Dirac Concentrations in a Chemostat Model of Adaptive Evolution?

Alexander LORZ Beno?t PERTHAME Cécile TAING

(In honor of the immense scientific influence of Ha?m Brezis)

1 Introduction

We survey several methods developed to study concentration effects in parabolic equations of Lotka-Volterra type.Furthermore,we extend the theory to a coupled system motivated by models of chemostat,where we observe very rare mutations for a long time.These equations were established with the aim of describing how speciation occurs in biological populations,taking into account competition for resources and mutations in the populations.There is a large literature on the subject where the mutation-competition principles are illustrated in various mathematical terms:for instance,in[23,28,35]for an approach based on the study of the stability of differential systems,in[29–30,45]for the evolutionary games theory,in[14]for the study of stochastic individual based models,or in[6,36,42]for the study of integrodifferential models.We choose here the formalism using parabolic partial differential equations,widely developed in[5,7,21,41]to describe the competition dynamics in a chemostat.

The chemostat is a bioreactor to which fresh medium containing nutrients is continuously added,while culture liquid is continuously removed to keep the culture volume constant.This device is used as an experimental ecosystem in evolutionary biology to observe mutation and selection processes driven by competition for resources.From the mathematical point of view,the theoretical description of the population dynamics in a chemostat leads to highly nonlinear models and questions of long term behaviour and convergence to an evolutionary steady state naturally arise(see[1,19,24,39,44]).

Our aim is to study a generalization of the chemostat model introduced in[34]with a representation of mutationsby a diffusion term.In this model,each individual in the population is characterized by a quantitative phenotypic trait x ∈ Rd,and nε(t,x)denotes the population density at time t with the trait x.We study the following equations:

where the function R(x,Sε)represents a trait-dependent birth-death rate and Sεdenotes the nutrient concentration which changes over time with rate Q.Hereεis a small parameter which allows to consider very rare mutations and large time of orderε?1.The idea of an ε?1rescaling in the space and time variables goes back to[31–32]to study propagation for systems of reaction-diffusion PDE.The parameter β,introduced first in[34],gives a time scale which,asleads to the equation Q(ρ,S)=0.In this case,under suitable assumptions,we deduce the existence of a function f by implicit function theorem,such that S=f(ρ)and the concentration results are known to hold(see[7,33]).

Such models can be derived from stochastic individual based models in the limit of large populations(see[16–17]).

A possible way to express mathematically the emergence of the fittest traits among the population is to prove thatconcentrates as a Dirac mass centered on a point(or a sum of Dirac masses)whenεvanishes.This means the phenotypic selection of a quantitative trait denoted byin long time.The main results of the paper can be summarized as follows.

Theorem 1.1For well-prepared initial data and two classes of assumptions(monotonic in one dimension or concavity in multi-dimensions),the solution nε(t,x)concentrates,i.e.,

where the pairbe determined thanks to a constrained Hamilton-Jacobi equation given later on.

In order to describe these concentration effects,following earlier works on similar issues[5,7,13],we will use the Hopf-Cole transformation defining uε(t,x)= εln nε(t,x),and derive a Hamilton-Jacobi equation.Then we obtain by passing to the limita constrained Hamilton-Jacobi equation,whose solutions have a maximum value of 0.The point is that the concentration locations in the limitε→ 0 can be identified among the maximum points of these solutions.This method,introduced in[24]and used for instance in[42–43],is very general and was extended to various systems(see[18,33,41]).

Singular perturbation problems in PDEs is a classical subject that was studied from different viewpoints.For instance,a seminal paper on parabolic equations involving measures is[11].Also the above rescaling in parabolic equations or systems was deeply studied in reaction diffusion equations(see[4,25])leading to front propagation where a state invades another as in the Fishher-KPP equation where the stable state nε=1 invades the unstable state nε=0.This is also the case of Ginzburg-Landau equations(see[8]),where the quadratic observabletakes asymptotically the value 1.This is different from our case,as one can see in the above theorem and since we essentially derive L1bounds from the presented model.

