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[EQUATION] We also define the corresponding perturbed generator [MATH] as [EQUATION] We recall the main result from for the reader’s convenience. |
Theorem 2.1 (Greiner) If [MATH] , then [MATH] is the generator of a strongly continuous semigroup [MATH] on [MATH] We now apply Theorem 2.1 to establish the existence of the governing linear semigroup. In our setting we have [MATH] , and [MATH] . Furthermore, to compute the adjoint [MATH] , we note that |
[EQUATION] if we let [EQUATION] Next we note that (see 32 , Sect.III.5] [MATH] if there exists an [MATH] such that [EQUATION] For any [MATH] integration by parts yields |
[EQUATION] for [MATH] , and if we let [MATH] . Hence it follows from the regularity assumptions on [MATH] and [MATH] that we have |
[EQUATION] Hence Theorem 2.1 implies that [MATH] generates a strongly continuous semigroup. Next we note that for [MATH] , the operator [MATH] is a continuous bijection from [MATH] onto [MATH] , hence its inverse |
[EQUATION] is continuous, for [MATH] . In particular, a straightforward calculation shows that in our setting we have [MATH] , and therefore for [MATH] large enough [MATH] is invertible and positive. Hence by Lemma 1.4 in |
we have that [MATH] is positive for [MATH] large enough. Since [MATH] is a bounded multiplication operator, we draw the following conclusion. |
Corollary 2.2 [MATH] generates a positive strongly continuous semigroup of bounded linear operators on [MATH] 3. Asymptotic behaviour |
In this section we study the asymptotic behaviour of solutions of the linear model ( 1.1 )-( 1.3 ). In particular, as we will see, we are going to characterise the spectral bound of the linear semigroup generator [MATH] implicitly via the spectral radius of an associated (bounded) integral operator. We will then obtain... |
As we have pointed out earlier one of the advantages of the perturbation theorem of Greiner (Theorem 2.1 ) is that it allows us to establish a desirable regularity property of the semigroup in a straightforward fashion. To this end, for the rest of the paper we further assume that [MATH] satisfies the following regular... |
[EQUATION] Note that, for example if [MATH] is continuous on the square [MATH] , then condition ( 3.14 ) clearly holds. We recall now from |
the result we are going to apply, for the readers convenience. In particular, Proposition 3.1 in reads as follows. Proposition 3.3 |
If [MATH] generates a semigroup and [MATH] is compact then [MATH] . In particular, [MATH] contains only poles of finite algebraic multiplicity of the resolvent [MATH] |
In the proposition above [MATH] stands for the component of the resolvent set of [MATH] , which is unbounded to the right. We apply now Proposition 3.3 in our setting. In what follows, with a slight abuse of notation, we will denote the semigroup [MATH] by [MATH] |
Proposition 3.4 Assume that ( 3.14 ) holds. Then the spectrum of the governing linear semigroup [MATH] may contain only elements of the form [MATH] , where [MATH] is an eigenvalue of its generator [MATH] |
Proof. We note that [MATH] is bounded and [MATH] generates a nilpotent semigroup. Hence, utilising Proposition 3.3 , it is only left to show that [MATH] is compact. Let [MATH] denote the unit sphere of [MATH] . We have to show that [MATH] is relatively compact in [MATH] . Using the Fréchet-Kolmogorov criterion of compa... |
55 , Ch.X.1] , it is enough to show that on the one hand we have [EQUATION] On the other hand we have [EQUATION] therefore we have |
[EQUATION] uniformly in [MATH] We shall point out that we really needed to utilise Proposition 3.3 by Greiner, to obtain that the asymptotic behaviour of the semigroup is determined by the leading eigenvalue of its generator (if it exists). This is due to the distributed structuring with respect to the telomere length ... |
for a classic size-structured models, that the semigroup is eventually compact, and henceforth the spectral mapping theorem holds true. |
As we have seen earlier the semigroup [MATH] [MATH] is clearly positive, but it may not necessarily be irreducible. To see this, recall from |
, that the semigroup [MATH] is irreducible if and only if for every [MATH] [MATH] , we have that [MATH] , i.e. the resolvent is strictly positive, for some [MATH] |
Let [MATH] , and note that the solution of the resolvent equation [MATH] is [EQUATION] Applying the boundary operator [MATH] on both sides of equation ( 3.16 ), it is easily shown that [MATH] satisfies the inhomogeneous integral equation |
[EQUATION] Above in ( 3.17 ) we introduced the notation [EQUATION] Note that since telomere length is preserved during the lifetime of an individual, it is intuitively clear that the semigroup is irreducible if offspring of all telomere length is produced by some individuals. In other words, if there were individuals o... |
