H ε ( ϕ ε ) ≥ Z R K ( z ) exp (cid:18)Z z ∂ x ϕ ε ( t, x + θz ε ) dθ (cid:19) dz − Z R K ( z ) dz. The BV bound on ∂ x ϕ ε shows that this function admits right and left limitsat all x ∈ R . Let us denote ∂ x ϕ ε ( t, x + ) the limit on the right and ∂ x ϕ ε ( t, x − )the limit on the left. As ∂ xx ϕ ε is bounded from below, we know in additionthat ∀ x, ∂ x ϕ ε ( t, x + ) ≥ ∂ x ϕ ε ( t, x − ) . By differentiating once more Z z ∂ x ϕ ε ( t, x + θz ε ) dθ ≥ z ∂ x ϕ ε ( t, x + )+ Z z Z θ z ε ∂ xx ϕ ε ( t, x + θ ′ θzε ) dθ ′ dθ ≥ z ∂ x ϕ ε ( t, x + ) − C ε z , again as ∂ xx ϕ ε is bounded from below. Finally H ε ( ϕ ε ) ≥ Z R K ( z ) exp( z ∂ x ϕ ε ( t, x + ) − C ε z ) dz − Z R K ( z ) dz ≥ H ( ∂ x ϕ ε ( t, x + )) − C ε, where H is defined as in (1.7) and since K is compactly supported. Becausewe assumed that R R zK ( z ) dz = 0, we have H ( p ) ≥ p , which endsthe proof of Lemma 3.1. 14 .2 Passing to the limit in ϕ ε From the assumptions in Theorem 1.1, Lemma 3.1 gives uniform bounds on ϕ ε .Therefore up to an extraction in ε (still denoted with ε ), there exists a func-tion ϕ on [0 , T ] × R such that ∂ t ϕ ∈ L ∞ ([0 , T ] × R ), ∂ x ϕ ∈ L ∞ ([0 , T ] , BV loc ∩ L ∞ ( R )), ∂ tx ϕ ∈ L ∞ ([0 , T ] , M ( R )) and ∂ xx ϕ uniformly lower bounded on[0 , T ] × R , satisfying ϕ ε −→ ϕ uniformly in C ( K ) for any compact K of [0 , T ] × R ,∂ x ϕ ε −→ ∂ x ϕ in any L p loc ([0 , T ] , R ) , p < ∞ . (3.1)The first convergence follows from Arz´ela-Ascoli theorem. For the secondconvergence, observe that k ∂ x ϕ ε k L ∞ ([0 ,T ] ,BV loc ( R )) + k ∂ tx ϕ ε k L ∞ ([0 ,T ] ,M ( R )) ≤ C T implies that ∂ x ϕ ε is uniformly bounded in L ∞ ([0 , T ] × R ) ∩ BV loc ([0 , T ] × R ).The convergence in L p loc follows by compact embedding. We also have ϕ ≤ R R u ε ( t, x ) dx would be contradicted.As the I εi are bounded, it is possible to extract weak-* converging subse-quences (still denoted with ε ) to some I i ( t ).Now, we write again H ε ( ϕ ε ) = Z R K ( z ) (cid:18) exp (cid:18)Z z ∂ x ϕ ε ( t, x + ε z θ ) dθ (cid:19) − (cid:19) dz. From the L ∞ bound on ∂ x ϕ ε and its strong convergence, one deduces that H ε ( ϕ ε ) −→ H ( ∂ x ϕ ) in L loc . (3.2)Therefore one may pass to the limit in (1.5) and obtain (1.8) (for the momentin the sense of distribution; the equality a.e. will follow from the convergenceof I εi in L p ([0 , T ]), proved below).In addition by following [15] or [2], one may easily show that ψ ( t, x ) = ϕ ( t, x ) − P ki =1 R t I i ( s ) ds η i ( x ) is a viscosity solution to (1.15). We refer thereader to [15] or [2] for this technical part.It remains to obtain (1.12), the approximate right-continuity of I i for all time t and the convergence of I ǫi to I i in L p ([0 , T ]) for p < ∞ . This requires somesort of uniform continuity on the I εi which is the object of the rest of theproof. 15 .3 Continuity in time for the I εi First of all note that, as suggested by the simulations of [15], there areexamples where the I i have jumps in time at the limit. So we will onlybe able to prove their right-continuity.This regularity in time comes from the stability of the equilibrium definedthrough (1.12) and Prop. 1.1. Therefore let us define¯ I i ( t ) = ¯ I i ( µ ( { ϕ ( t, . ) } )) , where ¯ I i and µ are given by Prop. 1.1 and ϕ is the uniform limit of ϕ ε astaken in the previous subsection.Our first goal is the following result. Lemma 3.2
For any fixed s , there exist functions σ s , ˜ σ ∈ C ( R + ) with σ s (0) = ˜ σ (0) = 0 s.t. Z ts | I εi ( r ) − ¯ I i ( s ) | dr ≤ ( t − s ) σ s ( t − s ) + ˜ σ ( ε ) . Remark.
