Fragmentation-Coagulation Models of Phytoplankton

BULLETIN OF THE POLISH ACADEMY OF SCIENCES MATHEMATICS Vol. 54, No. 2, 2006 PROBABILITY THEORY AND STOCHASTIC PROCESSES Fragmentation-Coagulation Models of Phytoplankton by Ryszard RUDNICKI and Radosªaw WIECZOREK Presented by Andrzej LASOTA We present two new models of the dynami s of phytoplankton aggregates. The rst one is an individual-based model. Passing to innity with the number of individuals, we obtain an Eulerian model. This model des ribes the evolution of the density of the spatial-mass distribution of aggregates. We show the existen e and uniqueness of solutions of the evolution equation. Summary. In [4℄ the authors built a model of the phytoplankton dynami s, where the individual is an aggregatea group of phytoplankton ells living together. Aggregates are stru tured by their size, whi h hanges due to three pro esses: growth aused by ell division, fragmentation and oagulation. The size distribution of aggregates satises the equation ∂u ∂ (1) = [g(m)u] + Φu + Cu, ∂t ∂m where m is the size of an aggregate, g(m) is the growth rate, and Φ and C are the operators of fragmentation and oagulation, respe tively. The authors proved the existen e and uniqueness theorem for equation (1) and he ked the long-time behaviour of the distribution of size for some spe ial ases. In the present paper we onstru t an individual-based model whi h is additionally spatially stru tured and ontains a pro ess of random movement of aggregates. Our aim is to show that the limit passage in the model, when the number of individuals goes to innity whereas the mass of a single ell tends to zero, leads to a transport equation of type (1) with a diusion term. In many papers su h a limit is a sto hasti pro ess with values in the spa e of measures, alled a superpro ess (see [9, 13, 16, 2, 15℄). The measures whi h 1. Introdu tion. 2000 Mathemati s Subje t Classi ation : Primary 60K35; Se ondary 47J35, 92D40. Key words and phrases : phytoplankton dynami s, measure-valued pro esses, fragmentation- oagulation equation. [175℄ 176 R. Rudni ki and R. Wie zorek are values of this superpro ess des ribe the distribution of parti les in spa e. In our model we also obtain a limit but it is deterministi . In fa t, we derive the evolution of the distribution density a ording to the equation ∂ ∂u (2) = D(m)∆ u + [λ(m)u] + Φ u + Cu, ∂t ∂m where Φ and C are operators responsible for fragmentation and oagulation (for their form, see (29) and (30) in Se tion 6). Finally, we prove the existen e and uniqueness of solutions of our equation. The approa h resembling ours was applied to a model of oagulation with diusion by Norris [24℄ and in a dierent setting of intera ting parti le systems by Kolokoltsov [20℄. Measure-valued limits of intera ting parti le systems leading to so- alled generalized Smolu howski equations were also onsidered in [6, 14℄. Similar equations, but used in a dierent ontext, appear e.g. in [1, 5, 7, 25℄, while in [11℄ one an nd a survey of oagulation equations. Other results on erning this subje t an be found in [3, 22℄ and the papers quoted therein. For the biologi al models that use similar methods we also refer to [10, 21, 23, 30℄. We exploit methods that were developed by Dawson ( f. [9℄) and other probabilists working on superpro esses (see also [16, 17, 13℄). The s heme of this paper is as follows. In the next se tion we introdu e our model, whi h is mathemati