Positive semigroups and perturbations of boundary conditions

We present a generation theorem for positive semigroups on an $L^1$ space. It provides sufficient conditions for the existence of positive and integrable solutions of initial-boundary value problems. An application to a two-phase cell cycle model is given.


Introduction
We study well-posedness of linear evolution equations on L 1 of the form where Ψ 0 , Ψ are positive and possibly unbounded linear operators on L 1 , the linear operator A is such that equation (1.1) with Ψ = 0 generates a positive semigroup on L 1 , i.e., a C 0 -semigroup of positive operators on L 1 . We provide sufficient conditions for the operator A to be the generator of a positive semigroup. Our result is an extension of Greiner [12] by considering unbounded Ψ and positive semigroups. For a general theory of positive semigroups and their applications we refer the reader to [2,5,8,26]. An overview of different approaches used in studying initial-boundary value problems is presented in [7]. We also present a result about obtain stationary solutions of (1.1). We illustrate our general results with an age-size-dependent cell cycle model generalizing the discrete time model of [15,18,30]. This model can be described as a piecewise deterministic Markov process (see Section 5). Our approach can also be used in transport equations [3,16].

General Results
Let (E, E, m) and (E ∂ , E ∂ , m ∂ ) be two σ-finite measure spaces. Denote by L 1 = L 1 (E, E, m) and L 1 ∂ = L 1 (E ∂ , E ∂ , m ∂ ) the corresponding spaces of integrable functions. Let D be a linear subspace of L 1 . We assume that A : D → L 1 and Ψ 0 , Ψ : D → L 1 ∂ are linear operators satisfying the following conditions: (i) for each λ > 0, the operator Ψ 0 : D → L 1 ∂ restricted to the nullspace N (λI − A) = {f ∈ D : λf − Af = 0} of the operator (λI − A, D) has a positive right inverse, i.e., there exists a positive operator Ψ(λ) : (ii) the operator Ψ : D → L 1 ∂ is positive and there exists ω ∈ R such that the operator I ∂ − ΨΨ(λ) : L 1 ∂ → L 1 ∂ is invertible with positive inverse for all λ > ω, where I ∂ is the identity operator on L 1 ∂ ; (iii) the operator A 0 = A with D(A 0 ) = {f ∈ D : Ψ 0 f = 0} is the generator of a positive semigroup on L 1 ; (iv) for each nonnegative f ∈ D By assumption (i), AΨ(λ)f ∂ = λΨ(λ)f ∂ and Ψ 0 Ψ(λ)f ∂ = f ∂ for f ∂ ∈ L 1 ∂ . This together with condition (2.1) implies that for all nonnegative f ∂ ∈ L 1 ∂ , completing the proof of (2.5). Let I be the identity operator on Combining this with (ii) we conclude that I − BR(λ, A) is invertible with positive inverse (I − BR(λ, A)) −1 for all λ > ω. Hence, the spectral radius of the positive operator BR(λ, A) is strictly smaller than 1 for some λ > 0. It follows from [29,Theorem 1.4] that the part of (A + B, D(A)) in is densely defined and generates a positive semigroup on L 1 . Finally, the operator (A + B, D(A)) is resolvent positive with resolvent given by R(λ, A + B) = R(λ, A)(I − BR(λ, A)) −1 for λ > ω. Hence, the formula for R(λ, A Ψ ) is also valid.
Remark 2.2. If we assume that (A, D) is closed, Ψ 0 is onto and continuous with respect to the graph norm f A = f + Af , then Ψ(λ) exists for each λ > 0 and is bounded, by [12, Lemma 1.2]. Remark 2.3. Greiner [12, Theorem 2.1] establishes that (A Ψ , D(A Ψ )) is the generator of a C 0 -semigroup for any bounded Ψ provided that (A, D) is closed, (A 0 , D(A 0 )) is the generator of a C 0 -semigroup, Ψ 0 is onto and continuous with respect to the graph norm, and that there exist constants γ > 0 and λ 0 such that This is condition (2.1) of Greiner [12,Theorem 2.1]. Some extensions of this result are provided in [22] and [13] for unbounded Ψ.
