Influence of mutations in phenotypically-structured populations in time periodic environment

We study a parabolic Lotka-Volterra equation, with an integral term representing competition, and time periodic growth rate. This model represents a trait structured population in a time periodic environment. After showing the convergence of the solu-tion to the unique positive and periodic solution of the problem, we study the in(cid:29)uence of di(cid:27)erent factors on the mean limit population. As this quantity is the opposite of a certain eigenvalue, we are able to investigate the in(cid:29)uence of the di(cid:27)usion rate, the period length and the time variance of the environment (cid:29)uctuations. We also give biological interpretation of the results in the framework of cancer, if the model represents a cancerous cells population under the in(cid:29)uence of a periodic treatment. In this framework, we show that the population might bene(cid:28)t from a intermediate rate of mutation.


Motivations
Evolution is a complex phenomenon, which intervenes in various scales of time and population sizes.In this article, we study an integro-dierential model of a trait-structured population in a changing environment.This model aims at analysing the eect of environmental oscillations on the heterogeneity of a population.This question has emerged from the observation of phenotypical and genetic diversity inside solid tumours [4,31].It is conjectured that this heterogeneity might be a consequence of the variations in the external conditions during tumour growth: oxygen and nutrients availability [13], immune system response and presence of chemotherapy [24] are varying during time.This phenomenon has been coined as bet hedging [18], in the sense that a large heterogeneity allows a tumour to better react to a non constant environment.The model we will study arises from more general models of adaptive evolution of phenotype-structured populations, however we will often come back to the cancer cells model to give biological insights on the theoretical results.
The study of evolving populations under constant environment has been carried out in several mathematical frameworks.Game theory and adaptive dynamics for example have been used in several studies [3,15,22,23].Replicator-mutator models [1] focus on the frequency of phenotypes in a group, without studying the actual size of the population.Stochastic models are relevant for small size populations, and integro-dierential models can be derived from them in the limit of large size populations [9,10].The integro-dierential equations framework has been studied especially in the case of small mutations [7,12], to model the evolution of species on very large scales of time.Still in the case of constant environment, it has been studied with non local competition [11], or for specic growth terms [32].
We will study here an integro-dierential model with a time-periodic environment, and investigate the role of mutations in the nal outcome.
Consider Ω an open, connected bounded domain.We consider the following model: = Ω n(t, y)dy for all t ≥ 0, ∂ ν n(t, x) = 0 on (0, +∞) × ∂Ω, n(0, x) = n 0 (x) for all x ∈ Ω. (1.1) Here, n(t, x) represents the density of individuals of trait x at time t.We suppose that mutations occur randomly and are reversible, which is represented by the Laplace term of diusion in the traits space, with a mutation rate coecient D. The term R(t, x) is a timeperiodic function of period T , which represents the growth rate of individuals subject to a varying environment.The term ∂ ν n is the normal derivative of the function n on the boundary ∂Ω.This model with periodic coecients is very similar to models studied in [14,26,2].However, the means and scopes of our article are dierent.
In [14], the authors consider a general growth rate which is periodic in t.They prove similar existence and large-time behaviour results for the solutions of (1.1), but their approach is slightly dierent since they consider traits belonging to the full space x ∈ R k (as in [2,26]), while we consider traits in a bounded domain x ∈ Ω, but this is mostly a small technical dierence.But our approach diverges after this rst step, since the authors of [14] are mostly interested in the asymptotic D → 0, and investigate the inuence of the time-heterogeneity for the asymptotic problem they derive.In the present paper, one of our aims is to show that, in time-periodic media, it is sometimes more advantageous for the population to keep a positive mutation rate D > 0, in the sense that it could give a larger mean population

