
schematic representation of the variables on an E. coli bacterium
.Citation: Hidetaka Hamasaki, Yasuteru Hamasaki. Nuts for Physical Health and Fitness: A Review[J]. AIMS Medical Science, 2017, 4(4): 441-455. doi: 10.3934/medsci.2017.4.441
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In structured population dynamics, finding the structuring variable(s) which best describes a phenomenon is a crucial question. For a population of proliferating cells or bacteria the variables usually considered are age, size (see [27,12,18]) or a combination of both (see [1,24,10] for modeling and [26,12,10,6] for mathematical analysis). Recent experimental work highlighted the limits of these models to describe bacteria, and a new variable to trigger division emerged: the size-increment, namely the size gained since the birth of the cell (see [22] and references therein for a review of the genesis of the related model). This so called 'adder principle ensures homeostasis with no feedback from the bacteria and explains many experimental data. In this model, bacteria are described by two parameters: their size-increment and their size, respectively denoted by
{∂tn(t,a,x)+∂a(g(x)n(t,a,x))+∂x(g(x)n(t,a,x))+B(a)g(x)n(t,a,x)=0,t≥0, x>a>0,g(x)n(t,0,x)=4g(2x)∫∞0B(a)n(t,a,2x)da,t≥0, x>0. |
The function
∂tn(t,a,x)+∂a(g(x)n(t,a,x))+∂x(g(x)n(t,a,x))+B(a)g(x)n(t,a,x)=0,t≥0, x>a>0, | (1a) |
g(x)n(t,0,x)=∫10g(xz)∫∞0B(a)n(t,a,xz)da dμ(z)z,t≥0, x>0. | (1b) |
It appears that this model is a particular case of the one proposed in the pioneer work [10] for plants growing in a single dimension, mixing age and size control. Indeed, in this paper the authors noticed that in the case of a deterministic and positive growth rate, a size/age model is equivalent to a size/birth-size through the relation
First, we want the sum of the daughters sizes to be equal to the size of the mother. This rule, called mass conservation, prescribes
∫10zdμ(z)=1. | (2) |
We also assume that the division does not produce any arbitrarily small daughter by imposing that the support of
θ:=infsuppμ>0and∃η∈(θ,1), suppμ⊂[θ,η]. | (3) |
In particular, these assumptions imply that the mean number of daughters
∃b≥0,suppB=[b,∞), | (4) |
see [8] for instance. It will be useful in our study to define the associated survivor function
Ψ(a)=e−∫a0B(z)dz. |
For a given increment
∃k0>0,Ψ(a)=+∞O(a−k0). | (5) |
This assumption on the decay at infinity of the survivor function enables a wide variety of division rates. For instance, it is satisfied if there exists
∀a≥A,B(a)≥k0a. |
The function
Φ=BΨ=−Ψ′ | (6) |
which is the probability distribution that a cell divides at increment
∬xn(t,a,x)dadx=et∬xn0(a,x)dadx. | (7) |
This implies that if we look for a solution with separated variables
∂a(xN(a,x))+∂x(xN(a,x))+(1+xB(a))N(a,x)=0,x>a>0, | (8a) |
N(0,x)=∫10∫∞0B(a)N(a,xz)da dμ(z)z2,x>0, | (8b) |
N(a,x)≥0,x≥a≥0, | (8c) |
∫∞0∫x0N(a,x)dadx=1. | (8d) |
It is convenient to define the set
Theorem 1.1. Let
N:(a,x)∈X↦Ψ(a)x2f(x−a) | (9) |
where
f∈L1(R+,xldx) |
for all
suppf=[bθ,∞) |
with
The fast decay of the function
Notice also that for the function
The article is organised as follows. In Section 2 we reduce the Perron eigenvalue problem with two variables to a fixed point problem for an integral operator in dimension one. Section 3 is dedicated to proving the existence and uniqueness of the fixed point by using functional analysis and Laplace transform methods. In Section 4 we go through the usefulness of knowing
Our study consists in constructing a solution to the eigenproblem (8) from the solution of a fixed point problem. First, we notice that the size
M(a,s):=N(a,a+s). | (10) |
Thanks to this relation, it is equivalent to prove the existence of an eigenvector for the increment-size system or for the increment/birth-size system. To determine the equation verified by
∂a((a+s)M(a,s))+(1+(a+s)B(a))M(a,s)=0. |
Writing the non-local boundary condition (8b) with the new variables takes less calculation and more interpretation. In (8b) the number of cells born at size
M(0,s)=∫ηθ∫sz0B(a)M(a,sz−a)da dμ(z)z2 |
since there is no mass for
∂a((a+s)M(a,s))+(1+(a+s)B(a))M(a,s)=0,a,s>0, | (11a) |
M(0,s)=∫ηθ∫sz0B(a)M(a,sz−a)da dμ(z)z2,s>0, | (11b) |
M(a,s)≥0,a,s≥0, | (11c) |
∫R2+M(a,s)dads=1. | (11d) |
Considering the variable
M(a,s)=Ψ(a)(a+s)2s2M(0,s). |
Having this expression in mind, we note that for any nonnegative function
Mf:(a,s)↦Ψ(a)(a+s)2f(s) |
is a solution of (11a) and satisfies (11c). Then it remains to choose the appropriate function
Tf(s)=∫ηθ∫sz0Φ(sz−a)f(a)dadμ(z), | (12) |
where
Lemma 2.1. The function
Proof.