ally formulated in Se tion 3. Se tion 4 onerns the res aling of the individual model and the limit passage; the proof of the onvergen e theorem is given in Se tion 5. In Se tion 6 we derive the evolution equation that des ribes the behaviour of the limit pro ess, and we prove the existen e and uniqueness theorem. We onstru t an individual-based model of phytoplankton. In our model an individual is an aggregate that onsists of indistinguishable ells with equal masses joined by some organi glue. Cells in the aggregate may die or divide into two daughter ells, whi h auses the de rease or growth of the aggregate. An aggregate may shatter into two smaller aggregates or die (sink or be eaten). Thus the whole situation is des ribed by the following pro esses: • A single ell in the aggregate may die in a unit of time with probability λ (m) depending on the mass (number of ells) m of the aggregate or may divide into two new ells with probability λ (m). • A whole aggregate moves a ording to the ε-random walki.e. it skips by a ve tor of length ε in one of 2d dire tions (parallel to one of the axes, d is the dimension of the spa e) with probability (1/ε )D(m) (where D is a oe ient depending on the mass). • The aggregate may die in a unit of time with probability λ (m). x ∗ ∗ 2. Individual-based model of phytoplankton ells. m b 2 d Fragmentation-Coagulation Models of Phytoplankton 177 • The aggregate of mass m may split in a unit of time with probabilparts with masses m and m − m with probability ity λf (m) into twoP m (1) p (m, m) (where m=1 p(1) (m, m) = 1). We assume that after frag- mentation both new aggregates appear at the same lo ation as their parent. • Two aggregates may join up with probability k (1) depending on their masses and lo ations, and on the state of the whole population. More pre isely, let the rate of oagulation of the ith aggregate be P c(mi ). Then the probability that it joins the j th aggregate is c(mj )/ N k=1 c(mk ) and it is modied by a distan e-dependent oe ient v(xi − xj ), thus k (1) takes the form (3) c(mi )c(mj ) k (1) (mi , mj , xi − xj , ν) = P v(xi − xj ), k c(mk ) where the sum in the denominator extends over all living individuals. Our model of the oagulation pro ess is essentially dierent from standard physi al models (e.g. Smolu howski [29℄) where the probability of oagulation is proportional to the square of the number of parti les. We onsider the more biologi ally justiable ase, when the ability of oagulation of a single aggregate is not unbounded, but approximately onstant. Ability of oagulation depends on the on entration of some organi glue (TEP) [8, 26℄. This means that the probability of joining is a fun tion of produ tion of TEP by an aggregate, whi h depends on the mass of the aggregate. It should be noted that the probability of oagulation of two aggregates: 1) is proportional to the ability of both aggregates to oagulate, 2) depends on the distan e of the aggregates, 3) is symmetri al, i.e. it does not depend on the order of the aggregates. It seems di ult to nd another model of oagulation whi h has all the above features and, at the same time, has good mathemati al properties. 