Nadia Marchettini
email: marchettini@unisi.it

Emilio Del Giudice
email: emilio.delgiudice@mi.infn.it

Vladimir Voeikov

Enzo Tiezzi
email: tiezzienzo@unisi.it

Water: a medium where dissipative structures are produced by a coherent dynamics

Keywords: water, dissipative structures, coherent domains

The Belousov-Zhabotinsky phenomenon is analyzed in a framework where the dynamics of dissipative structures outlined by Prigogine is implemented through the collective dynamics produced in liquid water by Quantum Electrodynamics, which has received recently some experimental support. A mechanism allowing the appearance of self-produced oscillations is suggested.

Introduction

The problem how to account for the emergence of the highly ordered structures observable in Nature (in particular in the biosphere) out of the unordered crowd of independent atoms/molecules has been addressed in modern science along two pathways: the thermodynamics of irreversible processes (TIP) pioneered by Prigogine and the quantum field theory (QFT).

The first approach has produced the discovery of the dissipative structures that are highly complex structures which are open systems able to self-organize, thus producing order out of chaos, and give rise to dramatic A c c e p t e d m a n u s c r i p t 2 decreases of entropy. This discovery occurred in a conceptual framework that looks at the system as a whole and extracts from its behaviour in space and time a global principle that characterizes its dynamics.

The second approach has been able to show that quantum interaction among elementary objects, through the breakdown of the symmetries of the Lagrangian, is able to produce extended regions where the phases of the elementary components get correlated. This loss of independence curtails sharply the number of microstates producing the decrease of entropy.

The two above discoveries open the way to a fascinating perspective; could the extended coherent regions discovered by QFT be the dynamical basis of the dissipative structures discovered by TIP?

We wish to address this problem in the present paper in the particular case of an ordered system, the Belousov-Zhabotinsky (BZ) reaction [START_REF] Belousov | A Periodic Reaction and Its Mechanism in Sbornik Referatov po Radiatsionni Medtitsine[END_REF]. The BZ reaction can occur within many different set-ups, as illustrated in a recent paper [START_REF] Rossi | [END_REF] giving rise to the appearance of ordered space patterns exhibiting well-defined time evolution. The emergence of such structures is connected with the dynamically induced break-down of the symmetries existing in the original ensemble of elementary components. This break-down of symmetry is the consequence of the collective dynamics generated by the boundary conditions whose emergence is described in thermodynamics by the concept of negentropy introduced by Schrödinger [START_REF]Schrodinger What is Life[END_REF] and developed by Szent-Györgyi [START_REF] Szent-Györgyi | Bioenergetics[END_REF], Prigogine [START_REF] Prigogine | Thermodynamics of Irreversible Process[END_REF] and the Siena group [START_REF] Facchini | [END_REF].

The BZ reaction exhibits some peculiar features that, inside a more complex frame, are present also in the living process. An important recent result [7] is the discovery that the occurrence of the BZ reaction in a particular layout where lamellar lipid structures are involved demands the presence of liquid water whose weight should exceed a critical threshold (70% of the weight of the lipid) whereas neither water nor lipids are chemical reactants in the reaction. Their role should then be connected to the creation of the conditions of the selforganizing capability of the system [START_REF] Tiezzi | The Essence of Time[END_REF]. Below a threshold of hydration this capability could not arise; it arises above a critical threshold. The aim of this paper is to try to shed some light on this surprising result. However, this result could appear less surprising considering that the presence of water has been found long ago to be necessary in a number of chemical processes including the combustion [START_REF] Bon | [END_REF].

The analysis presented here mainly deals with the experimental layout used in Ref. [7]. Different layouts investigated in the literature could involve different dynamics. Consequently the proposal presented in this paper does not replace, but adds up to the dynamical ingredients governing the very complex phenomenon, known as BZ-reaction.

A c c e p t e d m a n u s c r i p t 3 Dissipative structures (DS) and QED coherence in liquid water

In this section we summarize some information on water needed for our analysis. Water is the most abundant substance on Earth and has been extensively studied; a number of model structures have been proposed and refined in the conventional framework of molecular physics governed by quantum mechanics. However, it remains an anomalous liquid in as far as no single model has been able to explain all of its properties. All these models share the property that water molecules are assumed to be interacting through static short-range forces.

However this tenet should not be universally true since extended regions of organized water had been detected in aqueous systems mainly at interfaces.

As early as in 1957 Albert Szent-Györgyi [START_REF] Szent-Györgyi | Bioenergetics[END_REF] wrote that by 1947 up to that date up to 175 papers were published devoted to water hydrating hydrophilic surfaces. They showed that the zone of hydrating water (interfacial water) may have the thickness of tens and hundreds of molecule layers and not just a few as usually believed by many physicists and chemists. This result emerged from the consideration of the luminescent properties of different dyes dissolved in water and depended strongly on the physical state of water, as structured by the different states of polarization of the surface. Based on the full set of these experimental results, Szent-Györgyi was able to suggest that the thick layers of "organized" water adjacent to the hydrophilic surfaces could allow very long lasting electronic excitations that could play a significant role in the energy transfer near the cell surfaces, where water resides in a state similar to a liquid crystal, as stressed also by Chandrasekhar [START_REF] Chandrasekhar | Liquid Crystals[END_REF]. The above considerations suggest strongly that a fruitful approach to self-organization is a joint task of chemists, biologists and physicists that help each other to get rid of the respective prejudices [START_REF] Tiezzi | The Essence of Time[END_REF].

Same decades later, just after 2000, the suggestion of Szent-Györgyi has been corroborated by G. Pollack and his Seattle group [START_REF] Zheng | [END_REF].

They were able to find, by using microspheres and dyes as probes, that very thick layers of water appear on hydrophilic surfaces, whose depth can reach many microns (sometimes up to 500!). In these regions dyes and other solutes cannot penetrate, so that they are called exclusion zones (EZ).

EZ water is more viscous than normal water and is able to release electrons in a significant amount, so that EZ water has a non negligible (up to hundreds of millivolts) redox potential with respect to bulk water or adjacent aqueous solution; EZ water is thus a reducing species whereas normal water is an oxidant. The layers of EZ water should play an important role in the chemical interactions occurring on their surface. Moreover differences of electric potential (about tens of millivolts) appear on the interface EZ water-normal water.

A c c e p t e d m a n u s c r i p t 4

These results find a theoretical corroboration in a recently proposed theory of liquids based on the concepts of Quantum Electrodynamics (QED). Referring the reader to the Refs. [START_REF] Preparata | QED Coherence in Matter[END_REF][START_REF] Del Giudice | [END_REF][14][15][16] we summarize here the main results of this approach:

1) At room temperature, liquid water consists of a mixture of coherent and incoherent water. At 0°C the percentages are 50% and 50%; at 30°C they are 40% coherent, 60% incoherent [14].