To prove the main convergence results of this paper,we adapt the method introduced in[5,7,34]to find BV estimates for the appropriate quantities as a first step.Then we use the theory of viscosity solutions to Hamilton-Jacobi equations(see[2–3,20,27]for general introduction to this theory)to obtain the Dirac locations.In the first part,we proceed with assumptions of weak regularity of the growth rate in a first instance,and then we resume the study under concavity assumptions.

This paper is organized as follows.We first state(see Section 2)the framework of the general weak theory and its main results.We start the study by establishing BV estimates onand Sεin Section 3.Section 4 is devoted to the analysis of the solutions to the constrained Hamilton-Jacobi equations.We first prove some regularity results for uε.Then we study the asymptotic behaviour of uεand deduce properties of the concentration points.In Section 5,we set the simple case of our results when the dimension d equals 1 and prove concentration effects.In Section 6,we review the d-dimensional framework where we assume uniform concavity of the growth rate and initial conditions.We establish again the BV estimates in this specific case and prove the uniform concavity of uε.The regularity obtained for uεallows us to derive the dynamics of the concentration points in the form of a canonical equation.We complete these results by numerics in Section 7.

2 The Weak Theory:Assumptions and Main Results

First of all,we give assumptions to set a framework for the general weak theory.We use the same assumptions as in[34].

For the Lipschitz continuous functions R and Q,we assume that there are constants S0>0,andsuch that

We complete the system with the initial conditionssuch that

whereρm, ρMand Smare defined below.

We add to these assumptions a smallness condition onβwhich can be written as

with the definition ofρMstated below.

Note that from(2.1),we directly obtain the bounds

First we recall the following lemma,whose proof is given in[34].

Lemma 2.1Under the assumptions(2.1)–(2.4),there are constants ρm,ρMand Sm>0,such that

where the value Sm

This result is required to prove the following theorem.

Theorem 2.1Assuming also(2.5), ρε(t)and Sε(t)have locally bounded total variation uniformly in ε.Consequently,there are limit functionssuch that after extraction of a subsequence,we have

and

The next section is devoted to the proof of Theorem 2.2.Contrary to what we could expect,the establishment of the BV estimates will be more complicated than in the previous works(see[7,33]),where the nutrients are represented by an integral term asHere the main challenge comes from the equation(1.2)that we also have to consider to obtain BV estimates on Sε.Another difficulty comes from the parameter β.For β large enough,it seems that we cannot derive BV estimates with our approach,but anyway we do observe the convergence of the solutions in the numerics we performed.This is not the case for inhibitory integrate-and-fire models for instance(see[12]),where delays generate periodic solutions.

In the following proofs,C denotes a constant which may change from line to line.

3 BV Estimates on and Sε(t)

3.1 Bound s forρε

We follow the lines of[34]to give the boundsρmand ρM.By integrating the equation(1.1)and using the assumptions(2.2)–(2.3),we arrive to the inequalities

and

Notice thatfrom the as sumptions in(2.1).By adding the equation(1.2)to the in equation above,we arrive to

It follows that,for C2,the root in lnρε+βSεof the right-hand side,

Hence we have the upper bound ρMfor ρε(t).

Thanks to this upper bound,we obtain the lower bound Smon Sε(t),since,by using the assumption(2.1)on Q,we remark that

Then there is a unique value Smsuch that Q(Sm,ρM)=0,and from the initial conditions(2.4),we deduce that Sm≤ Sε(t)for t≥ 0.

Next,let us look for the lower bound.It follows,from the integration of(1.1)as above,that we have

By subtracting(1.2)and still using(2.1),we obtain

Taking C3the root in lnρε?βSεof the right-hand side in(3.3),we have the lower bound

which ends the proof of Lemma 2.1.

3.2 Local B V estimates

To find local BV bounds forρεand Sεwhich are uniform in ε>0,we apply the method described in[34]that we explain in detail in this section.