Assumptions and application framework
In the model (1.1), the domain Ω ⊂ R k is the set of all possible traits for the population, we consider it to be connected, bounded and smooth.
We make very few assumptions on the growth rate R, except that it is T −periodic with respect to time and that it belongs to L ∞ (0, T ) × Ω .This will guarantee the regularity of solutions, without imposing a particular term.
We consider initial data n 0 ∈ L ∞ (Ω) that are non-negative and non null.
As announced in the introduction, we will often consider the framework of tumour growth to put the theoretical results in a biological perspective.For this, we will consider Ω = (0, 1) and R of the following form: In this case, the phenotype x will denote a proliferation trait and a trait of resistance to a certain chemotherapy.The function p(x) is then the proliferation rate of the cells of phenotype x, α(x) the eciency of the treatment on those cells, and C the concentration of the treatment.The question we will ask then, is how one should allocate a dose of treatment during each period of time?In particular, if one can give a quantity M of treatment during each period, we want to investigate the inuence of τ on the outcome, where A treatment schedule with a small time of drug administration τ can be linked to MTD (maximal tolerated dose) protocols, where drugs are given at a very high dose for short periods of time.A time of administration τ close to the time period T can be linked to metronomic treatments, where smaller doses of drugs are used but for longer periods of time.

Main results and their biological interpretation
We rst state a proposition on the regularity of solutions of (1.1).
Theorem 1.1 (Regularity).There exists a unique weak solution n of (1.1), with This theorem is obtained from classical analysis arguments, see for example [11,17].Before stating the next theorem, we dene λ 1 (R, D) as the rst eigenvalue of the linear time-periodic operator L dened by: with Neumann boundary condition on (0, ∞) × ∂Ω.The existence of λ 1 (R, D) is demonstrated in Lemma 14.3 of [17], in which useful properties of λ 1 (R, D) can be found.
The long time behaviour of solutions is described by the following theorem: Theorem 1.2 (Existence, Uniqueness and attractiveness).
If λ 1 (R, D) ≥ 0 then all solutions of the Cauchy problem for equation (1.1) converge towards 0.
If λ 1 (R, D) < 0, then there exists a unique positive periodic solution N of (1.1).Moreover, this solution attracts all the solutions of the Cauchy problem with non-negative bounded initial data, and 1 T t+T t ρ n (s)ds converges to −λ 1 (R, D) as t → +∞.
To study the long time behaviour of solutions of (1.1), we thus have to study the periodic solution N in the case λ 1 (R, D) < 0. Especially, we are interested in the variation of ρN = 1 T T 0 Ω N (t, x)dxdt, the mean limit total population, with respect to the dierent parameters of (1.1).As Theorem 1.2 yields that we have to study the inuence of D, R and other parameters on λ 1 (R, D).
We then derive from earlier works on the optimization of principal eigenvalues the following results.Proposition 1.3 (Minimization of ρN ).The mean limit population ρN is a convex function of R. Consecutively, for a given R(x) = 1 T T 0 R(t, x)dt, ρN is minimal for a constant in time R.
The proof of this proposition is a direct application of Proposition 2.10 in [27].
Remark 1.In the framework of cancer treatment, i.e. if R is as described in (1.2), this proposition gives us a method of minimization of the nal mean tumour burden.Indeed, for a given quantity of drug M to be delivered during each treatment period, the protocol minimizing ρN is C(t) ≡ M T .Moreover, if two treatments C 1 , C 2 are of the form: , the mean nal total population ρN,2 associated to C 2 is smaller than the mean nal total population ρN,1 associated to C 1 .In other words, concentrating the same quantity of treatment on half the time of administration will make the nal mean population of cells higher.It is also true for a concentration on an administration time of τ /n, for any n ∈ N. We conjecture it to be true for any real factor of concentration.