2) Coherent water is organized in coherence domains, whose size is about 1000 Å (0.1 μm), each containing about 5.5 millions molecules that oscillate, in phase with an electromagnetic field trapped in the domain, between two configurations of the molecular electron cloud: the fundamental one in which electrons are strongly bound (ionization potential 12.60 eV) and an excited configuration (12.06 eV) in which one electron per molecule is "quasi-free" and is therefore susceptible of further excitation.

The coherent state is the superposition of two molecule states, the molecule ground state having the statistical weight 0.87 and the excited state having the statistical weight 0.13. This means that at any time there are 0.13 quasi-free electrons per molecule, i.e. 700,000 quasi-free electrons, in the coherence domain.

The external surface of the domain is permanently polarized with the negative charges on the outside, so that for an isolated domain, an electrical "membrane" potential of about 100 millivolts is generated! If domains coalesce, as in normal water, these potentials neutralize each other.

When there are isolated domains, electrons have a chance to come out through the quantum tunnel effect or, possibly, a mild external excitation. This property could account for the huge release of electric charge from the water droplets suspended in the clouds, which is at the origin of lightning.

3) In bulk water the electromagnetic oscillations and the thermal fluctuations couple mutually, so that each molecule can shift in time from the coherent to the non-coherent regime and viceversa. The array of CDs of bulk water is then flickering and CDs cannot exhibit in a permanent way their features.

However, near a hydrophilic surface, the attraction CD-surface protects the CDs from the thermal assaults stabilizing them. Water gets totally coherent (glass state) at about 200 K, so that an attraction corresponding to a decrease of the effective temperature by one hundred of Kelvins would be enough [16]. By the way, the Pollack's estimate [START_REF] Zheng | [END_REF] of the effective temperature existing in the EZ zone is much lower than that of bulk water.

4) If coherent and incoherent water are separated in space, a redox pile up to hundreds of millivolts is created, in which coherent water, the electron donor, is the negative pole (reducing element) and incoherent water the positive pole (oxidizing element) [17].

5) By virtue of its quasi-free electrons, a coherence domain of water has unique properties with respect to all other liquids. It has a rich spectrum of excited states corresponding to vortices of quasi-free electrons. These property explains why turbulent water can be discovered by animals and humans, whose nervous systems are known to be very sensitive detectors of magnetic fields). In the presence of external fields, such as the magnetic field of the Earth, these vortices line up and are summed.

Since electrons move coherently, they do not collide, which means that the vortices are cold, and in the absence of internal friction caused by collisions, their life is very long (weeks, months, years). Because of their persistence, it is possible to sum a large number of excitations, producing energies as large as one wishes starting from individual low-energy excitations, provided that they are below the ionization threshold. The coherence domain of water is therefore a structure capable of transforming low-grade (high entropy) incoherent environmental energy into high-grade (low entropy) coherent energy, that can excite specific chemical reactions.

When the energy stored in the electromagnetic field of a coherence domain becomes equal to (resonates with) the energy of activation of a specific non aqueous molecule captured in the domain by the fact that it resonates at the oscillation frequency of the domain (the criterion for biomolecules), the domain discharges, transferring its energy to the single molecule that receives a specific excitation (enzyme activation) that makes it capable of performing a specific chemical reaction (coupling of electromagnetic and chemical modes).

If the domain is immersed into an energy field, it recharges again and a permanent oscillation is created in which charging is electromagnetic and discharge is chemical. As a result the dissipative structure introduced by Prigogine is supported by the QFT also [18].

The above QED predictions have found important experimental corroboration. In reference [19] the properties of liquid water at low temperature, when the liquid is in a glassy state, are successfully accounted for quantitatively. Moreover evidence has been found [20] of the coexistence in liquid water of two phases having different densities. Finally evidence of the existence in the aqueous surroundings of biomolecules of e.m. fields playing a significant dynamical role has been reported by the Nobel Prize Luc Montagnier [21]. He was able to detect the emission of electromagnetic signals in the aqueous structures surrounding highly diluted bacterial DNA, showing therefore that water plays an active dynamical role.

Analysis of the BZ reactions

Let us address now the problem of the BZ reaction, describing first the TIP framework and discussing at the end the possible dynamical role of water.

The basic mechanism of morphogenesis, namely differentiation of tissues from stem cells, was proposed 50 years ago by Turing [22], who pointed out that if morphogens obey a reaction-diffusion equation, they may ,...) , (...,

2 j i i i i i C C f C D t C + ∇ = δ δ (1)
where i C is the concentration of the i th species participating in the reaction, i f is the nonlinear kinetics equation describing the reaction and i D is the diffusion coefficient of species i. Assuming that eq.( 1) has a homogeneous steady state solution, ( ) 0 i s f C = , let us consider the evolution of a small perturbation s c around s C and separate it in Fourier space,

r k i t k sk s k e a c ⋅ + ¦ = λ ( 2 )
where k λ is the growth rate of the mode with a wave vector k. Substituting eq.( 2) in eq.( 1) and retaining only the linear term,we obtain an eigenvalue equation for k λ for the linear operator

ij i ij ij k D F L δ 2 - = (3)
where ij F is the Jacobian matrix of the kinetic function Target patterns where pulses are emitted periodically from the same leading centre (pacemaker) have been observed in a host of reaction-diffusion systems that generate chemical oscillations. These systems occasionally produce more complex structures, e.g. outward rotating spirals [START_REF]Chemical Waves and Patterns[END_REF].

The simultaneous presence of propagating waves and stationary patterns was found when the BZ reaction was carried out in the anisotropic environment of a lipid/water binary system. In order to mimic the biological complexity of a cell environment, the Siena Ecodynamics group studied the BZ oscillator in the aqueous compartment of different phospholipid/water lamellar phases [7,[START_REF] Marchettini | [END_REF][25][26][27].

All the lipids studied are major components of plasma membranes. Lipids in water are a good model for biological environments, since different aqueous domains in living systems are usually separated by a membrane where the basic structural unit is a single phospholipid bilayer where both surfaces facing water are hydrophylic.

In this system we get the appearance of a time dependent symmetry breaking pattern of reactant concentrations through the amplifications, produced by different rates of diffusion, of the gradients present at a given time.

However, we have discussed in the previous section that on hydrophilic surfaces there are gradients of energy able to drive the motion of different species in water. We ask ourselves if, in the layout of Ref. [7], where lamellar structures water-lipids are present, these gradients of energy could induce a self-organizing dynamics able to contribute, possibly in cooperation with other dynamics, to the emergence of a pattern of reactant concentrations.

Figure 1 shows a typical Turing structure obtained in the Siena laboratory.