Let us first defineWith these definitions,we have the equations

Definingαεandγεas

respectively,we differentiate both equations above,then we obtain the following equations on Jεand Pε:

However,at this stage,we cannot obtain directly the BV bounds on ρεand Sεwhich we expect.Thus we consider a linear combination of Pεand Jε.Let με(t)be a function which we will determine later.By combining the equalities above,we obtain the following equation on Pε+ μεJε:

First we prove the following result.

Lemma 3.1Considering the solution μεof the differential equation

there exist constants 0< μm< μM,such that,choosing initially μm< με(0)< μM,we have

Furthermore,we have the following estimate concerning the negative part of the linear combination:

ProofOur goal is to choose a function με(t)which solves the differential equation

We use the same argument as in[34].Therefore,we concentrate on the main ideas.

Note that,because the solution might blow up to?∞in finite time,we need to prove that solutions of(3.9)remain strictly positive for all time.To do so,we first notice that the zeroes ofare

and from the smallness condition(2.5),both zeros are positive.

We need to find two constants0< μm< μM,such that,choosing initially μm< με(0)< μM,then we have for all time,

This condition is satisfied with the following constants:

andμmdefined as

which defines a positive constant because of the smallness condition forβ(2.5).

Coming back to(3.7),we arrive to

and we conclude that,for all t≥0,

which concludes the proof of Lemma 3.1.

From the estimate of the Lemma 3.1,we can deduce the local BV bounds uniform inε.We start with Pε.Addingto(3.5)and using(2.3)and Lemma 2.1,we find

Notice thatBy considering the negative parts of Pεand using(2.2)and(3.8),we arrive to the inequality

With this inequality,the BV bounds follow.SinceεPεis bounded,by integrating the inequality above,we have

Consequently,we obtain

Since ρε(t)is bounded,we have finally thathas local bounded variations.Therefore,up to an extraction,there exists a functionon(0,∞)satisfying

Since we have the lower bound ρε≥ ρmby Lemma 1.1,we obtain the bound for the negative part of the derivative ofρε

Finally,it remains to study Sε.To do so,we rewrite(3.6)as

With our assumptions(2.1)on the Lipschitz function Q,we have

and

The term εJεis bounded because of our assumptions on Q.So,integrating this equation,we have,for T>0,

And then,since Sεis uniformly bounded,we conclude that there exists a function S(t),such that,after extraction of a subsequence,

To conclude,it follows thatconverges in measure to 0 asεvanishes,and thus

4 Concentration and Constrained Hamilton-Jacobi Equation

We complete assumption(2.4)on the initial data with

with A,B>0.

We prove in this section the following result.

Theorem 4.1Under the assumptions(2.1)–(2.5)and(4.2),then after extraction of a subsequence(uε)εconverges locally uniformly to a Lipschitz continuous viscosity solution u to the constrained Hamilton-Jacobi equation

In the simple case,when dimension d is equal to 1 and when R(x,S)is monotonic in x for all S,n concentrates in one single point.

We first prove that uεis equi-bounded,then the equi-continuity,and finally we explain how to pass to the limit in(4.1).

4.1 Local bounds and equi-continuity in space

Sincefrom initial data(4.2),we conclude thatis a super-solution andfor all t∈[0,T].

Next we prove that uεis uniformly Lipschitz continuous in space on[0,T]×Rd.We define for h small enough,.Since the initial conditionare uniformly continuous,givenδ>0,for h small enough,we haveFrom(4.1),we arrive to

Thus by the maximum principle,we deduce that

We conclude that uεis uniformly Lipschitz in space on[0,T]×Rd,and set

for 0< ε < ε0,ε0small enough and r large enough.We also have from Lemma 3.1 that ρε≥ ρm,then for 0< ε< ε0and r large enough,we obtain

This implies

Using the Lipschitz bound(4.5),we obtain

Hence we have the local lower bound on uε.

4.2 The equi-continuity in time

For given T,ηand r>0,we fixand define

where E and D are constants to be determined.We prove in this section the uniform continuity in time.The idea of the proof is to find constants E and D large enough such that,for anyand for allε< ε0,

and

Then by taking y=x,we have the uniform continuity in time on compact subsets of[0,∞)×Rd.We prove here the inequality(4.6),and the proof of(4.7)is analogous.