Theorem 1.4 (Inuence of mutations).Decompose R in the following way: Then if S is not zero and R is not spatially uniform, for γ large enough, there exists D 0 > 0 such that, in the neighbourhood of D 0 , the mean limit population ρN is an increasing function of the mutation rate D.
Remark 2. A biological interpretation of this theorem is that bet hedging tumours are more successful in some conditions.Indeed, if R satises the conditions of Theorem 1.4, then a tumour with a higher mutation rate D will have a higher mean nal population.We identify here the heterogeneity with a high plasticity.
Remark 3. Proposition 1.5 presents some conditions in which the result presented in Theorem 1.4 does not hold.Theorem 1.5 in [29] presents a similar result: their proof follows the same type of reasonning.We will provide here our own proof for sake of completeness.
Proposition 1.6 (Minimization of the minimal size among time).Assume that R(t, x) = p(x) − α(x)C(t) for all (t, x) ∈ (0, ∞) × Ω.Let C max > 0, T > 0 and 0 < σ < 1.Consider the solution n hom of (1.1) associated with the constant function C(t) = C max σ for all t ∈ (0, +∞) and the solution n het of (1.1) associated with the T −periodic function Assume that max x∈Ω p(x) − α(x)C max < 0 and min x∈Ω p(x) − α(x)C max σ > 0. Then if T is large enough, one has Remark 4. Proposition 1.6 gives, in the framework of cancer treatment, an interesting comparison between constant and "bang-bang" treatments.Indeed, it demonstrates that in some situations, the population ρ n (t) will reach regularly a smaller size if subjected to a "bang-bang" protocol than if the same amount of treatment is given constantly, even if the mean limit population ρN is higher, as demonstrated by Proposition 1.3.Biologically, if ρ n (t) reaches a very small size, it is very likely that the population is in fact extinct, and thus ρ n (t) = 0 afterwards.Thus, while the constant treatment reduces the global mean tumoural charge, the "bang-bang" treatments increases the chances of eradicating the tumour.
The next Proposition is indeed an immediate corollary of Theorem 1.1 of [25], but we state it here since its biological interpretation is meaningful.Proposition 1.7 (Inuence of the period).Assume that R is 1−periodic in t and cannot be written as We refer to [25] for the proof of their theorem, but will not present here the proof of this proposition, since it is a very straightforward corollary.
Remark 5.The condition for this theorem to hold does not have a straightforward biological interpretation, but the outcome is interesting.Indeed, if T is large, the environment R T is changing slowly, allowing the population n to approach equilibrium values if R is constant in time on some interval.On the contrary, if T is short, the population faces a fast changing environment, which does not give it time to adapt to any new situation.Hence, n achieves a smaller total mean population ρN if T is short than if T is large.Theorem 1.8 (Competition).Let (n, m) be a solution of: Neumann conditions for both functions on the border of Ω. (1.4) We consider that n 0 ≡ 0 and m 0 ≡ 0 are non-negative functions on Ω. Suppose that where M is the T -periodic solution of (1.1) with D = D 2 .Remark 6.This theorem ensures that, if two populations with dierent plasticities are in competition, the one with the largest equilibrium population will dominate.This article is divided as follows: section 2 is devoted to the demonstration of results of existence, i.e.Theorems 1.1 and 1.2, and Proposition 1.5, which proof is closely linked to these theorems.Theorem 1.4 is demonstrated in section 3, along with Proposition 1.6, as both results give interesting insight on treatment protocol choice, when R is of the form (1.2).Section 4 is devoted to the demonstration of Theorem 1.8 on competing species.Finally, section 5 presents numerical simulations of the model, illustrating dierent phenomenons described earlier.