< FIGURE 1 >

The intriguing result obtained in the Siena laboratory was the disappearance of all the BZ patterns and structures when the (water phase)/(lipid) ratio was low. For water concentrations below 70% weight/weight with respect to lipid, no self-organization phenomena were observed. Since all known forms of life and all physiological solutions have a water content in the range 60-99% (i.e. water is the molecule of life), our findings could open new perspectives in the understanding of life mechanisms, especially with respect to the role of water in self-organizing systems (dissipative structures) [START_REF] Rossi | [END_REF][START_REF] Prigogine | Thermodynamics of Irreversible Process[END_REF]7,[START_REF] Marchettini | [END_REF].

A c c e p t e d m a n u s c r i p t 8

The big question is whether there is a unified theory for the ways in which elements of a system organize themselves to produce behaviours typical of large classes of systems. In complex systems, we accept that processes occurring simultaneously at different scales or levels are important and coupled together, and the intricate behaviour of the whole system depends on its units in a nontrivial way. Description of the whole system's behaviour requires a qualitatively new theory [START_REF] Tiezzi | The Essence of Time[END_REF] because the laws governing this behaviour are qualitatively different from those governing the individual units.

Dissipative Structures maintain their non-equilibrium thermodynamic state thanks to a continuous dissipation of energy toward the environment. The order produced by this dissipation generates new order and new organization (autocatalytic structures) but if the dissipation is interrupted or diminished, the structure can collapse and may not return to its initial state (irreversibility) [START_REF] Prigogine | Thermodynamics of Irreversible Process[END_REF][START_REF] Tiezzi | The Essence of Time[END_REF].

The system thus self-organizes in virtue of internal non-linear processes that guarantee a balance between the entering energy and the exiting energy [START_REF] Tiezzi | The Essence of Time[END_REF], maintained by the variation of entropy occurring within the dissipative structure. The low grade energy picked up from the environment is transformed with a corresponding decrease of entropy within the dissipative structure into high-density energy able to perform "useful" work, such as collective molecular excitations. This energy produces work on the environment, inducing in the same time an increase of entropy in it that compensates the decrease of entropy which occurred within the dissipative structure.

Let us now discuss how the presence of water and in particular of coherent-EZ water described in the previous Section could contribute to the self-organization of the system.

The BZ-system has initially a number of interfaces (the Petri dish, the interfaces among different component phases present in the emulsion and so on), many of them hydrophilic, so that a pattern of EZ thick layers is present.

Correspondingly, on the surface of the EZ-layers gradients of electric potential should appear according to the QED theory and to the experimental findings of the Pollack group [START_REF] Zheng | [END_REF]. The formation of the EZ-layers, that have a well defined depth dependent on the polarization of the hydrophilic surface, demand a well defined amount of water, that should correspond just to the observed critical threshold in the hydration. The theoretical framework we are suggesting should allow to estimate the value of the critical threshold. We hope to do that in future research.

Let us outline now the dynamics occurring on the surface of EZ-coherent water. We recall from Ref.

[14] and [16] the main physical parameters of coherent water.

Coherent fraction F c (T)=0.4 at T= 300K

= =

Volume of the (spherical) CD at T0 = 4 3 ư (R 0 -į) 3 Within the coherent region an e.m. field is trapped, whose energy density

U M Q A M Q F ∇ - ∝ ∇ - = 2 2 2 G (6)
that is repulsive and inversely proportional to M. Thus in a molecule electrons are repelled tens of thousands times more strongly than nuclei. Molecules thus get stretched and acquire an electric dipole moment, whose negative charge is outwards.

As a consequence a charge double layer appears on the surface of coherent water where the external layer is negatively charged. In order to estimate the amount of electric charge per unit surface, we recall (Ref. [29]) that the quasi-free electron settles in the molecule 0.35 Å out of the electron core in the ground state, so that we can assume that during the dynamical process of excitation-deexcitation this distance could be included in the range 0. We can write the fundamental relation between the surface electric potential and the density of energy released to the water

According to the definition of the coherent fraction

U const V ∇ = ∇ . ( 11
)
In this way the excitation of coherent water induced by the output of chemical reactions increases the electric potential barrier, reducing the inflow of reactants and a fluctuating rate of chemical reactions appears.

Moreover the molecules present in the region settle according to the profiles of the electric potential and the possibility of patterns arises, supporting the Turing dynamics.

The inflow of the BZ reactants occurs now not in a homogeneous space, but in a volume where the existence of the EZ-water on the surfaces produces a spatial pattern of the non-coherent fraction of water, where solutes are dissolved. This pattern depends on electric potential and redox potential gradients that are affected in turn by the energy produced in the chemical reactions. This energy is mainly absorbed by the coherent water (the energy transfer toward coherent regions is much faster than the transfer toward non-coherent objects), whose inner structure changes, by modifying in turn the potential gradients.

We get thus a non linear self-consistent dynamics.

Reactants flow in the system not in a diffusive random way but along the paths corresponding to the steepest gradients, then react chemically; the energy produced, absorbed by the EZ-water, modifies the coherent structure changing the pattern of the potential gradients, that in turn change the rate of flow of the reactants, that consequently change the rate of reaction. A scheme for ordered fluctuations thus arises, both in space and time.

Let us explore an extreme case. Suppose that the rate of the BZ-reaction, enhanced by the electron excitations available on the EZ-water, becomes so high that the EZ-water is put out of coherence and loses its potential structure. Then the rate of reaction would decrease dramatically because the inflow of reactants is no longer driven by the potential gradients but left to diffusion.

Consequently the inflow of energy that pushed water out of coherence is stopped and QED dynamics can restore in a very short time the original coherence and the rate of reaction starts again to increase. We have got an oscillation!

A c c e p t e d m a n u s c r i p t 12

The suggested interplay between the quantum electrodynamics of interfacial water and the chemical dynamics of the BZ reactants opens a promising outlook. We conclude by observing that the patterns of electric and redox potentials on the water interfaces that constrain the independence of single molecules and funnel them along preferred pathways are nothing else that the electrodynamical implementation of the thermodynamic concept of negentropy whose formulation is the combination of the insights of the physicist Schrödinger, the biologist Szent-Györgyi and the chemist Prigogine.

Water and spatially ordered structures

In this section we discuss how a liquid emerging from the QFT dynamics outlined above could produce spatially ordered (Turing) structures. A CD of water is an extended region where all molecules are in a minimum entropy state as a consequence of a common oscillation on a frequency whose value is defined by the electrodynamics of molecules, chemically non interacting. The coherent feature of this state guarantees that its entropy is fairly low.