First we prove thatξε(t,y)>uε(t,y)on[s,T]×?B(0,r),for allη,D and.Since uεare locally uniformly bounded according to Sections 4.1 and 4.3,by taking E large enough,such that

we obtain

Next we prove that,for E large enough,for all y ∈ B(0,r).We argue by contradiction.Assume that there existsη>0,such that for all constants E>0,there exists yE∈B(0,r),such that

This implies

where M is a uniform upper bound forForwe haveSince uεare uniformly continuous in space,this is a contradiction.

Finally,from assumption(2.3),if D is large enough,ξεis a super-solution to(4.3)in

With the proof of(4.7)which is similar,we deduce that the sequence uεis uniformly continuous in time on compact subsets of[0,∞)× Rd.

4.3 Passing to the limit

We proceed as in[5]to prove the convergence of(4.1)to(4.3)asεgoes to 0.Considering the regularity results above,the point at this step is to pass to the limit in the term R(x,Sε).To avoid the complications of the discontinuity,we define

and it follows that?εsatisfies the equation

As Sε(t)converges tofor all t≥0 and R(x,I)is a Lipschitz continuous function,we have

for all t≥0.Furthermore,the limit functions

are locally uniformly continuous.

Then u is a solution to the following equation in the viscosity sense:

From Subsection 4.1,we have for 0< ε< ε0and for some r>0 large enough,

Furthermore,recall that we have

Then it follows that,for r large enough,

It is an open problem to know if the full sequence uεconverges,and it is equivalent to the question of uniqueness of the solution to the Hamilton-Jacobi equation.We consider in Section 5 a special case where uniqueness holds.

In the next section,we derive some properties of the concentration points that also hold in the concavity framework(Section 6),and is useful in what follows.

4.4 Properties of the concentration points

We prove in the rest of this section the following theorem.

Theorem 4.2Let(2.3)hold.For anythe solution of(4.3)is semi-convex in x for any t>0,i.e.,there exists a C(t)such that,for any unit vectorξ∈Rd,we have the following inequality:

Consequently,u(t,·)is differentiable in x at maximum points,and we have

whereis a maximum point of u(t,·).

Furthermore,for all Lebesgue points ofwe have

Step 1 The semi-convexityTo increase readability,we use the notationuξξfor a unit vector ξ.We obtain from(4.1),

and

Notice thatbecauseTherefore,the function w:=uξξsatisfies

from the assumption(2.3).The semi-convexity follows from the comparison principle with the subsolution given by the solution to the ODE

Step 2The semi-convexity implies that u is differentiable at its maximum points.Therefore,we have for t>0,

Moreover,we also have the property that,for any sequence(tk,xk)of x-differentiability point of u which converges towe have

In fact,we deduce that,for h,r>0,h,

and

We obtain these convergence results by applying Lebesgue’s dominated convergence theorem to the integral

given by a change of variable,combined with the local Lipschitz continuity of u.

Step 3 Proof ofWe first integrate the equation on rectanglesWe obtain

By the semi-convexity,we have

and also u(t+h,y)≤0.We deduce

Therefore,we obtain

We conclude that at any Lebesgue point ofwe have

Next,we prove the opposite inequality.By integrating on the rectanglewe have

and

Hence,we have that,at any Lebesgue point of

Hence the statement of Theorem 4.2 holds.

5 The M onomorphic Case in Dimension d=1

In the case when dimension d equals 1 and R(x,S)is monotonic in x for each S,we have the expected convergence toward a single Dirac mass under the additional assumption(which holds for instance when R is monotonic in x)

Theorem 5.1Assume(2.1)–(2.5)and(5.1)hold,and thatare uniformly continuous in Rd.Then,the solution nεof(1.1),still after extraction of a subsequence,converges in the weak sense of measures

and we also obtain the relations

Moreover,the full sequence nεconverges when R has one of the following form,for some functions b>0,d>0,F>0,

or

We do not prove this result in detail.It is a consequence of the following observation.As the measure n defined in(5.2)satisfies the condition supp n(t,·) ? {u(t,·)}from the properties obtained in the previous section(see details in[5,7]),n is monomorphic.Indeed,from the condition(5.1),the set{u(t,·)}is reduced to an isolated point for all t ≥ 0.The uniqueness of the solution when R is written as(5.3)or(5.4)is entirely explained in[7].The idea of the proof is to consider for instance the function

and,by noticing that?satisfies the equation

to derive an estimate on the derivative of the difference between two different solutions?1and ?2with the same initial data.By considering the different quantities at the maximum points of u(t,·),we see that there exists a constant C>0,such that

and the uniqueness follows.