2
Existence of a T -periodic solution and treatment optimization Proof of Theorem 1.1.Consider the operator T that associates with ρ ∈ L ∞ (0, ∞) the func- (2.5) Clearly, as n 0 ≥ 0 by hypothesis, one has n ≥ 0.Moreover, it follows from classical L p regularity for parabolic equations that the operator Assume that there exist σ ∈ (0, 1) and ρ ∈ L ∞ (0, ∞) such that ρ = σT (ρ) = σρ n , with ρ n (t) := Ω n(t, x)dx ≥ 0. Integrating the equation satised by n, one gets from which we easily derive σρ n (t) ≤ max{R max , σ Ω n 0 } for all t > 0. Hence, 0 ≤ ρ ≤ max{R max , σ Ω n 0 } and the set of all such ρ is bounded.We can thus apply the Shaefer xed point theorem and get the existence of solution n of (1.1), with ρ n ∈ L ∞ (0, ∞).
We now prove the uniqueness of such a solution.Let n 1 , n 2 be two solutions of the system (1.1) with the same initial condition n 0 .Then the function satises the following equation: By uniqueness of the solution of (2.6), we deduce that: (2.7) Moreover, Ω n(t, x)dx > 0 for all t ≥ 0 since n 0 ≡ 0, thus ρ n i > 0. By integrating (2.7) on Ω and derivating in time, we get that ρ n 1 satises the following dierential equation: Since ρ n 2 also satises (2.8) with the same initial condition, we deduce that ρ n 1 = ρ n 2 , and thus that n 1 = n 2 on [0, +∞) × Ω by (2.7).
Proof of Theorem 1.2.We rst prove Theorem 1.2 in the case λ 1 (R, D) < 0. The proof is organized in three parts: we rst prove the existence of a periodic solution, then its uniqueness, and nally its attractiveness.
Existence Let φ be the eigenfunction of (1.3) associated to λ 1 (R, D), with normalization Ω φ(0, x)dx = 1.By denition, φ > 0. We dene: Consider the following dierential equation: Since ρ φ ρ φ is a T -periodic function, equation (2.9) admits a single T -periodic solution, namely )s ds Notice that this function does not depend on our choice of normalization for the eigenfunction φ.We dene the following functions: By direct calculations one sees that N is a T -periodic in time solution of (1.1), with This concludes the proof of existence of a periodic solution to problem (1.1).
Uniqueness Let N 1 and N 2 be two T -periodic solutions of (1.1).We dene the following functions: where We denote Attractiveness Let n 0 ≥ 0 be non null, and n be the solution to the Cauchy problem (1.1).
By derivating the logarithm of the integral of the denition of n, we see that ρ n satises Thus, since ρ n (0) > 0, there exists A 0 > 0 such that Similarly, there exists B 0 > 0 such that Thus: Since there exists a > 0 such that ρ φ 0 (t) ≥ a and ρ n (t) ≥ a for all t ≥ 0, we can see that: We now treat the numerator: we will show that We rst decompose A: set µ > 0 such that µ < −λ 1 (R, D), and write: Noting that we conclude that all terms of A are in fact o(e −λ 1 (R,D)t ), and thus ρ n (t)−ρ From this, it is easy to check that ρ n (t) − ρ N (t) −→ 0 when t → +∞.Thus, φ 0 = N , and we conclude that n(t, x) − N (t, x) −→ 0 as t → +∞.
The case λ 1 (R, D) ≥ 0. First note that n is a subsolution of the parabolic equation satised by Cφ(t, x)e −λ 1 (R,D)t , where C is large enough so that Cφ ≥ n 0 .Hence, in the case where λ 1 (R, D) > 0, as φ is periodic in t, one gets n(t, x) → 0 as t → +∞ uniformly with respect to x ∈ Ω.
Proof of Proposition 1.5.Consider the periodic principal eigenfunction φ associated with λ 1 (R, D), that is, the unique positive and T −periodic solution, up to multiplication, of Let rst show that this function is decreasing with respect to x.Let ψ := ∂ x φ.As ∂ x R ≤ 0 and ψ > 0 by hypothesis, one has Consider now z := ψ/φ.One has and z is T −periodic.If z admits a positive maximum over R × [0, 1], then this maximum is reached at an interior point, and the parabolic maximum principle, together with the T −periodicity, would imply that z is constant, meaning that ψ = ∂ x φ would be proportional to φ.But as ∂ x φ(t, 0) = ∂ x φ(t, 1) = 0 for all t ∈ (0, T ), one would then necessarily have ∂ x φ ≡ 0, which is impossible since R(t, •) is not constant with respect to x for a non-negligible set of t ∈ (0, T ).We have thus reached a contradiction, which proves that z ≤ 0.Moreover, if z(t, x) = 0 for some (t, x) ∈ R × (0, 1), then this point is an interior maximum point and the parabolic maximum principle would again give z ≡ 0 and a contradiction.Hence, z(t, x) < 0 in R × (0, 1), that is, ∂ x φ(t, x) < 0 for all (t, x) ∈ (0, T ) × (0, 1).Now, we know from Lemma 2.3 of [20] that where φ is the adjoint principal eigenfunction, that is, the positive T −periodic solution of normalized by 1 T (0,T )×Ω φ φ = 1.Indeed, one could prove as above that, as ∂ x R ≤ 0, one has ∂ x φ(t, x) < 0 for all (t, x) ∈ (0, T ) × (0, 1).It immediately follows that which yields the result since ρ N = −λ 1 (R, D).