Molecules of species other than water can be admitted within the CD provided that they are able to oscillate on the same frequency, without changing the self-governing electrodynamical structures of the CD. This is possible only if these host molecules are a small number, say less than 1% of the water molecules, so that their QED regime within the CD can be determined by perturbation theory. Different than the water molecules, these other molecules can bind each other by several and different chemical or electrostatic forces, so that their oscillation frequency depends on the strength of such bindings, that usually depends in turn on the geometry. The prescription that the host molecules should resonate with the water molecules in order to minimize entropy implies that the geometry should be well-defined. A dynamical condition, the electrodynamical coherence, produces then geometrical order! Moreover, the existence of a rich spectrum of excited levels of CDs, each one with its specific oscillation frequency, implies that the geometrical order could evolve in time and produce also regular oscillations according to the oscillations of the CDs between their configurations. The role of the water CDs is thus fundamental and the necessity of its presence for the occurrence of the Turing structures can be understood, along the above lines.

Thus water gives rise to the emergence of order out of chaos, as in the case of the above described BZ experiments.

A c c e p t e d m a n u s c r i p t 13

By virtue of the above oscillations, a set of coherence domains may in turn become coherent (coherence of coherence domains), giving rise to a living (pulsing) organism consisting maily of water molecules organized in coherence domains, resonating with a small number of biomolecules residing in less organized water.

Conclusion

The discussion of the present paper applies to the layout of Ref. [7], whereas different layouts could involve different dynamics, as described in the existing literature. The experimental finding in Ref. [7] of a critical relationship between the amounts of water and lipids forming lamellar structures indicates that both the formation of EZ-water due to the presence of hydrophilic surfaces and existence of less organized aqueous phase in the same reaction system is the condition for the appearance of order out of chaos in chemical processes proceeding in aquatic systems.

A c c e p t e d m a n u s c r i p t 5 vortices

 5 have angular momentum (quantized at integer values) and similarly quantized magnetic moments (this

A c c e p t e d m a n u s c r i p t 6 undergo

 6 symmetry-breaking transitions. This, in turn, generates spatially organized states, now known as Turing structures, which may explain the initial stages of cell aggregation and development and epigenesis. In particular, if we consider a generic reaction-diffusion system:

A c c e p t e d m a n u s c r i p t 7 time

 7 nonzero mode k, Turing instability occurs when the real part of the eigenvalue k λ of operator[START_REF]Schrodinger What is Life[END_REF] becomes positive, so that the homogeneous steady state becomes unstable and the system undergoes a transition from the homogeneous state to a patterned state. Turing structures are characterized by a constant wavelength and -independent behaviour. They are generated when the diffusion coefficients of the morphogens (in the simplest case one activator and one inhibitor) are very different.

A c c e p t e d m a n u s c r i p t 9 Density

 9

2 A

 2 the square of the vector potential A JG of the trapped e.m. field (see Ref. [14]).Out of the coherent region the e.m. field falls off exponentially so that on the boundary there is a strong gradient ∇ and consequently, for eq. U ∇ ≠ 0.This gradient gives rise to a force, acting on the present particles having charge Q and mass M (ponderomotive force)[28]

  recalling eqs. (4), į=20Å that the depth 2į of the boundary region is 40 Å.The capacity of the interface layer per unit surface

  potential difference of the double layer is enclosed in the range: agreement with the observed membrane potentials is intriguing.

A c c e p t e d m a n u s c r i p t 11 In

 11 this way the energetic excitations of the coherent interfacial water give rise to a variable gradient of the electric potential, that in turn governs the rate of inflow of the reactants.

A c c e p t e d m a n u s c r i p t A c c e p t e d m a n u s c r i p t 15 Figure

 15 Figure captions.

Figure 1 .A c c e p t e d m a n u s c r i p t

 1 Figure 1. Lipid/BZ system: Turing structures

R. Arani, I. Bono, E. Del Giudice, and G. Preparata, Int. J. Mod. Phys. B, 1995, 9, 1813-1841.


Sz Horvát
email: szhorvat@phys.ubbcluj.ro

A Derzsi
email: aderzsi@phys.ubbcluj.ro

Z Néda

A Balog

A spatially explicit model for tropical tree diversity patterns

Keywords: neutral model, spatial distribution of species, population dynamics, lattice models, Monte Carlo simulations

A simple two-parameter model resembling the classical voter model is introduced to describe macroecological properties of tropical tree communities. The parameters of the model characterize the speciation-and global dispersion rates. Monte Carlo type computer simulations are performed on the model, investigating species abundances and the spatial distribution of individuals and species. Simulation results are critically compared with the experimental data obtained from a tree census on a 50 hectare area of the Barro Colorado Island (BCI), Panama. Fitting to only two observable quantities from the BCI data (total species number and the slope of the log-log species-area curve at the maximal area), it is possible to reproduce the full species-area curve, the relative species abundance distribution, and a more realistic spatial distribution of species.

Introduction

Macroecology [1,[START_REF] Pielou | Mathematical Ecology[END_REF][START_REF] Bell | Neutral macroecology[END_REF] studies the relationships between organisms and their environment at large spatial scales in order to characterize and explain universal statistical patterns of abundance, distribution, and diversity [START_REF] May | Patterns of species abundance and diversity[END_REF][START_REF] Harte | Scaling and self-similarity in species distribution: implications for extinction, species richness, abundance and range[END_REF][START_REF] Harte | Self-similarity in the distribution and abundance of species[END_REF]. A variety of models and methods from statistical physics are appropriate for the study of ecosystem dynamics, in which the main "entities" are either many individual organisms within populations, or many species within local, regional or continental communities [START_REF] Dewdney | A dynamical model of communities and a new species-abundance distribution[END_REF][START_REF] Chave | Neutral theory and community ecology[END_REF][START_REF] Enquist | Modeling macroscopic patterns in ecology[END_REF][START_REF] Norris | Neutral theory: A new, unified model for ecology[END_REF][START_REF] Chave | Comparing classical community models: Theoretical consequences for patterns of diversity[END_REF].

The remarkable regularities in patterns of data on how species originate, persist, assemble in groups, and eventually go extinct, suggest the existence of general mechanisms from which the biodiversity and the structure of ecological communities originate [1,[START_REF] Hubbell | The Unified Neutral Theory of Biodiversity and Biogeography[END_REF][START_REF] Tokeshi | Species coexistence[END_REF]. The difficulties in explaining the biological diversity of such systems originate from the very different spatial and temporal scales, starting with the evolution and biogeographic distribution of species, and ending with individual births and deaths in local communities. Present approaches for describing the dynamics of population genetics and ecology consider either the importance of the genetic fitness or the influence of random events governing birth, death and migration phenomena. There are thus two main, but conflicting perspectives on the nature of ecological communities: the niche-assembly and the dispersal-assembly perspective [1,[START_REF] Hubbell | The Unified Neutral Theory of Biodiversity and Biogeography[END_REF]. Here we intend to use the latter approach, constructing a simple spatially explicit stochastic model for describing the relative species abundance and the spatial diversity patterns of trees in a tropical forest [START_REF] Condit | Beta-diversity in tropical forest trees[END_REF][START_REF] Volkov | Neutral theory (communication arising): The stability of forest biodiversity[END_REF][START_REF] Volkov | Patterns of relative species abundance in rainforests and coral reefs[END_REF]. The present work is motivated by a recent study [START_REF] Zillio | Spatial scaling in model plant communities[END_REF], where a one-parameter, spatially explicit model is applied for describing tropical tree diversity patterns.