6 The Concavity Framework in R d

In this section,we are going to assume more regularity in order to prove the convergence of nεto a Dirac mass in the sense of measure.The specific feature of this framework is that uniform concavity of the growth rate and initial data induce uniform concavity of the solutions uεto the Hamilton-Jacobi equations,which implies that uεhas only one maximum point.The main technical difficulty is that uniform bounds are not possible because of the quadratic growth at infinity.Therefore,following the work[33],we start with assumptions on R∈C2as follows:

We also need the uniform concavity of the initial data

and we add some compatibility conditions

For this section,we need

We keep the same assumptions on Q and Sεas in the previous section.Next we are going to prove the following result.

Theorem 6.1Under assumptions(6.2)–(6.8)and the assumptions on Q,ρεand Sεhave locally bounded total variations uniformly inε.Therefore,there exist functionsandsuch that,after extraction of a subsequence,we have

Furthermore,we have weakly in the sense of measures for a subsequence nε,

and the pairalso satisfies

As a first step,we give estimates on uε.Next,we adapt the proof of Section 3 to give BV estimates on ρεand Sε,and then pass to the limit asεgoes to 0.Finally,we prove the following theorem.

Theorem 6.2Assuming that(6.1)–(6.11)hold,-function,and its dynamics is described by the equation

with u(t,x)given below in(6.29)and(6.11).Furthermore,is a-function.From this equation,it follows thatis a decreasing function and

6.1 Uniform concavity of uε

Again we use the Hopf-Cole transformation defining uε= εln nε,and we obtain the same equation as in Section 4

We focus now on the study of the properties of the sequence uε.

We first prove the following lemma.

Lemma 6.1Under assumptions(6.2)and(6.8),we have for t≥0 and for x∈Rd,

ProofFirst we achieve an upper bound for uε.By definingwithwe obtain from assumptions(6.2),(6.6)and(6.8)thatand

Then by a comparison principle,we conclude thatfor all t≥0 and x∈Rd.

Next for the lower bound,we defineThus,we haveand

Consequently,we obtain thatfor all t≥0 and x∈Rd.Hence we have the estimates on uε(t,x).

The next point is to show that the semi-convexity and the concavity of the initial data are preserved by(1.1).In other words,we are going to show the following lemma.

Lemma 6.2Under assumptions(6.2)–(6.8),we have for t≥ 0 and x ∈ Rd,

ProofFor a unit vector ξ,we use the notationandto obtain

By usingand the definitionwe arrive at the inequality

Finally by a comparison principle and assumptions(6.7)–(6.8),we obtain

Hence the uniform semi-convexity of uεis proved.

To prove the uniform concavity,we first recall that,at every point(t,x)∈R+×Rd,we can choose an orthonormal basis,such that D2uε(t,x)is diagonal.Thus,we can estimate the mixed second derivatives in terms of uξξ,and consequently,we have

By a comparison principle and assumption,we obtain the estimate

which ends the proof of Lemma 6.2.

6.2 B V estimates on and their limits

We use exactly the same proof as in Section 3 to obtain BV estimates onand Sε.To obtain these estimates,an important point is the bounds on εPε.We need to confirm that εPεis bounded,which is clear in Section 3 thanks to the bounds on the growth rate.Here the growth rate has a quadratic decrease at infinity,which does not give an immediate lower bound on εPε.Furthermore,we do not have a lower bound on ρεeither because of the same argument and we cannot obtain directly a BV estimate on Sεas in Subsection 3.2.However,we derive a lower bound for εPε,and we use the uniform concavity of uεfor that purpose.