Of course the whole point is that σ s and ˜ σ are uniform in ε . It isalso crucial for the following that ˜ σ does not depend on s . Step 0: ϕ has compact level sets. Observe that ϕ ε ( t = 0 , x ) → −∞ when x → ±∞ since R R u ε ( t = 0 , x ) dx < ∞ and ∂ x ϕ ( t = 0) is bounded. Because of the uniform convergence of ϕ ε ( t =0) to ϕ on R , one deduces that ϕ ( x ) → −∞ when x → ±∞ .Since ∂ x ϕ ∈ L ∞ ([0 , T ] , R ) and I i ( t ) ∈ [0 , ∂ t ϕ ∈ L ∞ ([0 , T ] , R ) and thus ϕ ( t, x ) → −∞ when x → ±∞ for all t ≥ { ( t, x ) ∈ [0 , T ] × R : ϕ ( t, x ) ≥ − } is compact. Step 1: One basic property of { ϕ = 0 } . Let us start by the following crucial observation ∀ s, ∃ τ s ∈ C ( R + ) with τ s (0) = 0 , s.t. ∀ t ≥ s, ∀ x ∈ { ϕ ( t, . ) = 0 } , ∃ y ∈ { ϕ ( s, . ) = 0 } with | y − x | ≤ τ s ( t − s ) . (3.3)16his is a sort of semi-continuity for { ϕ = 0 } . It is proved very simply bycontradiction. If it were not true, then ∃ s, ∃ τ > , ∃ t n → s, t n ≥ s, ∃ y n ∈ { ϕ ( t n , . ) = 0 } ,d ( y n , { ϕ ( s, . ) = 0 } ) ≥ τ , where d ( y, ω ) = inf x ∈ ω | x − y | is the usual distance.Since all the y n belong to the compact set Ω of Step 0, we can extracta converging subsequence y n → y . As ϕ is continuous, ϕ ( s, y ) = 0 or y ∈{ ϕ ( s, . ) = 0 } . On the other hand one would also have d ( y, { ϕ ( s, . ) = 0 } ) ≥ τ which is contradictory. Step 2: The functional.
Denote µ s = µ ( { ϕ ( s, . ) = 0 } , as given by Prop. 1.1. We look at the evolution of F ε ( t ) = Z R log u ε ( t, x ) dµ s ( x ) = 1 ε Z R ϕ ε ( t, x ) dµ s ( x ) , for t ≥ s . Compute ddt F ε ( t ) = 1 ε Z R k X i =1 I εi ( t ) η i ( x ) − ! dµ s ( x ) + 1 ε Z R H ε ( ϕ ε ( t )) dµ s . Now write1 ε Z R k X i =1 I εi ( t ) η i ( x ) − ! dµ s ( x ) = ddt Z R u ε ( t, x ) dx − ε Z R k X i =1 I εi ( t ) η i ( x ) − ! ( u ε ( t, x ) dx − dµ s ( x )) . As P ki =1 ¯ I i ( s ) η i ( x ) − µ s ,1 ε Z R k X i =1 I εi ( t ) η i ( x ) − ! dµ s ( x ) = ddt Z R u ε ( t, x ) − A ( t ) ε − ε Z R k X i =1 ( I εi ( t ) − ¯ I i ( s )) η i ( x ) ! ( u ε ( t, x ) dx − dµ s ( x )) , A ( t ) = Z R k X i =1 ¯ I i ( s ) η i ( x ) − ! u ε ( t, x ) dx. Notice that Z R k X i =1 ( I εi ( t ) − ¯ I i ( s )) η i ( x ) ! ( u ε ( t, x ) dx − dµ s ( x )) = − k X i =1 ( I εi ( t ) − ¯ I i ( s )) I εi ( t ) ¯ I i ( s ) . So we deduce1 ε Z ts k X i =1 ( I εi ( r ) − ¯ I i ( s )) I εi ( r ) ¯ I i ( s ) dr = Z R log u ε ( t, x ) u ε ( s, x ) dµ s − Z R ( u ε ( t, x ) − u ε ( s, x )) dx + Z ts A ( r ) ε dr − ε Z ts Z R H ε ( ϕ ε ( r )) dµ s . (3.4) Step 3: Easy bounds.