Neutral models

Recently it has been argued in the scientific community that systems of ecologically similar species whose interaction with each other is primarily competitive can be successfully approached by using neutral models. One of the first and most widely used neutral models is that of Stephen Hubbell [START_REF] Hubbell | The Unified Neutral Theory of Biodiversity and Biogeography[END_REF][START_REF] Volkov | Neutral theory and relative species abundance in ecology[END_REF], who assumed in his theory that within groups of ecologically similar species, individuals fill the landscape up to a point of saturation and the dynamics of all these species is governed by the same birth-, death-, migration-and mutation-rates. This simple neutrality principle allows us to study a wide variety of systems by means of stochastic computer simulations, and to use the methods of statistical mechanics to get analytical results [START_REF] Alonso | The merits of neutral theory[END_REF][START_REF] Azaele | Linear one-step processes with artificial boundaries[END_REF][START_REF] Etienne | Neutral community theory: How stochasticity and dispersal-limitation can explain species coexistence[END_REF][START_REF] Pigolotti | Stochastic model for the species abundance problem in an ecological community[END_REF][START_REF] Volkov | Organization of ecosystems in the vicinity of a novel phase transition[END_REF].

Tropical forests [START_REF] Condit | Beta-diversity in tropical forest trees[END_REF][START_REF] Volkov | Neutral theory (communication arising): The stability of forest biodiversity[END_REF][START_REF] Volkov | Patterns of relative species abundance in rainforests and coral reefs[END_REF][START_REF] Condit | Tropical Forest Census Plots: Methods and Results from Barro Colorado[END_REF] contain many more tree species than temperate or boreal ones (2 orders of magnitude more), and it is believed that such systems are much closer to the neutrality assumption. Since the number of mature individual trees in equal-area samples of each forest type is almost the same, the increase in the abundance of a particular species needs to be compensated by a decrease in another species's population.

In order to develop an appropriate model, one needs to take into account the whole set of factors influencing community composition: births, deaths, immigration, and (on longer time scales) speciation. These processes interact in a complex manner and produce the commonly observed empirical patterns of diversity and distribution of abundances.

The model should prescribe, for each generation, a method for choosing the species identity of the individuals that die and those that will occupy (by birth or immigration) the vacancies created. A simple method which describes fairly well the above mentioned processes is the one developed by Hubbell and it is called "zero-sum ecological drift" [START_REF] Hubbell | The Unified Neutral Theory of Biodiversity and Biogeography[END_REF][START_REF] Volkov | Neutral theory and relative species abundance in ecology[END_REF]. This method implies that the replacing species are drawn at random from the existing community of species.

The "ecological drift" is connected to the random replacement process and assumes that all individuals, regardless of species, have equal probabilities of giving birth, dying, immigrating or acquiring a mutation to generate a speciation event. The ecological drift does not imply an equal chance for each species to fill a given vacancy. Obviously, the greater probability of being drawn into a vacancy belongs to the more abundant species. The basic assumption is that individuals are equal, but species, as collective entities, are not. In Hubbell's model [START_REF] Hubbell | The Unified Neutral Theory of Biodiversity and Biogeography[END_REF][START_REF] Volkov | Neutral theory and relative species abundance in ecology[END_REF], the ecological drift, without any additional mechanisms, is enough to produce the patterns of species abundance and diversity observed in nature. In particular, the community of tropical trees is very well described by these neutral models and neutrality will be also the basis of the approach considered in this work. Contrary to earlier modelling efforts [START_REF] Hubbell | The Unified Neutral Theory of Biodiversity and Biogeography[END_REF][START_REF] Volkov | Neutral theory and relative species abundance in ecology[END_REF][START_REF] Volkov | Density dependence explains tree species abundance and diversity in tropical forests[END_REF], here we continue the novel idea introduced by Zillio et al. [START_REF] Zillio | Spatial scaling in model plant communities[END_REF], and a spatially explicit model resembling the classical voter model will be studied. In our approach, however, a slightly more complex, two-parameter version is considered which is not suitable for an analytical study. We argue why it is important to extend model considered in [START_REF] Zillio | Spatial scaling in model plant communities[END_REF], and we determine the best parameters for optimally reproducing the measured macroecological patterns.

A simple spatially explicit model

Most of the previous models used for understanding relevant ecological measures are mean-field like approaches in the sense that the spatiality of the individuals is lost, and the only relevant quantity that characterizes the system is the number of individuals belonging to each species [START_REF] Dewdney | A dynamical model of communities and a new species-abundance distribution[END_REF][START_REF] Chave | Neutral theory and community ecology[END_REF][START_REF] Enquist | Modeling macroscopic patterns in ecology[END_REF][START_REF] Norris | Neutral theory: A new, unified model for ecology[END_REF][START_REF] Chave | Comparing classical community models: Theoretical consequences for patterns of diversity[END_REF][START_REF] Condit | Beta-diversity in tropical forest trees[END_REF][START_REF] Volkov | Neutral theory (communication arising): The stability of forest biodiversity[END_REF][START_REF] Volkov | Patterns of relative species abundance in rainforests and coral reefs[END_REF][START_REF] Zillio | Spatial scaling in model plant communities[END_REF][START_REF] Volkov | Neutral theory and relative species abundance in ecology[END_REF][START_REF] Alonso | The merits of neutral theory[END_REF][START_REF] Azaele | Linear one-step processes with artificial boundaries[END_REF][START_REF] Etienne | Neutral community theory: How stochasticity and dispersal-limitation can explain species coexistence[END_REF][START_REF] Pigolotti | Stochastic model for the species abundance problem in an ecological community[END_REF][START_REF] Volkov | Organization of ecosystems in the vicinity of a novel phase transition[END_REF][START_REF] Condit | Tropical Forest Census Plots: Methods and Results from Barro Colorado[END_REF][START_REF] Volkov | Density dependence explains tree species abundance and diversity in tropical forests[END_REF].

Recently Zillio et al. [START_REF] Zillio | Spatial scaling in model plant communities[END_REF] took a step beyond the mean-field approach by considering a simple, but still realistic spatially explicit model. Their model consists of individuals placed on a lattice, and apart from speciation, only first neighbour interactions are taken into account. The community of tropical trees, where the coordinates of individuals are fixed, is especially suited for such an approach. This one-parameter model is successful in reproducing realistic relative species abundance (RSA) curves, but as it will be seen in section 6 (Fig. 6a), it produces unrealistically compact homogeneous islands of tree species.

