Individual based model with competition in spatial ecology

INDIVIDUAL BASED MODEL WITH COMPETITION IN SP A TIAL EC OLOGY ∗ Dmitri Fink elsh tein † Y uri Kondratiev ‡ Oleksandr Kutov iy § No v em b er 6, 2018 Abstract W e analyze an intera cting p article system with a Marko v evolution of birth-and-d eath type. W e hav e shown that a local comp etition mec hanism (realized via a density depend ent mortalit y) leads to a globally regular b ehavior of th e p opulation in course of the stochastic evolution. Key wo rds. Con tinuous systems, spatial birth-and-death pro cesses, correlation functions, ind ividual based mo dels, spatial p lant ecology AMS sub ject classification. 60J80; 60K35; 82C21; 82C22 1 In tro duction Complex systems th eory is a q uickly growi ng interdisciplinary area with a very broad sp ectru m of motiv ations and applications. Having in mind biological applications, S. Levin (see [2 6 ]) c haracterized complex adaptive systems by such prop erties as di- versi ty and individu alit y of components, localized interactions among comp onents, and the outcomes of in teractions used for replication or enh an cement of components. In the stu dy of th ese sy stems, proper language and tec hniqu es are delivered by th e inter- acting p article models which form a ric h and p ow erful d irection in mo dern sto chastic and infinite dimensional analysis. Interacting particle systems have a wide use as mod els in condensed matter p hysics, chemical kinetics, p op u lation biology , ecology (individual b ased mo dels), so ciology and economics (agent b ased mo dels). In th is pap er w e consider an individual based mod el (IBM) in spatial ecology introduced by Bolker and Pa cala [4, 5], D iec kmann and La w [6] (BDLP mo del). A p opulation in this mo del is represented by a configuration of motionless organisms ∗ The financial supp ort of DFG through the SFB 701 (Bielef el d Universit y) and German- Ukrainian Pro ject 436 UKR 113/80 and 436 UKR 113/94 i s gratefully ac knowledged. This wo rk was partially supported b y FCT, POCI2010, FEDER. Y u.K. is very thankful to R. Law for fruitful and stim ulating discussions. † Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, Ukraine ( fdl@ima th.kiev.ua ) . ‡ F akult¨ at f ¨ ur Mat hematik, Unive rsi t¨ at Bielefeld, 33615 Bielefeld, Germany ( kondrat @math.uni-b ielefeld.de ); Reading Unive rsity . § F akult¨ at f ¨ ur Mat hematik, Unive rsi t¨ at Bielefeld, 33615 Bielefeld, Germany ( kutoviy @math.uni-b ielefeld.de ). 1 2 D. Finkelsht ein, Y u. Kondratiev, O. Kuto viy (plants) located in an infinite habit (an Euclidean space in our considerations). The habit is considered to b e a continuous space as opp osed to discrete spatial lattices used in the most of mathematical mo dels of interacting particle systems. W e need the infinite habit to av oid b ound ary effects in the p opulation evol ution and t he latter moment is q uite similar to the necessit y to work in the th ermody namic limit for models of statis tical physics. Let us also mention a recen t paper [2] in which a modification of the BDLP mo del for th e case of moving organisms (e.g., branching diffusion of the plankton) w as considered. A general IBM in th e plant ecology is a stochastic Marko v pro cess in th e configu- ration space with events comprising birth and death of the configuration p oints, i.e., w e are d ealing with a birth-and-death pro cess in the contin uum. In the particular case of th e BDLP model, eac h plant p rod uces seeds indep endently of oth ers and then these seeds are distributed in th e space accordingly to a disp ersion k ernel a + . This part of the process ma y b e considered as a k ind of the spatial branching. I n th e same time, the mo del includes also a mortalit y mechanism. The mortality intensit y consists of tw o parts. The first one corresp onds to a constant intrinsic mortalit y va lue m > 0 s.t. any plant d ies indep endently of others after a random time (exp onentially distributed with p arameter m ). The second part in the mortality rate is density dep end ent. The latter is expressed in terms of a competition kernel a − whic h describes an add itional mortalit y rate for any giv en p oint of the configuration coming from the rest of the p opulation, see Section 3 for th e precise d escription of the mo del, in particular, (3.6). The latter form ula gives the heuristic form of the Marko v generator in the BDLP mod el. Assuming the existence of th e corresponding Marko v process, we deriv e in Section 5 an evolution equation for correlation fun ctions k ( n ) t , n ≥ 1 , of th e considered mod el. In [4, 5], [6] this system was called the system of sp atial moment equations for plant compet ition and, actually , this system itself was taking as a definition of the dynamics in th e BDLP model. The mathematical structure of the correlation functions evolution equation is close to oth er w ell-know n h ierarchical sy stems in mathematical physics, e.g., BBGKY hierarc hy for the Hamiltonian dynamics (see, e.g. [3 ]) or the diffusion hierarc hy for th e gradient stochastic dynamic in th e con tinuum (see e.g. [21]). As in all hierarchica l chains of equations, w e can not ex p ect the explicit form of t h e solution, and even more, the existence problem for these equ ations is a highly delicate question. There is an appro ximative approac h to pro duce an informati on ab out the b eh a vior of the solutions to the h ierarc hical chains. This approach is called the closure p rocedu re and consists of the follo wing steps. The first step is to cut all correlation functions of the higher orders and th e second