1 Introduction
In their recent paper [1], O. Arino and R. Rudnicki considered a model of phytoplankton at the level of aggregates of cells. The aggregates are structured by their size and the phytoplankton system consists of aggregates of all possible sizes. The aggregate size can change due to the usual birth and death of individual cells, but also there are two other mechanisms acting at the level of aggregate: splitting of an aggregate into several parts and combining of two or more aggregates into a bigger one. The latter two are known in physics and chemical engineering as fragmentation–coagulation processes and describe a variety of phenomena ranging from polymerization/polymer degradation, droplets break-up and build-up, through rock crushing and grinding, solid drugs break-up in organisms, to blood cell aggregation and fragmentation. In phytoplankton, the major role in fragmentation and coagulation processes is played by the substance called TEP (Transparent Exopolymer Particles) that is a by-product of the growth of phytoplankton, and its stickiness causes the cells to remain together [2–5]. On the contrary, a low level of concentration of TEP results in fragmentation of the aggregate due to external causes, like currents or turbulence on one hand, and internal unspecified forces of biotic nature on the other.
In [1], the authors considered a relatively simple model of binary fragmentation and coagulation with bounded fragmentation and coagulation rates, as their aim was to investigate the long-time behaviour of the solution, and they succeeded in proving the existence of a time-invariant distribution to which the population of aggregates converges as time tends to infinity, whatever the initial population might be.
Our aim in this paper is to analyze more closely the inter-relation between the growth and fragmentation of aggregates so that we shall disregard the coagulation part. By the very nature of the model, the fragmentation process itself should be conservative, that is, the total amount (mass, the number of particles or cells) of the described quantity, say Q, contained in all the aggregates before and after a fragmentation event should be the same. Thus, if in some system the fragmentation occurs alongside another process of growth or decay determined by a certain law, then the evolution of the total amount of Q should follow this law due to the conservativity of the fragmentation process. If this is the case, then such a process is said to be honest. However, for pure fragmentation models and models combining fragmentation of clusters with their dissolution in the surrounding solute, it has been known for some time [6–9], that if the fragmentation rate of small clusters is large enough, then there appears an unexpected leakage of Q from the system, that is, the amount of Q in the system is strictly smaller than predicted by the laws of nature used to build the model.
In the existing physical literature, op. cit., this unaccounted for loss of Q (in this case, mass-loss), termed shattering fragmentation, is attributed to a phase transition and formation of a ‘dust’ of particles with zero size and non-zero mass (a similar but in some sense opposite process of forming an ‘infinitely large’ particle is known in coagulation as a gelation). For some relatively simple models, shattering fragmentation was analyzed in [4,7] by probabilistic methods. In a series of recent papers [10–14], the shattering and non-shattering fragmentation was fully characterized by the properties of the generator of the semigroup describing the evolution and the theory was applied to a wide range of processes providing a comprehensive classification of fragmentation models.
In particular, in [10], a model where fragmentation occurs together with a continuous mass loss due to dissolving of the substance has been analyzed and conditions ensuring conservativity and shattering have been provided. A crucial rôle in the analysis is played by the theory of substochastic, that is, positivity preserving and contractive semigroups. In this paper, we shall show that the model introduced by Arino and Rudnicki, though obviously not substochastic due to the appearance of the growth term, can be nevertheless transformed into one, and treated by a generalization of the theory developed in [10] yielding similar results, that is, the process is honest for rates of fragmentation bounded at 0, otherwise shattering fragmentation occurs irrespective of the growth rate (within the limits of the model).
It is, however, fair to admit that shattering fragmentation, as related to the creation of infinitesimally small aggregates, is not really a biological (or physical) phenomenon as in the real world there is always a lowest size of objects beyond which we cannot reach without encountering quantum effects. If one adopts such a point of view, then our results can be restated as saying that the models with fragmentation rates that are unbounded at 0 are non-biological.
2 The model
Following [1], we consider the following fragmentation model with mass loss:
(2.1) |
(2.2) |
(2.3) |
The fragmentation is characterized by two functions: p and k. The function p is the fragmentation rate, that is, the number of fragmentation events of aggregates of size x per unit time. We assume that and a.e. Further, k is a non-negative measurable function that describes the distribution of particle masses x spawned by the fragmentation of a particle of mass y. Formal balance of mass in fragmentation requires:
(2.4) |
(2.5) |
The typical choices for k used in the literature are: the power law with , and its generalization
(2.6) |
Integrating (2.1) multiplied by x, we obtain the formal equation governing the evolution of the total size of the system:
(2.7) |
3 Transport semigroup
In this section we consider the differential part of Eq. (2.1), that is, the Cauchy problem:
(3.1) |
(3.2) |
Denoting by B a fixed antiderivative of , say, we see, due to for , that:
(3.3) |
(3.4) |
(3.5) |
Using the above we can prove the following result for the the semigroup solving (3.1).