Advantage of a large mutation rate
This section is devoted to the proof of Theorem 1.4 and Proposition 1.6.The proof of Theorem 1.4 is very similar to the proof of Theorem 2.2 in [20]: for sake of clarity, we state it here, but we would like to refer to it for other corollaries, and to [17] for the crucial part of the proof.We also present an interpretation of the theorem in the case where R is as described in (1.2).
Parameter γ in the proof of 1.4 is a measure of the amplitude of changes in the environment during one period of time.Let us now consider the particular case where R is dened as (1.2).The quantities r, r and γS can be expressed by: Parameter γ can not be isolated simply in this situation.However, we can conclude from Theorem 1.4 that there exist p, α such that for certain treatment schedules C that are not constant in time, the nal population ρN is around some D 0 increasing in D. In other words, under non constant treatments, more plastic populations might realise a larger nal mean population.
Since R hom (t, x) := p(x) − α(x)C max σ does not depend on time, the corresponding limit population N hom is also constant in time, and thus Notice that does not depend on T .
On the other hand, integrating the equation satised by n het , we nd that for any k ∈ N, where m := − max x∈Ω p(x) − C max α(x) > 0. Hence, it follows that On the other hand, we know that ρ n het ≤ max ρ n 0 , max (0,T )×Ω R .Hence, This shows that, for T large enough, one will get

Competition
This section is devoted to the demonstration of Theorem 1.8.We begin with a preliminary lemma.
Let ψ be the T -periodic eigenfunction satisfying: and φ the similar eigenfunction associated with λ 1 (R, D 1 ).Then one can easily prove, with the same arguments as in the proof of Lemma 4.1, that Similarly, one gets Adding these two inequalities, using the Harnack inequality and the fact that φ and ψ are both uniformly positive and bounded, one gets the existence of a positive constant, that we still denote C > 0, such that Thus τ < +∞ and If t 0 = 0, this provides a bound on τ .Otherwise, one can assume that t 0 is such that sup x∈Ω m(t 0 , x) + n(t 0 , x) = ε/|Ω|, and thus τ ≤ − ln C/ε.This bound being independent of t 0 , we thus eventually obtain, by using the parabolic regularity and the Harnack inequality, inf t∈(t 0 ,t 0 +τ ) inf x∈Ω m(t, x) + n(t, x) > Cε for some positive constant C.
In other words, we have proved that there exist ε > 0 and C > 0 such that for all t > 1 such that inf x∈Ω n(t Proof of Theorem 1.8.Consider the sequences n k (t, x) := n(t+kT, x) and m k (t, x) := m(t+kT, x).
Parabolic regularity yields that one can assume, up to extraction, that these two sequences converge to some limits n ∞ and m ∞ in W 1,p/2;2,p loc (R, Ω), which are time-global solutions of: Assume rst that m ∞ ≡ 0. Then we know from Lemma 4.1 that n ∞ cannot be identically equal to 0 nor N .We now use Proposition 2.7 of [19], which yields that the equation admits a unique time-global positive solution u up to multiplication.Using the change of variables for some constant C > 0.
Integrating in x, taking the log and derivating in t, we obtain, as above, that We know (see for instance [27]) that this equation admits a unique time-global solution which is uniformly positive and bounded.As ρ N is also a solution, we get ρ n∞ ≡ ρ N and thus C = 1 and n ∞ ≡ N .This is a contradiction with Lemma 4.1.Hence, m ∞ ≡ 0. As this is true for any extraction of the sequence (m k ), we have thus even proved that inf R×Ω m ∞ > 0.
Next, using the change of variables and, again, by Proposition 2.7 of [19], we get where C 1 > 0 and ψ is the T -periodic eigenfunction satisfying: Assume now by contradiction that n ∞ ≡ 0. The parabolic strong maximum principle gives n ∞ > 0. Then we could prove similarly as above that The right-hand side goes to +∞ as t → +∞ since φ and ψ are both positive and periodic, and λ 1 (R, D 2 ) > λ 1 (R, D 1 ).But the left-hand side is bounded since n ∞ is bounded and inf R×Ω m ∞ > 0. We have thus reached a contradiction.Hence, n ∞ ≡ 0.
We then easily deduce from the above identities that m ∞ ≡ M , which concludes the proof.We present in this section some numerical illustrations of the properties exposed in the previous sections.The simulations were done with R of the form (1.2).We considered functions p and α of the following form: These functions were chosen arbitrarily to illustrate our results.They are represented in gure 1.In this case, phenotypes around x = 0 represent more proliferative but more sensitive cells, while phenotypes around x = 1 represent resistant cells, which have a decit in proliferation.

