On the long-time behaviour of an age and trait structured population dynamics

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On the long-time behaviour of an age and trait structured population dynamics

Tristan Roget

To cite this version:

Tristan Roget. On the long-time behaviour of an age and trait structured population dynamics. 2017.

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On the Long-Time Behaviour of an Age and Trait Structured Population Dynamics

Tristan Roget November 21, 2017

Abstract

We study the long-time behaviour of a population structured by age and a phenotypic trait under a selection-mutation dynamics. By analysing spectral properties of a family of positive operators on measure spaces, we show the exis- tence of eventually singular stationary solutions. When the stationary measures are absolutely continuous with a continuous density, we show the convergence of the dynamics to the unique equilibrium.

1 Preliminaries and Main Results

1.1 Introduction

Our ultimate goal is the understanding of the long-time behaviour of a population where the individuals differ by their physical agea^PR and some hereditary variable x^PS ^€R^dcalled trait. The population evolves as follows. An individual with trait x^PS and agea^PR has a death rateD^px, a^q cN where Dis the intrinsic death rate, N is the population size and c ^¡ 0 the competition rate. This individual gives birth at rate B^px, a^q. At every birth, a mutation occurs with probability p ^{P s}0,1^r and the trait of the newborn y ^P S is choosen according a distribution k^px, a, y^qdy. Otherwise, the descendant inherits of the traitx^PS. In his thesis [19], Tran introduced an individual-based stochastic model to describe such a discrete population. The population is described by a random point measure

Z_t^K 1 K

N_t^K

¸

i1

δpx_i^pt^q,a_i^pt^qq (1) which evolves as a càdlàg Markov process with values in the set M ^pSR q of positive finite measures on SR and each jump corresponds to birth or death of individuals. When the order K of the size of the population goes to infinity such that Z₀^K approximates a deterministic measure n0 ^PM ^pSR q, it is shown (see [19],[20]) that the process approximates the unique weak solution (in the sense given by (10))^pnt^qt^¥0^PC^pR ,M ^pSR qqof the partial differential equation

#

Btn_t^px, a^{q B}_an_t^px, a^q

D^px, a^q c

³

SR n_t^py, α^qdydα n_t^px, a^q,

n_t^px,0^qFrn_t^spx^q, ^pt, x, a^qPR SR , (2)

where

F^rn_t^spx^q (3)

p1p^q

»

R

B^px, α^qnt^px, α^qdα p

»

SR

B^py, α^qk^py, α, x^qnt^py, α^qdydα.

Recently, the well-posedness of measure solutions for a large class of partial differ- ential equations including (2) has also been established in [5], using a deterministic method. At our knowledge, nothing has been done about its long-time behaviour.

In [4], the stationary problem is solved in L^1pSR q for a similar dynamics with a pure mutational kernel (p1). The present paper is also motivated by [1]. The authors study the long-time behaviour of a selection-mutation dynamics with trait structure (and no age) and p ^Ps0,1^r. They show the existence of stationary mea- sures which can admit dirac masses in some traits and they analyse the long-time behaviour of the solutions when the stationary measure admits a bounded density.

In this paper, we extend these facts to an age and trait structured population. We show the existence of non-trivial stationary measures for Equation (2) (see Theorem 1.3) which can be singular. When these measures are absolutely continuous with a continuous density, we show that the solutions of (2) converge to the (unique) equilibrium (see Theorem 1.5). The method is based on the analysis of the lin- ear dynamics. Indeed, the stationary states of (2) are eigenvectors for the direct eigenvalue problem

#

BaN^px, a^qpD^px, a^q λ^qN^px, a^q0

N^px,0^q FrN^spx^q. (4)

The solutions of the dual problem

Baφ^px, a^qpD^px, a^q λ^qφ^px, a^q G^rφ^spx, a^q0, (5) where

Grφ^spx, a^qB^px, a^q

p1p^qφ^px,0^q p

»

S

φ^py,0^qk^px, a, y^qdy

, (6)

give us some useful invariants and allow us to apply a method based on [11],[17]

leading to obtain an exponential rate of convergence for the linear dynamics to the stable distribution.