The operator T defined by the formal expression:
Proposition 3.1
on the domain:
where
, generates a positive semigroup, say
, satisfying for any
:
where
is defined in
(2.2)
.
(3.6)
Let us consider the resolvent equation of (3.1): Proof
Solving the above equation, we see that a good candidate for the resolvent is:
where is a fixed antiderivative of . Direct integration gives:
where we used the fact that is non-increasing, and (3.5). Further, we have:
so that(3.7)
where we again used monotonicity of and (3.5).
Next we observe that for ,
(3.8) |
From this proposition it follows that the operator
(3.9) |
(3.10) |
4 Substochastic semigroups
In this section we shall summarize relevant facts from substochastic semigroup theory as developed in [10]. To avoid confusion, we shall use the same notation for the abstract operators as for the particular application discussed in this paper, however the theory is fairly general and requires only that the assumptions (A1)–(A3) be satisfied.
Let be a measure space and let . If is a subspace, then denotes the cone of nonnegative elements of Z and for the symbols denote the positive and negative part of f, that is, and . Let be a strongly continuous semigroup on X. We say that is a substochastic semigroup if for any , and , and a stochastic semigroup if additionally for .
Accordingly, we consider linear operators in X: with , and K, that have the following properties:
- (A1) generates a substochastic semigroup ;
- (A2) and for ;
- (A3)
for all
(4.1)
Under the above assumptions, there exists a smallest substochastic semigroupgenerated by an extensionof the operator. This semigroup, for arbitraryand, satisfies:Theorem 4.1
[11,17]
can be obtained as a strong limit in X of semigroupsgenerated byas; if, then the limit is monotonic.(4.2)
The generator of is characterized by:
(4.3) |
Formula (4.3) does not provide any explicit information as to how large an extension of the generator is and this problem is closely related to the behaviour of . To make this remark precise, we adapt the concept of honesty and dishonesty from the theory of Markov processes [18].
Firstly, note that (4.1) can be written as:
(4.4) |
(4.5) |
We say that a substochastic semigroup (generated by an extension of the operator ) is honest if c is finite on , and, for any , the solution of (4.2) satisfies: Definition 4.1
(4.6)
The definition of honesty is not restricted to contractive semigroups and is valid even if c in (4.4) is of undetermined sign. In fact, for the original model (2.1) we shall be using this definition with a positive right-hand side in (4.6). However, for a general c, the existence part of the theory is usually not a trivial matter and this is why we prefer to present a complete theory for substochastic semigroups, and then apply it to a wider class of models that can be transformed to a substochastic case.Remark 4.1
It can be proved that the honesty of , (4.6) is equivalent to its integral version: is honest if and only if for any and :
(4.7) |
(4.8) |
For any fixed, there iswithsuch that:Theorem 4.2
Moreover, c extends to a nonnegative continuous linear functional on, given again by (4.5).(4.9)
The properties of and its relation to are summarized in the proposition below.
The following holds:
Proposition 4.1
henceis honest if and only iffor any (some) ;
An important characterization of honesty is given in the following theorem.
The following are equivalent:
Theorem 4.3
(4.10)
The problem with the characterization results given above is that they require the knowledge of the generator itself and therefore they are not immediately useful. To circumvent this problem, we shall be using certain extensions of the involved operators, that are defined below.
Define by the set of measurable functions that are defined on Ω and take values in the extended set of real numbers and by the subspace of consisting of functions that are finite almost everywhere. is a vector lattice with respect to the usual relation: ⩽ almost everywhere, with X and being sublattices of .
In what follows, we shall denote by and extensions of the operators , K, and , respectively. By we abbreviate . At this moment, we shall require only that all the extensions have domains and ranges in , that and are positive operators on their domains and that .
We shall present here a theorem giving a sufficient condition for dishonesty in terms of these extensions.
Assume that there exists
such that
Theorem 4.4
Then the semigroup
is dishonest.
