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Cellular automata with inertia species competitionspatial patterns and survival in ecotonesTo cite this article K Kramer et al 2010 J Phys Conf Ser 246 012040

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Cellular automata with inertia species competition

spatial patterns and survival in ecotones

K Kramer M Koehler and M G E da Luzlowast

Departamento de Fısica Universidade Federal do Parana CP 19044 81531-980 Curitiba-PRBrazil

E-mail lowastluzfisicaufprbr

Abstract We consider a two-dimensional CA model with three possible states for the systemindividual cells 0 and plusmn As for the dynamical rules only plusmn can exert pressure to change thecells actual states In this way the 0 state is neutral and in some sense competitively weakerthan the other two states We further assume an inner property the inertia which is an intrinsicresistance to changes in the system We evolve an ensemble of initial configurations for the CAuntil reaching steady states By calculating averages over some relevant quantities for the finalstationary configurations we discuss how certain features of the problem namely initial statespopulation and degree of aggregation as well as the values of inertia can determine the differentcharacteristics of the spatio-temporal pattern created by the CA evolution We finally discusshow our findings may be relevant in the understanding of structures formation due to speciescompetition in biology specially in the transition regions between different biomes the so calledecotones

1 IntroductionDuring the 1940rsquos John von Neumann with the help of Stanislaw Ulam introduced the idea ofcellular automata (CA) [1] The initial motivation was to mimic replication ie reproductionof living organisms by means of a simple mathematical construction the CA Such type ofsystems were then forgot until the 1970rsquos when John H Conway created the famous Game ofLife [2] again a CA which tries to simulate fundamental biological processes life and deathfrom very straightforward rules Around the same time Konrad Zuse [3] probably becomes thefirst person to propose CArsquos as useful tools to model different natural phenomena Since thenit has been an increasing interest in CArsquos specially after the detailed classification of the 1Dand 2D cases made by Stephen Wolfram [4] in the 1980rsquos

Given their simplicity and generality (even being possible realizations for universal Turingmachines [5]) nowadays CArsquos are considered in very broad contexts Indeed CArsquos are ableto reproduce the complex behavior seen in many systems in physics geology social sciencespsychology etc [6] But an area in which CArsquos find most applications (perhaps because theabove mentioned historical initial developments) is still biology [7] specially when related tospatio-temporal pattern formation [8] as eg in the study of spread of epidemics [9] the shapeof sea shells [10] and the organization of social insects [11]

In very general terms spatio-temporal structures often arise from driving forces acting onan extended system formed by spatially distributed cell (or sub-system) elements The cellsevolve and interact with each other under the influence of such driving forces Thus the time

XI Latin American Workshop on Nonlinear Phenomena IOP PublishingJournal of Physics Conference Series 246 (2010) 012040 doi1010881742-65962461012040

ccopy 2010 IOP Publishing Ltd 1

evolution of the whole system will generate successive configurations giving rise to the patternwhich may or may not converge to a final stationary global state From the above descriptionand given the way CArsquos are constructed (see next Section) it is not just a coincidence the greatutility of CArsquos in generating spatio-temporal pattern particularly in biological processes [7]

However an aspect usually overlooked in CA models is the natural intrinsic resistance(hereafter called ldquoinertiardquo) which the cells or sub-systems may have to change their local statesFor instance bacterial populations can develop a certain reaction against extension of theexisting colonies (growth) when harsh conditions are present [12] It is true that some CArsquoswith resistance have been proposed mainly to address socio-economics problems [13] But theyhave a probabilistic rather than a deterministic (mechanicist) character with the latter beingthe more frequent kind of inertia in pattern formation

Thus in the present contribution we introduce a CA where a term associated to inertia isexplicitly considered We moreover assume three different possible states for the CA cells Onlytwo of them (+ and minus) are dynamically active hence the ones playing a competitive role inthe model The third state (0) is neutral in terms of any dynamical pressure to change theother two We study how the inertia (an inner state associated to each CA cell) can alterthe emerging pattern in the system Our analyses are based on the evolution of an ensemble ofinitial configurations for the CA until reaching steady states We determine how certain relevantquantities characterizing the final structures (obtained by averaging over the evolved replicas)depend on the parameters of the initial ensemble and the inertia values We discuss how ourgeneral findings may be of interest to the problem of maintenance and generation of biodiversitypattern [14] Finally we address the phenomenon of survival of less fitted species in ecotones[15] ie transition regions between distinct biomes We show that the weaker species 0 can livein such transition zones (but not within the biomes) provided it has a certain small resistanceto the competition pressure exerted by the stronger species + and minus