one is to subscribe t he rest correlation functions by the prop erly factorized correlation functions of the low er orders. As result, one obtains a finite sy stem of non-linear equations instead of th e original linear but infinite sy stem of a hierarchical typ e. This closure pro cedure is essentia lly n on-unique, see [7]. The aim of this paper is to study the moment eq uations for the BD LP mod el by metho d s of functional analysis and analysis on the configuration spaces dev elop ed in [13], [14], [15] and already applied to th e non-equilibrium birth-and - death t yp e conti nu- ous space sto chastic dynamics in [16], [18]. W e obtain some rigoro us results concerning the existence and prop erties of the solution for different classe s of initial conditions. One of t he main qu estion we clarify in t h e pap er concerns the role of th e comp etition mec hanism in the regulation of the spatial stru ct u re of an evolving p op u lation. More precisely , considering the mo del without comp etition, i.e. , the case a − ≡ 0, we arrive in the situation of the so-called continuous contact model [9], [17], [22 ]. In the ecolog- ical framew ork, th is mo del describ es free gro wth of a plan t p opulation with the given IBM Mo del with Co mpe tition in Spatial E cology 3 constant mortalit y . W e note th at (ind ep endently on the v alue of the mortality m > 0) the considered contac t model exhibits ve ry strong clustering that is reflected in the b ound (3.5) on t he correlation functions at any moment of time t > 0. Note that this effect on the lev el of the computer simulation w as disco vered already in [2] and n ow it has t h e rigorous mathematical fo rmulation and clarification. A d irect consequence of the comp etition in th e mo del is the suppression of such clustering. Namely , assuming the strong enough comp etition and th e big intrinsic mortali ty m , w e prov e the sub- P oissonian b ou n d for the solution to the moment equations provided such b ound was true for the initial state. Moreo ver, we clarify specific influences of th e constant and the densit y dep endent mortality in tensities separately . More precisely , th e big enough intrinsi c m ortality m gives a uniform in time b ound for each correlation funct ion and the strong comp etition results ensure the regular spatial distribut ion of the typical configuration for any momen t of time th at is reflected in th e sub-Po issonian b oun d . Join t influence of the in trinsic mortality an d the competition leads to the existence of th e unique inv arian t measure for our mo del which is just Dirac measure concen- trated on the empty confi gu ration. The latter means that the corresponding sto chasti c evol ution of the p opulation is asymp t otically exhausting. W e w ould like to mention also the work [10] in which the BDLP mo del was studied in the case of the b ounded habit in t he stochastic analysis framew ork. The latter case differs essen tially from the mo del we consider in the present pap er as well as main problems studied in [10], which are related to the scaling limits for th e considered processes. 2 General facts and n otations Let B ( R d ) b e the family of all Borel sets in R d . B b ( R d ) den otes th e system of all b ounded sets in B ( R d ). The sp ace of n -p oin t configuration is Γ ( n ) 0 = Γ ( n ) 0 , R d := n η ⊂ R d ˛ ˛ ˛ | η | = n o , n ∈ N 0 := N ∪ { 0 } , where | A | den otes the cardinality of the set A . The space Γ ( n ) Λ := Γ ( n ) 0 , Λ for Λ ∈ B b ( R d ) is defined analogously to the space Γ ( n ) 0 . As a set, Γ ( n ) 0 is equiv alent to the symmetrizatio n of ^ ( R d ) n = n ( x 1 , . . . , x n ) ∈ ( R d ) n ˛ ˛ ˛ x k 6 = x l if k 6 = l o , i.e. ^ ( R d ) n /S n , where S n is the p ermutation group ov er { 1 , . . . , n } . Hen ce one can in- trod uce the corresponding top ology and Borel σ - algebra, whic h w e den ote by O ( Γ ( n ) 0 ) and B (Γ ( n ) 0 ), resp ectively . Also one can define a measure m ( n ) as an image of the prod uct of Leb esgue measures dm ( x ) = dx on ` R d , B ( R d ) ´ . The sp ace of fi nite configurations Γ 0 := G n ∈ N 0 Γ ( n ) 0 is equipp ed with the top ology whic h has structure of d isjoin t u n ion. Theref ore, one can defin e the corresp onding Borel σ -algebra B (Γ 0 ). 4 D. Finkelsht ein, Y u. Kondratiev, O. Kuto viy A set B ∈ B (Γ 0 ) is called b ounded if th ere exists Λ ∈ B b ( R d ) and N ∈ N such that B ⊂ F N n =0 Γ ( n ) Λ . The Leb esgue—Pois son measure λ z on Γ 0 is defi ned as λ z := ∞ X n =0 z n n ! m ( n ) . Here z > 0 is the so called activity parameter. The restriction of λ z to Γ Λ will b e also denoted by λ z . The confi guration space Γ := n γ ⊂ R d ˛ ˛ ˛ | γ ∩ Λ | < ∞ , for all Λ ∈ B b ( R d ) o is equipp ed with the v ague top ology . It is a P olish space (see e.g. [15]). The cor- respond ing Borel σ -algebra B (Γ) is defined as the smallest σ - algebra for which all mappings N Λ : Γ → N 0 , N Λ ( γ ) := | γ ∩ Λ | are measurable, i.e., B (Γ) = σ “ N Λ ˛ ˛ ˛ Λ ∈ B b ( R d ) ” . One can also show that Γ is the pro jective limit of the spaces { Γ Λ } Λ ∈B b ( R d ) w.r.t. the pro jections p Λ : Γ → Γ Λ , p Λ ( γ ) := γ Λ , Λ ∈ B b ( R d ). The Poisson measure π z on (Γ , B ( Γ)) is given as t he pro jective limit of the family of measures { π Λ z } Λ ∈B b ( R d ) , where π Λ z is t he measure on Γ Λ defined by π Λ z := e − z m (Λ) λ z . W e will use th e follo wing classe s of functions: L 0 ls (Γ 0 ) is th e set of all meas urable functions on Γ 0 whic h ha ve a local supp ort, i.e. G ∈ L 0 ls (Γ 0 ) if there exists Λ ∈ B b ( R d ) such th at G ↾ Γ 0 \ Γ Λ = 0; B bs (Γ 0 ) is the set of