## 1. Introduction

**Figure 1.**A schematic representation of the models considered in terms of two coordinate axis corresponding to physics an niche space.

## 2. Mean Field Competition between Many Species along a Niche Axis: Emergent Neutrality

#### 2.1. The Lotka-Volterra Competition Model and the MacArthur-Levins Niche Overlap Formula

_{i}(ξ) = exp[−(ξ − μ

_{i})

^{2}/(2σ

_{i}

^{2})] centered at μ

_{i}, corresponding to its mean size (i.e., its position on this niche axis ξ), and with a standard deviation σ

_{i}, which measures the width of its niche. The competition for finite resources among the n species can be described by the Lotka-Volterra competition (LVC) equations:

_{i}is its maximum per capita growth rate, K

_{i}is the carrying capacity of species i (the asymptotic population size it reaches when isolated from the other competing species) and the coefficient α

_{ij}is the coefficient of competition between species i and j. A measure of the intensity of this competition is provided by their niche overlap, i.e., the overlapping between Pi(ξ) and Pj(ξ). The rationale is that species that are far apart in the niche axis will interact less strongly than those that are closer, and that species with narrower niches will compete with less species than those with wider niches. Therefore the competition coefficients α

_{ij}can be computed by the MacArthur-Levins niche overlap (MLNO) formula [24]:

_{ij}given by Equation (2). Furthermore, I show that ultimately solving the linear problem is enough to get both the transient pattern for SRA—clumps and gaps between them—as well as the asymptotic equilibrium.

_{i}, K

_{i}and σ

_{i}—is not possible, I consider in Section 2.2 a series of simplifications. This simplifying assumptions allow to obtain an analytic expression for SRA, in terms of the dominant eigenvector of α, which provides a qualitatively good description of the system for not too short times and becomes quite good for asymptotic times [32]. Using simulations it can be shown that all these simplifications do not destroy EN: clumps and gaps for SRA remain in the case of a finite linear niche axis no matter whether the niche is non-periodic (i.e., it has borders), or the species are randomly distributed, or when r, K and σ change from species to species [33]. Indeed LVCNT with heterogeneous species-dependent parameters is able to predict quite well the number of lumps for several different ecosystems [30,31,33,34].

#### 2.2. An Analytical Proof of Self-Organized Similarity in a Simplified Case

- S1. The n species are evenly distributed along a finite niche axis of length L = 1, i.e., their mean sizes are given by μ
_{i}= (i − 1)/n (i = 1, ..., n). - S2. To avoid border effects, the niche is defined as circular, i.e., periodic boundary conditions (PBC) are imposed. This is done by just taking the smallest of |μ
_{i}− μ_{j}| and 1 − |μ_{i}− μ_{j}| as the distance between the niche centers. - S3. All species have the same niche width: σ
_{i}= σ ∀i. - S4. All species have the same per capita growth rate which we take equal to 1: r
_{i}= 1 ∀i. - S5. All species have the same carrying capacity: Ki = K ∀i.

_{i}= N

_{i}/K. Conditions S1 to S3 allow to write the competition coefficients α

_{ij}as:

_{i}− μ

_{j}|, 1 − |μ

_{i}− μ

_{j}|} and $\stackrel{\u25cb}{\left|i-j\right|}$= min{|i − j|, n − |i − j|} and α = ${\text{e}}^{-\hspace{0.17em}\frac{1}{4{n}^{2}{\sigma}^{2}}}$. Therefore α becomes a “circulant” matrix [37] whose rows are cyclic permutations of the first one:

_{i}* verifying ${{x}^{*}}_{i}\left(1-{\displaystyle \sum _{j=1}^{n}{\alpha}_{ij}{{x}^{*}}_{j}}\right)$ = 0. Linear stability analysis for this equilibrium starts by considering initially small disturbances y

_{i}(0) from the equilibrium values x

_{i}* and to study their fate as the time grows. Let us take x

_{i}* = x* ∀ i which, by virtue of conditions S1 and S2, is an exact equilibrium. The evolution equation for y

_{i}(t) can then be written as:

_{ij}are symmetric, in the eigenvector basis {

**v**

_{i}} it becomes diagonal with all its eigenvalues λ

_{i}real. Hence integrating Equation (6), and using that y

_{i}(0) is small, y

_{i}(t) can be approximated by:

**y**becomes proportional to the dominant eigenvector

**v**

^{m}, the one associated with the minimum eigenvalue of α, λ

_{m}, i.e.,

**y**(t) ∝ e

^{−x*λmt}

**v**

^{m}(for large times)

_{m}(n,σ) is negative (see below). Hence, from Equation (8),

**y**is amplified over time instead of decaying to zero (as it would happen in the case of a positive λ

_{m}). Therefore, for large times, Equation (8) implies that we can express the time derivative of

**x**as:

**x**/dt = −

**x**(t)λ

_{m}

**v**

^{m}

**x**(t) ≈ exp (−λ

_{m}

**v**

^{m}t) (for large times)

_{n}

_{− k + 2}= λ

_{k}, one corresponding to the sine and the other to the cosine eigenvector. Equation (11) can be used to find numerically the index k = m with the minimal eigenvalue λ

_{m}(n,σ) (as we have just seen, k = n − m + 2 has the same value). The specific value of this index m is important since, as turns out from Equation (12), the corresponding dominant eigenvector

**v**

^{m}has m − 1 peaks and m − 1 valleys.

_{m}was determined from Equation (11) from a grid of values of n and σ: 2 ≤ N ≤ 200, and 0.05 ≤ σ ≤ 0.5. The surface depicted in Figure 2 corresponds to λ

_{m}(n,σ), showing that λ

_{m}is negative except for small values of n (n < 8) or when σ is below a critical value σ

_{c}which is approximately 0.075. This means that for higher values of n the equilibrium in which all species have the same biomass is not stable, implying that a pattern can be formed.

**Figure 2.**The minimal eigenvalue λ

_{m}, determined from Equation (11), as a function of n and σ. The arrow denotes the point n = 200 and σ = 0.15.

**v**

^{m}can also be used to predict the species distribution in time, using Equation (10). In Figure 3 we can see that the clumps and gaps in SRA coincide, respectively, with the m − 1 peaks and valleys of

**v**

^{m}.

**Figure 3.**The coincidence of lumps and gaps of species relative abundances (SRA) with, respectively, the peaks and valleys of the dominant eigenvector for n = 200 and σ = 0.15. (

**A**) Results from numerical integration of Equations (3), with competition coefficients given by Equation (4), after t = 1000 generations. (

**B**) the components of the dominant eigenvector

**v**

^{m}.

_{m}= −0.3938 and ${v}_{j}^{m}$ is either sin(8πμ

_{j})/10 or cos(8πμ

_{j})/10. Panel (A) of Figure 4 is a plot of a typical population distribution, produced by simulation, for t = 1000 generations and the expected biomass based on the dominant eigenvector ${v}_{j}^{m}$ = cos(8πμ

_{j})/10 substituted in the linearized model Equation (10). Notice that the agreement is quite good and that the quality of the agreement improves with time (Figure 4B), until it becomes very good when the lumps are thinned to single lines.

**Figure 4.**Distribution of species for n = 200 and σ = 0.15. In black results from a simulation after t generations and in gray exp[λ

_{m}v

^{m}t]. (

**A**,

**B**): Species evenly spaced along the niche axis for t = 1000 and t = 10,000 generations, respectively. (

**C**,

**D**): Species randomly distributed along the niche axis for t = 1000 and t = 10,000. (v

^{m}is obtained now numerically from the matrix α).

_{∞}(σ) = m(σ) − 1

**Box 1.**Correspondence between niche axis and spherical ferromagnet.

**Above: Schematic representation of a niche axis for bird species.**

_{AB}, is proportional to their niche overlap: the closer they are in size the stronger their interaction.

**Below: Schematic representation of a ferromagnetic material.**

**S**

_{i}and

**S**

_{j}, i.e., their interaction energy ε

_{ij}, is also proportional to their “overlap” or scalar product

**S**

_{i}

**.**

**S**

_{j}.

**S**

_{i}

_{ij}↔ ε

_{ij}

**Glossary of statistical mechanics terms of Section 2**

**Ferromagnetism:**The basic mechanism by which certain materials, such as iron and nickel, form permanent magnets. Microscopically the ferromagnetism is explained in terms of the electrons contained in the material. Specifically, one of the fundamental properties of an electron is that it has a magnetic dipole moment, i.e., it behaves itself as a tiny magnet. When these tiny magnetic dipoles are aligned in the same direction, their individual magnetic fields add together to create a measurable macroscopic magnetic field.