Lemma 3.1 tells that − H ε ( ϕ ε ) ≤ C T ε. The total mass stays bounded in time so − Z R ( u ε ( t, x ) − u ε ( s, x )) dx ≤ Z R ( u ε ( t, x )+ u ε ( s, x )) dx ≤ C. And furthermore Z R log u ε ( t, x ) u ε ( s, x ) dµ s = 1 ε Z R ( ϕ ε ( t, x ) − ϕ ε ( s, x )) dµ s ≤ ε Z R ( ϕ ( t, x ) − ϕ ( s, x )) dµ s + 2 ε k ϕ ε − ϕ k L ∞ (Ω) , where the last bound comes from the fact that, by Prop. 1.1, µ s is supportedon { ϕ ( s, . ) = 0 } ⊂ Ω, where Ω is defined in Step 0. Since in addition weknow that ϕ ≤ Z R log u ε ( t, x ) u ε ( s, x ) dµ s ≤ ε k ϕ ε − ϕ k L ∞ (Ω) . ε Z ts k X i =1 ( I εi ( r ) − ¯ I i ( s )) I εi ( r ) ¯ I i ( s ) dr ≤ C + 2 ε k ϕ ε − ϕ k L ∞ (Ω) + Z ts A ( r ) ε dr. (3.5) Step 4: Control on A and the measure of { x, ϕ ε ∼ } . For some α ε to be chosen later, decompose Z ts A ( r ) dr = Z ts Z R k X i =1 ¯ I i ( s ) η i ( x ) − ! u ε ( r, x ) I ϕ ε ( r,x ) ≤− α ε dx dr + Z ts Z R k X i =1 ¯ I i ( s ) η i ( x ) − ! u ε ( r, x ) I ϕ ε ( r,x ) ≥− α ε dx dr. For the first part, note again that by (1.9), there exists R s.t. ∀| x | > R, k X i =1 ¯ I i ( s ) η i ( x ) ≤ / . Therefore we may simply dominate Z ts Z R k X i =1 ¯ I i ( s ) η i ( x ) − ! u ε ( r, x ) I ϕ ε ( r,x ) ≤− α ε dx dr ≤ C ( t − s ) e − α ε /ε . Concerning the second part, we constrain 1 / ≥ α ε ≥ k ϕ − ϕ ε k L ∞ (Ω) andmay therefore bound Z ts k X i =1 ¯ I i ( s ) η i ( x ) − ! u ε ( r, x ) I ϕ ε ( r,x ) ≥− α ε ≤ Z ts k X i =1 ¯ I i ( s ) η i ( x ) − ! u ε ( r, x ) I ϕ ( r,x ) ≥− α ε . Now P ki =1 ¯ I i ( s ) η i ( x ) − { ϕ ( s, . ) = 0 } and so k X i =1 ¯ I i ( s ) η i − ! I ϕ ( r,. )=0 ≤ C sup x ∈{ ϕ ( r,. )=0 } inf y ∈{ ϕ ( s,. )=0 } | y − x | ≤ C τ s ( t − s ) ,
19y Step 1 as the η i are uniformly Lipschitz. For two sets O and O , definein general δ ( O , O ) = sup x ∈ O inf y ∈ O | x − y | . By the same argument, one gets k X i =1 ¯ I i ( s ) η i − ! I ϕ ( r,. ) ≥− α ε ≤ C τ s ( t − s )+ C δ ( { ϕ ( r, . ) ≥ − α ε } , { ϕ ( r, . ) = 0 } ) . Inequality (3.4) now becomes Z ts k X i =1 ( I εi ( r ) − ¯ I i ( s )) I εi ( r ) ¯ I i ( s ) dr ≤ C ε + 2 k ϕ ε − ϕ k L ∞ (Ω) + C ( t − s ) e − α ε /ε + C Z ts τ s ( r − s ) dr + C Z ts δ ( { ϕ ( r, . ) ≥ − α ε } , { ϕ ( r, . ) = 0 } ) . (3.6) Conclusion.