2 The modelNext we describe our construction of a CA with inertia as well as explain the methodology wewill use to characterize the system

21 CA with inertiaAs previously mentioned a CA is a system composed by K cells labeled by k = 1 2 KTo each cell k it is ascribed a number sk (s = 1 2 S) describing its actual local stateat the discrete time instants n = 0 1 2 There is an update rule telling how the systemwill evolve over time so that sk(n + 1) is function of the system configuration at the earliertime n We will consider the 2D case and a square array of K = N times N elements thusk equiv (i j) (i j = 1 2 N ) For such array we take a wall-like (instead of the more commonperiodic) boundary condition more appropriate in biological applications as in fragment patternin ecosystems [14] and species proliferation in ecotones [16]

For the dynamics we assume the Moore neighborhood of radius 1 meaning that each cell willevolve under the influence of only its first surround neighbors ie three five and eight neighborsfor respectively the cells at corners edges and middle of the array We suppose three possiblestates [17 18] s = 0 and s = plusmn The former being the neutral (passive) state in the sense thatit does not attempt to change the state of the other two In fact most of the CArsquos proposedin the literature are defined so that all states exert competitive pressure over each other like inRef [19] which studies the spatio-temporal dynamics of communities Here as it is going tobe clear in the latter applications it is interesting to have a ldquoweakerrdquo player in the game thestate 0

We add a new ingredient into the CA by considering an extra ldquointernal staterdquo the inertiathat quantifies a resistance to changes in the actual state of a cell The inertia of cell k Ik


2

is a number between 0 and 8 where 0 means no inertia at all and 8 is equal to the maximumpossible number of first surrounding neighbors In the extreme case of I = 8 the cell neverchanges its initial state Intermediate Irsquos influence the cell time evolution as described below

Let us defineSk(n) =

summ isin neighborhood of k

sm(n) (1)

Then we define the CA rules of evolution as

bull The inertia ruleLet Ik(n) be the value of the inertia of the cell k at time n If Ik(n) ge |Sk(n)| the state skat n+1 remains the same that at n If Ik(n) lt |Sk(n)| we apply the dynamical rule below

bull Dynamical ruleIf Sk(n) gt 0 (Sk(n) lt 0) the cell k state changes to sk(n + 1) = + (sk(n + 1) = minus)Otherwise it remains the same

The above two rules are summarized by (for sign[x] the signal of x)

sk(n+ 1) =

sign[Sk(n)] if Ik(n) lt |Sk(n)|sk(n) if Ik(n) ge |Sk(n)|

(2)

Although models where the inertia is a function of time certainly would present very richdynamics here we concentrate only on time independent Irsquos

22 Characterizing the systemTo analyze the influence of the inertia in the formation of spatio-temporal pattern we shallfirst define few quantities helping to characterize the state configurations of the CA In factthere are different possible sophisticated topographical analysis to identify spatial point patternin extended systems (a nice overview is given for instance in Ref [20]) some even based onfractal structures [21] Nevertheless here we propose a direct way to do so which is simpleto implement and capable of measuring the degree of aggregation or fragmentation found inrealistic biological patterns [22]

Thus let ps(n) be the number of cells in the state s (the population of s) at the time instantn Of course always p0+ p++ pminus = K = N 2 Then we consider the ldquoclusterizationrdquo of the fullCA at n by setting the clustering functions crsquos which quantify the average number of elementsof a same state around each CA cell forming small local clusters They can be total c or forthe state s = 0plusmn cs given by

c(n) =1

p(n)

sumk

[c]k(n)

cs(n) =1

ps(n)

sumk

[cs]k(n) (3)

Here [c]k is the number of cells in the most populated state in the neighborhood of cell k (egif around the cell k four cells have state 0 one + and three minus then [c]k = 4) If the twohighest local populations have equal number of elements then [c]k = 0 [cs]k = [c]k if the mostpopulated state in the neighborhood of k is s otherwise we assume [cs]k = 0 Note then thatthe parameters c and cs measure the frequency of aggregation in the CA

Another important quantity for our CA is the convergence time τ When an initial CA issubmitted to an evolution rule it either may converge to a certain stationary final configuration(therefore invariant under further successive applications of the same rule) or may never entering