As we will see, the study of the problem (4) involves to understand spectral prop- erties of a family of positive operators on the space of continuous functions onS of the form

pr J^qf^px^qr^px^qf^px^q

»

S

K^py, x^qf^py^qdy (7) wherer is a continuous and positive function over S and K a continuous and non- negative kernel overS. In [6], Coville finds a useful non-integrability criterion on the parameterrwhich gives the existence of eigenfunctions associated with the principal eigenvalue of the operatorr J. When this criterion fails, he gives examples where there’s no eigenfunction. Nonetheless, he shows in [7] that there are always principal eigenvectors in the of Radon measures space. Other properties of the operator

are studied in [8]. In Section 1, we give a new, shorter and unified proof of all these results (see Theorem 2.3). Our approach is based on duality arguments (see Proposition 2.7 which is adapted from a result due to Krein-Rutman [14]) and allows us to obtain the existence of eigenvectors in a measures space. The criterion for the existence of principal eigenfunctions is also deduced. Our approach allows us to study at the same way the operatorr G defined by

pr G^qf^px^qr^px^qf^px^q

»

S

K^px, y^qf^py^qdy (8) which will be used for studying the dual problem (5).

In Section 1.3, we state our main results on the long-time behaviour of the solutions of (2). In Section 2, we study spectral properties of the operators of the form r J,r G defined by (7), (8) and of their analogous operators in measure spaces. In Section 3, we apply these results to the study of the long-time behaviour of the linear dynamics. In Section 4, we deduce from the previous sections the proofs of our main results.

Notations. LetX be a metric space.

• C^pX^q (resp. C ^pX^q) represents the sets of continuous functions from X to R (resp. R ). C_b^pX^q (resp. C_b ^pX^q) represents the sets of continuous and bounded functions fromXtoR(resp. R ). M^pX^q(resp. M ^pX^q) represents the set of finite Radon (resp. positive and finite Radon) measures on X.

M_loc^pX^q (resp M_loc^pX^q) represents the set of Radon (resp. positive Radon) measures on X. For any metric space Y, we denote by C^pX, Y^q the set of continuous functions fromX to Y.

• We denote by C_b^1,0,1 C_b^1,0,1pR XR q (resp. C_c^1,0,1) the set of continu- ous and bounded (resp. with compact support) functions fromR XR with continuous and bounded derivatives with respect to the first and third variables. We define similarlyC_b^0,1 C_b^0,1pXR qandC_c^0,1 C_c^0,1pXR q.

• For anyx^PXandǫ^PR, we denote byV^px, ǫ^q(resp. V^px, ǫ^q) the open (resp.

closed) ball centred inx with radiusǫ.

1.2 Preliminaries

We first give the main assumptions on the model. Then we recall some facts about topology of measure spaces and we conclude by giving some words about the well- posedness of the dynamics (1) and (2).

Assumptions 1.1.

S Ω, Ω^€R^d is open, bounded, connected with Lipschitz boundary, (A1) B, D^PC_b ^pSR q, D^px, a^q¥D^¡0, k^PC_b ^pSR S^q, (A2)

p^Ps0,1^r, c^¡0 (A3)

and there exists ǫ0 ^¡ 0 and I ^€ R with I 0 such that for all x ^P S and y^PV^px, ǫ0^qXS:

I ^€supp^pk^px, y, .^qqXsupp^pB^px, .^qq. (A4) Measure theory. We recall some classical definitions and facts about topology on measures spaces. The Jordan decomposition theorem ensures that for any µ ^P M^pSR q, there isµ , µ^PM ^pSR qmutually singular, such that µµ µ. The total variation measure is defined by^|µ^|µ µand the Total Variation norm by

}µ^}TV^|µ^|pSR q.

