A STRUCTURED COAGULATION-FRAGMENTATION EQUATION IN THE SPACE OF RADON MEASURES: UNIFYING DISCRETE AND CONTINUOUS MODELS

We present a structured coagulation-fragmentation model which describes the population dynamics of oceanic phytoplankton. This model is formulated on the space of Radon measures equipped with the bounded Lipschitz norm and unifies the study of the discrete and continuous coagulationfragmentation models. We prove that the model is well-posed and show it can reduce down to the classic discrete and continuous coagulation-fragmentation models. To understand the interplay between the physical processes of coagulation and fragmentation and the biological processes of growth, reproduction, and death, we establish a regularity result for the solutions and use it to show that stationary solutions are absolutely continuous under some conditions on model parameters. We develop a semi-discrete approximation scheme which conserves mass and prove its convergence to the unique weak solution. We then use the scheme to perform numerical simulations for the model. Mathematics Subject Classification. 35L60, 35Q92, 92D25. Received January 25, 2021. Accepted September 23, 2021.

models presented in [13,55] (1.1) Here, ( ) represents the number of particles of the th size at time , , represents the rate at which particles of the th and th size coagulate, represents the global fragmentation rate of particles of the th size, and , represents the rate at which particles of the th size fragment to particles of the th size. In subsequent works, Müller [49] extend the discrete coagulation terms to a continuous setting in the form of an integro-differential equation. In a similar fashion, Melzak [48] extended the fragmentation terms to a continuous setting. In [3], Ackleh and Fitzpartick introduced the coagulation equations in the context of size-structured population where biological processes including birth, death and growth were modeled, and the fragmentation equation were added to this size-structured model by Ackleh in [1]. These models take the form of a nonlinear nonlocal first-order hyperbolic differential equation with a nonlocal boundary condition. All together, these continuous models can be covered under the following partial differential equation: Analogous to the discrete equation, ( , ) represents the density of individuals of size at time , ( , ) represents the rate at which particles of size and coagulate, ( ) represents the global fragmentation rate of particles of size , and ( , ) represents the rate at which particles of size fragment into particles of size . Additionally, the functions , , and represent the growth, death, and birth functions respectively. Each of these functions is influenced by time and density of the population.
Throughout the literature, there are a variety of assumptions on the coagulation kernel. Common assumptions include: the kernel being bounded by some combination of linear functions [10,33]; some ratio of the kernel and the size of the individual particle tending to zero [38,50]; and, the kernel blowing up for small sizes [19]. Without some additional assumptions on either the kernel or initial condition, the above assumptions can cause the formation of particles of infinite size. This phenomenon is known as gelation and has been shown to happen in finite time [56]. Since gelation is not the focus of this paper, we will require more regularity on our coagulation kernel.
Most studies of coagulation-fragmentation equations focus on the case of binary fragmentation; in other words, when particles only fragment into two smaller units (see [45] and the references therein, as well as the previously mentioned works). Although the initial work [48] considers the more general case of multiple fragmentation, where particles can fragment into more than 2 smaller particles, it is difficult to find many results concerning this case. In the setting of density-based equations, the authors of [47,48] work with only an assumption of bounded kernels for both coagulation and fragmentation. Meanwhile, the work [40] allows for linear growth in the rate of fragmentation, but requires a bound on the coagulation kernel. The case where both the coagulation and fragmentation kernels are unbounded is studied in [28,29].
In this work we will extend the formulation of model (1.2) to the space of Radon measures. This extension allows the unification of the discrete model (1.1) and the continuous (density) model (1.2) under the same framework. In recent years, the space of Radon measures equipped with the bounded Lipschitz norm has been used in the study of population dynamics [17,18,31,35]. While many population models have been studied intensely in this setting, the study of coagulation-fragmentation equations in this space is sparse. Mild measure solutions to a coagulation-diffusion equation have been obtained in [50], where state-space of study was the space of finite measures with absolutely continuous first marginal. More so, the model considered does not include any biological processes (i.e. growth, birth, or death). Existence of solutions to a coagulation-fragmentation equation is obtained in [23] via probabilistic means. However, authors in [23] only prove existence of a measure solution in the topology of weak convergence and also do not consider any biological processes. The authors in [20] consider a growth-fragmentation equation with a multiple fragmentation kernel identical to that studied in [28]. They cite well-posedness of their model as a consequence of [18] and do not consider a coagulation term. In this paper, we adopt similar assumptions on our model ingredients as in [20], but will prove well-posedness using a fixed-point approach presented in [6]. Finally, for a structured model without coagulation or fragmentation, [35] proves that solutions are absolutely continuous to the left of the zero characteristic curve. Under similar assumptions, we will extend this result to structured coagulation-fragmentation equations.
The layout of the paper is as follows. In Section 2, we present notation used throughout the paper. In Section 3, we formulate the model, prove some useful properties of the model ingredients, and show the model is indeed well-posed. In Section 4, we analyze the interplay between the biological processes (growth, death and birth) and the physical processes (coagulation and fragmentation). In particular, we study their effects on the regularity of solutions to the structured model. In Section 5, we show that the density equation (1.2) and discrete equations (1.1) are indeed special cases of our model. In Section 6, we present a semidiscrete numerical scheme which we test against a few examples providing approximate error in the BL-norm and the numerical order. Finally, in Section 7 we will provide discussion of the results and some concluding remarks.