The model considered in the present study is discrete both in time and space and it is also inspired by the classical voter model [START_REF] Holley | Ergodic theorems for weakly interacting infinite systems and the voter model[END_REF]. Individuals are placed on a predefined lattice. For the sake of simplicity, a simple square lattice was considered. Each individual belongs to a given species, and the state variable characterizing each lattice site encodes these species. It is assumed that the lattice with size L × L is always completely filled up, corresponding to a constant ecological saturation of the territory. Thus the total number of individuals in the system is always N = L × L.

The number of possible states (species) is W, (W 1). The dynamics of the system is governed by two adjustable parameters: p and q (0 < p, q < 1).

In the beginning a randomly chosen Potts state (species) is assigned to each lattice site (individual). Starting from this random initial condition, at each discrete time step a lattice site is randomly chosen and its state is changed. This process models the disappearance (death) of a tree, and the immediate occupation of the freed-up space by a newly born sapling. The p and q parameters govern the selection of the species identity of the new individual that occupies this free position as follows:

• With probability 1qp, the state of the chosen site is changed to one of its 8 Moore-neighbours. This is the most probable outcome, and models the quite common case of the new individual being an offspring of a neighbouring tree.

• With probability q, a randomly selected species is assigned to the chosen site. This species is selected with a uniform probability from the ensemble of possible species. If W is big enough and q much smaller than one, this process is suitable for modelling speciation or immigration into the considered territory.

• With probability p, the species identity of the new tree is chosen to be the same as the species of a randomly selected individual from among the remaining N -1 ones. This rule models the diffusion of the seeds in the considered area, allowing seeds originating from faraway individuals to reach the location.

It will be seen later that in order to get realistic results, the parameters p and q need to be selected as 0 < q p < 1.

By settings p = 0 and choosing W N, the analytically tractable model described in [START_REF] Zillio | Spatial scaling in model plant communities[END_REF] is regained. However, when p > 0, the analytical treatment of the model becomes difficult due to the combined short and long-range interactions. The model was studied thus by Monte Carlo (MC) simulations. One MC step is defined as N updates of lattice sites. In order to minimize the effect of boundaries, periodic boundary conditions were used.

Experimental data: the Barro Colorado Island dataset

The experimental results used for testing the model are from a detailed 50 hectare tropical forest tree census in Barro Colorado Island (BCI), realized by the Smithsonian Tropical Research Institute, Center for Tropical Forest Science (CTFS) [START_REF] Condit | Tropical Forest Census Plots: Methods and Results from Barro Colorado[END_REF]. BCI is located in the Atlantic watershed of the Gatun Lake (Panama) and was declared a biological reserve in 1923. It has been administrated by the Smithsonian Tropical Research Institute since 1946. From the viewpoint of ecological studies, this island is ideal because it is covered with a rainforest that is still undisturbed by humans. The flora and fauna of BCI have been studied extensively and inventories have reported 1369 plant species, 93 mammal species (including bats), 366 avian species (including migratory), and 90 species of amphibians and reptiles. The tropical tree census was performed only on a small part of the island, precisely on a 1000 × 500 metre (50 hectare) area. The first census was completed in 1982, revealing a total of approximately 240,000 stems of 303 species of trees and shrubs. The importance of this CTFS programme consists in the fact that in each census all free-standing woody stems at least 1 cm diameter at breast height are identified, tagged, and mapped, and hence accurate statistics can be made. Data is publicly available for the years 1982, 1985, 1990 and 1995 [START_REF] Hubbell | Barro colorado forest census plot data[END_REF].

The BCI database contains more than 320,000 individuals in total, belonging to 316 different tree species [START_REF] Condit | Tropical Forest Census Plots: Methods and Results from Barro Colorado[END_REF].

The density of individuals for the 1995 census is illustrated in Fig. 1 with a greyscale code (darker shades correspond to higher densities). In the western part of the studied area lies a relatively large swampland (lighter region in the left side of Fig. 2, i.e. less trees). This area is ecologically quite different from the other parts (see for example the spatial distribution of the most abundant species, Hybanthus prunifolius, in Fig. 2). Therefore, in determining the relevant macroecological measures, only trees located in the eastern half of the region (500 × 500 metres, 25 ha) were considered.

Relevant macroecological measures. Results for the Barro Colorado Island dataset

In order to describe the statistics of species sizes in ecological communities, the main measure which is usually considered is the relative species abundance (RSA) distribution. Provided that our data is spatially accurate, and we know the location of each individual, we can also investigate the species-area scaling and the spatial auto-correlation function for the individuals of a given species. These measures are briefly discussed here.

Relative species abundance distribution (RSA)

In ecology, the abundance of a species is understood as the relative number of individuals belonging to that species in an ecosystem. The relative species abundance distribution is introduced for characterizing the frequency of species with a given abundance [START_REF] Volkov | Patterns of relative species abundance in rainforests and coral reefs[END_REF][START_REF] Preston | The canonical distribution of commonness and rarity: Part i[END_REF]. Three different types of plots are generally used for representing species-abundance distributions. Historically the first, and thus also the most widespread, representation of species abundances is due to F. W. Preston [START_REF] Preston | The canonical distribution of commonness and rarity: Part i[END_REF], who sorted the species of a sample into abundance intervals of consecutively doubling lengths [START_REF] Pielou | Mathematical Ecology[END_REF][START_REF] May | Patterns of species abundance and diversity[END_REF], [START_REF] May | Patterns of species abundance and diversity[END_REF][START_REF] Chave | Neutral theory and community ecology[END_REF], etc.), and plotted the number of species found within these "octaves". Preston's method of plotting is motivated by the fact that abundances can vary over several orders of magnitude and there are far fewer abundant species than rare ones. Using fixed-length abundance intervals would result in large statistical fluctuations at the tail of the curve. In Fig. 3a the shape of the RSA Preston plot for the Barro Colorado tropical tree census for year 1995 is illustrated.

([1, 2),
The second and mathematically most rigorous way of representing the species-abundance distribution is plotting the associated probability density function ρ(s), i.e. the probability for finding a species with a given s abundance. For most of the neutral-like communities this distribution function has a tilted J shape on a log-log scale. Fig. 3b shows this probability density for the Barro Colorado tropical tree census (year 1995). It is worth mentioning that ρ(s) can be derived from the Preston plot by dividing the number of species in each interval with the length of the interval, and plotting on log-log scale this quantity versus the mean abundance in the given interval.