By the definition of Pε,it follows from(6.2)and(6.17)that

And we have a bound for

We recall that(3.14)also holds true in this framework

Then,we integrate this inequality over[0,T]for T>0 and by the same arguments used in Subsection 3.1,it follows thathas local BV bounds.Therefore there exists a functionsuch that after extraction of a subsequence,

The next aim is to show that Sεhas local BV bounds.We go back to(3.6),and we recall

Then we have the following inequality:

and

By integrating this inequality over[0,T]for T>0,using

and sinceρεis bounded above,we deduce from(3.14)that

To conclude,we can extract a subsequence from Sεwhich locally converges into a limit function

6.3 The limit of the Hamilton-Jacobi equation

From the estimates obtained above on uεand D2uε,we can deduce that ?uεis locally uniformly bounded and thus from(4.1)forε< ε0that?tuεis also locally uniformly bounded.Therefore there exists a function u such that,after extraction of a subsequence(see[10,26]for compactness properties),we have for T>0,

and

Then,passing to the limit asin(4.1),we deduce that u satisfies in the viscosity sense the equation

In particular,u is strictly concave,therefore it has exactly one maximum.This proves that n stays monomorphic and characterizes the Dirac location by

This completes the proof of Theorem 6.1.

6.4 The canonical equation

In this section,we establish from the regularity properties proved in the previous sections a form of the so-called canonical equation in the language of adaptive dynamics(see[15,22])as follows:

This equation was formally introduced in[24]and holds true in our framework.The point of this differential equation is to describe the long time behaviour of the concentration point

Step 1 Bounds on third derivatives of uεFor the unit vectorsξandη,we use the notationandto derive

Let us define

Again,at every,we can choose an orthogonal basis,such thatis diagonal.And sincewe haveThen we obtain the following inequality:

As(6.10)givesa bound on M1(t=0),by using the Gronwall lemma,we obtain an L∞-bound on the third derivative uniform inε.

Step 2 Maximum point of uεWe denote the maximum point of uε(t,·)bySince we havewe obtain

Then the chain rule gives

By using(6.16),it follows that,for all t≥0,we have

Thanks to the uniform bound on D3uεand the regularity on R,we pass to the limit

As we haveand(6.2),is bounded inThen it implies from the canonical equation thatis bounded in W1,∞(R+),andis also bounded insinceis invertible by the implicit function theorem.We differentiate(6.13)and obtain the following differential equation:

Step 3 Long time behaviourUsing the canonical equation,we obtain

Since the left-hand side equals 0 from(6.13),it follows that

We deduce thatdecreases.Consequently,converges and subsequences ofalso converge,sinceis bounded.However,the possible limitshave to satisfyThen from(6.1),(6.3)and(6.13),we conclude that

which ends the proof of Theorem 6.2.

7 Numerical Results

We illustrate in this section the evolution of nε,ρεand Sεin time with different values of β.We choose the following initial data:

and growth rate R and Q as follows:

The numerics are performed in Matlab with parameters as follows.We consider the solution on interval[0,1].We use a uniform grid with 1000 points on the segment and denote byand Skthe numerical solutions at grid pointand at time tk=k?t.We choose as initial value of the nutrient concentration Sε(t=0)=5.We also choose β to be 2·103,the time step?t=10?4and Cmasssuch as the initial mass of the population in the computational domain is equal to 1.The equation is solved by an implicit-explicit finite-difference method with the following scheme:

We use Neumann boundary conditionsWe use an implicit explicit scheme for the growth term in order to maintain the positivity of the numerical solution.

Figure 1 shows the dynamics forε=1·10?3and Figure 2 forε=5·10?4.We observe that,sinceεis smaller in Figure 2,the concentration location of the population moves to the maximum point of fitness more quickly than in Figure 1,which illustrates the dynamics given by the canonical equation,and then the concentration point and the population density become stable.

Figure 1 Dynamics of(left)and dynamics of the density nεfor β =2·103 and ε=10?3.