(4.11)
5 Back to the growth–fragmentation equation
Let us look at the problem (2.1) from the point of view of the developed theory. Let us recall that we consider the operator K defined by the expression:
(5.1) |
There is an extension G of
given by
that generates a positive semigroup
. Moreover, the generator G is characterized by:
Proposition 5.1
for
and
.
(5.2)
The operator was constructed from T by subtracting the bounded operator . Let us consider the approximating semigroups , mentioned in Theorem 4.1. They are generated by , and Proof
in X, uniformly in t on bounded intervals. Define semigroups generated by . As multiplication by does not affect convergence, we see in (5.3) that converges strongly to the semigroup which is generated by and thus is an extension of , defined on the same domain as , .(5.3)
Formula (5.2) follows immediately from (4.3) by noting that since , we have for and the same holds for the resolvent of T. □
Formula (5.1) for takes the form:
(5.4) |
(5.5) |
(5.6) |
To proceed, we have to specify the extensions of the operators which we will be working with. Possibly the most general choice is as follows. For we denote:
(5.7) |
(5.8) |
(5.9) |
(5.10) |
We illustrate the usefulness of the concept of extensions in the following observation. Any functionis continuous on.Proposition 5.2
Let first and . Since extends to a positive integral operator on , by (5.2) the element where , is a well-defined element of as the series is increasing. However, as , it must be finite almost everywhere. From (5.10) we have: Proof
and as the functions B, A and b can have zeroes or singularities only at 0 and infinity, we see that is integrable over for any and therefore u is continuous with a possible exception at . If we take now arbitrary u, we see that , where are the positive and negative parts of f. For , the corresponding are also positive and hence are continuous on , which yields continuity of u. □
The following technical result can be proved as in [10 (Lemma 4.2)]
Let
and
be the extensions introduced above. If for some
, both g and
belong to
, where
, then:
Lemma 5.1
(5.11)
A crucial rôle in the following considerations is played by the next theorem.
If
, then there are sequences
and
as
such that:
Theorem 5.1
(5.12)
Using a similar argument to Proposition 5.2 we see that if is such that , then for any . Following [10 (Corollary 4.1)], we observe that if , there is , constructed as in the proof of Proposition 5.2, such that and: Proof
and, as , we have and, by Lemma 5.1,
for any sequences and converging to 0 and ∞, respectively. This can be extended to arbitrary u using the decomposition of Proposition 5.2.
Since we know that , we have . Thus, there is a sequence converging to ∞ such that . Similarly, we obtain a sequence that converges to 0 as , such that . Since for , we obtain the thesis. □
If
Theorem 5.2
then
, thus
is honest.
(5.13)
As in the previous proof, it is enough to consider , ; for such f we have also for some . Since , by (5.10) and Tonelli's theorem, we obtain: Proof
The function is continuous and non-negative, and the only points where it may be zero are at or as . As , the integral term tends to infinity, see (3.8). Since a is bounded at 0, the other term tends to 0 by (3.3) and the l'Hospital rule gives:
as and . Thus for any , and we can put in (5.11), and thus in (5.12), getting:
so that (5.6) is obviously satisfied. □
The theorem on dishonesty below is intended primarily as an example so that the regularity assumptions on the coefficients are not optimal. We shall also put as adding or subtracting a bounded operator does not change the domain of the generator; hence . Moreover, we restrict our attention to k given by (2.6): and satisfying:
(5.14) |
Assume that
with
,
Theorem 5.3
for some
,
on
and:
(5.15)
Then
is dishonest.
(5.16)
To simplify notation, we put . We use Theorem 4.4 so that we work with the operator extensions introduced at the beginning of this section and construct satisfying the assumptions of this theorem. Let us define: Proof
where and , see (5.15). Clearly and it is continuous on . Moreover, for any , and therefore we can pass to the limit with in the integral terms on the right-hand side of (5.11) (taking into account that ). Thanks to the continuity, we can repeat the argument of Theorem 5.1 getting: (5.17)
for some converging to zero, where we used the estimate (2.2) to pass to the limit in the last term.(5.18)
Consider first the interval where we have . Using , we have:
(5.19) |
For we have similarly to (5.19)
Acknowledgment
The paper was prepared while the author visited the Department of Mathematics of the University of Franche-Comté in Besançon, France, as an invited professor. The warm hospitality of Prof. Mustapha Mokhtar-Kharroubi and many stimulating discussions with him are greatly appreciated. The visit was partially supported by the National Research Foundation of South Africa under GUN 2053716.