**Hamiltonian:**The mathematical descriptor for the energy of a given interaction. The total hamiltonian describes all energies of all the interactions that affect the system.

## 3. The Parallelism between a Spatial Grazing Model and Liquid-Gas Phase Transition: Metastability, Catastrophic Shifts in Ecosystems and Early Warnings [38]

#### 3.1. Catastrophic Shifts beyond Mean Field Theory

- (i)
- How spatial heterogeneity of the environment and diffusion of matter and organisms affects the existence of alternative stable states.
- (ii)
- Whether emergent characteristic spatial patterns are really useful as early warnings and how they are connected with temporal signs of catastrophic shifts.
- (iii)
- The search for scaling laws underlying spatial patterns and self-organization.

#### 3.2. The Mean Field Ecological Model

_{c}= 3

^{3/2}≅ 5.196 only one stable solution exists for each c. As long as we consider quasi-stationary evolution for increasing c, the system will exhibit a smooth response. On the other hand, for K > Kc, the response curve is folded backwards at two saddle-node bifurcation points. For certain values of c the system can be found either in the upper or the lower stable branch. If the system starts on the upper branch and c increases slowly, X will vary smoothly until a threshold value is found, where a catastrophic transition to the lower branch occurs. If at this point we want to reverse this transition by decreasing c, the system would not be able to recover its original state. Instead, the system would remain on the lower branch, until we decrease c enough to reach another threshold value and “jump” to the upper branch. From an ecological management viewpoint, it would be desirable to anticipate these transitions.

_{B}[63], the locus of the points (c,K) such that the second derivative of the potential V vanishes and then an attractor pops out or in. It divides the phase space into two regions corresponding either to single stability or bistability of the system (see Figure 6).

_{M}[63]. On the Maxwell set the values of V at two or more stable equilibria are equal (see the inset of V for K = 7.5, c = 1.91 in Figure 6).

**Figure 6.**Bifurcation set (solid line) with a cusp point at c = 8/3

^{3/2}, K = Kc and the Maxwell set (dashed). The potential V is shown for selected values of c and K.

_{B}and S

_{M}are connected to two commonly applied criteria or conventions. Systems which remain in the equilibrium that they are in until it disappears are said to obey the delay convention. On the other hand, systems which always seek a global minimum of V are said to obey the Maxwell convention. Indeed these two conventions correspond to two extremes in a continuum of possibilities. Furthermore, real systems may obey either of these two conventions depending on the rate of change of the control parameters or on other external conditions. When the control parameters, and so also the shape of V, change very slowly the system tends to follow the delay convention. In contrast, when the control parameters change more quickly or when perturbations on the system are big enough, the Maxwell convention describes the dynamics better (more on this below).

#### 3.3. Spatial Model

^{2}are between 0.1 and 0.5.

- The spatial mean <X(t)>:$$<X(t)>=\frac{1}{{L}^{2}}{\displaystyle \sum _{i,j}X(i,j;t)}$$
- The spatial variance σ
^{2}_{X}:$${\sigma}_{X}^{2}=<X{(t)}^{2}>-<X(t){>}^{2}$$ - The temporal variance σ
^{2}_{t}, computed from mean values of X at different times, which is defined as:$${\sigma}_{t}^{2}=\frac{1}{\tau}{\displaystyle \sum _{t\prime =t-\tau}^{t}<X(t\prime ){>}^{2}-}{\left(\frac{1}{\tau}{\displaystyle \sum _{t\prime =t-\tau}^{t}<X(t\prime )>}\right)}^{2}$$ - The patchiness or cluster structure. Clusters of high (low) X are defined as connected regions of cells with X(i, j, t) > Xm (X(i, j, t) < Xm) where Xm is a threshold value. There are different criteria for defining Xm (see below).
- The two-point correlation function for pairs of cells at (i
_{1}, j_{1}) and (i_{2}, j_{2}), separated by a given distance R, which is given by:G2(R) = <X(i_{1}, j_{1})X(i_{2}, j_{2})> − <X(i_{1}, j_{1})><X(i_{2}, j_{2})>