The Bounded Lipschitz norm is defined for any µ^PM^pSR q by

}µ^}BLsup

#

»

SR

f^px^qµ^pdx^q

:f ^PW^1,8pSR q,^}f^}1,^8¤1

+

.

where W^1,8pSR q is the set of bounded Lipschitz functions from SR to R and^}f^}1,^{8 }}f^}8 Lip^pf^qwhere Lip^pf^qis the Lipschitz constant of f and ^}f^}8

sup^t|f^px, a^q|:^px, a^qPSR u. We recall (see [21]) that for any sequence µn ^P

M pSR qandµ^PM pSR q,^}µ_nµ^}BL ^Ñ

n^Ñ80 if and only if for all continuous and bounded function f fromSR to R,

nlim^Ñ8

»

SR

f^px^qµn^pdx^q

»

SR

f^px^qµ^pdx^q, i.e thatµ_n ^ѵweakly in ^pC_b^pSR qq

1 (^pC_b^pSR qq

1 represents the dual space of C_b^pSR q). We denote by C^pR ,M ^pSR qq the space of continuous maps from R to M pSR q with respect to the Bounded Lipschitz norm.

Well-posedness. We precise the link between the stochastic process (1) and the partial differential equation (2). We denote^xµ, f^y

³

SR f^px, a^qµ^pdx, da^q. Let us consider a sequence^pZ₀^Kq_K¥0 of M pSR qvalued random variables of the form

Z₀^K 1 K

N_t^K

¸

i1

δpxi,ai^q.

For each K ^P N, let ^pZ_t^Kq_t¥0 be defined as the càdlàg measure-valued process started at Z₀^K with infinitesimal generator L^K given, for any f ^P C_b^0,1 and µ ^P M ^pSR q by

L^KF_f^pµ^q

»

SR

Baf^px, a^qF^1pxµ, f^yqµ^pdx, da^q (9) K

»

SR

"

pF^pxµ δpx,0q

K , f^yqF^pxµ, f^yqqp1p^qB^px, a^q

»

S

pF^pxµ δpy,0^q

K , f^yqF^pxµ, f^yqqB^px, a^qpk^px, a, y^qdy

pF^pxµδpx,a^q

K , f^yqF^pxµ, f^yqqpD^px, a^q c^xµ,1^yq

*

µ^pdx, da^q

where F_f^pµ^q : F^pxµ, f^yq (we note that the set of functions of the form F_f is sufficient to characterise the infinitesimal generator, as it is proved in [9]). The following proposition allows to obtain the solutions of (2) as a large population limit of the stochastic processZ^K. We refer to [19] for the proof.

Proposition 1.2. Assume Assumptions 1.1. Assume that Z₀^K converges in law to n0 ^P M ^pSR q as K ^{Ñ 8}. Then, the sequence of processes ^pZ^KqK^¥0

converges in law (on finite time interval) to the unique weak solution ^pn_t^q_t¥0 ^P

C^pR ,M pSR qqof (2) which satisfies for all f ^PC_b^1,0,1 and t^PR ,

»

SR

ft^px, a^qnt^pdx, da^q

»

SR

f0^px, a^qn0^pdx, da^q (10)

»t

0

»

SR

Bsf_s ^B_af_s

D c

»

SR

n_s

f_s Grf_s^s

px, a^qn_s^pdx, da^qds whereG has been defined in (6).

1.3 Main Results

Let us introduce some notations. For any ^pλ, x, y, a^{q P s} D, ^8rS²R , we define:

$

'

&

'

%

R_λ^px, a^qexp ^³a₀D^px, α^qdα^qλa

, r_λ^px^qp1p^q

³

R B^px, a^qR_λ^px, a^qda, K_λ^px, y^qp

³

R B^px, a^qk^px, a, y^qR_λ^px, a^qda.

(11)

For anyλ^PsD, ^8r, we define the linear operator ˜rλ J˜λ :M^pS^qÑM^pS^q by

pr˜_λ J˜_λ^qµr_λ^px^qµ

»

S

K_λ^py, x^qµ^pdy^q

dx (12)

and we denote by ρ^pr˜λ J˜λ^q its spectral radius. We now give the main results of the paper. The first one shows the existence of stationary states for the dynamics (10), under some assumption on the spectral radius of the operators introduced above (similarly as in [4]). This assumption is related to the supercriticality of the associated linear dynamics.