Mf satisfies (11b)⟺f(s)s2=∫ηθ∫sz0B(a)Ψ(a)(sz)2f(sz−a)da dμ(z)z2⟺f(s)=∫ηθ∫sz0Φ(a)f(sz−a)da dμ(z)⟺f(s)=∫ηθ∫sz0Φ(sz−a)f(a)da dμ(z)⟺f(s)=Tf(s) |
The operator
Lemma 2.2. For all
Proof. We start with
∫βα|Tf(s)|slds≤∫ηθ∫βαsl∫sz0Φ(sz−a)|f(a)|dadsdμ(z)≤∫ηθ∫αz0|f(a)|∫βαΦ(sz−a)sldsdadμ(z)+∫ηθ∫βzαz|f(a)|∫βzaΦ(sz−a)sldsdadμ(z)≤∫ηθ∫αz0|f(a)|∫βz−aαz−aΦ(σ)(a+σ)lzl+1dσdadμ(z)+∫ηθ∫βzαz|f(a)|∫βz−a0Φ(σ)(a+σ)lzl+1dσdadμ(z)≤∫ηθzl+1∫αz0|f(a)|aldadμ(z)+∫ηθzl+1∫∞αz|f(a)|aldadμ(z)≤θl‖f‖L1(R+,slds), |
which gives the conclusion by passing to the limits
For the second part we begin with the proof that under condition (5), for any
∫∞0Φ(a)akda<∞. |
First, recall that
∫β0Φ(a)akda≤∫10Φ(a)da+∫β1Φ(a)akda≤1+k∫β1Ψ(a)ak−1da |
and the last integral converges when
∫β0|Tf(s)|skds≤∫ηθ∫β0sk∫sz0Φ(sz−a)|f(a)|dadsdμ(z)=∫ηθ∫βz0|f(a)|∫βz−a0Φ(σ)(a+σ)kzk+1dσdadμ(z)≤C∫ηθzk+1∫βz0|f(a)|ak∫βz−a0Φ(σ)dσdadμ(z)+C∫ηθzk+1∫βz0|f(a)|∫βz−a0Φ(σ)σkdσdadμ(z)≤Cηk(‖f‖L1(R+,skds)+‖Φ‖L1(R+,skds)‖f‖L1(R+)). |
In this section we prove the existence of a unique nonnegative and normalized fixed point of the operator
Let us first recall some definitions from the Banach lattices theory (for more details, see [9,23]). Let
f≥0 if and only if f(s)≥0 ν-a.e. on Ω. |
Furthermore, endowed with its standard norm, the space
To prove the existence of an eigenvector associated to the eigenvalue
Theorem 3.1. Let
Due to a lack of compactness of the operator
TΣf(s)=∫ηθ∫min(sz,Σ)bθΦ(sz−a)f(a)dadμ(z) | (13) |
={∫ηθ∫szbθΦ(sz−a)f(a)dadμ(z),bθ≤s<θΣ,∫sΣθ∫ΣbθΦ(sz−a)f(a)dadμ(z)+∫ηsΣ∫szbθΦ(sz−a)f(a)dadμ(z),θΣ≤s≤ηΣ,∫ηθ∫ΣbθΦ(sz−a)f(a)dadμ(z),ηΣ<s≤Σ. |
Defining the lower bound of the domain as
The following lemma ensures that the truncated operator
Lemma 3.2. If
(Tf)|[bθ,Σ]=TΣ(f|[bθ,Σ]). |
Lemma 3.2 is a straightforward consequence of the definition of operator
{∀ϵ>0, ∀ω⊂⊂Ω, ∃δ∈(0,dist(ω,cΩ)) such that ‖τhf−f‖L1(ω,ν)<ϵ, ∀h∈(−δ,δ), ∀f∈F | (14) |
{∀ϵ>0, ∃ω⊂⊂Ω, such that ‖f‖L1(Ω∖ω,ν)<ϵ, ∀f∈F | (15) |
where
Theorem 3.3 (from [3], corollary 4.27). If
Using Theorem 3.1, we prove the existence of an eigenpair
Proposition 1. Let
Applying Theorem 3.3, to
F={TΣf,f∈L1((bθ,Σ),slds),‖f‖L1((bθ,Σ),slds)≤1}, |
which is bounded in
Lemma 3.4. Let
Σ>max(b1−θ,1), |
the set
Proof of Lemma 3.4. The set
∫βα|TΣf(s+h)−TΣf(s)|slds≤∫θΣ−hα|TΣf(s+h)−TΣf(s)|slds=:(A)+∫θΣθΣ−h|TΣf(s+h)−TΣf(s)|slds=:(B)+∫ηΣ−hθΣ|TΣf(s+h)−TΣf(s)|slds=:(C)+∫ηΣηΣ−h|TΣf(s+h)−TΣf(s)|slds=:(D)+∫βηΣ|TΣf(s+h)−TΣf(s)|slds=:(E). |
since for
(C)=∫ηΣ−hθΣ|TΣf(s+h)−TΣf(s)|slds≤∫ηΣ−hθΣ|∫s+hΣθ∫ΣbθΦ(s+hz−a)f(a)dadμ(z)−∫sΣθ∫ΣbθΦ(sz−a)f(a)dadμ(z)|slds+∫ηΣ−hθΣ|∫ηs+hΣ∫s+hzbθΦ(s+hz−a)f(a)dadμ(z)−∫ηsΣ∫szbθΦ(sz−a)f(a)dadμ(z)|slds≤∫ηΣ−hθΣsl∫sΣθ∫Σbθ|Φ(s+hz−a)−Φ(sz−a)||f(a)|dadμ(z)ds=:(C1)+∫ηΣ−hθΣsl∫s+hΣsΣ∫ΣbθΦ(s+hz−a)|f(a)|dadμ(z)ds=:(C2) |
+∫ηΣ−hθΣsl∫ηs+hΣ∫szbθ|Φ(s+hz−a)−Φ(sz−a)||f(a)|dadμ(z)ds=:(C3)+∫ηΣ−hθΣsl∫ηs+hΣ∫s+hzszΦ(s+hz−a)|f(a)|dadμ(z)ds=:(C4)+∫ηΣ−hθΣsl∫s+hΣsΣ∫szbθΦ(s+hz−a)|f(a)|dadμ(z)ds=:(C5) |