A third way of representing the species abundance distribution is arranging the species in decreasing order by their abundances and plotting the species's rank versus its abundance (rank-abundance plot) on a log-linear scale [START_REF] Norris | Neutral theory: A new, unified model for ecology[END_REF][START_REF] Chave | Comparing classical community models: Theoretical consequences for patterns of diversity[END_REF].

This representation is inspired by several abundance studies in sociology and economics, leading to the very general Pareto-Zipf distribution. For the Barro Colorado tropical tree census the rank-abundance curve for the same, 1995 census year, is plotted in Fig. 3c. It is also easy to realize that the rank-abundance plot is related to the cumulative distribution function corresponding to the ρ(s).

In Figs. 3a-3c, for reference, the data is compared with the prediction of the Fisher log series distribution described in [START_REF] Fisher | The relation between the number of species and the number of individuals in a random sample of an animal population[END_REF], with which the probability density plot and the rank abundance plot show good agreement. Data for other census years give similar results.

+a/

Species-area relationship (SAR)

Generally, the number of detected species does not scale linearly by increasing the size of the sampled territory.

In order to characterize this dependence, the species-area relationship is studied. This is done by considering larger and larger territories, and counting the number of species present within these areas [START_REF] Harte | Scaling and self-similarity in species distribution: implications for extinction, species richness, abundance and range[END_REF][START_REF] Harte | Self-similarity in the distribution and abundance of species[END_REF][START_REF] Zillio | Spatial scaling in model plant communities[END_REF]. For a better statistics, an average species number is calculated on several territories with similar areas.

The dependence of the average species number as a function of the sampling area defines the SAR curve. In Fig. 4 the characteristic shape of this curve is shown for the 1995 census year BCI data.

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Figure 4: Species-area curve for the BCI data (census year: 1995). In the limit of large areas, the SAR is compared to a power-law trend with exponent 0.15 (dashed line). Both scales are logarithmic.

For the BCI dataset the species-area scaling is not obvious at all [START_REF] Plotkin | Predicting species diversity in tropical forests[END_REF]. We believe that the reason for this is that the data was collected from relatively small area. As it is visible in Fig. 4 a trend resembling a power-law appears only in the limit of larger areas.

The spatial auto-correlation function

If the individuals are restricted to a given spatial position, like trees for example, one can characterize the spatial distribution of the individuals from a given species using the spatial auto-correlation function. The way we construct the C(r) auto-correlation function is the following: first a check-board type uniform mesh is considered, and the territory is divided into small, square-like domains (labelled by coordinates i, j) and the number of individuals from the considered species is determined in each domain (N i, j ). The auto-correlation function for a relative coordinate p, q (C p,q ) is calculated as

C p,q = (N i, j -N )(N i+p, j+q -N ) i, j (1)
Here N denotes the average number of individuals of the considered species in the constructed domains: N = N i, j i, j . Since there is no reason to assume that the distribution of the individuals is non-isotropic, we can calculate the average of the C p,q values for all p, q values that are inside a ring with radius r and width Δr (r ≤ p 2 + q 2 ≤ r + Δr), considering a reasonably small Δr value:

C(r) = C p,q {r ≤ p 2 + q 2 ≤ r + Δr} p,q (2)
The C(r) auto-correlation function, characterizing the spatial distribution of the 10 most abundant species in the 1995 BCI dataset was calculated. For all of them, a power-law type decrease has been observed. As an example, in Fig. 5 the spatial auto-correlation function of the three most abundant species is plotted on a log-log scale. The linear trend suggests a power-law like decrease for C(r).

MC simulation results

Given enough simulation time, the model converges to a statistically stable stationary state. For 0 < q 1, several species will co-exist in the final stationary state. Simulations performed on several systems proved that in order for the final, statistically stable state to be realistic, both q 0 and p 0 are necessary. If q = 0, eventually one species will replace all others. Setting p = 0 (which reduces the model to the one studied by Zillio et al. [START_REF] Zillio | Spatial scaling in model plant communities[END_REF]) results in unrealistically compact islands of species (Fig. 6a), in obvious contradiction with the fractal-like intermixing observed in the BCI dataset (Fig. 6c). By choosing p > 0, it is possible to get a more realistic spatial distribution for the abundant species (Fig. 6b).

a.

b. c. One important aim of the present study was to find those model parameters that reproduce the experimental data best. In order to compare the simulation results with the measurements, a unit of area needed to be defined on the lattice. The unit of area was chosen so that the density of trees would be the same in the model as in the BCI data.

The 1995 census was used for all comparisons.

The parameters p and q were optimized to reproduce two quantities from the BCI dataset: (1) the number of species present on a 25 ha area; (2) the slope of the log-log SAR curve in the neighbourhood of the 25 ha area. By optimizing for these two quantities only, the RSA, as well as the SAR for the whole range of available area values could be reproduced.

An extensive search of the p-q parameter space by MC simulations suggested p = 0.3 and q = 1.3 × 10 -4 as optimal parameter values. In Fig. 7, the time evolution of the total species number (W s ) in the system is illustrated (thick continuous line). After 5000 MC steps the W s species number reaches a stable limit and fluctuates around W s ≈ 300. In the same figure, the time evolution of the population for two selected species is also shown, one which appears at a later time moment and prevails, and one that appears quicker but gets extinct during the simulation (thin dashed and continuous lines, respectively).

The total number of species fluctuates in time, so after the optimal p and q parameters have been found, a new set of simulations were run for the purpose of comparing the SAR along the full range of available area values: after reaching equilibrium, these simulations were run for a few more MC steps until the fluctuating species number on a 25 ha sub-lattice reached the same value that is observed in the 1995 census data, which is 273 species. The results are plotted in Fig. 8. Even though the simulation parameters were optimized to reproduce the measured species number and the slope of the log-log SAR at a 25 ha area only, the SAR is reproduced remarkably well over the complete range of areas.

The shape of the RSA curve is also in good agreement with the BCI results. The RSAs resulting from the measurement and the simulation are compared in Fig. 9. Both the Preston plot (Fig. 9a) and the probability density number of individuals (thin continuous and dashed lines) function (Fig. 9b) resembles the experimentally observed ones (Fig. 3).

Unfortunately, this spatially explicit model does not reproduce the observed power-law type decay (Fig. 5) of the spatial auto-correlation function. The auto-correlation functions for the abundant species that result from the model show an exponential decay for all reasonable p and q parameter values. The auto-correlation function of the most abundant species obtained for the above discussed optimal p and q values is shown in Fig. 10.

To test the inter-dependence of the two main spatial measures we used, the SAR and the spatial auto-correlation function, we calculated the SAR function for the zero-correlation case, i.e. when a set of trees is scattered completely randomly in an area. It can be shown that the zero-correlation species area relation is very well approximated by the formula S (a) = k (1e -(a/A)n k ), where A is the total area of the considered territory, and n k is the number of individuals belonging to species k. Summation is over all species in the territory.