3. Sto hasti pro ess des ribing the model. The state of our model is des ribed by the ve tor (k; x1 , m1 , . . . , xk , mk ), where k is the number of aggregates and xi , mi , for i = 1, . . . , k, denote, respe tively, the lo ation and mass of the ith aggregate. Sin e k (and so the number of variables) hanges during evolution, and the order of pairs xi , mi is not important, we need a spe ial state spa e. We use the set of measures N = k nX i=1 o δxi ,mi : k ∈ N, (xi , mi ) ∈ Rd × N , i.e. we denote the aggregate of size m at position x by the Dira delta 178 R. Rudni ki and R. Wie zorek measure δx,m at (x, m) ∈ Rd × N. The set N is a subspa e of the spa e M of all nite Borel measures on Rd × R+ with the topology of weak onvergen e. Constrained by the nature of N (whi h is not even a Bana h spa e), we use the formalism of D([0, ∞), N ) martingale problems. By D([0, ∞), N ) we denote the spa e of all àdlàg fun tions on N , i.e. right ontinuous fun tions with left hand limits. Let us re all Definition 1. Let B(N ) be the spa e of measurable and bounded fun tions on N and let L be a linear operator dened on a subspa e of B(N ) with values in B(N ). We say that a sto hasti pro ess X(t) solves the D([0, ∞), N ) martingale problem for L and the initial state ν0 if this pro ess has D([0, ∞), N )-traje tories, Prob(X(0) = ν0 ) = 1 and for every f from the domain of L, \ t f (X(t)) − f (X(0)) − Lf (X(s)) ds is a martingale with respe t to 0 \ s   Fbt = σ X(s), h(X(r)) dr : s ≤ t, h ∈ B(N ) , 0 where B(N ) denotes the set of bounded Borel fun tions on N . Throughout this paper we omit the D([0, ∞), E) and by the martingale problem we mean the D([0, ∞), E) martingale problem. We will also speak of the (L, δν0 )-martingale problem, where δν0 is the Dira delta at the initial point. We will refer to L as the generator of the sto hasti pro ess. For an extensive guidebook to sto hasti pro esses and martingale problems we refer to [17℄. We formulate an individual version of the model in the setting of pure jump pro esses. We dene a generator L(1) as a jump operator L(1) f (ν) N  d X D(mi ) X = [f (ν − δxi ,mi + δxi +εk ,m ) + f (ν − δxi ,mi + δxi −εk ,m )] ε2 (4) i=1 k=1 + mi λb (mi )f (ν − δxi ,mi + δxi ,mi +1 ) + mi λm (mi )f (ν − δxi ,mi + δxi ,mi −1 ) + λd (mi )f (ν − δxi ,mi )  mi X + λf (mi ) p(mi , m)f (ν − δxi ,mi + δxi ,m + δxi ,mi −m ) m=1 + N X i,j=1 c(mi )c(mj )v(xi − xj ) P f (ν − δxi ,mi − δxj ,mj + δ(xi +xj )/2,mi +mj ) k c(mk ) − λ(ν)f (ν), 179 Fragmentation-Coagulation Models of Phytoplankton where λ(ν) = N  X 2d D(mi ) i=1 + ε2 + mi λb (mi ) + mi λm (mi ) + λd (mi ) + λf (mi )  N X c(mi )c(mj )v(xi − xj ) P k c(mk ) i,j=1 and εi is a d-dimensional ve tor with ε at the ith pla e and zeros elsewhere. In this se tion we assume that ε = 1, but in the next se tion we use a modied form of the operator L(1) with ε = 1/N . We assume that the fun tions D(m), mλm (m), mλb (m), λf (m), λd (m) , c(m) and v(x − x) are bounded and ontinuous; moreover c(m) > 0 for all m ∈ [0, ∞). Sin e the probability of extin tion of the pro ess is nonzero, we must also assume that for ν = 0 we have L(1) f (ν) = 0 (this means that after extin tion the pro ess remains in the state ν(t) = 0). Proposition 1. For any initial state ν0 ∈ N there exists a unique solution {ν (1) (t)}t≥0 of the martingale problem for (L(1) , δν0 ). The operator L(1) given by (4) is a jump operator with unbounded jump rate (for the theory of jump pro esses see [18℄ or [17℄). To obtain the existen e of the pro ess generated by L(1) we onstru t an approximating sequen e of sto hasti pro esses that are solutions of stopped martingale problems with operators with bounded jump rates. For any n ∈ N dene N ≤n = {ν ∈ N : h1, νi ≤ n}. Noti e that the jump rate satises (5) λ(ν) ≤ Cn ≤n on N with some onstant C . That is why the solution of the stopped martingale problem for (L(1) , δν0 , N ≤n ) oin ides with the solution of the stopped martingale problem with the operator bounded by Cn. Moreover, the stopping time τn = inf{t ≥ 0 : ν(t) 6∈ N ≤n or ν(t−) 6∈ N ≤n } is su h that n Proof. τn ≥ X ∆k n→∞ −−−→ ∞, Cn k=1 where {∆n }n∈N is a sequen e of i.i.d. random variables, exponentially distributed with intensity one. We use Proposition 3.2 in Chapter 4 of [17℄ to end the proof. Remark 1. Although the des ription of the pro ess is now formulated in the language of jump pro esses, in the subsequent se tions we will use a different setting. That is why we will write the operator (4) in a dierent form. Compare it with the approa h used in papers on superpro esses ( f. [16℄, 180 R. Rudni ki and R. Wie zorek [17℄). Let Cb2 be the spa e of all bounded fun tions with bounded derivatives 2 = {g : Rd × R+ → R : g ∈ Cb2 and inf g > 0}. up to se ond order and Cb,pos 2 For a given g ∈ Cb,pos we dene a fun tion Fg ∈ Cb (M) (bounded and ontinuous on M) by the formula Fg (ν) = exp [hlog g, νi] . The generating operator L(1) on the fun tions Fg (ν) has the form (6)  (1)   L g + B (1) g + Φ(1) g L(1) Fg (ν) = exp [hlog g, νi] , ν + C(g, ν) , g where d D(mi ) X [g(x + εk , m) + g(x − εk , m) − 2g(x, m)] ε2 L(1) g(x, m) = k=1 is the operator responsible for the spatial movement; B (1) g(x, m) = m[λm (m)g(x, m − 1) + λb (m)g(x, m + 1) − (λm (m) + λb (m))g(x, m)] is the operator of birth and death inside aggregates; (1) Φ g(x, m) = λf (m) m hX (1) g(x, m − m)g(x, m)p (m, m) − g(x, m) m=1 i + λd (m)(1 − g(x, m)) is responsible for the fragmentation and death of whole aggregates; and C(g, ν) = \\\\ c(m)c(m) TT v(x − x) c(m) ν(dx dm)   g((x + x)/2, m + m) − 1 ν(dx dm) ν(dx dm) × g(x, m)g(x, m) is the oagulation term. Now, we onstru t a sequen e of res aled pro(1) (t)} esses t≥0 , N ∈ N, based on {ν t≥0 that will approximate some ontinuous model. Assume that the number of parti les at time 0 in reases to innity as N → ∞ and assume that the mass of ea h ell is 1/N . The N th pro ess ν (N ) has values in the spa e 4. The limit passage. {ν (N ) (t)} NN =     k 1 X 1 ni d ∈R × N . δxi ,ni /N : k ∈ N, xi , N N N i=1 From now on we set m = n/N . The res aling means that the pro ess N ν (N ) behaves like ν (1) with appropriate oe ients. Namely: 181 Fragmentation-Coagulation Models of Phytoplankton • The birth or death of a ell means the hange of mass by a fa tor of 1/N . • We set the step of the random walk to be ε = 1/N . • The result of fragmentation of an aggregate of size m = n/N may have ; so we assume that the oe ients any mass in 1/N, 2/N, Pn. . . , (n−1)/N p(N ) (n/N, n/N ) = 1; moreover, we assume p(N ) are su h that n=1 that there exists a ontinuous fun tion q : R+ × R+ → R+ su h that for all m, m ∈ R+ with m ≤ m and all sequen es (nN ), (nN ) of positive integers su h that nN /N → m and nN /N → m as N → ∞ we have N p(N ) (nN /N, nN /N ) → q(m, m) and this onvergen e is uniform, • the oagulation term remains un hanged. T m Noti e that the fun tion q satises T 0 q(m, m) dm = 1 for m > 0 and the probabilisti kernel P (m, A) := A q(m, m) dm will des ribe the distribution of the size of the aggregates after fragmentation if the aggregate before fragmentation has size m. So the operator governing this N th approximation has the form (7) L(N ) Fg (N ν) = exp [hlog g, N νi]    (N ) L g + B (N ) g + Φ(N ) g , N ν + C(g, N ν) , × g with L(N ) equal to L(1) (at m = n/N instead of n) and with            n n n n−1 n+1 B (N ) g x, = n λm g x, + λb g x, N N N N N        n n n − λm + λb g x, , N N N   n Φ(N ) g x, N  X         n n n n−n n (N ) n n = λf g x, g x, p , − g x, N N N N N N n=1     n n 1 − g x, . + λd N N The sequen e of res aled pro esses onverges weakly to some measurevalued sto hasti pro ess (governed also by a martingale problem), but it turns out that the limit pro ess des ribes a deterministi behaviour. Theorem 1. Let w ν (N ) (0) → ν0 . The sequen e of pro esses verges weakly in distribution to the deterministi ν (N ) on- measure-valued pro ess 182 R. Rudni ki and R. Wie zorek uniquely determined by the equation t \ (8) hh, ν(t)i − hh, ν0 i = [h(L + B + Φ)h, ν(s)i + C(h, ν(s))] ds for all h ∈ Cb2 , with 0 (9) Lh(x, m) = D(m)∆x h(x, m), (10) Bh(x, m) = m(λb (m) − λm (m)) (11) ∂ h(x, m), ∂m  h m \ Φh(x, m) = λf (m) 2 h(x, m)q(m, m) dm − h(x, m) 0 − λd (m)h(x, m), (12) C(h, ν) = \\\\c(m)c(m)v(x − x) TT c(m) ν(dx dm) × (h(x + x/2, m + m) − h(x, m) + h(x, m)) × ν(dx dm) ν(dx dm). The limit pro ess ν has values in the spa e M. 5. Proof of Theorem 1. The s heme of the proof is as follows. Firstly we dene a new operator L (see (16)). Next we prove that if the pro ess {ν(t)}t≥0 solves the (L, ν0 )-martingale problem then it is a deterministi evolution of measure given by (8); moreover it is unique (i.e. there exists at most one solution of the (L, ν0 )-martingale problem). Then we he k that the sequen e ν (N ) onverges to the solution of this problem. In the proof we will use the following auxiliary theorem of Kurtz and Ethier: Proposition 2 ([17, Corollary 8.16, Chapter 4℄). Let (E, r) be a Polish spa e , A ⊂ Cb (E) × Cb (E) be an operator (possibly multivalued ), and P0 be a probability measure on E . Suppose that the martingale problem for (A, P0 ) has at most one solution. For N = 1, 2, . . . , suppose that YN is a progressive Markov pro ess in a Polish spa e EN orresponding to a measurable ontra tion semigroup with generator AN and ηN : EN → E is Borel measurable. Let XN = ηN ◦ YN . Assume that : the distribution of XN (0) onverges weakly to P0 as N → ∞, XN satises the ompa t ontainment ondition , and the losure of the linear span of D(A) ontains an algebra that separates points. If , moreover , for all (f, g) ∈ A and T > 0 there exist sequen es of fun tions (fN , gN ) ∈ AN and sets ΓN ⊂ EN su h that : (i) lim Prob({YN (t) ∈ ΓN , 0 ≤ t ≤ T }) = 1, N →∞ (ii) sup kfN k < ∞, N 183 Fragmentation-Coagulation Models of Phytoplankton (iii) lim sup |f ◦ ηN (y) − fN (y)| = lim sup |g ◦ ηN (y) − gN (y)| = 0, N →∞ y∈ΓN N →∞ y∈ΓN then there exists a solution X of the X. onverges weakly in distribution to (A, P0 ) martingale problem Here k · k is the maximum XN norm in and Cb (E). Remark every ε>0 2. By the T >0 and ompa t there is a ontainment ompa t set ondition we mean that for Γε,T su h that inf Prob{XN (t) ∈ Γε,T , 0 ≤ t ≤ T } ≥ 1 − ε. N Moreover we need some lemmas: Lemma lem (where 1. If the pro ess L(N ) {ν(t)}t≥0 (L(N ) , δν0 )-martingale prob- is given by (7)) then Prob( sup h1, ν(t)i ≥ a) ≤ (13) solves the 0≤t≤T h1, ν0 i exp[T (kλf − λd k + kcvk)]. a The proof is based on that of Lemma 4.1 in [17, Chapter 9℄. Although it requires some al ulation, it is not very interesting, so we omit it here. Lemma 2. The operator