Figure 2 Dynamics of(left)and dynamics of the density nεforβ =2·103 and

In Figure 3,we show the numerical results corresponding to the same data as in Figure 1,except that we chooseβ =2·102.We can observe oscillations ofρεand Sεin the first case(β =2·103),whereas there are very few variations of these quantities when β is smaller.Indeed the parameterβcan be considered as a measure of the ecological dynamics:Asβgoes to 0,we approach the case of the quasi-stationary state of the resource level,and we then observe mostly the dynamics of the concentration location.However,as explained in the next section,the convergence to the quasi-stationary solutions asβgoes to 0 cannot be proved with our approach and remains an open problem.

Figure 3 Dynamics of(left)and dynamics of the density nεforβ =2·102 and ε=10?3.

In Figure 4,we show the numerical results for the 2-dimensional model withβ =2·102,ε =1·10?2and Sε(t=0)=5.We choose the time step ?t to be 5·10?3and ?x to be 1·10?2.We also choose the initial condition and the functions

As confirmed by the analysis we conducted,the population density concentrates at the maximum point of the growth rate.

Figure 4 Dynamics in dimension 2 of(at top left),the initial condition for nε(at top right)and the stationary state nεconverges to(at bottom),with β =2·102 and ε=10?2.

8 Discussion

The weak assumptions provide a generic framework to study the asymptotic behaviour of uε,but do not enable us to derive a canonical equation describing the dynamics of the concentration points,and we can observe jump phenomena of the concentration location.Indeed,the lack of regularity can produce a discontinuity ofρ(t)and local maxima of u(t,x)can become global maxima and jumps from a given concentration location to another one can occur,which means the extinction of a population quickly invaded by another growing one(see[7]for further details).The concavity assumptions are suitable to avoid the jump cases because these assumptions preserve regularity,and they ensure that the global maximum of u is the only maximum.The canonical equation derived in this framework describes the evolution of the selected trait in an evolutionary time scale.

Many models are studied to illustrate the diversity of evolutionary problems.For instance,the problem of coevolution was tackled in[7,18,33].The branching phenomenon where a monomorphic population at some point becomes dimorphic is described in[24,41].In the chemostat model,the spatial component is neglected here with the hypothesis that the content of the chemostat is well-mixed,it was taken into account in[9,37–38].

The inclusion of mutations in structured population models is necessary to generate phenotypic variability in a given population,which is a fundamental ingredient of the selection process.It implies the separation of the ecological time scale and the evolutionary one.In the presented model,the mutation term has little phenotypic effects due to the parameterε.Especially in the canonical equation form,we observe that the pressure of mutants on the dynamics ofis small and,asεgoes to 0,it does not change the convergence ofto the maximum point of fitnessIt means that only the mutations with positive effects on the phenotypic trait can influence the dynamics:Mutants emerging with a better fitness than the residents can invade,while the other mutants go to extinction.

However,some open questions arise from the present study.First it seems that the method developed in this work does not give TV bounds for the full range[0,β0]for some smallβ0,since the estimates providing the uniform BV estimates onρ2in Subsection 3.2 are local in time and then it is not possible to prove uniform convergence of S(t)ason[0,∞)at this stage.Thus we cannot obtain the asymptotic behaviour of the limit functions asβgoes to 0,while the convergence ofεto 0 describes the dynamics of the presented system on a larger time scale,therefore local estimates are enough.

As mentioned in Section 4,the uniqueness of the solution of the Hamilton-Jacobi equation(6.29)has up to now been an open problem,apart from very particular cases(see for instance[6]),and the issue of the convergence of the full sequence uεhas remained unsolved.However a recent work of Mirrahimi and Roquejoffre[40]has shown uniqueness of the constrained Hamilton-Jacobi equation related to the following selection-mutation model in the concavity framework:

and generalizes a result on a selection model with spatial structure(see[38]),where the proof relies on the uniqueness of the solution of the corresponding constrained Hamilton-Jacobi equation.The proof of the uniqueness property in our chemostat model is a forthcoming work.

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