Proposition 3.5 The semigroup [MATH] [MATH] is irreducible if and only if for any set [MATH] of positive Lebesgue measure, such that its complement set [MATH] is also a set of positive Lebesgue measure, we have |
[EQUATION] Proof. Note that by virtue of the positivity of the semigroup [MATH] , the resolvent operator [MATH] is positive, for [MATH] large enough. Hence the solution [MATH] of equation ( 3.17 ) is necessarily non-negative almost everywhere. Assume now that ( 3.18 ) does not hold, i.e. [MATH] for some sets [MATH] of ... |
Proposition 3.4 implies that the asymptotic behaviour of solutions of model ( 1.1 )-( 1.3 ) is determined by the eigenvalues of the generator (if there are any), hence we study this eigenvalue problem now. In particular, the solution of the eigenvalue problem |
[EQUATION] is given by [EQUATION] Applying the boundary operator [MATH] on both sides of the equation above we have [EQUATION] where we defined |
[EQUATION] Hence [MATH] is an eigenvalue of [MATH] if and only if, for the given [MATH] , the integral equation ( 3.21 ) has a non-trivial solution [MATH] . Then, the eigenvector corresponding to [MATH] |
is given by ( 3.20 ). We are in particular interested in the leading eigenvalue (if it exists), which is the spectral bound of the generator, since the semigroup [MATH] is positive. This dominant real eigenvalue, together with the corresponding eigenspace, determines the asymptotic behaviour of solutions. Also note tha... |
[EQUATION] With this, the characteristic equation ( 3.21 ), which is notably a functional equation, in contrast to a scalar equation in case of a model with age-structure only; can be viewed as an eigenvalue problem for a bounded linear integral operator. More precisely, [MATH] is an eigenvalue of the generator [MATH] ... |
[MATH] has eigenvalue [MATH] . Note that, since [MATH] defined in ( 3.22 ) is strictly positive, the integral operator [MATH] (for every [MATH] ) is irreducible if and only if condition ( 3.18 ) holds 45 , Ch.V] . Hence, rightly so, the irreducibility conditions of the semigroup [MATH] and the integral operator [MATH] ... |
Also note that the function [MATH] is continuous and strictly monotone decreasing. These properties can be established by using perturbation results from |
and , respectively; see also for similar developments. Also note that if [MATH] satisfies condition ( 3.14 ) then [MATH] is shown to be compact exactly in the same way as the operator [MATH] was shown to be compact earlier. Hence the spectrum of [MATH] may contain only eigenvalues and [MATH] |
Therefore, in the case when the spectrum of [MATH] is not empty, we have the following complete characterisation of the asymptotic behaviour of solutions of model ( 1.1 )-( 1.3 ). |
(1) If [MATH] , then solutions of model ( 1.1 )-( 1.3 ) decay exponentially. (2) If [MATH] , then solutions of model ( 1.1 )-( 1.3 ) grow exponentially. Moreover, if [MATH] satisfies condition ( 3.18 ), then solutions exhibit asynchronous exponential growth. |
(3) If [MATH] , then for any eigenvector [MATH] corresponding to the spectral radius [MATH] of [MATH] , the function [MATH] in ( 3.20 ) determines a one-parameter family of positive steady states of model ( 1.1 )-( 1.3 ). If [MATH] satisfies ( 3.18 ), then there is only one such family of steady states, moreover they a... |
After the previous general analysis of the asymptotic behaviour of our model next we intend to study the effect of the well-known capacity of telomere restoring of cancer cells on the dynamics. In particular we are going to show, that at least for some general classes of the model ingredients, the telomere length resto... |
[EQUATION] This in particular implies that the eigenvalue problem ( 3.21 ) now reads [EQUATION] where we also introduced the notation [MATH] , for simplicity. Let us assume now that the the function [MATH] is separable, i.e. [MATH] holds for some functions [MATH] . For example we may assume that [MATH] is continuously ... |
[EQUATION] together with the initial condition [EQUATION] The solution of ( 3.26 ) is [EQUATION] which, utilising ( 3.27 ), leads to the following characteristic equation |
[EQUATION] It is clear that equation ( 3.29 ) does not admit any solution [MATH] , which, together with the positivity of the semigroup, implies that the spectrum of [MATH] does not contain any eigenvalue with a corresponding eigenvector [MATH] with a continuously differentiable [MATH] . On the other hand it is clear f... |
Next we consider what happens if we account for the telomere restoring capacity of cancer cells. In particular we are going to show that in this case even exponential growth of the cancer cell population is possible. As before, we start with the case of a separable [MATH] , i.e. we assume that [MATH] for some functions... |