L+B +Φ generates a strongly ontinuous semi- C0 (Rd × R+ ). group on Proof. The operator L + B generates a strongly ontinuous semigroup on C0 (Rd × R+ ) ( f. [28℄) and Φ is a bounded operator on Cb (Rd × R+ ), so the Phillips perturbation theorem [12℄ gives the result. Let us write b Cν(A) = C(h, ν) as b , hh, Cνi \\\\c(m)c(m)v(x − x) TT where c(m) ν(dx dm) × [1A ((x + x)/2, m + m) − 1A (x, m) − 1A (x, m)] ν(dx dm) ν(dx dm). One an prove that for every measure Lemma 3. Let h ∈ C(Rd × R+ ) ν∈M we have be su h that b ∈ M. Cν khk ≤ 1 and let ν, µ ∈ M. Then (14) Here kνkTV [17℄). Proof. Set b − Cµi| b |hh, Cν ≤ kµ − νkTV . denotes the total variation norm of the measure TT α(ν) = c(m) ν(dx dm) and   x+x b h=h , m + m − h(x, m) − h(x, m). 2 ν ( f. e.g. 184 R. Rudni ki and R. Wie zorek Fix µ ∈ M \ {0} and let ε = kck−1 α(µ). Then α(ν) ≤ 2α(µ) for kν − µkTV ≤ ε. Moreover b − Cµi| b |hh, Cν \\\ (α(µ) − α(ν)) \ c(m)c(m)v(x − x)b h ν(dx dm) ν(dx dm) = α(ν)α(µ) \\\ 1 \ c(m)c(m)v(x − x)b h (ν + µ)(dx dm) (ν − µ)(dx dm) + α(µ) |α(µ) − α(ν)| α(ν)α(ν) ≤ 3kvk α(ν)α(µ) \ 3kvk kckα(µ + ν) \ fh,µ,ν (x, m) (ν − µ)(dx dm), + α(µ) where fh,µ,ν (x, m) = \\c(m)c(m)v(x − x) 3kvk kckα(µ + ν) b h(ν + µ)(dx dm). T Noti e that fh,µ,ν is bounded by 1. Therefore fh,µ,ν d(µ − ν) ≤ kµ − νkTV . Thus, going on with the above al ulations, for kν − µkTV ≤ ε we have (15) 2α(µ) b − Cµi| b |hh, Cν ≤ 3kvk kck kν − µkTV α(µ) 3α(µ) + 3kvk kck kν − µkTV α(µ) ≤ 15kvk kck kν − µkTV . Let us now take arbitrary measures µ, ν ∈ M \ {0}. Let µt = (1 − t)µ + tν and ε = kck−1 inf 0≤t≤1 α(µt ). Choose an n su h that kν − µkTV /n < ε. From inequality (15) it follows that b i/n − Cµ b (i−1)/n i| ≤ 15kvk kck kµi/n − µ(i−1)/n kTV |hh, Cµ for i = 1, . . . , n. Therefore b − Cµi| b |hh, Cν ≤ ≤ n X i=1 n X b i/n − Cµ b (i−1)/n i| |hh, Cµ 15kvk kck kµi/n − µ(i−1)/n kTV i=1 ≤ 15kvk kck kν − µkTV . Proof of Theorem 1. (16) Dene the operator L[exp[−hh, νi]] = exp[−hh, νi][h−Lh − Bh − Φh, νi + C(h, ν)] with L, B , Φ and C given by (9)(12) in Se tion 4. 185 Fragmentation-Coagulation Models of Phytoplankton Assume that {ν(t)}t≥0 solves the (L, ν0 )-martingale problem. This means that (17) \ t h i E e−hh,ν(t)i − e−hh,ν(s)i − L[e−hh,ν(r)i ] dr Fs = 0 s for all h ∈ Cb2 . Take h = θh and dierentiate with respe t to θ; setting now θ = 0 we get \ h i t E hh, ν(s)i − hh, ν(t)i + [h(L + B + Φ)h, ν(r)i + C(h, ν(r))] dr Fs = 0. s That means that (18) hh, ν(t)i = hh, ν0 i \ t + [h(L + B + Φ)h, ν(s)i + C(h, ν(s))] ds + M (t), 0 where M (t) is a Pν0 -martingale. From the It formula (see e.g. [19℄) we have \ t e−hh,ν(t)i − e−hh,ν0 i − e−hh,ν(s)i [hLh + Bh + Φh, ν(r)i + C(h, ν(r))] dr \ 0 t = e 0 −hh,ν(s)i \ t 1 −hh,ν(s)i dM (s) + e dhM i(s), 2 0 where hM i is the quadrati variational pro ess of M . We know that the left hand side is a martingale with mean value 0 and the rst integral on the right hand side has the same property. Therefore the integral \ t e−hh,ν(s)i dhM i(s) 0 is also a martingale with mean value 0. But it is the integral of a nonnegative, nontrivial fun tion with respe t to a quadrati variational pro ess, whi h is in reasing. Thus, sin e its mean value is 0, we know that hM i(s) = 0. This means that also M (s) = 0. Therefore (18) implies that ν(t) satises (8) for all h ∈ Cb2 . Now we prove that this solution is unique. Assume that the (nonrandom) right ontinuous family {ν(t)} of measures satises (8) and ν(0) = ν0 . It follows from (8) that hh, ν(t)i