Inuence of the mutation rate
In gure 2, we represent the nal mean population ρN as a function of the mutation rate D for dierent treatment schedules.All schedules deliver the same mean quantity of drug M = 1 T T 0 C(t)dt = 2 over each period of time, but the length of the administration time varies.We recall that in this situation, the quantities lim D→0 ρN and lim D→+∞ ρN will not depend on τ : for our particular choice of p, α and M they are equal to: We observe, as stated in Propositions 1.3 and 1.5, that if R is constant in time (case τ = T ), then ρN is a decreasing function of D. However, if τ is shorter, ρN is in some range of D an increasing function of D.Moreover, for τ = 1/4, the maximum of ρN is reached for D = 0.3 > 0. In this case, a population with a positive mutation rate will be favoured.
On gure 3, we represent the phenotype repartition for three particular populations A, B and C, corresponding to D = 0.3, D = 4 * 10 −3 , and D = 2 respectively, all illustrated for τ = 1/4.As we compare A and B, we can argue that ρN (D = 0.3) > ρN (D = 4 * 10 −3 ) because the high mutation rate is so that, at the beginning of the treatment, a larger population is already present around x = 1.Thus during treatment, the resistant part of the population will reach higher levels.However, if D is too large as in population C, then it does not prot enough of the high growth rate at x = 0 before treatment starts.

Inuence of the time of administration
We stated in the introduction that for any κ ∈ N, a convexity argument proves that if 0 < κτ < T and We conjectured that this is true for any κ ∈ [1, T /τ ], in other words, that if C τ is dened by then the nal mean population (ρ N ) associated to R τ = p − αC τ is a non-increasing function of τ .This is illustrated in gure 4.    On gure 4a, we observe that for dierent values of the diusion D, the nal mean population ρN is a decreasing function of τ .On gure 4b, we observe the population over time under two types of treatments: one with τ = 0.7T , the other with τ = 0.1T .We see there that for a short τ , the population oscillates during each period between two extreme values, while for a larger τ the population uctuates less.
On gure 5, we illustrate Proposition 1.7, which concerns the inuence of the period length T on the nal mean population ρN .As the function R we chose does satisfy the conditions of application of Proposition 1.7, we observe on gure 5a that ρN is a non- decreasing function of T .Furthermore, we depict on gure 5b the time evolution of two particular populations, where we only changed the period length T .We observe that when T is large, the population ρ n (t) approaches, at each period, a maximal plateau during the time without treatment (C = 0).On the contrary, if T is smaller, ρ n (t) does not have time to reach these values before treatment is applied again.However, if T is large, the population ρ n (t) reaches at each period very low values.As discussed in Proposition 1.6, biologically, there exists at these moments a possibility of extinction of the population.