Preliminaries and notation
In this section, we will provide some preliminary notation. The space of finite Radon measures over R + := [0, ∞) is denoted by ℳ(R + ). The non-negative cone of ℳ(R + ) will be denoted ℳ + (R + ). Unless otherwise stated, both of these spaces will always be equipped with the Bounded-Lipschitz norm given by Here, 1,∞ (R + ) is the usual Sobolev space over R + with codomain R equipped with the usual norm ‖ ‖ 1,∞ := ‖ ‖ ∞ + ‖ ′ ‖ ∞ . In the literature, the BL-norm has had a few names such as the flat norm [25,26], the Dudley norm [22,24], and the Fortet-Mourier norm [27,41]. Another norm commonly associated with measures is the total variation norm given by It should be noted that while over nonnegative measures they are equivalent, the BL-norm and TV-norm are different on the space of signed measures. In particular, for ∈ ℳ(R) We refer the reader to [32] and the references therein for more information.
We say a sequence ( ) of Radon measures is tight if In ℳ + (R + ), we additionally have that the BL-norm metrizes weak convergence. That is ( ) converges weakly to ∈ ℳ + (R + ) if for every ∈ (R + ), as −→ ∞. For more detail, see [30,31]. It is often convenient to use the operator notation in place of integration. That is for a function , we say Finally, we say the flow of a Lipschitz vector field ( , ) is a function , ( ) which satisfies d d , ( ) = ( , , ( )), , ( ) = . (2.1)

Structured coagulation-fragmentation equation
In this section, we formulate a structured coagulation-fragmentation population model on the space of Radon measures. We then establish existence and uniqueness of solutions for the structured coagulation-fragmentation model. Finally, we prove a stability result which will be useful in later sections.