[EQUATION] From ( 3.30 ) we can see that depending on the functions [MATH] and [MATH] , the spectral radius [MATH] may be greater than [MATH] . For example in the case of constant functions [MATH] , and setting [MATH] (note that we can always normalise the maximal age and telomere length), we have |
[EQUATION] To obtain estimates for the spectral radius of [MATH] in the more general and difficult non-separable case, note that the Krein-Rutman theorem asserts that if [MATH] is compact and positive, and its spectral radius is positive, then it has a positive (not necessarily strictly positive) eigenvector correspond... |
that if the operator is also irreducible then its spectral radius is strictly positive. As we noted earlier if [MATH] satisfies condition ( 3.14 ) then [MATH] is shown to be compact in the same way as the operator [MATH] . Assuming now that the spectral radius is positive, let [MATH] denote the positive eigenvector cor... |
[EQUATION] which yields [EQUATION] This observation allows us to obtain immediately the following estimates for the spectral radius |
[EQUATION] To obtain different estimates, in particular when [MATH] may not be irreducible we are going to utilise some minimax principles established in |
. Recall that if [MATH] is a Banach lattice with positive cone [MATH] , and with dual space [MATH] and dual cone [MATH] , respectively; then a set [MATH] is called [MATH] -total if and only if from [MATH] it follows that [MATH] . Then for any positive linear endomorphism [MATH] on a Banach lattice [MATH] and for any [M... |
[EQUATION] Recall that by Lemma 3.1 in for any [MATH] -total set [MATH] we have [EQUATION] Moreover, Lemma 3.3 in asserts that for any [MATH] we have [MATH] If we let [MATH] , then we have for any [MATH] |
[EQUATION] Similarly, recall from that if we define [EQUATION] then for every [MATH] we have [MATH] . Again, choosing [MATH] , we have for any [MATH] |
[EQUATION] Hence we obtain the following estimates for the spectral radius of [MATH] [EQUATION] Note that the estimates ( 3.33 ) and ( 3.38 ) are quite different, in general. |
We next provide hypotheses on [MATH] that yield specific growth behavior of the solutions in the presence or absence of highest telomere class renewal. It is known that in the discrete telomere length case, the cell population can have polynomial growth or decay with cells with shortest telomere length having the highe... |
). Similar results hold in the continuum telomere length case if we divide the population into telomere length classes. We first consider the case of no self-renewal within any class, that is, all cell divisions result in daughter cells in a shorter telomere length class. |
Proposition 3.6 Assume there exists [MATH] such that for [MATH] [MATH] for [MATH] . Assume that [MATH] [MATH] , for all [MATH] [MATH] , and [MATH] , for all [MATH] . Let |
[EQUATION] Let [MATH] be the solution of ( 1.1 )-( 1.3 ), such that [MATH] , and let [EQUATION] Then [EQUATION] [EQUATION] [EQUATION] |
and in general [EQUATION] Proof. From ( 1.1 ) and ( 1.2 ), for [MATH] [EQUATION] (since [MATH] for [MATH] [MATH] - see Figure (a)). Thus, [MATH] |
A similar calculation to ( 3.43 ) yields [EQUATION] (since [MATH] for [MATH] [MATH] - see Figure (b)). Thus, [MATH] satisfies ( 3.40 ). |
A similar calculation to ( 3.44 ) yields [EQUATION] since [MATH] for [MATH] [MATH] - see Figure (c). Thus, [MATH] satisfies ( 3.41 ). The general case ( 3.42 ) is proved by induction following similar steps as above. |
We next provide sufficient conditions for a class of cells of longest telomeres to have sufficient self-renewal capacity for them to attain proliferative immortality. We assume that the division rate [MATH] and mortality rate [MATH] are constant in this class, and the fraction of dividing cells in this class with self-... |
Proposition 3.7 Let [MATH] , such that [MATH] and [MATH] , for [MATH] and [MATH] . Let [MATH] , and let [MATH] be such that [MATH] for [MATH] and assume that |
[EQUATION] Let [MATH] be the solution of ( 1.1 )-( 1.3 ), such that [MATH] [MATH] . There exists a constant [MATH] (depending on [MATH] ) such that |
[EQUATION] (Note that ( 1.4 ) and ( 3.46 ) imply that [MATH] , which automatically holds if [MATH] is normalised to 1. The hypothesis on [MATH] means that the fraction of daughter cells with telomere length between [MATH] and [MATH] produced by mother cells in this class is greater than [MATH] .) |
Proof. Let [MATH] From ( 1.1 )-( 1.3 [EQUATION] [EQUATION] From the method of characteristics (see e.g. [EQUATION] Let [MATH] satisfy |