#### 3.4. Alternative Stable States and Early Warnings

**Mean and Variances**

^{2}

_{X}and σ

^{2}

_{t}in terms of increasing c with <K> = 7.5, d = 0.1 and the initial condition for each X(i, j) in the interval [0, <K>]. The position of the peak for the spatial variance, c

_{m}≅ 2.08, is earlier than the position of the peak for the temporal variance in nearly 110 time steps. So σ

^{2}

_{X}works better than σ

^{2}

_{t}as a warning signal for the upcoming transition. The reason for this is clear. When estimating the temporal variance one must consider past values in the time series, which correspond to situations where the ecosystem is far from undergoing a transition. The spatial variance considers only the present values, hence a signal announcing the shift is not blurred by averages including situations where these indications are absent. However, notice that when the peak in σ

^{2}

_{X}occurs, <X> has already experienced a decrement of almost 50% over its initial value.

**Figure 7.**<X>, σ

^{2}

_{X}and σ

^{2}

_{t}for d = 0.1, <K> = 7.5. The peak of σ

^{2}

_{X}occurs at c

_{m}≅ 2.08 and the peak of σ

^{2}

_{t}at c ≅ 2.30.

^{2}

_{X}is always narrower for the backward transition than for the forward transition. Second, the width of the hysteresis loop decreases with d, so diffusion tends to make the transition more abrupt.

**Figure 8.**<X> (black curves) and σ

^{2}

_{X}(blue curves) for <K> = 7.5 and δK = 2.5, computed for forward and backward changes of the control parameter c. Results for d = 0 (above), d = 0.1 (middle) and d = 0.5 (below).

**Correlation Function**

**Patchiness: Cluster Structure**

_{m}as a reference for the grid values X(i, j). For <K> = 7.5 and d = 0.1 the maximum in σ

^{2}

_{X}is given at c

_{m}≅ 2.08 (Figure 7). The value of <X> corresponding to c

_{m}is <K> c

_{m}≅ 2.89 and we will take it as the threshold. In the first column of Figure 10 we include snapshots of typical patch configurations for c = c

_{m}− 0.1, c = c

_{m}and c = c

_{m}+ 0.1 and in the second column a binary representation, i.e., dark red (blue) cells correspond to cells for which X > <X> c

_{m}(X << X> c

_{m}). The plots in the third column are the corresponding cluster distributions. At c = cm the patch-size distribution follows a power law over two decades—with exponent γ ≈ −1.1 for d = 0.1 and γ ≈ −0.9 for d = 0.5—which disappears for smaller or greater value of c. Therefore this particular distribution may be considered as a signature of an upcoming catastrophic shift in the system.

**Figure 10.**First column: a portion of 50 × 50 cells from the original 800 × 800 lattice is shown, grids representing the value taken by X(i, j) at each cell for <K> = 7.5, d = 0.1. The rows correspond to c = 1.98, c = 2.08 and c = 2.18. Second column: same as the first, for binarized data (blue: cells with low vegetation density, red: cells with high vegetation density). Third column: number of clusters versus area on a logarithmic scale.

#### 3.5. Usefulness of the Spatial Early Warnings

^{2}

_{X}, I have performed calculations over sample grids of different sizes L

_{s}< L. In Figure 11 we observe that the signal does not depend qualitatively on the number of points on the grid that are considered for estimating σ

^{2}

_{X}.

^{2}

_{X}still exhibits a noticeable peak. Of course, the quality of the signal improves with the size of the sample.

**Figure 11.**σ

^{2}

_{X}for <K> = 7.5, d = 0.1 and δK = 2.5, calculated for lattices of size L

_{s}= 3 (dotted line), 10 (dashed line) and 400 (full line).

^{2}

_{X}, c* = c

_{m}≅ 2.08 (for <K> = 7.5), its usefulness depends on the value of d. For d small (d = 0.1) the decay in <X(t)> stabilizes soon to a value above 2, i.e., the system remains in a mixed state.

**Figure 12.**<X> (black) and σ

^{2}

_{X}(blue) for <K> = 7.5 in the case of a remedial action consisting in keeping constant the control parameter after it reaches some threshold value c*. The red line indicates a threshold c* coinciding with the peak of σ

^{2}

_{X}, c* = c

_{m}≅ 2.08. Full (dashed, dash–dotted) curves correspond to d = 0.1 (d = 0.5). The green line corresponds to a value of c* before c

_{m}, c* = 1.9.