Remark 2.5. Condition (iv) ensures that the operator (A 0 , D(A 0 )) satisfies for all nonnegative f ∈ D(A 0 ). If, additionally, (v) (A 0 , D(A 0 )) is densely defined and resolvent positive, then (A 0 , D(A 0 )) is the generator of a substochastic semigroup on L 1 , i.e., a positive semigroup of contractions on L 1 . This is a consequence of the Hille-Yosida theorem, see e.g. [2,26]. Thus it is enough to assume condition (v) instead of (iii). It should be noted that in [26,Theorem 4.6] the assumption that the domain D(A Ψ ) of the operator A Ψ is dense is missing. Making use of Theorem 2.1, we get the following result.
then (A Ψ + B, D(A Ψ )) is the generator of a substochastic semigroup.

A model of a two phase cell cycle in a single cell line
The cell cycle is the period from cell birth to its division into daughter cells. It contains four major phases: G 1 phase (cell growth before DNA replicates), S phase (DNA synthesis and replication), G 2 phase (post DNA replication growth period), and M (mitotic) phase (period of cell division). The Smith-Martin model [28] divides the cell cycle into two phases: A and B. The A phase corresponds to all or part of G 1 phase of the cell cycle and has a variable duration, while the B phase covers the rest of the cell cycle. The cell enters the phase A after birth and waits for some random time T A until a critical event occurs that is necessary for cell division. Then the cell enters the phase B which lasts for a finite fixed time T B . At the end of the B-phase the cell splits into two daughter cells. We assume that individual states of the cell are characterized by age a ≥ 0 in each phase and by size x > 0, which can be volume, mass, DNA content or any quantity conserved trough division. We assume that individual cells of size x increase their size over time in the same way, with growth rate g(x) so that dx/dt = g(x), and all cells age over time with unitary velocity so that da/dt = 1. We assume that the probability that a cell is still being in the phase A at age a is equal to H(a), so the rate of exit from the phase A at age a is ρ(a) given by where h is a probability density function defined on [0, ∞), describing the distribution of the time T A , the duration of the phase A. We make the following assumptions: is Borel measurable and the function H in (3.1) satisfies: The growth rate function g : (0, ∞) → (0, ∞) is globally Lipschitz continuous and g(x) > 0 for x > 0. The Smith and Martin hypothesis ( [28]) states that h is exponentially distributed with parameter p > 0, so that ρ(a) = p for all a > 0. However, this does not agree with experimental data, see e.g. [11,34] for recent results. The generation time of a cell, i.e. time from birth to division, can be written as T = T A + T B . Thus the distribution of generation time has a probability density of the form Cell generation times can have lognormal or bimodal distribution (see [27]), exponentially modified Gaussian ( [10]), or tempered stable distributions ( [23]).
To describe the growth of cells we define dr, x > 0, wherex > 0 orx = 0, if the integral is finite. The value Q(x) has a simple biological interpretation. Ifx is the size of a cell, then Q(x) is the time it takes the cell to reach the size x. It follows from assumption (II) that the function Q is strictly increasing and continuous. We denote by Q −1 the inverse of Q. Define for t ≥ 0 and x 0 > 0. Then π t x 0 satisfies the initial value problem 3) extends to all t ∈ R and x 0 > 0. We also set π t 0 = 0 for t > 0 in this case. If Q(0) = 0 then Q −1 is defined only on (0, ∞) and we set π t 0 = Q −1 (t) for t > 0. We can extend formula (3.3) to all negative t satisfying Q(x 0 ) + t > 0; otherwise we set π t x 0 = 0. Note that at time t = T , the generation time, a "mother cell" of size π T x 0 divides into two daughter cells of equal size 1 2 π T x 0 . In the probabilistic