[EQUATION] Again from the method of characteristics, [MATH] satisfies [EQUATION] From ( 3.51 ), for [MATH] [EQUATION] where the last equality is obtained by a change of the variable of integration. Let [MATH] . Then ( 3.53 ) implies |
[EQUATION] which implies [EQUATION] Integrating from [MATH] to [MATH] to obtain [EQUATION] which implies [EQUATION] Then [EQUATION] |
Let [MATH] Then for [MATH] , from ( 3.49 ) and ( 3.53 [EQUATION] Then ( 3.55 ) implies [EQUATION] which integrating from [MATH] to [MATH] implies [MATH] Then ( 3.47 ) follows from ( 3.54 ) and ( 3.50 ). |
Remark 3.8 We note that a similar result can be established for other classes of cells with self-renewal capability, of telomere length in a specific [MATH] -interval. Such cell populations can arise from a single mutant cell, which generates more daughter cells with this mutation than competitor cells, and thus expand... |
4. Incorporating crowding effect Next we introduce a nonlinearity in model ( 1.1 )-( 1.3 ) by incorporating crowding/competition effects via imposing extra mortality pressure on cells. We follow the same approach as employed previously in |
for similar cell population models. Our equations now read [EQUATION] In equation ( 4.56 [MATH] is a non-negative function, and it also satisfies some smoothness assumptions. For example it suffices to assume that [MATH] is continuously differentiable. Equation ( 4.56 ) is still semilinear, hence global existence of so... |
, where this approach was in fact applied to establish global existence of solutions of a similar model. In the simplest case, when [MATH] is a linear function, our model ( 4.56 )-( 4.58 ) fits into the exact framework studied in |
, if some additional hypotheses are fulfilled. In particular, if we assume that there exists a [MATH] and a bounded linear operator [MATH] (a projection onto the finite-dimensional eigenspace corresponding to the spectral bound of [MATH] ), such that [MATH] and [MATH] holds; then the nonlinear semigroup governing ( 4.5... |
[EQUATION] In the formulas above we introduced the notation [MATH] for the bounded linear integral operator [MATH] , with domain [MATH] |
Note that some asymptotic results for more general classes of nonlinearities were already obtained in the earlier paper . In particular for a continuous, non-negative and monotone increasing function [MATH] it was proven, under the same assumptions as above, that [MATH] implies that solutions corresponding to initial c... |
Here we are mainly interested how the asymptotic behaviour of the nonlinear model changes compared to the linear model ( 1.1 )-( 1.3 ), for a general nonlinear function [MATH] . In particular, we are interested if the linear model with exponential growth (accounting for the telomere restoring capacity of cancer cells) ... |
The existence and uniqueness of the positive steady state of the nonlinear model ( 4.56 )-( 4.58 ) is established using the techniques we developed in the previous section to study the asymptotic behaviour of the linear model. In particular solving equation ( 4.56 ) for a positive steady state we obtain |
[EQUATION] Next we substitute this solution into the boundary condition ( 4.57 ) to arrive at an integral equation of the form [EQUATION] |
Hence, we define a parametrised family of bounded positive integral operators [MATH] for [MATH] as follows [EQUATION] with domain D [MATH] . Note that for any fixed [MATH] we have [MATH] Hence if there exists a [MATH] such that the integral operator [MATH] has eigenvalue [MATH] with a corresponding normalised positive ... |
[EQUATION] determines a positive steady state of the nonlinear model ( 4.56 )-( 4.58 ), where the constant [MATH] is chosen such that [MATH] holds. Making use of the results we established in the previous section about the spectral radius of the operator [MATH] , we summarize our finding in the following proposition. |
Proposition 4.9 Assume that [MATH] satisfies condition ( 3.18 ). Then, if [MATH] is a monotone increasing function, and either of the following conditions holds |
[EQUATION] the nonlinear model ( 4.56 )-( 4.58 ) has a unique strictly positive steady state. On the other hand, if [MATH] is a monotone decreasing function, then either of the following conditions |
[EQUATION] implies that the nonlinear model ( 4.56 )-( 4.58 ) has a unique strictly positive steady state. Note that if [MATH] is not monotone, for example if the competition induced mortality exhibits an Allée-type effect, then we can still establish the existence of the positive steady state by using the estimates ( ... |
Next we investigate the stability of the steady states of the nonlinear model ( 4.56 )-( 4.58 ). Note that our model is a semilinear one (moreover the nonlinear operator is differentiable), hence stability can be studied indeed via linearisation, see e.g. |