b oun ded measurable functions with b ounded supp ort, i.e. G ↾ Γ 0 \ B = 0 for some b ounded B ∈ B (Γ 0 ). On Γ w e consider the set of cy linder fun ct ions F L 0 (Γ), i.e. th e set of all measurable functions G on ` Γ , B (Γ) ) ´ whic h are measurable w.r.t. B Λ (Γ) for some Λ ∈ B b ( R d ). These functions are characterized by th e follo wing relation: F ( γ ) = F ↾ Γ Λ ( γ Λ ). The follo wing mapping b etw een functions on Γ 0 , e.g. L 0 ls (Γ 0 ), and funct ions on Γ, e.g. F L 0 (Γ), plays the key role in our further considerations: K G ( γ ) := X η ⋐ γ G ( η ) , γ ∈ Γ , (2.1) where G ∈ L 0 ls (Γ 0 ), see e.g. [13, 24, 25]. The summation in t h e latter exp ression is taken o ver all finite sub configurations of γ , whic h is denoted by t h e symb ol η ⋐ γ . The map p ing K is linear, p ositivit y preserving, and inv ertible, with K − 1 F ( η ) := X ξ ⊂ η ( − 1) | η \ ξ | F ( ξ ) , η ∈ Γ 0 . (2.2) Let M 1 fm (Γ) b e the set of all probability measures µ on ` Γ , B (Γ) ´ whic h ha ve finite local moments of all orders, i.e. R Γ | γ Λ | n µ ( dγ ) < + ∞ for all Λ ∈ B b ( R d ) and n ∈ N 0 . A measure ρ on ` Γ 0 , B (Γ 0 ) ´ is called locally fi nite iff ρ ( A ) < ∞ for all b ound ed sets A from B (Γ 0 ). The set of such measures is denoted by M lf (Γ 0 ). One can define a transform K ∗ : M 1 fm (Γ) → M lf (Γ 0 ) , whic h is dual to the K - transform, i.e., for every µ ∈ M 1 fm (Γ), G ∈ B bs (Γ 0 ) w e hav e Z Γ K G ( γ ) µ ( dγ ) = Z Γ 0 G ( η ) ( K ∗ µ )( dη ) . IBM Mo del with Co mpe tition in Spatial E cology 5 The measure ρ µ := K ∗ µ is called th e correlation measure of µ . As show n in [13] for µ ∈ M 1 fm (Γ) and any G ∈ L 1 (Γ 0 , ρ µ ) the series (2.1) is µ -a.s. absolutely converge nt. F urthermore, K G ∈ L 1 (Γ , µ ) and Z Γ 0 G ( η ) ρ µ ( dη ) = Z Γ ( K G )( γ ) µ ( dγ ) . (2.3) A measure µ ∈ M 1 fm (Γ) is called locally absolutely continuous w.r.t. π z iff µ Λ := µ ◦ p − 1 Λ is absolutely con tinuous with resp ect to π Λ z for all Λ ∈ B Λ ( R d ). In th is case ρ µ := K ∗ µ is absolutely contin uous w.r.t λ z . W e d enote k µ ( η ) := dρ µ dλ z ( η ) , η ∈ Γ 0 . The functions k ( n ) µ : ( R d ) n − → R + (2.4) k ( n ) µ ( x 1 , . . . , x n ) := ( k µ ( { x 1 , . . . , x n } ) , if ( x 1 , . . . , x n ) ∈ ^ ( R d ) n 0 , otherwis e are th e correlation functions well known in statistical physics, see e.g [28], [29 ]. W e recall n o w the so-called Minlos lemma which plays very imp ortant role in our calculations (cf., [20]). Lemma 2.1. L et n ∈ N , n ≥ 2 , and z > 0 b e given. Then Z Γ 0 . . . Z Γ 0 G ( η 1 ∪ . . . ∪ η n ) H ( η 1 , . . . , η n ) dλ z ( η 1 ) . . . dλ z ( η n ) = Z Γ 0 G ( η ) X ( η 1 ,...,η n ) ∈P n ( η ) H ( η 1 , . . . , η n ) dλ z ( η ) for al l me asur able functions G : Γ 0 7→ R and H : Γ 0 × . . . × Γ 0 7→ R with r esp e ct to which b oth sides of the e quality make sense. Her e P n ( η ) denotes the set of al l or der e d p artitions of η i n n p arts, which may b e empty. 3 Description of the mo del In the p resent paper we study the sp ecial case of th e general birth-and -death pro- cesses in contin uum. The spatial birth -and-death pro cesses describ e evolution of con- figurations in R d , in which p oints of confi gu rations (particles, individu als, elements) randomly app ear (b orn) and d isapp ear (die) in the space. Among all b irth-and-d eath processes w e will distinguish those in which new p articles app ear only from existing ones. These pro cesses corresp ond to the mod els of th e spatial ecology . The simplest ex ample of such processes is the so-called “free growth” dynamics. During th is sto chas tic evolution the p oints of configuration indep en d ently create new ones distributed in the space according to a dispersion probabilit y k ernel 0 ≤ a + ∈ L 1 ( R d ) whic h is an even function. A ny existing p oin t has an infinite life time, i. e. they do not die. Heuristically , the Mark ov pre-generator of this birth pro cess has the follo wing form: ( L + F )( γ ) = κ + X y ∈ γ Z R d a + ( x − y ) D + x F ( γ ) dx, 6 D. Finkelsht ein, Y u. Kondratiev, O. Kuto viy where D + x F ( γ ) = F ( γ ∪ x ) − F ( γ ) , and κ + > 0 is some p ositive constant. The existence of the pro cess associated with L + can b e shown using the same technique as in [9], [22]. Let µ t b e the corresp onding ev olution of measures in time on M 1 fm (Γ). By k ( n ) t , n ≥ 0 we denote th e dyn amics of th e corresp onding n -th order correlation functions (provided they exist). Note, that each of such functions describes the den sit y of the system at the moment t . Then, u sing (2.3), for any contin uous ϕ on R d with b ounded supp ort, we obtain d dt Z R d ϕ ( x ) k (1) t ( x ) dx = d dt Z Γ h ϕ, γ i dµ t ( γ ) = Z Γ L + h ϕ, γ i dµ t ( γ ) = κ + Z Γ h a + ∗ ϕ, γ i dµ t ( γ ) = κ + Z R d ( a + ∗ ϕ )( x ) k (1) t ( x ) dx = κ + Z R d ϕ ( x )( a + ∗ k (1) t )( x ) dx, where ∗ denotes the classical conv olution on R d . Hence, k (1) t gro ws exp onentially in t . In p articular, for the t ranslation inv ariant case one h as k (1) 0 ( x ) ≡ k (1) 0 > 0 and as a result k (1) t = e κ + t k (1) 0 . (3.1) One of the p ossibilities to prevent th e densit y growth of the system is to include the death mechanism. The simplest one is described by the indep endent death rate (mortalit y) m > 0. This means that any element of a p op u lation h as an ind ep endent exp onentially distributed with parameter m random life time. The indep end ent death together with the indep end ent creation of new particles b y already existing ones de- scribe the so-call ed c ontact mo del in the con tinuum, see e.g. [22]. The pre-generator of such mo del