The integrals
(C1)=∫ηΣ−hθΣsl∫sΣθ∫Σbθ|Φ(s+hz−a)−Φ(sz−a)||f(a)|dadμ(z)ds=∫η−hΣθ∫Σbθ|f(a)|∫ηΣ−hzΣ|Φ(s+hz−a)−Φ(sz−a)|sldsdadμ(z)=∫η−hΣθzl+1∫Σbθ|f(a)|∫ηΣ−hz−aΣ−a|τhzΦ(σ)−Φ(σ)|(a+σ)ldσdadμ(z)=∫η−hΣθzl+1∫Σbθ|f(a)|al∫ηΣ−hz−aΣ−a|τhzΦ(σ)−Φ(σ)|dσdadμ(z)≤θlsupε∈[θ,η]‖τhεΦ−Φ‖L1(R+). |
These integrals are as small as needed when
(C2)=∫ηΣ−hθΣsl∫s+hΣsΣ∫ΣbθΦ(s+hz−a)|f(a)|dadμ(z)ds=∫θ+hΣθ∫Σbθ|f(a)|∫zΣθΣΦ(s+hz−a)sldsdadμ(z)+∫η−hΣθ+hΣ∫Σbθ|f(a)|∫zΣzΣ−hΦ(s+hz−a)sldsdadμ(z)+∫ηη−hΣ∫Σbθ|f(a)|∫ηΣ−hzΣ−hΦ(s+hz−a)sldsdadμ(z)≤∫ηθ∫Σbθ|f(a)|∫zΣzΣ−hΦ(s+hz−a)sldsdadμ(z)=∫ηθzl+1∫Σbθ|f(a)|∫Σ−aΣ−hz−aΦ(σ+hz)(a+σ)ldσdadμ(z)≤∫ηθzl+1∫Σbθ|f(a)|al∫Σ−aΣ−hz−aΦ(σ+hz)dσdadμ(z)≤θlsup|I|=hθ∫IΦ(a)da |
which is small when
(C4)=∫ηΣ−hθΣsl∫ηs+hΣ∫s+hzszΦ(s+hz−a)|f(a)|dadμ(z)ds=∫ηθ+hΣ∫zΣ−hθΣsl∫s+hzszΦ(s+hz−a)|f(a)|dadsdμ(z)=∫ηθ+hΣ∫zΣ−hθΣsl∫0−hzΦ(hz+a′)|f(sz−a′)|da′dsdμ(z)=∫ηθ+hΣ∫0−hzΦ(hz+a′)∫zΣ−hθΣ|f(sz−a′)|sldsda′dμ(z)=∫ηθ+hΣzl+1∫0−hzΦ(hz+a′)∫Σ−hz−a′θΣz−a′|f(σ)|(σ+a′)ldσda′dμ(z)≤∫ηθ+hΣzl+1∫0−hzΦ(hz+a′)da′dμ(z)≤θl(1−Ψ(hθ)), |
and the continuity of
(B)=∫θΣθΣ−h|∫s+hΣθ∫ΣbθΦ(s+hz−a)f(a)dadμ(z)+∫ηs+hΣ∫s+hzbθΦ(s+hz−a)f(a)dadμ(z)−∫ηθ∫szbθΦ(sz−a)f(a)dadμ(z)|slds≤∫ηθ∫Σbθ|f(a)|∫θΣθΣ−h[Φ(s+hz−a)+Φ(sz−a)]sldsdadμ(z)≤∫ηθzl+1∫Σbθ|f(a)|al∫θΣz−aθΣ−hz−a[Φ(σ+hz)+Φ(σ)]dσdadμ(z)≤2θlsup|I|=hθ∫IΦ(a)da |
and again the last term vanishes as
∫αbθ|TΣf(s)|slds |
≤∫αbθsl∫ηθ∫szbθΦ(sz−a)|f(a)|dadμ(z)ds≤∫ηθ∫bθzbθ|f(a)|∫αbθΦ(sz−a)sldsdadμ(z)+∫ηθ∫αzbθz|f(a)|∫αzaΦ(sz−a)sldsdadμ(z)≤∫ηθzl+1∫bθzbθ(Ψ(bθz−a)−Ψ(αz−a))|f(a)|aldazdμ(z)+∫ηθ∫αzbθz(1−Ψ(αz−a))|f(a)|aldazdμ(z)≤θl(1−Ψ(α−bθθ)), | (16) |
since for
As done before, we choose a
∫Σβ|TΣf(s)|slds≤∫ηθ∫Σbθ|f(a)|∫ΣβΦ(sz−a)sldsdadμ(z)≤∫ηθzl+1∫Σbθ|f(a)|al∫Σz−aβz−aΦ(σ)dσdadμ(z)≤θlsup|I|=Σ−βθ∫IΦ(a)da, | (17) |
which is small when
We have checked the assumptions of Theorem 3.3 for the family
To prove the irreducibility of the operator
TΣf(s)={∫sηbθf(a)∫ηθΦ(sz−a)dμ(z)da+∫sθsηf(a)∫saθΦ(sz−a)dμ(z)da,bθ≤s<θΣ,∫sηbθf(a)∫ηθΦ(sz−a)dμ(z)da+∫Σsηf(a)∫saθΦ(sz−a)dμ(z)da,θΣ≤s≤ηΣ,∫Σbθf(a)∫ηθΦ(sz−a)dμ(z)da,ηΣ<s≤Σ. |
Lemma 3.5. Let
Proof. Let
TΣfω(s)≥∫s0+ζs0∫θ+ξθΦ(sz−a)dμ(z)fω(a)da. |
For
Proof of Proposition 1. Lemma 3.4 shows that the set
We now want to show that up to a subsequence,
Proposition 2. Under hypotheses (2)- (5) there exists a nonnegative and normalized fixed point
f∈L1(R+,(sk+sl)ds) |
for all
First, we will show that the sequence
Lemma 3.6. If
1−Ψ((1η−1)Σ)≤ρΣ≤1−Ψ(Σθ−bθ). | (18) |
Proof. Integrating the equality
ρΣ∫ΣbθfΣ(s)ds=∫θΣbθ∫ηθ∫szbθΦ(sz−a)f(a)dadμ(z)ds=:(A)+∫ηΣθΣ∫sΣθ∫ΣbθΦ(sz−a)f(a)dadμ(z)ds=:(B)+∫ηΣθΣ∫ηsΣ∫szbθΦ(sz−a)f(a)dadμ(z)ds=:(C)+∫ΣηΣ∫ηθ∫ΣbθΦ(sz−a)f(a)dadμ(z)ds=:(D) |
(A)=∫ηθ∫bθzbθfΣ(a)∫θΣbθzΦ(sz−a)dsdadμ(z)+∫ηθ∫θΣzbθzfΣ(a)∫θΣzaΦ(sz−a)dsdadμ(z)=∫ηθz∫bθzbθfΣ(a)[Ψ(bθz−a)−Ψ(θΣz−a)]dadμ(z)+∫ηθz∫θΣzbθzfΣ(a)[1−Ψ(θΣz−a)]dadμ(z)(B)=∫ηθfΣ(a)∫Σbθ∫ηΣzΣΦ(sz−a)dsdadμ(z)=∫ηθz∫ΣbθfΣ(a)[Ψ(Σ−a)−Ψ(ηΣz−a)]dadμ(z) |