Theorem 1.3. Assume Assumptions 1.1 andρ^pr˜0 J˜0^q¡1. There exists a non-zero solution n^PM ^pSR qof:

f ^PC_b^0,1,

»

SR

Baf

D c

»

SR

n

f G^rf^s

px, a^qn^pdx, da^q 0, (13) which is given by

n^px, a^qµ_λ^pdx^qR_λ^px, a^qda

where λ ^¡0 is solution of the equation ρ^pr˜_λ J˜_λ^q 1 and µ_λ ^PM ^pS^q is an eigenvector of ˜r_λ J˜_λ associated with the eigenvalueρ^pr˜_λ J˜_λ^q1.

Let us now introduce an additional regularity assumption which allows us to obtain the continuity of the solutions with respect to the initial conditions (see Lemma 3.10).

Assumptions 1.4. B, D^PW^1,8pSR q and k^PW^1,8pR SR q.

We now focus on the case where there exists a stationary measurenwhich admits a continuous density (we keep the same notationnfor the density).

Theorem 1.5. Assume Assumptions 1.1, 1.4. Assume that ρ^pr˜0 J˜0^q¡1 and that there exists a solution n^PC^pS, L^1pR qq of (13). Assume that there existB, k ^¡0 such that B ^¥B and k ^¥k. Let^pn_t^q_t¥0 ^PC^pR ,M pSR qqbe the solution of (10) started atn0^PM ^pSR qzt0^u. Then we have

tlim^Ñ8

}n_tn^}TV0 and nis the unique stationary measure.

2 Spectral Properties of some Positive Operators

2.1 Position of the Problem and Results

Let us consider a subset S of R^d which satisfies Assumption (A1). We analyse the spectral properties of the operatorsr J, r G:C^pS^qÑC^pS^qdefined respectively by (7), (8) and of their analogous operators on measure spaces ˜r J ,˜ r˜ G˜:MpSqÑ

M^pS^q defined similarly by

pr˜ J˜^qµr^px^qµ

»

S

K^py, x^qµ^pdy^q

dx,

pr˜ G˜^qµr^px^qµ

»

S

K^px, y^qµ^pdy^q

dx.

Assumptions 2.1.

r^PC^pS^q is positive, (A5)

K^PC^pSS^q is non-negative, (A6)

Dǫ0, c0 ^¡0, inf

x^PS

y^PVpinfx,ǫ0^qXSK^px, y^q

¡c0. (A7) Remark 2.2. Assume Assumption (A7), then we have

xinf^PS

y^PVpinfx,ǫ0^qXSK^py, x^q

¡c0. (14)

Indeed, letx^PS and y^PV^px, ǫ0^qXS. Thenx^PV^py, ǫ0^qand K^py, x^q¡c0.

The following result deals with the spectral properties of the operators introduced above. We denote byρ^pr J^qthe spectral radius of the operatorr J (and similarly forr G). The reader can refer to Appendix A for terminology and recalls about spectral theory. The following theorem is proved in Section 2.2.

Theorem 2.3. Assume Assumptions 2.1. There existsµ^PM ^pS^q such that ρ^pr G^qµ^pr˜ J˜^qµ

which satisfies µ^pA^{q ¡} 0 for any Borel subset A of S such that Leb^pA^{q ¡} 0 (Leb denotes the Lebesgue measure on S). Moreover, let us denote r sup_xPSr^px^q and Σ^tx^PS :r^px^qr^u. Then we have:

(i) r^¤ρ^pr G^q.

(ii) r ρ^pr G^qif and only if there exists u^PC^pS^q,u^¡0 such that µu^px^qdx.

In this case,ρ^pr G^qis an eigenvalue ofr J with algebraic multiplicity equals to one, associated with the eigenfunctionu.

(iii) If Leb^pΣ^q¡0 or if, Leb^pΣ^q0 and _r¹

r ^RL^1pS^q, we have r ρ^pr G^q. (iv) Ifρ^pr G^q r, we have µµ^s h^px^qdx with h ^PL^1pSq and either µ^s0,

orµ^s0 and supp^pµ^sq€Σ.

(v) ρ^pr J^qρ^pr G^qρ^p˜r J˜^qρ^pr˜ G˜^q.

Moreover, the same results are true exchanging J and G, ˜J and ˜G.

Proof. See Section 2.2.