Eq. (3.4) indeed gives Lemma 3.2 if one defines σ s ( t − s ) = 1 t − s Z ts τ s ( r − s ) ds, ˜ σ ( ε ) = Cε + 2 k ϕ ε − ϕ k L ∞ (Ω) + C T e − α ε /ε + C Z T δ ( { ϕ ( r, . ) ≥ − α ε } , { ϕ ( r, . ) = 0 } ) dr. Of course σ s is continuous and, as τ s (0) = 0, then trivially σ s (0) = 0. Since { ϕ ( r, . ) ≥ − α ε } and { ϕ ( r, . ) = 0 } are subsets of Ω, ˜ σ ( ε ) is bounded for ε ≤
1, and thus, in order to complete the proof of Lemma 3.2, we only haveto check that ˜ σ ( ε ) → ε → α ε . If we take α ε ≥ k ϕ ε − ϕ k L ∞ (Ω) converging to 0 slowly enough to have α ε /ε → + ∞ , weonly have to prove that C Z T δ ( { ϕ ( r, . ) ≥ − α ε } , { ϕ ( r, . ) = 0 } ) dr −→ ε → . By dominated convergence it is enough that for any rδ ( { ϕ ( r, . ) ≥ − α ε } , { ϕ ( r, . ) = 0 } ) −→ . Just as in Step 1 this is a direct consequence of the continuity of ϕ .20 .4 Compactness of the I εi and the obtention of (1.12) First notice that simply passing to the limit in Lemma 3.2
Lemma 3.3 ∃ σ s , ∈ C ( R + ) with σ s (0) = 0 s.t. ∀ i Z ts | I i ( r ) − ¯ I i ( s ) | dr ≤ ( t − s ) σ s ( t − s ) . This means that at any point of Lebesgue continuity of I i , one has I i = ¯ I i .We recall that a.e. point is a Lebesgue point for I i . As the I i were definedonly almost everywhere anyhow (they are weak-* limits), we may identify I i and ¯ I i . This proves (1.12) and that ¯ I i is approximately continuous on theright for any time t (and not only a.e. t ).Now let us prove the compactness in L of each I εi . We apply the usualcriterion and hence wish to control Z T h Z s + hs | I εi ( t ) − I εi ( s ) | dt ds. Decompose Z T h Z s + hs | I εi ( t ) − I εi ( s ) | dt ds ≤ Z T h Z s + hs | I εi ( t ) − I i ( s ) | dt ds + Z T h Z s + hs | I i ( t ) − I εi ( s ) | dt ds + Z T h Z s + hs | I i ( t ) − I i ( s ) | dt ds. The first and third terms are bounded directly from Lemmas 3.2 and 3.3, forexample by Cauchy-Lipschitz Z T h Z s + hs | I εi ( t ) − I i ( s ) | dt ds ≤ Z T (cid:18) h Z s + hs | I εi ( t ) − I i ( s ) | dt (cid:19) / ds ≤ Z T ( σ s ( h ) + ˜ σ ( ε ) /h ) / ds. Z T h Z s + hs | I εi ( s ) − I i ( t ) | dt ds = Z T + h h Z t max(0 ,t − h ) | I εi ( s ) − I i ( t ) | ds dt ≤ Z T + h (cid:18) h Z t max(0 ,t − h ) | I εi ( s ) − I i ( t ) | ds (cid:19) / dt ≤ Z T + h ( σ t ( h ) + ˜ σ ( ε ) /h ) / dt. So finally we bound Z T h Z s + hs | I εi ( t ) − I εi ( s ) | dt ds ≤ Z T + h ( σ s ( h ) + ˜ σ ( ε ) /h ) / ds ≤ Z T + h p σ s ( h ) ds + 3( T + h ) p ˜ σ ( ε ) /h. Since of course the functions σ s ( · ) can be chosen uniformly bounded inLemma 3.2, again by dominated convergence, this shows that ∀ τ > ∃ h , ∃ ε ( h ) s.t. ∀ ε < ε ( h ) Z T h Z s + hs | I εi ( t ) − I εi ( s ) | dt ds ≤ τ. This is enough to get compactness of the I εi in L and then in any L p loc with p < ∞ , which concludes the proof of Theorem 1.1. References [1] G. Barles, S. Mirrahimi, B. Perthame, Concentration in Lotka-Volterraparabolic or integral equations: a general convergence result.
MethodsAppl. Anal. , No. 3, 321–340 (2009).[2] G. Barles, B. Perthame, Concentrations and constrained Hamilton-Jacobi equations arising in adaptive dynamics. Recent developments innonlinear partial differential equations , 57–68, Contemp. Math., , Amer. Math. Soc., Providence, RI,
Adv. Appl. Prob. , , 227–251 (1996).[4] Calsina, A., Cuadrado, S. Small mutation rate and evolutionarily stablestrategies in infinite dimensional adaptive dynamics. J. Math. Biol. , ,135–159 (2004).[5] Carrillo, J. A., Cuadrado, S., Perthame, B. Adaptive dynamics viaHamilton-Jacobi approach and entropy methods for a juvenile-adultmodel. Math. Biosci. , 137–161 (2007).[6] N. Champagnat. A microscopic interpretation for adaptive dynamicstrait substitution sequence models.