(C)=∫ηθ∫θΣzbθfΣ(a)∫zΣθΣΦ(sz−a)dsdadμ(z)+∫ηθ∫ΣθΣzfΣ(a)∫zΣzaΦ(sz−a)dsdadμ(z)=∫ηθz∫θΣzbθfΣ(a)[Ψ(θΣz−a)−Ψ(Σ−a)]dadμ(z)+∫ηθz∫ΣθΣzfΣ(a)[1−Ψ(Σ−a)]dadμ(z)(D)=∫ηθ∫ΣbθfΣ(a)∫ΣηΣΦ(sz−a)dsdadμ(z)=∫ηθz∫ΣbθfΣ(a)[Ψ(ηΣz−a)−Ψ(Σz−a)]dadμ(z) |
Then notice that for
bθz−a≤bθz−bθ=bθ(1z−1)≤bθ(1θ−1)=b, |
so as in the computations leading to (16),
ρΣ∫ΣbθfΣ(s)ds=∫ΣbθfΣ(s)ds−∫ηθz∫ΣbθΨ(Σz−a)fΣ(a)dadμ(z). | (19) |
Using the fact that the function
Now we show that up to a subsequence,
Σ0:=inf{Σ>max(11−θb,1) such that ρΣ>12}. |
Lemma 3.7. Under hypotheses (2), (3), (4) and (5), the set of eigenfunctions
Proof. Let
‖τhfΣ−fΣ‖L1(ω,(sk+sl)ds)≤2‖τhTΣfΣ−TΣfΣ‖L1(ω,(sk+sl)ds)≤2(βk+αl)‖τhTΣfΣ−TΣfΣ‖L1([α,β])≤2(βk+αl)‖τhTΣfΣ−TΣfΣ‖L1([α,Σ]). |
The last quantity is small when
‖fΣ‖L1((bθ,α),(sk+sl)ds)=1ρΣ∫αbθTΣfΣ(s)(sk+sl)ds≤2∫αbθTΣfΣ(s)slds+2αk∫αbθTΣfΣ(s)ds≤2θl(1−Ψ(α−bθθ))+2αk(1−Ψ(α−bθθ))≤2(θl+αk)(1−Ψ(α−bθθ)) |
which is again independent of
ρΣ∫ΣβfΣ(a)da=∫θΣβTΣfΣ(a)da+∫ηΣθΣTΣfΣ(a)da+∫ΣηΣTΣfΣ(a)da. |
For the first integral, we compute
∫θΣβTΣfΣ(a)da=∫ηθz∫βzbθ[Ψ(βz−a)−Ψ(θΣz−a)]fΣ(a)dadμ(z)+∫ηθz∫θΣzβz[1−Ψ(θΣz−a)]fΣ(a)dadμ(z). |
The two other integrals correspond to the integrals
ρΣ∫ΣβfΣ(a)da=∫ηθz∫βzbθΨ(βz−a)fΣ(a)dadμ(z)+∫ηθz∫ΣβzfΣ(a)dadμ(z)−∫ηθz∫ΣbθΨ(Σz−a)fΣ(a)dadμ(z). |
We deal with the last integral using (19) and obtain after interverting integrals
ρΣ∫ΣβfΣ(a)da=∫βηbθfΣ(a)∫ηθzΨ(βz−a)dμ(z)da+∫βθβηfΣ(a)∫βaθzΨ(βz−a)dμ(z)da+∫βθβηfΣ(a)∫ηβazdμ(z)da+∫ΣβθfΣ(a)da+ρΣ∫ΣbθfΣ(a)da−∫ΣbθfΣ(a)da⟺∫βηbθfΣ(a)da=∫βηbθfΣ(a)∫ηθzΨ(βz−a)dμ(z)da+∫βθβηfΣ(a)∫βaθzΨ(βz−a)dμ(z)da+ρΣ∫βbθfΣ(a)da−∫βθβηfΣ(a)∫βaθzdμ(z)da⟺∫βηβfΣ(a)∫ηθz[1−Ψ(βz−a)]dμ(z)da+(1−ρΣ)∫βbθfΣ(a)da+∫βθβηfΣ(a)∫βaθz[1−Ψ(βz−a)]dμ(z)da=∫βbθfΣ(a)∫ηθzΨ(βz−a)dμ(z)da |
Since
∫rββfΣ(a)∫ηθz[1−Ψ(βz−a)]dμ(z)da≤∫βbθfΣ(a)∫ηθzΨ(βz−a)dμ(z)da, |
then
(1−Ψ((1η−r)β))∫rββfΣ(a)da≤Ψ((1η−1)β), |
and finally
∫rββfΣ(a)(al+ak)da≤(βl+(rβ)k)Ψ((1η−1)β)(1−Ψ((1η−r)β))≤4(rβ)kΨ((1η−1)β) |
for
∫∞βfΣ(s)(sk+sl)ds=∞∑j=0∫rj+1βrjβfS(s)(sk+sl)ds≤4rk∞∑j=0(rjβ)kΨ((1η−1)rjβ) |
≤4Crk∞∑j=0(rjβ)k((1η−1)rjβ)−k0≤Ck,k0,η,rβk0−k |
due to hypothesis (5), for
We are now ready to prove the existence and uniqueness of a fixed point for the operator
Proof of Proposition 2. We have proved in Lemma 3.7 that the set of eigenfunctions
‖f−Tf‖L1(R+,(sk+sl)ds)≤‖f−fΣ‖L1(R+,(sk+sl)ds)+(1−ρΣ)+‖TΣfΣ−Tf‖L1(R+,(sk+sl)ds). |
The first term of the right-hand side tends to zero as
‖TΣfΣ−Tf‖L1(R+,(sk+sl)ds)≤‖TΣfΣ−TfΣ‖L1(R+,(sk+sl)ds)⏟=0+‖T(f−fΣ)‖L1(R+,(sk+sl)ds)≤‖f−fΣ‖L1(R+,(sk+sl)ds) |
due to Lemma 3.2 and to the continuity of
To prove uniqueness of the fixed point, we consider
f(s)=∫ηθΦ∗f(sz)dμ(z). | (20) |
Since
L[f−f1](y)=∫ηθL[f−f1](zy)L[Φ](zy)zdμ(z). | (21) |
The Laplace transform
L[f−f1](0)=∫∞0f(s)ds−∫∞0f1(s)ds=0. |
We now define the functions
¯L(y)=supx∈[0,y]L[f−f1](x) and L_(y)=infx∈[0,y]L[f−f1](x). |