Remark 2.4. In [6],[7], Coville studies some spectral properties of the operators introduced above. To do so, he introduces the generalised principal eigenvalue

λp^pr J^qsup^tλ^PR|Dϕ^PC^pS^q, ϕ^¡0 such that^pr J^qϕ λϕ^¤0^u which generalises the Perron-Frobenius eigenvalue for irreducible matrices with non- negative coefficients. The point (iii) is similar to the criterion obtained (in a more general setting) in [6]. The point (iv) is contained in the results of [7]. The point (v) is new. It is crucial for the proof of Propositions 3.5 and 3.6.

2.2 Proof of Theorem 2.3

The proof of Theorem 2.3 is given at the end of this section. We start by proving a lemma in which some well known facts about the operators introduced previously are recalled. We give a proof for the convenience of the reader. We denote by ρ_e^pr J^q the essential spectral radius ofr J (see Appendix A).

Lemma 2.5. Assume Assumptions 2.1.

(i) The operators r J and r G are positive and irreducible onC^pS^q. (ii) The operatorsJ and Gare compact on C^pS^q.

(iii) rρ_e^pr J^qρ_e^pr G^q.

Proof. (i): Since r is positive and K is non-negative, r J and r G are positive endomorphisms of C^pSq. To prove irreducibility, it suffices to prove that there is m ^P N such that for all f ^P C ^pS^q and x ^PS, ^pr J^qmf^px^{q ¡} 0 (see Definition A.6 and Lemma A.11). Since the setS is compact, there existn^PN and ^pBi^qn

i1 a family of balls with radius ǫ0^{4 such thatS €Y

n

i1B_i. Let f ^PC ^pSq be non-zero and letI be an open subset ofS such thatf is positive on I. For allx^PS we have

pr J^qnf^px^q¥

»

I

dx1f^px1^qK^px1, x2^q

»

S

dx2. . .

»

S

dx_nK^px_n, x^q

¥C

»

I^XBi1^XS

dx1K^px1, x2^q

»

Bi2^XS

dx2. . .

»

B_in^XS

dxnK^pxn, x^q whereC ^¡0 and ^pi1, . . . , i_n^{q P}J1, nKⁿ satisfies: B_i1^XI has non-empty interior; for any k^PJ1, n1K,u^PB_ik^XS,v^PB_{ik 1}^XS,K^pu, v^q¡c0 and for anyu^PB_n^XS, K^pu, x^q¡c0. It comes that

pr J^qnf^px^q¡0 and r J is irreducible. The proof is similar forr G.

(ii): Let M be a bounded subset of C^pS^q and let C_M be a positive constant such that for allf ^PM,^}f^}8

 C_M. Applying Ascoli’s criterion, we prove that J^pM^q is relatively compact in C^pS^q. Let x^PS and f ^PM, we have

|J f^px^q|¤}f^}8sup

z^PS

»

S

K^py, z^qdy

¤C_Msup

z^PS

»

S

K^py, z^qdy.

Then the set ^tJ f^px^q, f ^P M^u is bounded and so relatively compact in R. We check the equi-continuity condition. Let ǫ ^¡ 0, since K is uniformly continuous on SS, there exists δ ^¡ 0 such that if ^}x1 x2^{} }}y1 y2^}   δ, we have

|K^px1, y1^qK^px2, y2^q|   ǫ CMLebpSq

. Let y ^P S such that ^}xy^}   δ. For all f ^PM, we have

|J f^py^qJ f^px^q|

»

S

f^pz^qpK^pz, y^qK^pz, x^qqdz

¤CM

»

S

|K^pz, y^qK^pz, x^q|dz ǫ

which allows us to conclude for the compactness. The proof is similar for G.

(iii): Let us note from Lemma A.12 that the essential spectrum ofr is^tr^px^q, x^PS^u. Moreover J is compact. We deduce that ρ_e^pr J^qρ_e^pr^q and that rρ_e^pr J^q. The proof is similar for ρ_e^pr G^q.

The following lemma makes a duality link between the operators introduced above. It is crucial for the proof of Theorem 2.3.