Varying the treatment dosage
In this part, we numerically investigate the role of M = T 0 C(t)dt, the total drug used during each period, on the nal mean population ρN .
On Figure 6, we represent ρN for a treatment of the form C(t) = 1 0≤t≤τ M/τ with τ = 1/4, for dierent values of the diusion D and on the drug dosage M .
If M is small, ρN reaches its maximum for D = 0. Indeed, if M is too small, R is decreasing in x for all t ≥ 0, and thus ρN is a decreasing function of D, as stated in Proposition 1.5.But if M increases, around M = 3 we see that bet hedging occurs, in the sense that ρN is in some region increasing in D, illustrating the phenomenon described in

Random uctuations in the environment
The results we presented in this article address periodic changes of the environment, corresponding in the framework of cancer treatment to a regular chemotherapy schedule.But, as we stated in the introduction, chemotherapy is not the only reason why the environment changes: tumour vascularisation, immune system reaction and other phenomena can vary over time, with a less regular timing.We present here a numerical simulation where the environment no longer changes periodically, but randomly.More precisely, still using the same p, α, M dened in Section 5.1, for each time unit ∆t we have: R(t, x) = p(x) with probability 1 − γ, p(x) − α(x) M γ with probability γ, for a certain γ ∈ (0, 1).Notice that the expected value of the growth rate E(R) does not depend on γ: this way, we are doing something similar as in 5.1, with γ being an analogue of τ T .We are interested in the mean value of ρ n over time, namely the following quantity: Figure 7 presents simulations for dierent values of γ, and an initial condition n 0 (x) = 1 for all x ∈ Ω.If γ = 1, we are in fact in a situation of constant environment, and thus retrieve previous results: ρn is a non-increasing function of D, and its limit values for D → 0 and D → +∞ are known.If γ decreases, we observe that for the same D the mean population ρn increases, and for γ small enough, D → ρn seems to no longer be a non-increasing function.
As far as the authors know, the inuence of stochastic uctuations of the environment has only been investigated numerically for an ODE in [16].Simulation suggest that theoretical results on periodic uctuations might be extended to stochastic ones.It would be of great interest to investigate this, as biological phenomena often present stochastic uctuations.

Conclusion
We demonstrated in this article some properties of trait-structured populations in time periodic environment.Especially, we showed that in some situations, a population might benet from a large mutation rate.Moreover, in such an environment, a more plastic population would replace a less plastic one.This motivates the study of this equation in the regime of intermediate mutations, thus of large diusion rates.

Figure 1 :
Figure 1: Proliferation and sensitivity functions chosen for the simulations

10 −3 10 −Figure 2 :
Figure 2: Final mean populations ρN represented as functions of the mutation rate D for various treatment schedules.The treatments are of the form C(t) = 1 0≤t≤τ M/τ , with M = 2 and τ varying between 1/4 and T = 2. Populations A, B and C are detailed in gure 3.

Figure 3 :
Figure 3: Populations A, B and C of gure 2 are detailed for τ = 1/4.We represent the population repartition in phenotypes just before treatment, just after treatment and the mean population.

Figure 4 :
Figure4: Illustration of the conjecture that ρN is a decreasing function of the time of drug administration τ .These numerical simulations were performed for T = 2.

Figure 5 :
Figure 5: Illustration of Proposition 1.7 on the inuence of T on the population size.These numerical simulations were performed for τ = T /2 and a xed diusion coecient D = 0.1.

Figure 6 :
Figure 6: Mean limit population ρN as a function of both diusion and drug dosage per period M .

8 Figure 7 :
Figure 7: Populations for random changing of environment with same mean value.