is dierentiable as a fun tion of time, therefore this equation an be rewritten as (19) ∀h∈C 2 b d b hh, ν(t)i = h(L + B + Φ)h, ν(s)i + hh, Cν(s)i. dt 186 R. Rudni ki and R. Wie zorek Fix h0 ∈ C02 su h that khk ≤ 1. By Lemma 2 the evolution equation (20)   dh = (L + B + Φ)h, dt  h(0) = h0 , has a unique solution. Noti e that this solution satises kh(t)k ≤ kh0 keat ≤ eat for some a > 0 that is independent of h0 . Sin e h0 ∈ D(L + B + Φ) ⊂ C02 , we also have h(t) ∈ C02 for all t > 0. Thus for any ν ∈ M we an write (21) d hh(t), νi = h(L + B + Φ)h(t), νi. dt Using (19) and (21) we an write ∂ b hh(t − s), ν(s)i = hh(t − s), Cν(s)i. ∂s Integrating both sides of (22) with respe t to s we get (22) \ t (23) b hh0 , ν(t)i = hh(t), ν0 i − hh(t − s), Cν(s)i ds. 0 Now assume that {ν(t)}t≥0 and {µ(t)}t≥0 satisfy (8) with the same initial ondition ν(0) = µ(0) = ν0 . Then, using the above al ulations and Lemma 3, we have \ t b b − Cν(s)i ds hh0 , ν(t) − µ(t)i = hh(t − s), Cµ(s) 0 \ t =e at b b hh(t − s)e−at , Cµ(s) − Cν(s)i ds \ 0 t ≤ eat kµ(s) − ν(s)kTV ds. 0 Re all that this is valid for any h0 ∈ C02 . Hen e \ t (24) kν(t) − µ(t)kTV ≤ eat kµ(s) − ν(s)kTV ds, 0 and from the Gromwall inequality µ(t) = ν(t) for all t ≥ 0. Our aim now is to prove that the sequen e of the pro esses ν (N ) onverges to a solution of the (L, ν0 )-martingale problem. In order to do it, we he k the assumptions of Proposition 2. We have already he ked that this martingale problem has at most one solution. To prove the ompa t ontainment of the sequen e we will hange our spa e a little: namely we repla e Rd × R+ by its ompa ti ation Eb = (Rd × R+ ) ∪ {∞} and so the pro esses XN take values c = M(E) b of all nite Borel measures on the ompa ti ation in the spa e M Fragmentation-Coagulation Models of Phytoplankton 187 of Rd × R+ . Observe how our situation ts into the frame of Proposition 2: c and we an onsider EN = NN as subsets of E b so that In our ase E = M (N ) XN = µ oin ides with YN and ηN is just the identity. For the ompa t ontainment ondition we use the fa t that the set {µ : h1, µi ≤ M } is ompa t in Eb . So by Lemma 1, Prob(νN (t) ∈ {µ : h1, µi ≤ M } for 0 ≤ t ≤ T ) h1, ν0 i T (kλf −λd k+kcvk) e , ≥1− a whi h proves the ompa t ontainment. The family of fun tions {e−hh,νi : h ∈ Cb2 } is ri h enough to separate c. Fix h ∈ Cb,pos . We now onstru t fun tions FN su h that FN points in M onverges to exp[−hh, νi] and L(N ) FN onverges to L exp[−hh, νi]. Namely, let FN (ν) = exp[hN log(1 − h/N ), νi] (for N su iently large 1 − h/N > 0) and ΓN = NN . Obviously XN (t) ∈ ΓN for all t ≥ 0 and FN are uniformly bounded. Then sup |FN (ν) − exp[−hh, νi]| ν∈NN ≤ sup exp[−h1, νi inf h]|hh + N log(1 − h/N ), νi| ν∈NN ≤ sup exp[−h1, νi inf h]h1, νikh − log(1 − h/N )−N k ν∈NN N →∞ ≤ sup Ckh − log(1 − h/N )−N k −−−→ 0, ν∈NN where C is some onstant. Similar al ulations show that (25) N →∞ sup |L(N ) FN (ν) − L[exp(−hh, νi)]| −−−→ 0, ν∈NN whi h ompletes the proof. Consider the solution ν(t) of (8) and assume that it is absolutely ontinuous with respe t to the Lebesgue measure, i.e. ν(t)(dx dm) = u(t, x, m) dx dm. 6. Equation on densities. Remark 3. Sin e (8) implies uniqueness ( f. proof of Theorem 1) and Theorem 2 will give us an absolutely ontinuous solution for any initial density, it su es to assume that the initial measure ν0 in (8) is absolutely ontinuous. Then by simple al ulations one an he k that (8) is the mild version of the