is given by the follo wing exp ression: ( L CM F )( γ ) = m X x ∈ γ D − x F ( γ ) + ( L + F )( γ ) = m X x ∈ γ D − x F ( γ ) + κ + X y ∈ γ Z R d a + ( x − y ) D + x F ( γ ) dx, where D − x F ( γ ) = F ( γ \ x ) − F ( γ ) . The Marko v pro cess asso ciated with the generator L CM w as constructed in [22]. This construction was generalized in [9] for more general classes of functions a + . Let us note, that th e contact model in the contin uum may b e used in the epidemiology to mod el th e infection spreading pro cess. The val ues of th is p rocess represent the states of the infected population. This is analog of the contact pro cess on a lattice. Of course, suc h interpretation is not in the spatial ecology concept. On the oth er hand, conta ct pro cess is a spatial branching pro cess with a given mortalit y rate. The dynamics of correlation functions in the contact mo del w as considered in [17]. Namely , taking m = 1 for correctness, we ha ve for any n ≥ 1, t > 0 the corre lation IBM Mo del with Co mpe tition in Spatial E cology 7 function of n -th order has the follo wing form k ( n ) t ( x 1 , . . . , x n ) = e n ( κ + − 1) t " n O i =1 e tL i a + # k ( n ) 0 ( x 1 , . . . , x n ) (3.2) + κ + Z t 0 e n ( κ + − 1)( t − s ) " n O i =1 e ( t − s ) L i a + # × n X i =1 k ( n − 1) s ( x 1 , . . . , ˇ x i , . . . , x n ) X j : j 6 = i a + ( x i − x j ) ds, where L i a + k ( n ) ( x 1 , . . . , x n ) = κ + Z R d a + ( x i − y ) h k ( n ) ( x 1 , . . . , x i − 1 , y , x i +1 , . . . , x n ) − k ( n ) ( x 1 , . . . , x n ) i dy and the symbol ˇ x i means t hat the i -th co ordinate is omitted. N ote that L i a + is a Mark ov generator and the corresponding semigroup (in L ∞ space) preserves p ositivity . It was also sho wn in [17], th at if there exists a constant C > 0 (indep end ent of n ) such that for any n ≥ 0 and ( x 1 , . . . , x n ) ∈ ( R d ) n k ( n ) 0 ( x 1 , . . . , x n ) ≤ n ! C n , then for an y t ≥ 0 and almost all (a.a.) ( x 1 , . . . , x n ) ∈ ( R d ) n (w.r.t. Lebesgue measure) the follo wing estimate holds for all n ≥ 0 k ( n ) t ( x 1 , . . . , x n ) ≤ κ + ( t ) n (1 + a 0 ) n e n ( κ + − 1) t ( C + t ) n n ! (3.3) Here a 0 = k a k L ∞ ( R d ) , κ + ( t ) := max h 1 , κ + , κ + e − ( κ + − 1) t i . F or the translation inv ariant case the val ue κ + = 1 is critical. Namely , from (3.2) w e deduce that k (1) t = e ( κ + − 1) t k (1) 0 . (3.4) Therefore, for κ + < 1 the den sit y will ex p onential ly decrease to 0 (as t → ∞ ), for κ + > 1 the density will ex p onentiall y increase to ∞ , and for κ + = 1 the density will b e a constan t. One can easily see from the estimate (3.3) that, in the case κ + < 1, the correlation functions of all orders decrease to 0 as t → ∞ . On the oth er hand , for fix ed t , th e estimate (3.3) implies factorial b ound in n for k ( n ) t . As result, we may exp ect the clustering of our system. T o sho w clustering we start from the Poi sson distribution of p articles and obtain an estimate from b elo w for the time evol utions of correlations b etw een particles in a small region. Hence, let κ + < 1, k ( n ) 0 = C n . Let B is some b ounded domain of R d such that α := inf x,y ∈ B a + ( x − y ) > 0 . Let β = min { α κ + , C } . F or any { x 1 , x 2 } ⊂ B , formula (3.2) implies k (2) t ( x 1 , x 2 ) ≥ 2 C κ + α Z t 0 e 2( κ + − 1)( t − s ) ds ≥ 2 β 2 te 2( κ + − 1) t . 8 D. Finkelsht ein, Y u. Kondratiev, O. Kuto viy W e consider t ≥ 1. One can prov e by induction that for any { x 1 , . . . , x n } ⊂ B , n ≥ 2 k ( n ) t ( x 1 , . . . , x n ) ≥ β n e n ( κ + − 1) t n ! (3.5) Indeed, for n = 2 this statement has b een prov ed. Su pp ose that ( 3.5 ) holds for n − 1. Then, by (3.2), one has k ( n ) t ( x 1 , . . . , x n ) ≥ κ + Z t 0 e n ( κ + − 1)( t − s ) nβ ( n − 1) e ( n − 1)( κ + − 1) s ( n − 1)!( n − 1) αds ≥ β n n ! e n ( κ + − 1) t Z t 0 e − ( κ + − 1) s ds ≥ β n e n ( κ + − 1) t n ! . As it w as mentioned b efore, th e later bound shows the clustering in th e contact model. All previous con sideration may be extend ed for the case m 6 = 1: w e should only replace 1 by m in the previous calculations. As a conclusion w e hav e: the presence of mortalit y ( m > κ + ) in the free grow th mod el preven ts the growth of density , i. e. the correla tion fun ctions of all orders deca y in time. But it doesn’t influ ence on the clustering in the system. One of the p ossibilities to preven t such clustering is to consider the so-called densit y d ep endent death rate. Namely , let us consider the follo wing p re-generator: ( LF )( γ ) = X x ∈ γ 2 4 m + κ − X y ∈ γ \ x a − ( x − y ) 3 5 D − x F ( γ ) (3.6) + κ + Z R d X y ∈ γ a + ( x − y ) D + x F ( γ ) dx. Here 0 ≤ a − ∈ L 1 ( R d ) is an arbitrary , even function such that Z R d a − ( x ) dx = 1 (in other wo rds, a − is a probability density) and κ − > 0 is some p ositive constant. It is easy to see that the op erator L is w ell-defined, for examp le, on F L 0 (Γ). The generator (3.6) describ es the Bolk er—Dieckmann—La w—Paca la ( BDL P) mo del, whic h w as in tro duced in [4, 5, 6]. During the correspondin g stochastic evolution the birth of individuals o ccurs in dep endently and the death is ruled not only b y th e global regulation (mortalit y) but also by the lo cal regulation with the kernel κ − a − . This regulation may b e describ ed as a competition (e.g., for resources) betw een individu als in the p op u lation. The main result of this article is presented in Section 5, Theorem 5.1. It ma y b e informally stated in the follo wing wa y: If the mortality m and the c omp etition kernel κ − a − ar e lar ge enough, then the dynamics of c orr elation functions