model of [15,18,30,31] a sequence of consecutive descendants of a single cell was studied. Let f be the probability density function of the size distribution at birth at time t 0 of mother cells and let t 1 > t 0 be a random time of birth of daughter cells. Then the probability density function of the size distribution of daughter cells is given by ( [18,30]) The iterates P 2 f, P 3 f, . . . denote densities of the size distribution of consecutive descendants born at random times t 2 , t 3 , . . . . The operator P defined by (3.4) is a positive contraction on L 1 (0, ∞), the space of Borel measurable functions defined on (0, ∞) and integrable with respect to the Lebesgue measure. Here we extend the probabilistic model to a continuous time situation by examining what happens at all times t and not only at t 0 , t 1 , t 2 , . . .. We denote by p 1 (t, a, x) and p 2 (t, a, x) the densities of the age and size distribution of cell in the A-phase and in the B-phase at time t, age a, and size x, respectively. Neglecting cell deaths the equations can be written as with boundary and initial conditions 2 (a, x). (3.8) In this model, cells in the A-phase enter the B-phase at rate ρ. This is taken into account by the boundary condition (3.7). All cells stay in the B-phase until they reach the age T B . Then they divide their size into half (3.6). The model is complemented with initial conditions (3.8). The model we propose is different as compared to mass/maturity structured models [9,14,24,32] where a cell leaves the phase A with intensity being dependent on maturity, not age. In the case of T B = 0 there is only one phase present; a maturity structured model being a continuous time extension of [17] is studied in [20], while age and volume/maturity structured population models of growth and division were studied extensively since the seminal work of [6] and [19,25]. We refer the reader to [21] for historical remarks concerning modeling of age structured populations and to [27,33] for recent reviews.
We look for positive solutions of (3.5)-(3.8) in the space , m is the product of the two-dimensional Lebesgue measure and the counting measure on {1, 2}, and E is the σ-algebra of all Borel subsets of E. We identify L 1 = L 1 (E, E, m) with the product of the spaces L 1 (E 1 ) and L 1 (E 2 ) of functions defined on the sets E 1 and E 2 , respectively, and being integrable with respect to the twodimensional Lebesgue measure. We say that the operator P has a steady state in L 1 (0, ∞) if there exists a probability density function f such that P f = f . Similarly, a semigroup {S(t)} t≥0 has a steady state in L 1 if there exists a nonnegative f ∈ L 1 such that S(t)f = f for all t > 0 and f 1 = 1 where · 1 is the norm in L 1 . We give the proof of Theorem 3.1 in the next section. Theorem 3.1 combined with [4] implies the following sufficient conditions for the existence of steady states of (3.5)-(3.8).  If the cell growth is exponential so that we have g(x) = kx for all x > 0, where k is a positive constant, then it is known [15,30,31] that the operator P has no steady state. We now consider a linear cell growth and assume that g(x) = k for all x > 0. We see that Q(x) = x/k, the operator P is of the form (see [31] or the last section)

Proof of Theorem 3.1
We will show that Theorem 3.1 can be deduced from Theorems 2.1 and 2.8. To this end, we introduce some notation. Let us define where π t is given by (3.3). Then t → π(t, a 0 , x 0 ) solves the system of equations a ′ (t) = 1 and x ′ (t) = g(x(t)) with initial condition a(0) = a 0 and x(0) = x 0 . Recall that E 1 is an open set. For any x 0 , a 0 ∈ E 1 we define t − (a 0 , x 0 ) = inf{s > 0 : π(−s, a 0 , x 0 ) ∈ E 1 } and the incoming part of the boundary ∂E 1 Γ − 1 = {z ∈ ∂E 1 : z = π(−t − (y), y) for some y ∈ E 1 with t − (y) < ∞}. Observe that t − (a 0 , x 0 ) = a 0 for all (a 0 , x 0 ) ∈ E 1 and that Γ − 1 = {0} × (0, ∞). We consider on Γ − 1 the Borel measure m − 1 being the product of the point measure δ {0} at 0 and the Lebesgue measure on (0, ∞). We define the operator T max on L 1 (E 1 ) by ( [1]) where the differentiation is understood in the sense of distributions. Then it follows from [1] that for f ∈ D(T max ) the following limit exists for almost every z ∈ Γ − 1 with respect to the measure m − 1 on Γ − 1 . According to [1,Theorem 4.4] the operator T 0 = T max with domain is the generator of a substochastic semigroup on L 1 (E 1 ) given by By [1, Proposition 5.1], the operator (T, D(T)) defined by is the generator of a substochastic semigroup on L 1 (E 1 ) of the from . Note that we can identify the space L 1 (E 2 ) with the subspace and Γ − 2 = {z ∈ ∂E 2 : z = π(−t − (y), y) for some y ∈ E 2 with t − (y) < ∞}. We also define the exit time from the set E 2 by t + (a 0 , x 0 ) = inf{s > 0 : π(s, a 0 , x 0 ) ∈ E 2 } and the outgoing part of the boundary ∂E 2 We define the Borel measure m − 2 on Γ − 2 as the measure m − 1 and the m + 2 on Γ + 2 as the product of the point measure at T B and the one dimensional Lebesgue measure. Since U 0 (t)(L 1 (E 2 )) ⊆ L 1 (E 2 ), the part of the operator (T 0 , D(T 0 )) in L 1 (E 2 ) is the generator of a substochastic semigroup {U 2 (t)} t≥0 in L 1 (E 2 ). Moreover, the pointwise following limits exist for almost every z ∈ Γ ± 2 with respect to the Borel measure m ± 2 on Γ ± 2 .
{2}, E ∂ be the σ-algebra of Borel subsets of E ∂ and m ∂ be the product of the Lebesgue measure on the line {0} × (0, ∞) and the counting measure on {1, 2}. To simplify the notation we identify L 1 ∂ = L 1 (E ∂ , E ∂ , m ∂ ) with the product space L 1 (0, ∞) × L 1 (0, ∞). We define operators A 1 and A 2 by We take operators Ψ 0 , Ψ : To this end, we check that assumptions (i)-(iv) of Theorem 2.1 from Section 2 are satisfied. We first show that conditions (iii) and (iv) hold. The operator since {U 1 (t)} t≥0 and {U 2 (t)} t≥0 are semigroups on the spaces L 1 (E 1 ) and L 1 (E 2 ) with the corresponding generators. The semigroup {S 0 (t)} t≥0 is substochastic. For By [1,Proposition 4.6], this reduces to implying that condition (iv) holds.
Hence, we see that the right inverse of Ψ 0 when restricted to the nullspace of λI − A is given by Thus f 1 ∈ D 1 . Similarly, f 2 ∈ D 2 . Hence, condition (i) holds.

Final remarks
Our model can be described as a piecewise deterministic Markov process {X(t)} t≥0 . We considered three variables (a, x, i), where i = 1 if a cell is in the phase A, i = 2 if it is in the phase B, the variable x describes the cell size, and a describes the time which elapsed since the cell entered the ith phase. Let t 0 = 0. If we observe consecutive descendants of a given cell and the nth generation time is denoted by t n , then t n+1 = s n + T B where s n is the time when the cell from the nth generation enters the phase B, n ≥ 0. A newborn cell at time t n is with age a(t n ) = 0 and with initial size equal to x(t − n )/2, where x(t − n ) is the size of its mother cell. The cell ages with velocity 1 and its size grows according to the equations x ′ (t) = g(x(t)) for t ∈ (t n , s n ). If the cell enters the phase B then its age is reset to 0 and its size still grows according to x ′ (t) = g(x(t)) for t ∈ (s n , s n + T B ). We have (5.1) a(s n ) = 0, x(s n ) = x(s − n ), i(s n ) = 2, and at the end of the second phase the cell divides into two cells, so that we have Thus the process X(t) = (a(t), x(t), i(t)) satisfies the following system of ordinary differential equations a ′ (t) = 1, x ′ (t) = g(x(t)), i ′ (t) = 0, between consecutive times t 0 , s 0 , t 1 , s 1 , . . ., called jump times. At jump times the process is given by (5.1) and (5.2). If the distribution of X(0) has a density f then X(t) has a density S(t)f , i.e., Pr(X(t) ∈ B i × {i}) = g(π −a z) g(z) 1 (0,∞) (π −a z)