Stoch. Proc. Appl. , , 1127-1160(2006).[7] N. Champagnat, R. Ferri`ere, G. Ben Arous. The canonical equation ofadaptive dynamics: a mathematical view. Selection , , 71–81.[8] N. Champagnat, R. Ferri`ere, S. M´el´eard. From individual stochasticprocesses to macroscopic models in adaptive evolution, Stoch. Models , suppl. 1, 2–44 (2008).[9] N. Champagnat, S. M´el´eard. Polymorphic evolution sequence and evo-lutionary branching. To appear in Probab. Theory Relat. Fields (pub-lished online, 2010).[10] R. Cressman, J. Hofbauer. Measure dynamics on a one-dimensionalcontinuous trait space: theoretical foundations for adaptive dynamics.
Theor. Pop. Biol. , , 47–59 (2005).[11] L. Desvillettes, P.-E. Jabin, S. Mischler, G. Raoul. On selection dy-namics for continuous structured populations. Commun. Math. Sci. , , 729–747 (2008).[12] Dieckmann, U. and Law, R. The dynamical theory of coevolution: aderivation from stochastic ecological processes. J. Math. Biol. , , 579–612 (1996).[13] O. Diekmann, A beginner’s guide to adaptive dynamics. In Mathemati-cal modelling of population dynamics, Banach Center Publ. , , 47–86,Polish Acad. Sci., Warsaw (2004).2314] O. Diekmann, M. Gyllenberg, H. Huang, M. Kirkilionis, J.A.J. Metz,H.R. Thieme. On the formulation and analysis of general deterministicstructured population models. II. Nonlinear theory. J. Math. Biol. ,157–189 (2001).[15] O. Diekmann, P.E. Jabin, S. Mischler, B. Perthame, The dynamics ofadaptation: An illuminating example and a Hamilton-Jacobi approach. Theor. Popul. Biol. , 257–271 (2005).[16] Genieys, S., Bessonov, N., Volpert, V. Mathematical model of evolu-tionary branching. Math. Comput. Modelling , , 2109–2115(2009).[17] Geritz, S. A. H., Metz, J. A. J., Kisdi, E., Mesz´ena, G. Dynamics ofadaptation and evolutionary branching. Phys. Rev. Lett. , , 2024–2027 (1997).[18] Geritz, S. A. H., Kisdi, E., Mesz´ena, G., Metz, J. A. J. Evolution-ayr singular strategies and the adaptive growth and branching of theevolutionary tree. Evol. Ecol. , , 35–57 (1998).[19] J. Hofbauer, R. Sigmund. Adaptive dynamics and evolutionary stabil-ity. Applied Math. Letters , , 75–79 (1990).[20] P.E. Jabin, G. Raoul, Selection dynamics with competition. To appear J. Math Biol. .[21] Metz, J. A. J., Nisbet, R. M. and Geritz, S. A. H. How should wedefine ’fitness’ for general ecological scenarios?
Trends in Ecology andEvolution , , 198–202 (1992).[22] Metz, J. A. J., Geritz, S. A. H., Mesz´ena, G., Jacobs, F. A. J. andvan Heerwaarden, J. S. Adaptive Dynamics, a geometrical study ofthe consequences of nearly faithful reproduction. In: van Strien, S.J. & Verduyn Lunel, S. M. (ed.), Stochastic and Spatial Structures ofDynamical Systems , North Holland, Amsterdam, pp. 183–231 (1996).[23] S. Mirrahimi, G. Barles, B. Perthame, P. E. Souganidis, SingularHamilton-Jacobi equation for the tail problem.
Preprint .2424] Perthame, B., Gauduchon, M. Survival thresholds and mortality ratesin adaptive dynamics: conciliating deterministic and stochastic simu-lations.
IMA Journal of Mathematical Medicine and Biology , to appear(published online, 2009).[25] Perthame, B., G´enieys, S. Concentration in the nonlocal Fisher equa-tion: the Hamilton-Jacobi limit.
Math. Model. Nat. Phenom. , ,135–151 (2007).[26] F. Yu. Stationary distributions of a model of sympatric speciation, Ann. Appl. Probab. ,17