^{2}

_{X}reaches its maximum at cm, for c* = 1.9. It was checked that, for moderate or high diffusion (up to d = 0.5), this recipe of management works if c* is taken between the line corresponding to S

_{M}and the right fold line of S

_{B}(closer to the first than to the second one). So a possible criterion for choosing c* is as a point belonging to S

_{M}.

#### 3.6. Comparison with the Boiling Phase Transition: From the Delay to the Maxwell Convention

_{B}. Within this spinodal curve, infinitesimally small fluctuations in composition and density will lead to rapid phase separation via the mechanism known as spinodal decomposition [65]. Outside of the curve, the solution will be at least metastable with respect to fluctuations. An important remark is that this power law behavior for domains or patches exhibited as early-warning indicators in regime shifts is nothing to do with critical point phenomena. A critical point in statistical mechanics or thermodynamics occurs under conditions, such as specific values of temperature and pressure, at which no phase boundaries exist (e.g., at the critical point of water the properties of liquid and vapor become indistinguishable) and is associated with second order phase transitions. Instead, the transition here is first-order, the two distinguishable phases coexist during the transition, and this behavior arises because the ecosystem is approaching the spinodal line. That is, the transformation from one phase to another at a first-order phase transition usually occurs by a nucleation process. Nucleation near a spinodal appears to be very different from classical nucleation. Droplets appear to be fractal objects and the process of nucleation is due to the coalescence of these droplets, rather than the growth of a single one [66]. As a consequence the domain size grows with a power law for spinodal decomposition [65].

**Box 2.**Mapping from the grazing model to the van der Waals equation of state.

- Modality: the fluid is bimodal within the coexistence region, having well defined liquid and gas states. Hence in this aspect both systems are similar.
- Sudden jumps: in the case of the fluid it is certainly true that sudden jumps occur, since there is an abrupt increase in volume when a liquid transforms into vapor. However, this large change in volume occurs when a slight change in the temperature and pressure moves the fluid from one side of the coexistence curve to the other. Hence, the liquid-vapor coexistence curve can be identified with S
_{M}and the water changes of state obey in general the Maxwell convention. On the other hand, the shift in the ecological model considered always obeys the delay convention: the ecosystem remains in the higher attractor (higher values of X) until the bifurcation set is completely traversed. However as mentioned before that, when perturbations are big enough to allow the switching between equilibriums on different stability branches, the system may follow the Maxwell convention. Hence we will consider the effect of a sudden perturbation of the environment, represented here by a sharp decrease of the average carrying capacity <K> followed by a slow recovery. Figure 13 shows the evolution of the system for such a perturbation in <K>. Instead of remaining close to the initial attractor (upper branch of K = 7.5), the system rapidly falls to the lower branch of K = 6.0 (which corresponds to the minimum value of the potential V). Next it approaches slowly to the lower branch of K = 7.5 until it arrives at it for c ≅ 1.915. So one can conclude that this type of perturbation on the system produces a change of convention: from delay to Maxwell.**Figure 13.**The effect on <X> of a global perturbation on <K> which suddenly decreases from <K> = 7.5 to 6 and slowly recovers later. Thin lines represent “iso-K” curves for K = 7.5 and K = 6.0. - Hysteresis: in everyday situations one does not observe hysteresis in the liquid-gas phase transition of water—the liquid usually boils at the same temperature as the vapor condenses at. In other words, water changes of state obey in general the Maxwell convention. Nevertheless, a careful experimentalist can obtain a hysteresis cycle by first raising the temperature and superheating the liquid, and after evaporation, cooling the gas below the condensation point. Indeed the coexistence curve is surrounded by two spinodal lines which determine the limits to superheating and supersaturation. These spinodal or fold lines can then be identified with S
_{B}. - Anomalous variance: when a fluid condenses (boils) from its gas (liquid) to its liquid (gas) state, small droplets (bubbles) are formed. As a consequence, the variance of the volume may become large, which is similar to what happens for the ecosystem. This study illustrates well that the ultimate cause of the wide variations in patch size, giving rise to scale invariance, is spatial heterogeneity both in the initial conditions and the physical environment (i.e., in K).

**Glossary of Statistical Mechanics terms of Section 3**

**Anomalous variance**

**:**a particularly large variance at the onset of a catastrophic shift. A waving flags for such shifts is a sudden increase in the variance of certain relevant variable characterizing the state of the system (like the vegetation density).