Formulation of the model on the space of Radon measures
The coagulation term we propose is the measure given by where ( , ) represents the rate at which individuals of size coalesce with individuals of size . The first term in (3.1), + , represents the inflow of individuals due to coagulation. The second term in (3.1), − represents the number of individuals lost due to coagulation. Notice that ± [ ] are measures which can be described in a distribution sense by and We claim these terms are generalizations of the coagulation terms of the continuous coagulation equation (1.2) given by Indeed, multiplying + ( ) by a test function and integrating we see that Notice by formally taking = ∑︀ ∈N we can arrive at the discrete coagulation terms given in (1.1) from the traditional Smoluchowski equations [55].
The fragmentation term we propose is given by Here, ( ) represents the global fragmentation rate of individuals of size and ( , ·) is a measure supported on [0, ] such that ( , ) represents the probability a particle of size fragments to a particle with size in the Borel set . The positive term, + , represents the inflow of individuals due to fragmentation, and the negative term, − , represents the number of individuals lost due to fragmentation. Similar to the coagulation terms, ± [ ] are measures given explicitly by As before these terms are a generalization of the multiple fragmentation terms from (1.2): Here, following [23], we allow ( , ·) = ( , d ) to be a non-negative measure supported in [0, ]. With these generalized terms, we propose the following model: where As in the density model (1.2), the functions , , and are nonnegative and represent the growth, death, and birth functions, respectively. They are assumed to be influenced by both time, , and the solution to the population model, ( ). In applications (e.g. see [2,8,18,21]), it is common to choose , and to depend on a weighted mean of the population in the following form: and similar expressions for and , for given maps : [0, ] × R + × R + → R + and : R + → R + . Common physically motivated model functions utilize BevertonHolt type [12] or Ricker type [52] nonlinearities with respect to the weighted mean of the population and of a Von Bertalanffy type [51] model with respect to structure . In the boundary condition, d ( ) represents the Radon-Nikodym derivative of with respect to the Lebesgue measure, d , evaluated at .
We impose the following assumptions on the growth, death and birth functions: (A1) For any > 0, there exists > 0 such that for all ‖ ‖ TV ≤ and ∈ [0, ∞) ( = 1, 2) the following hold We assume that the coagulation kernel satisfies the following assumption: (K) is symmetric, nonnegative, bounded by a constant , and globally Lipschitz with Lipschitz constant .

Well-posedness of the structured coagulation-fragmentation equation
Here, we aim to prove model (3.8) is well-posed. To this end, we present the following propositions which describe useful properties of the source terms.
where¯, is a constant depending only on , , and .
Proof. To prove (3.12) notice that Since is symmetric, Taking the supremum over all such gives In the same way Combining these two results we see that Next we have the following proposition concerning the fragmentation term: (3.14) and and The following proposition is immediate from assumptions (A1) and (A2).
-For any ≥ 0 and for any , with ‖ ‖ TV , ‖ ‖ TV ≤ , We are now ready to show model (3.8) is well posed. More precisely, we have the following result: Moreover, if 0 has finite total mass in the sense that ∫︀ In particular, if = = = 0 then mass is conserved in the sense that ∫︀ Then equation (3.8) reads + ( ( , ) ) = ( , ) +¯( , ·, ) .
Remark 3.5. In applications the smallest size will not be of size 0 but rather some 0 > 0. Model (3.11) and the theorem above can be adjusted for such applications by shifting the Dirac measure at 0 to 0 , requiring ( , )( 0 ) > 0, and requiring ( , ·) to be supported on [ 0 , ). In this case, the mass conservation equation would be
It then follows from Theorem 3.4 that (3.20) has a unique solution Under some additional assumptions on the coefficients of (3.20) we can extract from a subsequence converging to a solution of (3.8).
In particular, it follows that ( ) is tight for any ∈ [0, ]. Moreover for 0 ≤ < ≤ , and any ∈ 1,∞ , ‖ ‖ 1,∞ ≤ 1, we have using (3.21) that Thus, ‖ − ‖ BL ≤¯( − ) so that the sequence ( ) ⊂ ([0, ], ℳ(R + )) is uniformly equicontinuous. By the Arzela-Ascoli theorem, for any > 0, we therefore have a convergent subsequence (not relabeled) of the in ([0, ], ℳ + (R + )) which converges to some ∈ ([0, ], ℳ + (R + )). A diagonal argument gives that → in ([0, ], ℳ + (R + )) for any > 0. Since is bounded Lipschitz, we can pass to the limit . Sending → ∞ gives that for any > 0, We now want to pass to the limit → ∞ in the equation satisfied by , namely Let ∈ (R + × R + ). We pass to the limit in the right-hand side using that → for any ≥ 0. Since → uniformly on compact sets, ( , 1) ≤ , and ⊗ → ⊗ weakly, we can pass to the limit Since |( [ ], )| ≤ , we obtain by dominated convergence that Similarly, we can pass to the limit in ( [ ], ) in the same way. Finally, in view of (S4), (3.21) and since has compact support we have for any ≥ 0 that Since moreover we obtain by the Dominated Convergence Theorem that We treat the terms with and in the same way.