By continuity in
∀y≥0,¯L(y)≥0, L_(y)≤0. |
From (21), we obtain the inequality
L[f−f1](y)≤¯L(ηy)∫ηθL[Φ](zy)zdμ(z)≤¯L(ηy), |
since
L[f−f1](x)≤¯L(ηx)≤¯L(ηy), |
from which we deduce
¯L(y)≤¯L(ηy). | (22) |
Iterating (22), we obtain for all
¯L(y)≤¯L(ηjy). |
Letting
It remains to prove that
We are now ready to prove the main theorem of the paper.
Proof of Theorem 1.1. Combining Lemma 2.1 and Proposition 2, we construct a solution to (11) using
M(a,s):=ψ(a)(a+s)2f(s). |
It remains to prove uniqueness of the solution in the appropriate space. This solution belongs to
∫∞bθ∫∞0M(a,s)(1+s2)dads=∫∞bθ∫∞01(a+s)2f(s)Ψ(a)dads+∫∞bθ∫∞0s2(a+s)2f(s)Ψ(a)dads≤∫∞bθf(s)s−2∫∞0Ψ(a)dads+∫∞bθf(s)∫∞0Ψ(a)dads=‖f‖L1((bθ,∞),(1+s−2)ds)<∞ |
because
M1(a,s)=Ψ(a)(a+s)2f1(s). |
For
∫∞0f1(s)ds=1β−α∫∞0∫βα(a+s)2Ψ(a)M1(a,s)dads≤2(β2+1)(β−α)Ψ(β)‖M1‖L1(R+,(1+s2)ds), |
and this ensures that
Now that we have solved the eigenvalue problem, we would like to characterize the asymptotic behaviour of a solution
|n(t,a,x)|≤CetN(a,x),t≥0, x>a>0. | (23) |
It is usually ensured by the hypothesis
H[n]=∫∞bθ∫x−bθ0xN(a,x)H(n(a,x)N(a,x))dadx |
which satisfies the following entropy property.
Proposition 3. If
ddtH[n(t,⋅,⋅)e−t]=−D[n(t,⋅,⋅)e−t], | (24) |
with
D[n]=∫∞bθx2N(0,x)[∫ηθ∫xz−bθ0H(n(a,xz)N(a,xz))dνx(a,z)−H(∫ηθ∫xz−bθ0n(a,xz)N(a,xz)dνx(a,z))]dx |
where
Before proving this proposition, we make a remark about the conservative problem (i.e. when only one daughter out of two is kept after division). In this case, the dominant eigenvalue is
∂∂a(x2N)+∂∂x(x2N)=−x2BN, | (25) |
which might also be obtained multiplying (8a) by
Proof. Easy computations lead to
∂∂tne−tN+x∂∂ane−tN+x∂∂xne−tN=0, |
where
∂∂t(xNH(ne−tN))+∂∂a(x2NH(ne−tN))+∂∂x(x2NH(ne−tN))=−x2BNH(ne−tN), | (26) |
and integrating (26) over
ddt∬(bθ,∞)×(0,x−bθ)xNH(ne−tN)dadx=∫∞bθx2N(0,x)H(n(t,0,x)e−tN(0,x))dx−∫∞bθx2N(x−bθ,x)H(n(t,x−bθ,x)e−tN(x−bθ,x))dx+∫∞0(a+bθ)2N(a,a+bθ)H(n(t,a,a+bθ)e−tN(a,a+bθ))da−∫∞bθ∫x−bθ0x2BNH(ne−tN)dadx=∫∞bθx2N(0,x)H(e−tN(0,x)∫ηθ∫xz−bθ0B(a)n(t,a,xz)dadμ(z)z2)dx−2∫ηθ∫∞bθ∫x−bθ0x2BNH(ne−tN)dadxzdμ(z)=∫∞bθx2N(0,x)H(∫ηθ∫xz−bθ0n(t,a,xz)e−tN(a,xz)dνx(a,z))dx−2∫ηθ∫bθzbθ∫xz−bθ0x2B(a)N(a,xz)H(n(t,a,xz)e−tN(a,xz))dadxdμ(z)z2−2∫ηθ∫∞bθ∫xz−bθ0x2B(a)N(a,xz)H(n(t,a,xz)e−tN(a,xz))dadxdμ(z)z2=∫∞bθx2N(0,x)[H(∫ηθ∫xz−bθ0n(t,a,xz)e−tN(a,xz)dνx(a,z))−∫ηθ∫xz−bθ0H(n(t,a,xz)e−tN(a,xz))dνx(a,z)]dx, |
since for
Appropriate choices of the function
λj=1+2ijπlog2,Nj(a,x)=x1−λjN(a,x),ϕj(a,x)=xλj, |
so we expect a behavior as in [2], i.e. the convergence of
∑j∈Z⟨n0,ϕj⟩e2ijπtlog2Nj(a,x), |
where
We have proved the existence and uniqueness of a solution of the eigenproblem (8) in the special yet biologically relevant case of linear growth rate with a self-similar fragmentation kernel. Hypotheses on both this kernel and the division rate are fairly general.