**Control parameter**

**:**a parameter whose variation controls the qualitative properties of the solutions of a differential equation.

**Droplets**

**:**tiny drops formed by the condensation of a vapor or by atomization of a larger mass of liquid.

**Hysteresis**

**:**is the dependence of a system not only on its current state but also on its history. This dependence arises because the system can be in more than one equilibrium state i.e., it has alternative stable states. To predict its future development, in addition to its present state, its history must be known.

**Long Range order:**characterizes physical systems in which remote portions of the same sample exhibit correlated behavior. LRO occurs for example in second order phase transitions (see below) and implies power law behavior (however the reciprocal is not true). LRO can be expressed as a correlation function or two point correlation.

**Nucleation process**

**:**is the extremely localized budding of a distinct thermodynamic phase. Some examples of phases that may form by way of nucleation in liquids are gaseous bubbles (boiling) or liquid droplets (condensation) in saturated vapor.

**Phase transitions**

**:**are changes of phase of some substance when the thermodynamic variables reach some critical values, e.g., the transition from liquid to gas in the boiling of water occurs at atmospheric pressure at sea level when the temperature reaches 100 °C.

**Order of a phase transition**

**:**Phase transitions can be divided in two large groups. First order transitions, in which the phases are distinguishable (for instance liquid and gas have very different densities) and coexist (like boiling in ordinary conditions in which the liquid and vapor coexist) and the transition absorbs or releases energy known as latent heat (e.g., to boil water in a pan the stove provides this latent heat). In second order transitions, on the other hand, the phases are no longer distinguishable and they involve no latent heat. Long range order is an important phenomenon of second order PT.

**Spinodal lines**

**:**are lines denoting, in a P-V diagram, the boundary of instability of pure phases, liquid or gas, to decomposition into a liquid-gas mixture. That is, the isotherms or lines of constant temperature in the P-V plane (blue curves) produced by the van der Waals equation have a non-physical piece, called “kink”, in which the volume V grows when the pressure is simultaneously P growing. The spinodal lines (red dotted lines) are the locus of the end points (big red dots) delimiting kinks in the van der Waals isotherms. From a mathematical point of view spinodals are the locus of bifurcations.

**Turing instability**

**:**Reaction-diffusion systems are mathematical models which explain how the concentration of one or more substances distributed in space changes under the influence of two processes: local chemical reactions in which the substances are transformed into each other, and diffusion which causes the substances to spread out over a surface in space. An important idea that was first proposed by Alan Turing is that a state that is stable in the local system should become unstable in the presence of diffusion.

**van der Waals equation:**The simplest equation of state for a gas is the ideal gas state equation relating the pressure P, the temperature T and the density of the gas n (number of moles per volume V) by P = nRT, where R is the gas constant. The ideal gas denies the interaction between molecules, which is crucial to yield a phase transition from gas to a liquid state. The van der Waals equation represents a step further since it takes into account molecular interactions and then it is able to model the liquid-gas phase transition (for example water boiling in a pan).

## 4. Nonequilibrium Dynamics in Cellular Automata Model for the Dynamics of Tropical Forests

#### 4.1. Three Main Theories for Biodiversity—Classical, Neutral and Maxent—and the Use of Statistical Mechanics Methods

#### 4.2. Describing Tropical Forests by the Transient Regime of Spatial LVCNT

_{∞}of surviving species (e.g., n = 200, σ = 0.1 >> 0.005 = 1/n, then n

_{∞}= 4). Therefore it seems natural to analyze non-equilibrium communities of trees through the transient regime of LVCNT. We will focus on LVCNT and its predictions for tropical forests when analyzed in the transient regime.