Interplay between growth, coagulation, and fragmentation processes
In the recent paper [35], it was shown that the steady state solution of a size-structured population model (i.e. model (3.8) with ≡ ≡ 0) with positive model ingredients is absolutely continuous with respect to the Lebesgue measure. This leads naturally to studying the effect the physical processes of coagulation and fragmentation would have on such regularity. To this end, we denote by 0 ( ) the solution to Before we present the main theorem of the section, we will establish the following useful lemma: Proof. The bijection property of Φ follows from the uniqueness of trajectories and the definition of 0 ( ). As for (4.1), taking the derivative with respect to in d d , (0) =˜( , , (0)) yields d d .
In particular for any bounded measurable function : [0, ∞) → R, We now make use of the following definition: With this definition and the above lemma, we are now in position to establish the main result of this section. The proof that we propose is inspired by Lemma 3.5 of [56] and Lemma 2.6 of [42]. However the presence of the growth term adds new difficulties and novel techniques are adopted to handle these difficulties. Proof. Recall that the solution was obtained as a fixed point of the map Γ defined in (3.17) namely Notice due to the positivity of the model functions Given some > 0 and ∈ [0, ], let be the family of subsets of [0, 0 ( )) of the form where ∈ N 0 , ≤ 1 ≤ . . . ≤ ≤ , 1 , . . . , ≥ 0, and ⊂ [0, 0 ( )) is a Borel subset with | | < . It is implicitly understood that at each step of the construction of we take the intersection with [0, ∞). Define then where we extend to (−∞, 0) by 0. Notice that ♯ 0 is supported in [ 0 ( ), ∞) and that any ∈ is a subset of [0, 0 ( )). It follows that for any ∈ of the form (4.6) we have by (4.5) that For any 0 ≤ ≤ ≤ and any subset ⊂ [0, ∞) we have by (4.2) and assumption (A2) that Using the translation invariance of Lebesgue measure we then have that the measure of given by If we assume that the family { ( , ·)} ≥0 is uniformly equi-integrable then sup ≥0, | |≤ ( )( ) goes to 0 as → 0. We denote (1) any quantity going to 0 as → 0 uniformly in ∈ [0, ] and . Coming back to (4.7) we thus obtained so far that To bound the coagulation term in the right-hand side recall the definition of + : Coming back to (4.8) we obtain Since this holds for any ∈ and any ≤ we deduce which yields by Gronwall inequality ℰ( ) = (1).
This leads us to the following corollary about the regularity of a steady state solution to model (3.8). Proof. The proof follows from similar arguments of Proposition 2.6 in [35] with making use of ( )( ) > 0 for all . Indeed, since ( )( ) > 0 for all we have

From measure equation to discrete and continuous equations
It is often claimed that one of the many benefits of population models set in measure spaces is the unification of the study of discrete and continuous structure. In this section, we demonstrate this property by showing that model (3.8) includes as special cases the discrete Smoluchowski equations [55] and the continuous Müller model [49].

Continuous density model
In this subsection, we briefly demonstrate how model (3.8) reduces to model (1.2) and hence encompasses continuous density models studied in [1,3,15,49] from (3.8). This follows naturally under the following assumptions: (B1) 0 is absolutely continuous with respect to the Lebesgue measure, (B2) ( , ·) is absolutely continuous with respect to the Lebesgue measure.
Then by undoing the derivations of (3.1) and (3.6), one arrives at the density equations (3.4) and (3.7) covered in the aforementioned works. In particular, we can recover the binary fragmentation kernels studied in [1,15,37] by taking where the function ( , ) models the rate at which a particles of size + fragment into particles of size and .