As possible future work we can imagine to extend the result to general growth rates. In this case the Perron eigenvalue is not explicit and it has to be determined in the same time as the eigenfunction, as in [26,12,6]. If we denote by
Pλ(s)=∫10e−λ∫szsdug(u)(Φ∗Pλ)(sz)dμ(z)z |
with
N:(a,x)↦Ψ(a)g(x)e−λ∫x0dαg(α)Pλ(x−a). |
Additionally for nonlinear growth rates, the function
The other natural continuation of the present work is the proof of the well-posedness and the long-time behavior of the evolution equation, as in [26,12]. To do so one can take advantage of the General Relative Entropy as in [14,4,2] or use general spectral methods [28,16].
The authors are very grateful to Marie Doumic for having suggested them the problem treated in this paper, and for the many fruitful discussions.
[1] |
Ros E (2010) Health benefits of nut consumption. Nutrients 2: 652-682. doi: 10.3390/nu2070652
![]() |
[2] | O'Neil CE, Keast DR, Fulgoni VL III, et al. (2010) Tree nut consumption improves nutrient intake and diet quality in US adults: an analysis of National Health and Nutrition Examination Survey (NHANES) 1999-2004. Asia Pac J Clin Nutr 19: 142-150. |
[3] |
O'Neil CE, Nicklas TA, Fulgoni VL III (2015) Tree nut consumption is associated with better nutrient adequacy and diet quality in adults: National Health and Nutrition Examination Survey 2005-2010. Nutrients 7: 595-607. doi: 10.3390/nu7010595
![]() |
[4] | Mukuddem-Petersen J, Oosthuizen W, Jerling JC (2005) A systematic review of the effects of nuts on blood lipid profiles in humans. J Nutr 135: 2082-2089. |
[5] |
Ros E, Mataix J (2006) Fatty acid composition of nuts--implications for cardiovascular health. Br J Nutr 96: S29-S35. doi: 10.1017/BJN20061861
![]() |
[6] |
Ros E (2015) Nuts and CVD. Br J Nutr 113: S111-S120. doi: 10.1017/S0007114514003924
![]() |
[7] |
Estruch R, Ros E, Salas-Salvadó J, et al; PREDIMED Study Investigators. (2013) Primary prevention of cardiovascular disease with a Mediterranean diet. N Engl J Med 368: 1279-1290. doi: 10.1056/NEJMoa1200303
![]() |
[8] |
Estruch R, Martínez-González MA, Corella D, et al; PREDIMED Study Investigators. (2016) Effect of a high-fat Mediterranean diet on bodyweight and waist circumference: a prespecified secondary outcomes analysis of the PREDIMED randomised controlled trial. Lancet Diabetes Endocrinol 4: 666-676. doi: 10.1016/S2213-8587(16)30085-7
![]() |
[9] |
Hernáez Á, Castañer O, Elosua R, et al. (2017) Mediterranean Diet Improves High-Density Lipoprotein Function in High-Cardiovascular-Risk Individuals: A Randomized Controlled Trial. Circulation 135: 633-643. doi: 10.1161/CIRCULATIONAHA.116.023712
![]() |
[10] |
Casas R, Sacanella E, Urpí-Sardà M, et al. (2016) Long-Term Immunomodulatory Effects of a Mediterranean Diet in Adults at High Risk of Cardiovascular Disease in the PREvención con DIeta MEDiterránea (PREDIMED) Randomized Controlled Trial. J Nutr 146: 1684-1693. doi: 10.3945/jn.115.229476
![]() |
[11] |
Macaluso F, Barone R, Catanese P, et al. (2013) Do fat supplements increase physical performance? Nutrients 5: 509-524. doi: 10.3390/nu5020509
![]() |
[12] |
Jeromson S, Gallagher IJ, Galloway SD, et al. (2015) Omega-3 Fatty Acids and Skeletal Muscle Health. Mar Drugs 13: 6977-7004. doi: 10.3390/md13116977
![]() |
[13] |
Maguire LS, O'Sullivan SM, Galvin K, et al. (2004) Fatty acid profile, tocopherol, squalene and phytosterol content of walnuts, almonds, peanuts, hazelnuts and the macadamia nut. Int J Food Sci Nutr 55: 171-178. doi: 10.1080/09637480410001725175
![]() |
[14] |
Ryan E, Galvin K, O'Connor TP, et al. (2006) Fatty acid profile, tocopherol, squalene and phytosterol content of brazil, pecan, pine, pistachio and cashew nuts. Int J Food Sci Nutr 57: 219-228. doi: 10.1080/09637480600768077
![]() |
[15] |
Curb JD, Wergowske G, Dobbs JC, et al. (2000) Serum lipid effects of a high-monounsaturated fat diet based on macadamia nuts. Arch Intern Med 160: 1154-1158. doi: 10.1001/archinte.160.8.1154
![]() |
[16] | O'Byrne DJ, Knauft DA, Shireman RB (1997)nLow fat-monounsaturated rich diets containing high-oleic peanuts improve serum lipoprotein profiles. Lipids 32: 687-695. |
[17] | Rajaram S, Burke K, Connell B, et al. (2001) A monounsaturated fatty acid-rich pecan-enriched diet favorably alters the serum lipid profile of healthy men and women. J Nutr 131: 2275-2279. |
[18] |