First, the SAR was calculated for the case of all the trees from the 1995 BCI dataset scattered randomly. This resulted in a SAR that is different from the actual measured one. Then we used an optimization algorithm based on simulated annealing to produce a set of 112,543 individuals belonging to 273 species (the same numbers that are found in the 1995 BCI census), so that the zero-correlation SAR would be as close as possible to the one measured in the BCI data. This resulted in a curve that is nearly indistinguishable from the measured one. The RSA of this artificially created set of trees matches the experimentally measured RSA well. From this it can be concluded that the

Discussion and Conclusions

A simple spatially explicit model, which was first considered in [START_REF] Zillio | Spatial scaling in model plant communities[END_REF], was extended here for explaining the statistical properties of individuals and species in a tropical tree community. The model is defined on a square lattice and it is inspired by the classical voter model [START_REF] Holley | Ergodic theorems for weakly interacting infinite systems and the voter model[END_REF]. It describes in a realistic manner the spatiality of individuals, birth and death processes, diffusion of seeds and speciation events. Within the model the dynamics of species is governed by two adjustable parameters, p and q. The parameter q defines the probability of speciation, or immigration events. The parameter p describes the probability for the global diffusion of seeds within the considered area.

The system was studied by large scale Monte Carlo simulations, and the p and q parameters were adjusted so that all experimentally studied macroecological measures would be reproduced optimally. For reasonable values of p and q, the model is successful in reproducing the shape of the species-area and RSA curves, as measured in the BCI dataset.

The species-area function is reproduced remarkably well, over the full range of areas. The model also generates a visually good spatial distribution of individuals within a species. The optimal values of the model parameters were found to be p = 0.3 and q = 1.3 × 10 -4 .

Despite the fact that the model is simple and ignores many relevant biological processes, it is successful in reproducing several measures of spatial distribution of tree species at the same time. Thus we believe it to capture the nature of the essential processes shaping these distributions, and to be a robust foundation for further complex models aiming to reproduce an even wider set of ecologically relevant measures.

It can be used with success to estimate the total number of species in an area and its fluctuations, from small scale samples. It can also predict qualitative effects of human impact that changes factors influencing the parameters p and q. As an example, factors that increase the value of p (which characterizes the rate of the global dispersal) lead to a lower total species number. Factors decreasing p may lead to the formation of compact islands.

Because of human influence, today species are disappearing with a higher rate than ever before, despite the numerous programs that have been started to stop this decline. In order to maintain forest biodiversity, it is important to identify the main processes that have an impact on it [START_REF] Plotkin | Predicting species diversity in tropical forests[END_REF]. Studies based on simple mathematical models, that take into account only a few primary mechanisms, are useful in such sense.

Beside all these successes, the model fails to reproduce the experimentally observed power-law decay of the spatial auto-correlation function of the individuals within a species as a function of separation distance. As one would naturally expect for lattice models where the local interactions dominate (p, q < 1), simulations lead to an exponential decay. This is an indication that in reality long-range interactions might be more important than it was considered in this simple approach. The logical next step in improving the model is including distance dependent dispersal. Such models would necessarily need to operate with more parameters, which diminishes the elegance of the model. Another way to make the model more precise would be to make use of not only dispersal-assembly but also niche-assembly theories [START_REF] Gravel | Reconciling niche and neutrality: the continuum hypothesis[END_REF][START_REF] Potts | Habitat heterogeneity and niche structure of trees in two tropical rain forests[END_REF].

As final conclusion, the present simple model is successful in reproducing the majority of the experimentally available macroecological measures.

A complete model is expected to be able to reproduce all measurable macroecological quantities. Focusing on global measures only, such as the SAR and RSA, is not sufficient. This study is such an attempt, and present a robust foundation for potential future modelling efforts. The circles show the SAR that is obtained by scattering the trees of the 1995 BCI census randomly, and the crosses show the same for a list of trees generated artificially to reproduce the experimentally measured SAR as well as possible. The solid line shows the SAR measured in the BCI data.

Note that the crosses overlap the solid line for all but the largest areas.

Figure 1 :Figure 2 :

 12 Figure 1: Density of trees in the BCI census (census year: 1995).

Figure 3 :

 3 Figure 3: RSA for Barro Colorado Island tropical tree census (census year: 1995). Three different ways of presenting the RSA: (a) the Preston plot, (b) the probability density function, and (c) the rank-abundance plot. Measurement data is in grey. The predictions of the Fisher log series for α = 33.64 are shown in black.

Figure 5 :

 5 Figure 5: The C(r) spatial auto-correlation function for the three most abundant species of the BCI census: Hybanthus prunifolius (white squares), Faramea occidentalis (black circles), and Trichilia tuberculata (grey triangles). Census year: 1995. The same function is plotted on a log-log scale on panel (a) and a linear-log scale on panel (b).

Figure 6 :

 6 Figure 6: Spatial distribution of the abundant species. In panels (a) and (b) simulation results are presented. Simulations were done on a 500 × 500 lattice with parameters q = 1.3 × 10 -4 , p = 0 (a) and p = 0.3 (b). In panel (c) the spatial distribution of the abundant Faramea occidentalis species in the BCI census is shown (census year: 1995).

Figure 7 :

 7 Figure 7: Time evolution of the total number of species W s (thick continuous line) and the population of two selected species (continuous and dashed thin line) on a 350 × 350 square lattice, q = 1.3 × 10 -4 , p = 0.3.

Figure 8 :

 8 Figure 8: Species-area curves: comparison of simulation (grey shaded area) and experimental (black line) results. The shaded area was computed from a set of 50 simulation runs, and represents the mean value ± 2 standard deviations. The white circles represent one simulation result that fits the experimental data best. Simulation parameters: L = 500, p = 0.3 and q = 1.3 × 10 -4 . Experimental data: BCI 1995.

Figure 9 :

 9 Figure 9: Simulation results (dark grey) for the RSA curve after 50,000 MC steps, compared with the measurement data (light grey), and the predictions of the Fisher log series for α = 33.64 (solid black lines). Panel (a) shows the Preston plot, (b) the probability density function, and (c) the rank-abundance plot. Simulation parameters: L = 500, p = 0.3 and q = 1.3 × 10 -4 .

Figure 10 :Figure 11 :

 1011 Figure10: The auto-correlation function of the most abundant species obtained after 50000 MC steps. Note that the scale for C(r) is logarithmic, while the scale for the radius is linear. Simulation parameters: L = 500, p = 0.3 and q = 1.3 × 10 -4 .

Acknowledgements

The authors wish to thank Dr. Joshua Plotkin for useful comments. Financial support was provided from programs co-financed by The sectoral operational programme human resources development, contract POSDRU 6/1.5/S/3-"doctoral studies: through science towards society and an MTA-KMEI research grant.