Lemma 2.6. Assume Assumptions 2.1. We have

pr J^q1˜r G˜ ^pr G^q1 r˜ J˜ (15) where^pr J^q1 is the adjoint operator of r J and ^pr G^q1 is defined similarly.

Proof. Letf ^PC^pS^q and µ^PM^pS^q,

»

S

µ^pdx^qpr J^qf^px^q

»

S

µ^pdx^q

r^px^qf^px^q

»

S

K^py, x^qf^py^qdy

»

S

f^px^qr^px^qµ^pdx^q

»

S

f^py^q

»

S

K^py, x^qµ^pdx^q

dy

»

S

f^px^q

r^px^qµ^pdx^q dx

»

S

K^px, y^qµ^pdy^q

»

S

f^px^qp˜r G˜^qµ^pdx^q

where we used Fubini’s Theorem. The proof is similar for^pr G^q1.

The next result is easily adapted from [18, Appendix §2.2.6] (it was originally in- troduced by Krein-Rutman [14]) and [18, Appendix §3.3.3]. Combined with Lemma 2.6, it is the main tool for the proof of Theorem 2.3.

Proposition 2.7. (i) Let T be a positive endomorphism of C^pS^q. The spectral radiusρ^pT^qis an eigenvalue ofT¹ associated with an eigenvector which belongs to M ^pS^q.

(ii) LetT be an irreducible endomorphism ofC^pS^q. Then the spectral radiusρ^pT^q is the only possible eigenvalue ofT associated with a non-negative eigenvector.

Moreover, if ρ^pT^q is a pole of the resolvent, it is an eigenvalue of T with algebraic multiplicity equals to one.

We give now a technical lemma.

Lemma 2.8. Assume Assumptions 2.1. Let x0 ^PΣ ^tx ^PS : r^px^q r^u. There exists a family ^prj^qj^¥0 ofC ^pS^q which satisfy for all j^¥0:

(i) For all x^PS,r_j^px^q¥r_j 1^px^q,

(ii) r_j^px0^qr_j r and Leb^pΣj^q¡0,where Σj ^tx^PS:r_j^px^qr_j^u, (iii) ^}r_jr^}8

Ñ

j^Ñ80.

Proof. Let x0 ^P Σ be fixed. For all ǫ ^¡ 0 sufficiently small, we define the closed set Aǫ V^px0, ǫ^qcYV^px0, ǫ^{2^qXS and a map gǫ ^P C^pAǫ^q by gǫ^px^q r^px^q if x ^P V^px0, ǫ^qc and g_ǫ^px^q r if x ^P V^px0, ǫ^{2^q. By Tietze Theorem, we extend g_ǫ in a continuous function h_ǫ on S such that ^}h_ǫ^}8

}g_ǫ^}8 We introduce r_ǫ ^P C^pS^q defined by rǫ^px^q max^phǫ^px^q, r^px^qq. It is straightforward to check that: 1) limǫ^Ñ0^}r_ǫ r^}8

0; Leb^pΣǫ^{q ¡} 0; 3) r_ǫ^px^{q ¥} r^px^q and sup_xPSr_ǫ^px^q r. We conclude by proving that we can extract a decreasing subsequence of the family r_ǫ which converges uniformly tor. To do so, we fixǫ0^¡small and we define a sequence

pǫ_k^q_k¥0 by ǫ_k 1 ǫk

2. We check that the sequence ^pr_ǫk^q_k¥0 is decreasing. Indeed, let k^¥0. If x ^PVpx0, ǫ_k 1^{2^q, then x^PVpx0, ǫ_k^{4^q and r_{ǫk 1}^px^q h_{ǫk 1}^px^qr hǫ_k^px^qrǫ_k^px^q. If x^PA^c_{ǫk 1} we have ǫ_k 1^{2 ^}xx0^} ǫ_k 1 ǫ_k^{2. So we have r_ǫk 1^px^q¤rh_ǫk^px^qr_ǫk^px^q. Finally, ifx^PV^px0, ǫ_k 1^qc,r_ǫk 1^px^qr^px^q¤r_ǫk^px^q. So we have proved that for all x^PS,r_{ǫk 1}^px^q¤r_ǫk^px^q.