Sabaté J, Fraser GE, Burke K, et al. (1993) Effects of walnuts on serum lipid levels and blood pressure in normal men. N Engl J Med 328: 603-607. doi: 10.1056/NEJM199303043280902
![]() |
[19] | Zambón D, Sabaté J, Muñoz S, et al. (2000) Substituting walnuts for monounsaturated fat improves the serum lipid profile of hypercholesterolemic men and women. A randomized crossover trial. Ann Intern Med 132: 538-546. |
[20] |
Nieman DC, Scherr J, Luo B, et al. (2014) Influence of pistachios on performance and exercise-induced inflammation, oxidative stress, immune dysfunction, and metabolite shifts in cyclists: a randomized, crossover trial. PLoS One 9: e113725. doi: 10.1371/journal.pone.0113725
![]() |
[21] |
Flores-Mateo G, Rojas-Rueda D, Basora J, et al. (2013) Nut intake and adiposity: meta-analysis of clinical trials. Am J Clin Nutr 97: 1346-1355. doi: 10.3945/ajcn.111.031484
![]() |
[22] |
Musa-Veloso K, Paulionis L, Poon T, et al. (2016) The effects of almond consumption on fasting blood lipid levels: a systematic review and meta-analysis of randomised controlled trials. J Nutr Sci 5: e34. doi: 10.1017/jns.2016.19
![]() |
[23] |
Del Gobbo LC, Falk MC, Feldman R, et al. (2015) Effects of tree nuts on blood lipids, apolipoproteins, and blood pressure: systematic review, meta-analysis, and dose-response of 61 controlled intervention trials. Am J Clin Nutr 102: 1347-1356. doi: 10.3945/ajcn.115.110965
![]() |
[24] |
Banel DK, Hu FB (2009) Effects of walnut consumption on blood lipids and other cardiovascular risk factors: a meta-analysis and systematic review. Am J Clin Nutr 90: 56-63. doi: 10.3945/ajcn.2009.27457
![]() |
[25] |
Mohammadifard N, Salehi-Abargouei A, Salas-Salvadó J, et al. (2015) The effect of tree nut, peanut, and soy nut consumption on blood pressure: a systematic review and meta-analysis of randomized controlled clinical trials. Am J Clin Nutr 101: 966-982. doi: 10.3945/ajcn.114.091595
![]() |
[26] |
Mayhew AJ, de Souza RJ, Meyre D, et al. (2016) A systematic review and meta-analysis of nut consumption and incident risk of CVD and all-cause mortality. Br J Nutr 115: 212-225. doi: 10.1017/S0007114515004316
![]() |
[27] |
Zelber-Sagi S, Salomone F, Mlynarsky L (2017) The Mediterranean dietary pattern as the diet of choice for non-alcoholic fatty liver disease: Evidence and plausible mechanisms. Liver Int 37: 936-949. doi: 10.1111/liv.13435
![]() |
[28] |
Luo T, Miranda-Garcia O, Adamson A, et al. (2016) Consumption of Walnuts in Combination with Other Whole Foods Produces Physiologic, Metabolic, and Gene Expression Changes in Obese C57BL/6J High-Fat-Fed Male Mice. J Nutr 146: 1641-1650. doi: 10.3945/jn.116.234419
![]() |
[29] | Mazokopakis EE, Liontiris MI (2017) Commentary: Health Concerns of Brazil Nut Consumption. J Altern Complement Med in press. |
[30] |
Stiefel G, Anagnostou K, Boyle RJ, et al. (2017) BSACI guideline for the diagnosis and management of peanut and tree nut allergy. Clin Exp Allergy 47: 719-739. doi: 10.1111/cea.12957
![]() |
[31] |
Tey SL, Robinson T, Gray AR, et al. (2017) Do dry roasting, lightly salting nuts affect their cardioprotective properties and acceptability? Eur J Nutr 56: 1025-1036. doi: 10.1007/s00394-015-1150-4
![]() |
[32] |
Schlörmann W, Birringer M, Böhm V, et al. (2015) Influence of roasting conditions on health-related compounds in different nuts. Food Chem 180: 77-85. doi: 10.1016/j.foodchem.2015.02.017
![]() |
[33] |
Masthoff LJ, Hoff R, Verhoeckx KC, et al. (2013) A systematic review of the effect of thermal processing on the allergenicity of tree nuts. Allergy 68: 983-993. doi: 10.1111/all.12185
![]() |
[34] |
Hamasaki H (2017) Exercise and gut microbiota: clinical implications for the feasibility of Tai Chi. J Integr Med 15: 270-281. doi: 10.1016/S2095-4964(17)60342-X
![]() |
[35] |
Yi M, Fu J, Zhou L, et al. (2014) The effect of almond consumption on elements of endurance exercise performance in trained athletes. J Int Soc Sports Nutr 11: 18. doi: 10.1186/1550-2783-11-18
![]() |
[36] |
Vedtofte MS, Jakobsen MU, Lauritzen L, et al. (2014) Association between the intake of α-linolenic acid and the risk of CHD. Br J Nutr 112: 735-743. doi: 10.1017/S000711451400138X
![]() |
[37] |
Schwingshackl L, Hoffmann G (2014) Monounsaturated fatty acids, olive oil and health status: a systematic review and meta-analysis of cohort studies. Lipids Health Dis 13: 154. doi: 10.1186/1476-511X-13-154
![]() |
[38] |
Cornish SM, Chilibeck PD (2009) Alpha-linolenic acid supplementation and resistance training in older adults. Appl Physiol Nutr Metab 34: 49-59. doi: 10.1139/H08-136