#### 4.3. A Cellular Automaton Model Based on Lotka-Volterra Competition Niche Theory

_{i}only interacts with the eight neighbors of its Moore neighborhood M

_{i}. The strength of its competition with a neighbor of species s

_{j}, α

_{ij}, is proportional to the niche overlap between species s

_{i}and s

_{j}. This overlap is determined by a symmetric version of MLNO Equation (2) (the symmetry of the competition coefficients is a neutral assumption in the absence of precise details of the interaction between pairs of species) for a linear niche [34]:

_{k}≡ μ(s

_{k}) and σ

_{k}≡ σ(s

_{k}). We further assume that all species were functionally and demographically equivalent by having the same niche width: σ

_{i}= σ (which could be regarded as an average niche width). Hence, the fitness f(s

_{i}) of a focal individual of species s

_{i}located at site i is given by $f({s}_{i})=8-{\displaystyle {\sum}_{j\ne i\hspace{0.17em}\in {M}_{i}}{\alpha}_{ij}}$, where the “8” corresponds to the numbers of neighbors of i and it ensures that f

_{i}is always non-negative. Thus, f(s

_{i}) has its maximum value when the focal species s

_{i}has minimal overlap with its eight neighbors, and minimal when their niche overlaps are maximal. The functional equivalence between species is consistent with the chosen normalization for the α

_{ij}(Equation (23)) to assure that the matrix α is symmetric.

_{i}, located at site i, is randomly chosen (with probability 1/N); (II) This focal individual is replaced with probability m, representing the non-local dispersal of seeds by wind or animals, by another randomly chosen individual of species s

_{k}from outside its neighborhood Ω

_{i}. And with probability (1 − m)*Pr(s; s

_{i}→ s

_{j}) by a randomly chosen neighbor of species s

_{j}where $\mathrm{Pr}({s}_{i}\to {s}_{j})={\left\{1+\mathrm{exp}\left[-\left(f({s}_{i})-f({s}_{j})\right)/T\right]\right\}}^{-1}$. The probability that the focal individual s

_{i}is replaced by its neighbor s

_{j}is greater as the difference between individual fitnesses f(s

_{j}) and f(s

_{i}) increases. If the stochasticity parameter T = 0, then the change from s

_{i}to s

_{j}is accepted if and only if f(s

_{i}) < f(s

_{j}) (I analyzed this update rule for a CA simulating a more general ecological context in which in addition to mutual competition there can be mutual cooperation and competition-cooperation relationships [108]). On the other hand, for very large values of T, Pr(s

_{i}→ s

_{j}) approaches to ½, i.e., a totally random process.

#### 4.3.1. Estimation of Parameters

_{s}is the abundance of species s), was used to decide when to stop simulations. That is, for each given pair (σ, m), 100 simulations (100 different initial conditions) were run until H was equal to the empirical value ${H}_{e}^{\mathrm{(1)}}$ for the first census with an accuracy of 1%. Replacements of species having only one individual were prevented in order to constrain the CA configuration corresponding to the first census to the observed ${S}_{e}^{\mathrm{(1)}}$ species. Among all the pairs σ-m it was chosen the one such that the coefficient of determination ${R}_{et}^{2}$ of the linear regression between the observed and predicted (average over the 100 simulations) RSA distributions, was the highest (in all the cases ${R}_{et}^{2}$ ≥ 0.95).

**Table 1.**CA parameters for BCI and Pasoh, values of empirical quantities (in bold) and predicted species richness.

Forest | L | n | σ, m, T | ${\mathit{H}}_{\mathit{e}}^{\mathbf{1}}$ | Species richness, S |
---|---|---|---|---|---|

Pasoh (Malaysia) | 580 | 823 | 0.085, 0.11, 0.5 | 0.842 | 823, 819, 811, 808 823, 821 ± 2, 815 ± 4, 808 ± 5 |

Barro Colorado (Panamá) | 500 | 320 | 0.077, 0.10, 3.0 | 0.694 | 320, 318, 303, 299, 292, 283320, 314 ± 4, 300 ± 5, 293 ± 6, 287 ± 7, 281 ± 7 |

_{BCI}> = 950,000 and <∆t

_{Pasoh}> = 435,000, i.e., an approximately 2:1 relationship. A higher <∆t> for BCI than for Pasoh reflects the fact that BCI is more dynamic than Pasoh [109]. The value of <∆t> for a forest plot should be proportional to the total number of trees N (the larger the number of trees, the larger the replacement attempts needed) as well as to the fraction of species that went extinct on average between consecutive censuses ∆S (idem), and inversely proportional to the average (over censuses) number of species with one individual Σ

^{1}(which represents the species most likely to become extinct). It turns out that N

_{BCI}∆S

_{BCI}/Σ

^{1}

_{BCI}= 250,000 × 9.8/20.3 = 120,690 [97] while N

_{Pasoh}∆S

_{Pasoh}/Σ

^{1}

_{pasoh}= 336,400 × 5.0/24.5 = 68,653 [97], which are absolutely consistent with the above 2:1 relationship.