asso ciate d with the pr e-gene r ator (3.6) pr eserves (sub-)Poisso nian b ound for c orr elation functions for al l times. In p articular, it prevents clustering in the mo del. In the next sections we explain h o w to pro ve this fact. In the last sectio n of the present paper we discuss the necessity to consider ”large enough” death. IBM Mo del with Co mpe tition in Spatial E cology 9 4 Semigroup for the sy m b ol of th e generator The problem of the construction of the corres p onding process in Γ concern s the p os- sibilit y t o construct the semigroup asso ciated with L . This semigroup determines the solution to the K olmogoro v equation, which formally (only in the sense of action of operator) has the follo wing form: dF t dt = LF t , F t ˛ ˛ t =0 = F 0 . (4.1) T o show that L is a generator of a semigroup in some reasonable fun ctional spaces on Γ seems to b e a difficult problem. This difficult y is hidden in the complex structure of the non-linear infinite dimensional space Γ. In v arious applications the evolution of the correspond ing correla tion functions (or measures) helps already to understand the behavior of th e process and gives candidates for inv arian t states. The evolution of correlation functions of the process is related heuristically t o the ev olution of states of our IPS. The latter evolutio n is formally given as a solution to the dual Kolmogoro v equ ation (F okker—Planck equation): dµ t dt = L ∗ µ t , µ t ˛ ˛ t =0 = µ 0 , (4.2) where L ∗ is the ad join t op erator to L on M 1 fm (Γ), provided, of course, th at it exists. In the recen t pap er [16], the auth ors prop osed th e analytic approac h for the con- struction of a non-equilibrium pro cess on Γ, which u ses deeply the harmonic analysis technique. In the present pap er w e follo w the sc heme prop osed in [16] in order to con- struct th e evolution of correlation functions. The existence problem for the evolution of states in M 1 fm (Γ) and, as a result, of the corresp onding pro cess on Γ is not realized in this pap er. It seems to b e a very technical question and remains op en. F ollowing the general scheme, first we should construct the evolution of functions whic h correspon d s to the symb ol ( K -image) ˆ L = K − 1 LK of the operator L in L 1 - space on Γ 0 w.r.t. the weigh ted Leb esgue—Poi sson measure. This weigh t is crucial for the correspondin g ev olution of correlation functions. It determines the growth of correlation functions in time and space. Belo w we start the detailed realization of the discussed scheme. Let us set for η ∈ Γ 0 E a # ( η ) := X x ∈ η X y ∈ η \ x a # ( x − y ) , where a # denotes either a − or a + . Proposition 4.1. The image of L under the K -tr ansform (or symb ol of the op er ator L ) on functions G ∈ B bs (Γ 0 ) has the fol lowi ng f orm ( b LG )( η ) := ( K − 1 LK G )( η ) = − “ m | η | + κ − E a − ( η ) ” G ( η ) − κ − X x ∈ η X y ∈ η \ x a − ( x − y ) G ( η \ y ) + κ + Z R d X y ∈ η a + ( x − y ) G (( η \ y ) ∪ x ) dx + κ + Z R d X y ∈ η a + ( x − y ) G ( η ∪ x ) dx. 10 D. Finkelsht ein, Y u. Kondratiev, O. Kuto viy F or th e pro of see [8]. With t h e help of Proposition 4.1, w e d erive the evolution equation for quasi- observables (functions on Γ 0 ) correspond ing t o the Kolmogoro v eq uation (4.1). It has t h e follo wing fo rm dG t dt = b LG t , G t ˛ ˛ t =0 = G 0 . (4.3) Then in the wa y analo gous to those in whic h the corresponding F okker-Planck equa- tion (4.2) w as determined for (4.1) w e get the evolution equation for the correl ation functions correspondin g to th e equation (4.3): dk t dt = b L ∗ k t , k t ˛ ˛ t =0 = k 0 . (4.4) The precise form of the adjoint op erator ˆ L ∗ will b e given in S ection 5. It is very imp ortant to emphasize that in the pap ers [4, 5] th e equation ( 4.4) w as obtained from quit heuristic arguments and, moreov er, it w as considered as the d efinition for the evol ution of the BDLP mo del. Let λ b e th e Leb esgue-Poisso n meas ure on Γ 0 with activity parameter equal to 1. F or arbitrary and fi x ed C > 0 we consider the op erator b L as a pre- generator of a semigroup in th e functional space L C := L 1 (Γ 0 , C | η | λ ( dη )) . (4.5) In t h is section, symbol k·k C stands for th e norm of the space (4.5). F or any ω > 0 we introduce th e set H ( ω , 0) of all densely defin ed closed op erators T on L C , th e resolven t set ρ ( T ) of which conta ins the sector Sect “ π 2 + ω ” := n ζ ∈ C ˛ ˛ ˛ | arg ζ | < π 2 + ω o , ω > 0 and for any ε > 0 || ( T − ζ 1 1) − 1 || ≤ M ε | ζ | , | arg ζ | ≤ π 2 + ω − ε, where M ε does n ot d ep end on ζ . Let H ( ω , θ ), θ ∈ R den otes the set of all op erators of the form T = T 0 + θ with T 0 ∈ H ( ω , 0). Remark 4.1. I t is wel l-known (se e e. g., [12]), that any T ∈ H ( ω , θ ) is a gener ator of a semigr oup U ( t ) which i s holomorphic i n the se ctor | arg t | < ω . The function U ( t ) is not ne c essarily uniformly b ounde d, but it i s quasi-b ounde d, i.e. || U ( t ) || ≤ const | e θt | in any se ctor of the form | arg t | ≤ ω − ε . Proposition 4.2. F or any C > 0 , m > 0 , and κ − > 0 the op er ator ( L 0 G )( η ) := − “ m | η | + κ − E a − ( η ) ” G ( η ) , D ( L 0 ) = n G ∈ L C ˛ ˛ ˛ “ m | η | + κ − E a − ( η ) ” G ( η ) ∈ L C o is a gener ator of a c ontr action semigr oup on L C . Mor e over, L 0 ∈ H ( ω , 0) f or al l ω ∈ (0 , π 2 ) . IBM Mo del with Co mpe tition in Spatial E cology 11 Pr o of. It is not difficult to sho w that L 0 is a densely defined and clo sed op erator in L C . Let 0 < ω < π 2 b e arbitrary and fi xed. Clear, that for all ζ ∈ S ect ` π 2 + ω ´ ˛ ˛ m | η | + κ − E a − ( η ) + ζ ˛ ˛ > 0 , η ∈ Γ 0 . Therefore, for any ζ ∈ Sect ` π 2 + ω ´ the inv erse operator ( L 0 − ζ 1 1) − 1 , the action of whic h is given by [( L 0 − ζ 1 1) − 1 G ]( η ) = − 1 m | η | + κ − E a − ( η ) + ζ G ( η ) , (4.6) is w ell defin ed on the whole space L C . Moreo ver, it is a b ounded op erator in th is space and || ( L 0 − ζ 1 1) − 1 || ≤ 8 < : 1 | ζ | , if R e ζ ≥ 0 , M | ζ | , if R e ζ < 0, (4.7) where th e constant M do es not dep en d on ζ . The case Re ζ ≥ 0 is a direct consequence of (4.6) and inequality m | η | + κ − E a − ( η ) + Re ζ ≥ Re ζ ≥ 0 . W e prov e n ow the b ound (4.7) in the case Re ζ < 0. Using (4.6), we have || ( L 0 − ζ 1 1) − 1 G || C = ‚ ‚ ‚ ‚ ‚ 1 ˛ ˛ m | · | + κ − E a − ( · ) + ζ ˛ ˛ G ( · ) ‚ ‚ ‚ ‚ ‚ C = = 1 | ζ | ‚ ‚ ‚ ‚ ‚ | ζ | ˛ ˛ m | · | + κ − E a − ( · ) + ζ ˛ ˛ G ( · ) ‚ ‚ ‚ ‚ ‚ C . Since ζ ∈ Sect ` π 2 + ω ´ , | Im ζ | ≥ | ζ | ˛ ˛ ˛ sin “ π 2 + ω ” ˛ ˛ ˛ = | ζ | cos ω . Hence, | ζ | ˛ ˛ m | η | + κ − E a − ( η ) + ζ ˛ ˛ ≤ | ζ | | Im ζ | ≤ 1 cos ω =: M and (4.7) is fulfilled. The rest of the statement of the lemma follow s directly from the t h eorem of H ille— Y osida (see e.g., [12]). W e define now ( L 1 G )( η ) := κ − X x ∈ η X y ∈ η \ x a − ( x − y ) G ( η \ y ) , G ∈ D ( L 1 ) := D ( L 0 ) . The lemma b elow implies that the op erator L 1 is well-defined. Lemma 4.3. F or any δ > 0 ther e exists C 0 := C 0 ( δ ) > 0 such that for al l C < C 0 the fol lowi ng estimate holds || L 1 G || C ≤ a || L 0 G || C , G ∈ D ( L 1 ) , (4.8) with a = a ( C ) < δ . 12 D. Finkelsht ein, Y u. Kondratiev, O. Kuto viy Pr o of. By mo d ulus prop erty || L 1 G || C = Z Γ 0 | L 1 G ( η ) | C | η | λ ( dη ) (4.9) can b e estimated by κ − Z Γ 0 X x ∈ η X y ∈ η \ x a − ( x − y ) | G ( η \ y ) | C | η | λ ( dη ) . (4.10) By Minlos lemma, (4.10 ) is equal to κ − Z Γ 0 Z R d X x ∈ η a − ( x − y ) | G ( η ) | C | η | +1 dy λ ( dη ) = = κ − Z Γ 0 C | η || G ( η ) | C | η | λ ( dη ) ≤ κ − m C || L 0 G || C . Therefore, (4.8) h olds with a = κ − C m . Clear, th at t aking C 0 = δ m κ − w e obtain that a < δ for C < C 0 . W e set n o w ( L 2 G )( η ) := ( L 2 , κ + G )( η ) = κ + Z R d X y ∈ η a + ( x − y ) G (( η \ y ) ∪ x ) d x , G ∈ D ( L 2 ) := D ( L 0 ) . The op erator ` L 2 , D ( L 2 ) ´ is we ll defined due to the lemma b elo w: Lemma 4.4. F or any δ > 0 the r e exists κ + 0 := κ + 0 ( δ ) > 0 such that for al l κ + < κ + 0 the fol lowi ng estimate holds || L 2 G || C ≤ a || L 0 G || C , G ∈ D ( L 2 ) , (4.11) with a = a ( κ + ) < δ . Pr o of. Analogously to the previous lemma we estimate || L 2 G || C = Z Γ 0 | L 2 G ( η ) | C | η | λ ( dη ) (4.12) by κ + Z Γ 0 Z R d X y ∈ η a + ( x − y ) | G (( η \ y ) ∪ x ) | C | η | dxλ ( dη ) . (4.13) By Minlos lemma, (4.13 ) is equal to κ + Z Γ 0 X x ∈ η Z R d a + ( x − y ) | G ( η ) | C | η | dy λ ( dη ) ≤ κ + m || L 0 G || C . T aking κ + 0 = δ m we prove the lemma. IBM Mo del with Co mpe tition in Spatial E cology 13 The op erator defined as: ( N G )( η ) = | η | G ( η ) , G ∈ D ( L 0 ) (4.14) is called the number op erator. Remark 4.2. W e pr ove d, in p articular, tha t for G ∈ D ( L 0 ) = D ( L 1 ) = D ( L 2 ) || L 1 G || C ≤ κ − C || N G || C , || L 2 G || C ≤ κ + || N G || C . Finally , we consider th e last part of the op erator b L : ( L 3 G )( η ) := κ + Z R d X y ∈ η a + ( x − y ) G ( η ∪ x ) dx, D ( L 3 ) := D ( L 0 ) . Lemma 4.5. F or any δ > 0 and any κ + > 0 , C > 0 such that κ + E a + ( η ) < δ C “ κ − E a − ( η ) + m | η | ” (4.15) the fol lowi ng estimate holds || L 3 G || C ≤ a || L 0 G || C , G ∈ D ( L 3 ) , (4.16) with a = a ( κ + , C ) < δ . Pr o of. Using the same tric ks as in the tw o prev ious lemmas we hav e || L 3 G || C = Z Γ 0 | L 3 G ( η ) | C | η | λ ( dη ) ≤ κ + Z Γ 0 Z R d X y ∈ η a + ( x − y ) | G ( η ∪ x ) | C | η | dxλ ( dη ) . (4.17) By Minlos lemma, (4.17) is equal to κ + C Z Γ 0 E a + ( η ) | G ( η ) | C | η | λ ( dη ) . The assertion of the lemma is now trivial. Theorem 4.6. Assume that the functions a − , a + and the c onstants κ − , κ + > 0 , m > 0 and C > 0 satisfy C κ − a − ≥ 2 κ + a + , (4.18) m > 2 ` κ − C + κ + ´ . Then, the op er ator b L i s a gener ator of a holomorphic semigr oup ˆ U t , t ≥ 0 i n L C . Pr o of. The statemen t of th e theorem follo ws directly from Remark 4.2, Lemma 4.5 and the theorem about th e p erturbation of holomorphic semigroup (see, e.g. [12]). F or the reader’s conv enience, b elo w we give its form ulation: F or any T ∈ H ( ω , θ ) and for any ε > 0 ther e exists p ositive c onstants α , δ such that if the op er ator A satisfies || Au || ≤ a || T u || + b || u || , u ∈ D ( T ) ⊂ D ( A ) , 14 D. Finkelsht ein, Y u. Kondratiev, O. Kuto viy with a < δ , b < δ , then T + A ∈ H ( ω − ε, α ) . In p articular, if θ = 0 and b = 0 , then T + A ∈ H ( ω − ε, 0) F ollowing the proof of this th eorem ( see, e.g. [12]) and taking into accoun t t he fact that L 0 ∈ H ( ω , 0) for any ω ∈ (0 , π 2 ), one can conclude in our case that δ can b e chos en equal to 1 2 . This is ex actly the reason of app earing multiplicand 2 at the l.h.s. of (4.18 ). 