equation (26) ∂u(t, x, m) = L∗ u(t, x, m) + B ∗ u(t, x, m) + Φ∗ u(t, x, m) + Cu(t, x, m), ∂t 188 R. Rudni ki and R. Wie zorek where (27) (28) (29) L∗ f (x, m) = D(m)∆x f (x, m), ∂ B ∗ f (x, m) = [m(λm (m) − λb (m))f (x, m)], ∂m h ∞ i \ Φ∗ f (x, m) = λf (m) 2 f (x, m)q(m, m) dm − f (x, m) m − λd (m)f (x, m), (30) \m\ d c(m − m)c(m)v(2(x − x)) TT 2 Cf (x, m) = c(m)f (x, m) dx dm Rd 0 × f (2x − x, m − m)f (x, m) dm dx − \∞\ Rd 0 c(m)c(m)v(x − x) 2 TT c(m)f (x, m) dx dm × f (x, m)f (x, m) dm dx, where ∆ is the Lapla e operator with respe t to the spatial variable x. 2 D(m) > 0 c(m) > 0 m ≥ 0 u ∈ u(t, x, m) ∈ L (R × R ) (26) L (R × R ) x Theorem 1 + d + su h that . Let and for . For any 1 + there exists a unique solution d + 0 of u(0, x, m) = u0 (x, m). The operator L + B generates a strongly ontinuous Markov semigroup of linear operators on X = L (R × R ), whi h an be written in the form (31) P (t)ϕ(x, m)  \ ∂  π m dx for λ(m) > 0, κ (π m, m, x, x)ϕ(x, π m)    ∂m      \ κ (t; m; x, x)ϕ(x, m) dx for λ(m) = 0, = ∗ Proof. ∗ 1 + d + −t −t −t Rd 0          Rd \− for λ(m) < 0. Noti e that, be ause λ is ontinuous, we an divide the half-line R into intervals where λ < 0 or λ > 0 and pla essingle points or intervalswhere λ = 0. The term π m is the solution of κ (π−t m, m, x, x)ϕ(x, π−t m) Rd ∂ π−t m dx ∂m + t d πt m = λ(πt m) dt with π m = m and λ(m) = m[λ 0 m (m) − λb (m)] . The fun tions κ +/0/− are 189 Fragmentation-Coagulation Models of Phytoplankton dened by 1 + κ (τ0 , τ1 , x, x) = q 4π Tτ1 D(τ ) τ0 λ(τ ) d dτ  exp − |x − x|2 4 Tτ1 D(τ ) τ0 λ(τ ) dτ  ,   |x − x|2 exp − , κ (t; m; x, x) = p d 4D(m)t 4πD(m) t   1 |x − x|2 − κ (τ0 , τ1 , x, x) = q Tτ0 D(τ ) d exp − 4 Tτ0 D(τ ) dτ . τ1 −λ(τ ) 4π τ1 −λ(τ ) dτ 0 1 The terms κ+ , κ0 , and κ− have the following natural interpretation. Fun tions κ+ and κ− are fundamental solutions of the non-autonomous, respe tively, forward and ba kward heat equation λ(τ ) ∂ u(τ, x) = D(τ )∆u(τ, x), ∂τ whereas κ0 is the fundamental solution of the autonomous heat equation with onstant diusion D(m). Sin e λf , λd and p are bounded, Φ∗ is a bounded linear operator on X . Thus, by the Phillips perturbation theorem, the operator L∗ + B ∗ + Φ∗ generates a strongly ontinuous semigroup of bounded positive operators on X . One an prove that the operator C is Lips hitzian on X+ = L1+ (Rd ×R+ ). This proof is based on that of Theorem 1 in [4℄ and is similar to the proof of Lemma 3. The rest of the proof of the existen e of the semigroup generated by (26) is a simple onsequen e of the method of variation of parameters (see e.g. [27℄). 4. We should underline that (26) is a fragmentation- oagulation equation, whi h, a ording to Theorem 2, has a unique solution that exists for all positive time. This feature distinguishes (26) from physi al oagulation equations whi h do not have global solutions. This surprising property of (26) results from the spe ial form of the oagulation term (30), whi h is homogeneous with respe t to f . 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Ryszard Rudni ki Institute of Mathemati s Polish A ademy of S ien es Bankowa 14 40-007 Katowi e, Poland and Institute of Mathemati s Silesian University Bankowa 14 40-007 Katowi e, Poland E-mail: rudni kius.edu.pl Radosªaw Wie zorek Institute of Mathemati s Polish A ademy of S ien es Bankowa 14 40-007 Katowi e, Poland E-mail: R.Wie zorekimpan.gov.pl Re eived May 5, 2006; re eived in nal form July 11, 2006 (7523)