#### 4.4. Results and Discussion

**Figure 14.**Observed (black) and predicted (gray) distributions of relative species abundances (RSA): % in log scale vs. species ranked abundance for all trees with dbh ≥ 1 cm in BCI for censuses corresponding to 1982 (

**a**), 1985 (

**b**), 1990 (

**c**), 1995 (

**d**), 2000 (

**e**) and 2005 (

**f**). S denotes the species richness (i.e., the number of coexisting species). The predicted values correspond to averages ± std of 100 model simulations for the best estimates of model parameters.

**Figure 15.**Observed (black) and predicted (gray) distributions of relative species abundances (RSA): % in log scale vs. species ranked abundance for all trees with dbh ≥ 1 cm in Pasoh for censuses corresponding to 1987 (

**a**), 1990 (

**b**), 1995 (

**c**) and 2000 (

**d**). S denotes the species richness. The predicted values correspond to averages ± std of 100 model simulations for the best estimates of model parameters.

**Figure 16.**Species-area relationships (SAR) and for selected censuses of tropical forests. Predicted curves correspond to averages over 100 simulations, and the error bars correspond to one std. Observed and predicted (grey line) number of tree species with dbh ≥ 1 cm for sampling areas of different sizes at BCI 1990 (∆) and Pasoh 1987 (×).

_{ij}> ≈ 0.25. Since, by construction, <α

_{ii}> = 1 it means that the average interspecific competition strength is one-quarter of the intraspecific competition strength. This factor of 0.25 corresponds to an intermediate value between the extreme claims of the neutral model, where species are functionally identical and have independent dynamics i.e., <α

_{ij}> = 0, and the classical niche-based model of community assembly, where interspecific competition is dominant i.e., <α

_{ij}> > 1.

**Table 2.**The ten most abundant species of BCI, dbh properties and the niche position according to our CA.

Genus | Species | <dbh> (cm) * | max dbh (cm) * | Empirical Abundance ** | Theor. Abundance | Niche position |
---|---|---|---|---|---|---|

Hybanthus | Prunifolius | 2.24 | 8.8 | 29,846 | 31,115 | 0.008 |

Faramea | Occidentalis | 4.54 | 23.2 | 26,038 | 29,560 | 0.998 |

Trichilia | Tuberculata | 5.49 | 65.3 | 11,344 | 13,711 | 0.995 |

Desmopsis | Panamensis | 2.59 | 13.1 | 11,327 | 13,152 | 0.012 |

Alseis | Blackiana | 5.64 | 91.1 | 7,754 | 8,013 | 0.993 |

Mouriri | Myrtilloides | 2.17 | 5.0 | 6,540 | 7,758 | 0.013 |

Garcinia | Intermedia | 5.68 | 41.5 | 4,602 | 4,707 | 0.988 |

Hirtella | Triandra | 4.71 | 48.3 | 4,566 | 4,193 | 0.984 |

Tetragastris | Panamensis | 4.64 | 75.9 | 4,493 | 3,744 | 0.981 |

Psychotria | Horizontalis | 1.77 | 6.3 | 3,119 | 3,443 | 0.021 |

**Figure 17.**An attempt to identify the most abundant species found at BCI in 2005 with it corresponding theoretical species (same ranking order) and then to locate them along the niche axis (see Table 2 ).

**Glossary of statistical mechanics terms of Section 4**

**MaxEnt (maximum entropy principle)**

**:**A principle that states that the probability distribution which best represents the current state of knowledge on a system is the one with maximizes its entropy subject to the constraints imposed by prior knowledge. It is based on a correspondence between statistical mechanics and information theory. The basic idea is that entropy of statistical mechanics and the information entropy of information theory are the same thing. Consequently, statistical mechanics should be seen just as a particular application of a general tool of logical inference and information theory.

**Stochastic Cellular Automata**

**:**Stochastic Cellular Automata are CA whose updating rule is a stochastic one, which means the new entities’ states are chosen according to some probability distributions. They are models of “noisy” systems in which processes do not function exactly as expected, like most processes found in natural systems. Stochastic CA are discrete-time random dynamical systems.

## 5. Concluding Remarks

## Acknowledgment

## Conflicts of Interest

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