5 Ev olution of correlation functions Let us consider the evolution equation (4.4), which corresp onds to the op erator ˆ L ∗ dk t dt = ˆ L ∗ k t , k t ˛ ˛ t =0 = k 0 . Using the general scheme, prop osed in [8] we find the precise form of ˆ L ∗ : ˆ L ∗ k ( η ) = − “ m | η | + κ − E a − ( η ) ” k ( η ) + κ + X x ∈ η X y ∈ η \ x a + ( x − y ) k ( η \ x ) + κ + Z R d X y ∈ η a + ( x − y ) k (( η \ y ) ∪ x ) d x − κ − Z R d X y ∈ η a − ( x − y ) k ( η ∪ x ) dx. The main questions whic h w e would like to study now are the ex istence and properties of the solution to the hierarc hical system of equations (4.4). The answers to these questions are given in the follo wing theorem Theorem 5.1. Supp ose that al l assumptions of The or em 4.6 ar e fulfil le d. Then for any i ni tial function k 0 fr om the class K C := n k : Γ 0 → R | k · C −| η | ∈ L ∞ (Γ 0 , λ ) o the c orr esp onding solution k t to (4.4 ) exists and wil l b e again the function fr om K C for any moment of time t ≥ 0 . Pr o of. F ollow ing the scheme prop osed in [16], we construct the corresp onding evol ution of the lo cally finite measures on Γ 0 . In order to realize t his construction w e consider the dual space K C to th e Banach space L C . The duality is giv en by the follo wing expression h h G, k i i := Z Γ 0 G · k dλ, G ∈ L C . (5.1) It is clear that K C is th e Banach sp ace with the norm || k || := || C −|·| k ( · ) || L ∞ (Γ 0 ,λ ) . Note also, th at k · C −|·| ∈ L ∞ (Γ 0 , λ ) means that the function k satisfies the b ound | k ( η ) | ≤ const C | η | for λ -a.a. η ∈ Γ 0 . IBM Mo del with Co mpe tition in Spatial E cology 15 The evolution on K C , which corresp onds to ˆ U t , t ≥ 0 constructed in Theorem 4.6, may b e determined in the follo wing w a y: h h G, k t i i := D D ˆ U t G, k 0 E E . W e denote ˆ U ∗ t k 0 := k t . Using the same arguments as in [16], it becomes clear that k t = ˆ U ∗ t k 0 is the solution to (4.4) in the Banach space K C .  It is important to emp hasize that in the case of a − ≡ 0 and κ + < m k ( n ) t → 0 , t → 0 , for any n ≥ 1 , see e. g. [17]. Therefore, we may exp ect that the correlation functions of our mo del satisfy this prop erty as wel l. 6 Stationary equation for the sy stem of correla- tion functions Let us consider for any α ∈ R the follo wing Banach sub space of K C : K α C := ˘ k ∈ K C ˛ ˛ k (0) ( ∅ ) = α ¯ . In this section we stud y the existence problem for the solutions to the stationary equation ˆ L ∗ k = 0 (6.1) in K 1 C . The main result is formulated in the follo wing wa y: Theorem 6.1. Supp ose that C κ − m + κ + m + 1 C < 1 (6.2) and κ − a − ≥ κ + a + then the solution k = ( k ( n ) ) n ≥ 0 to (6.1) is unique in K 1 C and such that k ( n ) = 0 , n ≥ 1 . Pr o of. Let “ ˆ L ⋆ k ” ( η ) = 0 . The latter means that “ m | η | + κ − E a − ( η ) ” k ( η ) = − κ − X x ∈ η Z R d k ( y ∪ η ) a − ( x − y ) dy + + κ + X x ∈ η X y ∈ η \ x a + ( x − y ) k ( η \ x ) + κ + Z R d X y ∈ η a + ( x − y ) k (( η \ y ) ∪ x ) dx. The last relation holds for any k ∈ K 1 C at t he p oint η = ∅ . Hence, one can consider it on K 0 C . 16 D. Finkelsht ein, Y u. Kondratiev, O. Kuto viy Let us denote for η 6 = ∅ ( S k ) ( η ) = − κ − m | η | + κ − E a − ( η ) X x ∈ η Z R d k ( y ∪ η ) a − ( x − y ) dy + + κ + m | η | + κ − E a − ( η ) X x ∈ η X y ∈ η \ x a + ( x − y ) k ( η \ x ) + + κ + m | η | + κ − E a − ( η ) Z R d X y ∈ η a + ( x − y ) k (( η \ y ) ∪ x ) d x and ( S k ) ( ∅ ) = 0 . Let k k k C = ess sup η ∈ Γ 0 | k ( η ) | C | η | , then k S k k C ≤ k k k C ess sup η ∈ Γ 0 \{∅} κ − C m | η | + κ − E a − ( η ) X x ∈ η Z R d a − ( x − y ) dy + k k k C C ess sup η ∈ Γ 0 \{∅} κ + m | η | + κ − E a − ( η ) X x ∈ η X y ∈ η \ x a + ( x − y ) + k k k C ess sup η ∈ Γ 0 \{∅} κ + m | η | + κ − E a − ( η ) Z R d X y ∈ η a + ( x − y ) dx ≤ k k k C C κ − m + k k k C κ + m + k k k C 1 C = k k k C „ C κ − m + κ + m + 1 C « , if κ + E a + ( η ) ≤ κ − E a − ( η ) + m | η | . As result, k S k ≤ 1 m C κ − + 1 m κ + + 1 C < 1 . The assertion of the th eorem is now obvious. Remark 6.1. F or any C > 1 one may chose κ − > 0 and m > 0 such that (6.2) i s satisfie d. The latter m e ans, that, asymptotic al ly, our syst em exhaust e d to the system with the stationary state δ ∅ ( dγ ) (the Dir ac me asur e c onc entr ate d on the empty c on- figur ation ∅ ). In other wor ds, the p opulation evolving due to the BD LP dynamics is asymptotic al ly de gener ate d. 7 F ur ther dev elopmen ts In Theorem 5.1 we have show n that functions k t is b ou n ded by C n for all t > 0, provided t hat k 0 satisfies initially the bound of the sa me type. Using appro ximation arguments (see e.g. [16], [18]) one ma y pro ve th at the corresp onding time evolution of the correlation function will b e also correlation function for some probability measure IBM Mo del with Co mpe tition in Spatial E cology 17 on Γ. W e supp ose to discuss this p roblem as wel l as other probabilistic asp ects of the BDLP model in a forthcoming pap er. The main aim of the present pap er is to analyze ev olution of correlation functions. Namely , w e ha ve shown that dyn amics of correlation functions stays in the space K C . This prop erty seems to b e very strong. T o sho w th at system of correlation functions evo lving in time stays in the same space is already d ifficult even for the contact mod el. Namely , (3.3) implies th at the evolution of correlation functions at some moment of time t ma y lea ve the space n k : Γ 0 → R | k · C −| η | · | η | ! ∈ L ∞ (Γ 0 , λ ) o . The reason is that C ma y dep end on t , which is true at least for the case κ + ≥ 1 ( m = 1 at the moment). Hence, w e may exp ect th at the d ynamics of correlation functions for the con tact process lives in some bigger space. Of course, this is possible only for κ + ≤ 1 since for κ + > 1 density tends t o infinity . Hence, let us consider the case κ + = 1. One cand idate for such bigger space is R C := n k : Γ 0 → R | k · C −| η | · ( | η | !) 2 ∈ L ∞ (Γ 0 , λ ) o . Note, that the inv ariant measure of the contact pro cess b elongs to t h is space (see [17, Theorem 4.2]), provided that d ≥ 3, a + has finite second moment w.r.t. the Leb esgue measure and the F ourier transform of a + is in tegrable on R d . Belo w we sho w that the evol ution of correlation functions at any moment of time t is a function from R C . Indeed, let κ + = 1 and supp ose th at there exists C > 0 suc h that for any n ≥ 1 and for any x 1 , . . . , x n ∈ R d k ( n ) 0 ( x 1 , . . . , x n ) ≤ 1 2 C n ( n !) 