. To this end note that the linearisation of equation ( 4.56 ) around a steady state [MATH] reads [EQUATION] where we set [MATH] . The linearised model ( 4.64 )-( 4.57 )-( 4.58 ) is also governed by a strongly continuous semigroup, since equation ( 4.64 ) is just a bounded perturbation (at least when [MATH] is [MATH] )... |
[EQUATION] where we set [MATH] . The solution of the first equation above is [EQUATION] Imposing the second equation of ( 4.65 ) leads to the following inhomogeneous integral equation |
[EQUATION] where we introduced the notation [EQUATION] Hence [MATH] is an eigenvalue of the linearised operator, if and only if the inhomogeneous integral equation ( 4.67 ) has a non-trivial solution [MATH] . As we can see the information pertaining [MATH] is rather implicit. However, in case of the extinction steady s... |
Proposition 4.10 Either of the conditions in ( 4.63 ) imply that the extinction steady state [MATH] is asymptotically stable. On the other hand, either of the conditions in ( 4.62 ) imply that the steady state [MATH] is unstable. |
Remark 4.11 Note the connection between the existence of a non-trivial steady state and the stability of the trivial one. In particular, for a monotone increasing [MATH] , either of conditions in ( 4.62 ) implies that a unique strictly positive steady state exists and the trivial one is unstable. On the other hand, if ... |
Next we study the stability of the positive steady state. First we note that in the special but interesting case, when [MATH] , the eigenvalue problem ( 4.67 ) (now a homogeneous integral equation) simply reads |
[EQUATION] That is, the eigenvalue problem ( 4.68 ) above can be written as [EQUATION] where [MATH] is defined earlier in ( 3.23 ). Note that, since the semigroup governing the linearised equation is positive, the spectral bound belongs to its spectrum, i.e. it is a dominant real eigenvalue, which determines the asympt... |
that the function [MATH] is strictly monotone decreasing, for [MATH] . Hence we conclude that [MATH] , and therefore [MATH] is the dominant eigenvalue of the linearised operator. In this case the governing linear semigroup is strongly stable, but not uniformly exponentially stable, see e.g. 19 , Ch.V] |
Next we consider the general case. We already noted that the information about the spectral values contained in ( 4.67 ) is rather implicit. Moreover, we note that a biologically relevant and meaningful function [MATH] would be of a logistic type, i.e. [MATH] and [MATH] (strictly) monotone increasing. In this case, as ... |
. The main advantage of this approach is that it does not rely on the positivity of the governing linear semigroup. We now introduce a notation for the generator of the semigroup governing the linearised problem. Let |
[EQUATION] Proposition 4.12 The stationary solution [MATH] of model ( 4.56 )-( 4.58 ) is asymptotically stable if [EQUATION] holds. |
Proof. Our goal is to show that there exists a [MATH] such that the operator [MATH] is dissipative ( [MATH] stands for the identity). That is, we need to show that there exists a [MATH] , such that we have |
[EQUATION] Then, invoking the Lumer-Phillips Theorem 19 , II.3] , we obtain that the semigroup [MATH] satisfies [MATH] , for [MATH] , i.e. it is exponentially stable. To this end, assume that for a given [MATH] and [MATH] [MATH] satisfies the equation |
[EQUATION] Then we are going to show that if condition ( 4.69 ) holds, then in fact there exists a [MATH] small enough, such that [MATH] holds, for all [MATH] . The main idea, as in |
, is to divide the interval [MATH] into a countable union of subintervals, now for any fixed [MATH] ; on each of which the function [MATH] is either positive or negative. That is we write |
[EQUATION] such that [MATH] is positive almost everywhere on each subinterval [MATH] , and negative almost everywhere on [MATH] , respectively; and vanishes at each end point except when [MATH] and [MATH] Over each of the subintervals we multiply equation ( 4.70 ) by sgn [MATH] , we integrate, and we sum up the integra... |
[EQUATION] This, together with [EQUATION] yields [EQUATION] Hence if condition ( 4.69 ) holds we can indeed choose a [MATH] such that the solution [MATH] of ( 4.70 ) satisfies [MATH] , for all [MATH] |
To verify that the range condition holds true (see 19 , II.3] ), note that for any [MATH] , the solution of the equation [MATH] is |
[EQUATION] where [EQUATION] Since [MATH] is bounded, it follows from the smoothness assumptions we imposed on [MATH] and [MATH] (in particular their boundedness), that for any [MATH] and [MATH] large enough, the right hand side of ( 4.74 ) belongs to [MATH] Therefore, [MATH] given by ( 4.73 ) clearly satisfies [MATH] ,... |
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