Discrete equation
In this subsection, we show under certain assumptions, model (3.8) will reduce to the discrete coagulationfragmentation equation (1.1) and hence covers models discussed in [10,55]. To obtain these equations, we set where the ( ), ∈ N, satisfy the discrete coagulation-fragmentation equation Proof. It is clear from Theorem 3.4 that (3.8) has a unique solution ∈ ([0, ∞), ℳ + (R + )). Moreover, according to the proof of Theorem 3.4, is a fixed-point of Γ defined in (3.17). Since = 0, , is the identity map. Thus Γ is simply given by and repeat the proof of Theorem 3.4 verbatim to obtain that is supported in ℎN for any ≥ 0. It follows that can be written as in (5.2). Equation (5.3) follows from (3.10) taking a 1 test-function, , constant in time and supported in ( ℎ − ℎ, ℎ + ℎ) such that ( ℎ) = 1.

Numerical methods and results
In this section, we present a semidiscrete scheme for a coagulation-fragmentation equation based on (5.3) and Theorem 5.1 as well as provide some numerical results based on this scheme. For the rest of this section, we assume that ( , ) = ( , ) ≡ 0.
It follows form this proposition that the assumption of Theorem 3.6 are satisfied. Thus, we deduce that ℎ converges along a subsequence ℎ → 0 to solution of equation (3.8). Since this equation has a unique solution, the whole sequence ℎ converges to : We can thus think of the system (6.2) as a semi-discrete scheme for solving equation (3.8). One could combine this semidiscrete scheme with any ordinary differential equation scheme (e.g. any Runge-Kutta Method) to arrive at a fully discrete scheme. Convergence for such a scheme then follows from a standard triangle inequality argument. In the next section we present some numerical experiments to evaluate the quality of such a scheme. Remark 6.3. One can easily include the case , > 0 as these terms do not affect the discrete structure of the solution. However, in the case of additionally assuming > 0, it is not true that the solution is discrete for all time. This result was shown for structured population models (without coagulation and fragmentation) in [35] and with coagulation-fragmentation in Section 4.

Mass conserving fragmentation term
To remedy the error generated in mass conservation of the scheme discussed in the previous section, we propose a new approximation of ( , d ) in the form ℎ ( , ·) = ∑︀ ∞

=1
( ) for which the following holds: A natural choice of ( ) is given by This approximation results in a mass conserving scheme at the expense of requiring a minimum positive size 0 . We have the following result: Proposition 6.4. Assume there is a positive minimum size 0 > 0 and therefore the points = 0 + ℎ. Then Proof. Taking ( ) ∈ 1,∞ (R) with ‖ ‖ 1,∞ ≤ 1 and letting := ( ) we have Since 0 < 0 ≤ the first term is bounded and making use of the Lipschitz property of we have Therefore by the same arguments in the section above, we can conclude that a scheme with this term will converge to the solution of equation (3.8) with = = = 0.
The standard kernel taken for a structure domain R + is given by ( , d ) = 2 d . For the domain [ 0 , ∞), an example of a kernel which satisfies assumption (F2) is given by Notice, that if 0 = 0, then the above kernel reduces to 2 d . It should be noted that it is important to calculate ( ) exactly when implementing the scheme. Otherwise, numerical integration error may be introduced resulting in lack of mass conservation.

Numerical results
In this section, we test the semidiscrete scheme against some commonly used examples. We begin by testing the coagulation and fragmentation portions of the scheme separately. We implement the semidiscrete scheme using MATLAB's ode45 function. In each example, we present the exact solution at time = 1 plotted against the structure variable, , the absolute value difference of the numerical and exact solution, and the relative mass between the numeric and exact solutions plotted against time. We remark that for examples with only coagulation, the semi-discrete scheme (6.2) conserves mass (i.e. (6.3)); therefore, any change of mass is due to simulating infinite domain problems over a finite interval. Where it is applicable, we provide a table calculating the BL-norm and numerical order of the scheme. The BL-norm is approximated by the algorithm provided in [34], while the numerical order of the scheme is calculated using the standard calculation:

Coagulation and fragmentation examples
In this section we presented several numerical example focused on coagulation and fragmentation processes.  [39] for more details. Numerical simulations for this example are presented in Figure 1 with ∆ = 1/40 and the BL error and order of conference are presented in Table 1. Simulation are performed over the finite domain ∈ [0, 20].
Example 6.6. Although our theory does not cover the phenomenon of gelation, we include a numerical example showing how the semi discrete scheme handles such kernels. In this example, we take ( , ) = with 0 = − / d . This has exact solution (see e.g. [39]).  Numerical simulations for this example are presented in Figure 2 with ∆ = 1/40 and the BL error and order of conference are presented in Table 2. For the order of convergence, the simulations are performed over the finite domain ∈ [10 −2 , 20].
Example 6.7. In this example we consider fragmentation. We let ( , ·) = 2 d and ( ) = . As given in [54], this problem has an exact solution of Numerical simulations for this example are presented in Figure 3 with ∆ = 1/40 and the BL error and order of conference are presented in Table 2. Although convergence for the mass conserving fragmentation scheme is only shown for positive minimum mass, it still seems to perform well for the simulations below. Solving the fragmentation terms exactly leads to an (ℎ 2 ) term in the last subinterval (where = := ∆ ). Explicitly, we have On the bottom, we present the relative mass against the initial condition.
However, we noticed that for this last interval truncating the second term ℎ 2 2 , which is of order (ℎ 2 ), we improve the scheme's performance. We present both the performance of the original scheme and the truncated scheme in Table 2. Simulations for Table 2 are performed over the finite domain ∈ [0, 20]. Example 6.8. In this example, take ( , ·) = 2 d and ( ) = 2 . Again, as given in [54], this problem has an exact solution of = (1 + 2 + 2 ) exp(− (1 + )) d .
Numerical simulations are presented for this example in Figure 4 with ∆ = 1/40. The BL error and order of convergence are presented in Table 3. Simulations for Table 3 are performed over the finite domain ∈ [0, 20].
Example 6.9. For this example, we demonstrate the performance of the scheme for a domain where the minimum size is positive. To this end, we truncate Example 6.7 above to the domain [10 −3 , 20] and use the kernel given by (6.4). Since the exact solution is not known for this equation, we compare to the solution given in Example 6.7. Though we do not compute any numerical orders of convergence, we point out the numerical and exact solutions in Figure 5 are very close. This simulation is again done with ∆ = 1 40 . On the bottom, we present the relative mass against the initial condition.  On the bottom, we present the relative mass against the initial condition.
Example 6.10. In this example, we demonstrate what a discrete system would look like in our current frame work as well as provide an example of the results show in Theorem 5.1. We also demonstrate the mass conservation property of the coagulation terms of the scheme. The simulation is performed over the interval [0, 20] however, for clarity we zoom into the interval [0, 4]. Take ( , ) ≡ 1 and 0 = 0.2 + 0.4 (Fig. 6).

Concluding remarks
In summary, we have presented a size-structured coagulation-fragmentation model formulated on the space of Radon measures endowed with the BL-norm. This model unifies the study of both the discrete and density based coagulation-fragmentation equations, both of which have been used in studying the dynamics of oceanic phytoplankton populations. We have shown, under biologically relevant assumptions (see e.g. [4] and the references therein), the model is well-posed using a fixed point approach discussed in recent papers [5,6]. We also established a regularity result that shows, under certain conditions on the model parameters, the solution to the model is absolutely continuous to the left of the characteristic curve emanating from the point (0, 0). This allows us to prove that any stationary solution of the model is absolutely continuous. This extends the result in [35] for structured population models without coagulation and fragmentation. Here, our proof differs from that in [35] since it relies on the implicit fixed point representation of the measure valued solution. Furthermore, we have shown how one obtains both the density and discrete coagulation-fragmentation equations from model (3.8). We also provided a semidiscrete method for approximating solutions to these equations and presented some numerical examples verifying our scheme. In these examples, we observed the semidiscrete scheme appears to have at best a second order convergence rate in the BL norm. In addition to the cases covered by our convergence proof, the scheme also seems to preform well in the case of a gelation coagulation kernel.
While the semidiscrete scheme presented in this paper is convergent and conserves mass, it does not take into account a growth term. In the future, we plan to develop and study fully discrete higher order schemes for the full model (3.8) that preserves solution non-negativity and mass (e.g. [14,46] in the space of integrable functions setting).