2 . Then, it is clear that k 0 ∈ R C . Now , su p p ose that k ( n − 1) t ≤ C n − 1 (( n − 1)!) 2 . W e prov e the corresp onding inequality for k ( n ) t using t he mathematical induction. By (3.3) w e hav e for any x 1 , . . . , x n ∈ R d k ( n ) t ( x 1 , . . . , x n ) (7.1) ≤ 1 2 C n ( n !) 2 + Z t 0 " n O i =1 e ( t − s ) L i a + # n X i =1 C n − 1 (( n − 1)!) 2 X j : j 6 = i a + ( x i − x j ) ds = 1 2 C n ( n !) 2 + C n − 1 (( n − 1)!) 2 n X i =1 X j : j 6 = i Z t 0 “ e 2( t − s ) L a + a + ” ( x i − x j ) ds, where for f ∈ L 1 ( R d ) L a + f ( x ) = Z R d a ( x − y )[ f ( y ) − f ( x )] dx, x ∈ R d . F or the b ound ab ov e we hav e u sed the fact, t h at for any 1 ≤ i 6 = j ≤ n L i a + a + ( x i − x j ) = L j a + a + ( x i − x j ) = ( L a + a + )( x i − x j ) , x i , x j ∈ R d . This relation can b e easily chec ked b y simple comp u tations. 18 D. Finkelsht ein, Y u. Kondratiev, O. Kuto viy Note, t hat L a + is a generator of t he Marko v semigroup which preserves p ositivit y in L 1 ( R d ). Hence, g t ( x ) := Z t 0 “ e 2( t − s ) L a + a + ” ( x ) ds ≥ 0 , x ∈ R d , t ≥ 0 , and g t ∈ L 1 ( R d ). Then we hav e g t ( x ) = | g t ( x ) | = 1 (2 π ) d ˛ ˛ ˛ ˛ Z R d e ipx b g t ( p ) dp ˛ ˛ ˛ ˛ ≤ 1 (2 π ) d Z R d Z t 0 e 2( t − s )( b a + ( p ) − 1) | b a + ( p ) | dsdp, where symbol b f denotes the F ou rier transform of the function f ∈ L 1 ( R d ). Therefore, g t ( x ) ≤ 1 (2 π ) d Z R d 1 − e 2 t ( b a + ( p ) − 1) 2(1 − b a + ( p )) | b a + ( p ) | dp ≤ 1 2(2 π ) d Z R d | b a + ( p ) | 1 − b a + ( p ) dp. It w as shown in [17] that und er the conditions p osed on function a + for the case of inv ariant measure D := Z R d | b a + ( p ) | 1 − b a + ( p ) dp < ∞ . Finally , if add itionally C ≥ D (2 π ) d , then we obtain from (7.1) k ( n ) t ( x 1 , . . . , x n ) ≤ 1 2 C n ( n !) 2 + 1 2 C n − 1 (( n − 1)!) 2 n ( n − 1) D (2 π ) d ≤ C n ( n !) 2 . As result, k t ∈ R C for all t ≥ 0. Therefore, the dynamics of correlation functions for th e contact model stays in R C , hence, this dynamics is really very clustering for κ + = m = 1. As b efore, we ma y extend our consideration on t h e case m 6 = 1. Summarizing previous results in this section w e claim t hat the presence of the big mortalit y and th e big comp etition kernel preven ts clustering in the system making it sub-Poisso nian distributed . But, is it reall y necessary to add “big” mortalit y and compet ition kernel? Belo w we discuss t h is p roblem. If we wan t to study the quasib ounded semigroup with t h e generator b L on L C for some C > 0 then, n atu rally , this generator should b e an accretive operator in L C . Hence, for some b ≥ 0 th e follo wing b ound sh ould b e tru e Z Γ 0 sgn ( G ( η )) · “ ` ˆ L − b 1 1 ´ G ” ( η ) dλ C ( η ) ≤ 0 , ∀ G ∈ D ( b L ) , since C | η | dλ ( η ) = dλ C ( η ) . Let us defin e the “diagonal” part of the op erator b L : ` ˆ L diag G ´ ( η ) := − m | η | G ( η ) − κ − E a − ( η ) G ( η ) + κ + Z R d X y ∈ η a + ( x − y ) G (( η \ y ) ∪ x ) dx and consider for some n ≥ 1 G = “ 0 , 0 , G ( n ) , 0 , 0 ” , G ( n ) ∈ L 1 (( R d ) n ) . IBM Mo del with Co mpe tition in Spatial E cology 19 Then ( b LG )( η ) = 8 > > > > > > < > > > > > > : κ + R R d P y ∈ η a + ( x − y ) G ( n ) ( η ∪ x ) dx, | η | = n − 1 − κ − P x ∈ η P y ∈ η \ x a − ( x − y ) G ( n ) ( η \ y ) , | η | = n + 1 “ ˆ L diag G ( n ) ” ( η ) , | η | = n 0 , otherwise . Note that sgn ( G ( η )) ≡ 0 if | η | 6 = n . Therefore, for arbitrary n ≥ 1 0 ≥ I n := Z Γ 0 sgn ( G ( η )) · “ ` ˆ L − b 1 1 ´ G ” ( η ) dλ C ( η ) = Z Γ ( n ) 0 sgn ( G ( η )) · “ ` ˆ L diag − b 1 1 ´ G ( n ) ” ( η ) dλ C ( η ) = C n n ! Z ( R d ) n sgn “ G ( n ) ` x ( n ) ´ ”“ ` ˆ L diag − b 1 1 ´ G ( n ) ” ` x ( n ) ´ dx ( n ) . Let us fix some t > 0 and Λ ∈ B b ( R d ). Set for n ≥ 1 G ( n ) ` x ( n ) ´ = t n n Y k =1 χ Λ ( x k ) = t n 1 1 Γ ( n ) Λ ` { x ( n ) } ´ ∈ L 1 (( R d ) n ) . Then, the equ alit y sgn “ G ( n ) ` x ( n ) ´ ” = n Y k =1 χ Λ ( x k ) implies 0 ≥ n ! t n C n I n = Z Λ n − mn n Y k =1 χ Λ ( x k ) − κ − E a − ` x ( n ) ´ n Y k =1 χ Λ ( x k ) + κ + Z R d n X j =1 a + ( x − x j ) Y k 6 = j χ Λ ( x k ) χ Λ ( x ) dx 1 A dx ( n ) − b Z Λ n n Y k =1 χ Λ ( x k ) dx ( n ) = − κ − Z Λ n E a − ` x ( n ) ´ dx ( n ) + κ + n X j =1 Y k 6 = j Z Λ n − 1 dx k Z Λ Z Λ a + ( x − x j ) dxdx j − ( b + mn ) | Λ | n = − κ − Z Λ n E a − ` x ( n ) ´ dx ( n ) + κ + n | Λ | n − 1 Z Λ Z Λ a + ( x − y ) dxdy − ( b + mn ) | Λ | n . W e supp ose, in fact, that for any n ≥ 1 I n ≤ 0 . 20 D. Finkelsht ein, Y u. Kondratiev, O. Kuto viy Since E a − ( η ) = 0 for | η | ≤ 1 we get 0 ≥ ∞ X n =1 I n = − m ∞ X n =1 n t n C n n ! | Λ | n − κ − ∞ X n =1 t n C n n ! Z Λ n E a − ` x ( n ) ´ dx ( n ) + κ + ∞ X n =1 t n C n n ! n | Λ | n − 1 Z Λ Z Λ a + ( x − y ) dxdy − b ∞ X n =1 t n C n n ! | Λ | n = − mtC | Λ | e C t | Λ | − κ − Z Γ Λ E a − ( η ) dλ C t ( η ) + κ + C te C t | Λ | Z Λ Z Λ a + ( x − y ) dxdy − b “ e C t | Λ | − 1 ” = − mtC | Λ | e C t | Λ | − κ − C 2 t 2 Z Γ Λ Z Λ Z Λ a − ( x − y ) dx dy dλ C t ( η ) + κ + C te C t | Λ | Z Λ Z Λ a + ( x − y ) dx d y − b “ e C t | Λ | − 1 ” = e C t | Λ | » C t „ κ + Z Λ Z Λ a + ( x − y ) dxdy − κ − C t Z Λ Z Λ a − ( x − y ) dx dy − m | Λ | « − b “ 1 − e − C t | Λ | ” – . Therefore, for any t > 0 and any Λ ∈ B b ( R d ) 0 ≥ κ + Z Λ Z Λ a + ( x − y ) dxdy − κ − C t Z Λ Z Λ a − ( x − y ) dx dy − m | Λ | − b “ 1 − e − C t | Λ | ” C t =: B . 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