![]() |
[39] |
Poulsen RC, Kruger MC (2006) Detrimental effect of eicosapentaenoic acid supplementation on bone following ovariectomy in rats. Prostaglandins Leukot Essent Fatty Acids 75: 419-427. doi: 10.1016/j.plefa.2006.08.003
![]() |
[40] |
Børsheim E, Kien CL, Pearl WM (2006) Differential effects of dietary intake of palmitic acid and oleic acid on oxygen consumption during and after exercise. Metabolism 55: 1215-1221. doi: 10.1016/j.metabol.2006.05.005
![]() |
[41] |
Henique C, Mansouri A, Fumey G, et al. (2010) Increased mitochondrial fatty acid oxidation is sufficient to protect skeletal muscle cells from palmitate-induced apoptosis. J Biol Chem 285: 36818-3627. doi: 10.1074/jbc.M110.170431
![]() |
[42] |
Mente A, de Koning L, Shannon HS, et al. (2009) A systematic review of the evidence supporting a causal link between dietary factors and coronary heart disease. Arch Intern Med 169: 659-669. doi: 10.1001/archinternmed.2009.38
![]() |
[43] |
Myles IA (2014) Fast food fever: reviewing the impacts of the Western diet on immunity. Nutr J 13: 61. doi: 10.1186/1475-2891-13-61
![]() |
[44] |
Francis H, Stevenson R (2013) The longer-term impacts of Western diet on human cognition and the brain. Appetite 63: 119-128. doi: 10.1016/j.appet.2012.12.018
![]() |
[45] |
Deer J, Koska J, Ozias M, et al. (2015) Dietary models of insulin resistance. Metabolism 64: 163-171. doi: 10.1016/j.metabol.2014.08.013
![]() |
[46] | Otten J, Stomby A, Waling M, et al. (2017) Benefits of a Paleolithic diet with and without supervised exercise on fat mass, insulin sensitivity, and glycemic control: a randomized controlled trial in individuals with type 2 diabetes. Diabetes Metab Res Rev 33. |
[47] |
O'Keefe JH, Gheewala NM, O'Keefe JO (2008) Dietary strategies for improving post-prandial glucose, lipids, inflammation, and cardiovascular health. J Am Coll Cardiol 51: 249-255. doi: 10.1016/j.jacc.2007.10.016
![]() |
[48] |
Villareal DT, Aguirre L, Gurney AB, et al. (2017) Aerobic or Resistance Exercise, or Both, in Dieting Obese Older Adults. N Engl J Med 376: 1943-1955. doi: 10.1056/NEJMoa1616338
![]() |
[49] |
Welch AA, MacGrego AJ, Minihane AM, et al. (2014) Dietary fat and fatty acid profile are associated with indices of skeletal muscle mass in women aged 18-79 years. J Nutr 144: 327-334. doi: 10.3945/jn.113.185256
![]() |
[50] |
Mickleborough TD (2013) Omega-3 polyunsaturated fatty acids in physical performance optimization. Int J Sport Nutr Exerc Metab 23: 83-96. doi: 10.1123/ijsnem.23.1.83
![]() |
[51] |
Aguilaniu B, Flore P, Perrault H, et al. (1995) Exercise-induced hypoxaemia in master athletes: effects of a polyunsaturated fatty acid diet. Eur J Appl Phytsiol Occup Physiol 72: 44-50. doi: 10.1007/BF00964113
![]() |
[52] |
Sureda A, Bibiloni MD, Martorell M, et al; PREDIMED Study Investigators. (2016) Mediterranean diets supplemented with virgin olive oil and nuts enhance plasmatic antioxidant capabilities and decrease xanthine oxidase activity in people with metabolic syndrome: The PREDIMED study. Mol Nutr Food Res 60: 2654-2664. doi: 10.1002/mnfr.201600450
![]() |
[53] |
Domínguez-Avila JA, Alvarez-Parrilla E, López-Díaz JA, et al. (2015) The pecan nut (Carya illinoinensis) and its oil and polyphenolic fractions differentially modulate lipid metabolism and the antioxidant enzyme activities in rats fed high-fat diets. Food Chem 168: 529-537. doi: 10.1016/j.foodchem.2014.07.092
![]() |
[54] |
Carey AN, Fisher DR, Joseph JA, et al. (2013) The ability of walnut extract and fatty acids to protect against the deleterious effects of oxidative stress and inflammation in hippocampal cells. Nutr Neurosci 16: 13-20. doi: 10.1179/1476830512Y.0000000023
![]() |
[55] |
Willis LM, Shukitt-Hale B, Cheng V, et al. (2009) Dose-dependent effects of walnuts on motor and cognitive function in aged rats. Br J Nutr 101: 1140-1144. doi: 10.1017/S0007114508059369
![]() |
[56] |
Liu Z, Wang W, Huang G, et al. (2016) In vitro and in vivo evaluation of the prebiotic effect of raw and roasted almonds (Prunus amygdalus). J Sci Food Agric 96: 1836-1843. doi: 10.1002/jsfa.7604
![]() |
[57] |
Nagel JM, Brinkoetter M, Magkos F, et al. (2012) Dietary walnuts inhibit colorectal cancer growth in mice by suppressing angiogenesis. Nutrition 28: 67-75. doi: 10.1016/j.nut.2011.03.004
![]() |
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