3

into a steady state changing its configuration at each time step When it converges we candefine τ as the number of iterates necessary to reach such stationary situation

To analyze the system dynamical evolution and the influence of varying the inertia to theformation of spatio-temporal pattern we will discuss most of our results in terms of averages overan ensemble of initial configurations for the CA (an approach already considered in the literaturebut with different purposes [23]) So we work with a large set of initial matrices all submitted tothe evolution of our CA We will restrict our study only to those initial configurations convergingto some steady state ie for which τ is finite Unless otherwise explicitly mentioned all thecalculated quantities result from proper means over the evolved ensemble

For our simulations we set N = 22 a good compromise of lattices already displaying richdynamics while keeping the computational efforts low We also fix p0(0) = 161 asymp N 23 andp+ gt pminus so that always before the evolution the majority state is + In total we work with36146 matrices in our ensemble (randomly generated) with a fairly homogeneous distributionof parameters values in the intervals 162 le p+(0) le 182 and 012 le c+(0) le 022 To calculatethe averages we proceed in the following way Suppose we want to determine the property γ interms of the initial population of the state + p+(0) Then we evolve all the initial matrices ofthe ensemble which have different c+(0)rsquos but a same p+(0) calculate for each of such matricesthe quantity γ at the steady state and finally obtain a simple mean It leads to γ = γ(p+) Ofcourse similar procedure gives γ say as a function of the initial clusterization of the state +c+(0)

3 ResultsAs already mentioned all the initial configurations considered for our CA are such that guaranteethe convergence to a certain stationary state As we are going to see the inertia parameter Ik = I(0 le I le 8) ndash assumed to be the same for all cells k in the lattice ndash is a very important factordetermining both the average features of the steady states reached by the CA and how long ittakes for the convergence Note that for I = 8 any initial configuration is already at the steadystate because then the dynamical rule cannot change the system configuration (Sec 2)

165 170 175 180150

200

250

300

350

Initial p+

Ave

rage

fina

lp +

I=8

I=7

I=6

I=5

I=4

I=3

I=2

I=1

I=0

Figure 1 Average final p+ as a function of the initial p+ for all values of inertia

First as a kind of test we consider a quantity whose dynamical evolution is intuitively easyto predict In Fig 1 we show the average final p+ (ie already at the steady configurations)as a function of the initial p+ assuming different values for the inertia Ik = I (I = 0 8)


4

The observed linear increase is straightforward to understand Indeed in a first (mean-field)approximation the local population of a state s is psN 2 But as the dynamical pressure tochange a cell state to the value s = plusmn is directly proportional to the local population of s thenif the initial p+ grows the dynamical pressure to create new +rsquos grows in the same proportion(given that + is initially the majority state) Thus the observed results follow Moreover notethat when I increases we have from Fig 1 that the difference ∆p+ = p+ minus p+(0) decreases Itis simply due to the stronger intrinsic resistance of each element to change its initial conditionfor greater Irsquos Actually for I = 6 7 the curves are very close to that for I = 8 In such casesbasically the dynamics no longer can change significantly the initial configurations

165 170 175 180

02

04

06

08

Initial p+

Ave

rage

fina

lc +

I=8

I=7

I=6

I=5

I=4

I=3

I=2

I=1

I=0

Figure 2 Average final c+ as a function of the initial p+ for all values of inertia

Figure 2 shows the curves for the final average clustering of + versus the initial population+ The behaviors are similar to those in Fig 1 As seen the larger the initial p+ the higherthe final p+ But for larger final populations of the state + naturally we should expect a higheraggregation of + explaining the observed increase in the final c+ Furthermore for larger Irsquoswe also find a decrease in the final value of the average c+ again a consequence of dynamicalpressure to create the majority states + to decrease with I

The only important qualitative difference between Figures 1 and 2 ndash although difficult to seeby direct visual inspection ndash is the way the lines slopes vary as a function of I Of course theyare all positive However in the former the slope of the curves decrease monotonically with Iwhereas in the latter they increase (very slowly) for I increasing up to I = 4 then they startto decrease (again very slowly) To understand that notice that if I 6= 0 the state of a cell kmay change to a state s only if its neighborhood has at last I + 1 cells at state s Hence suchneighborhood must have a relatively high clusterization to change the state of the k cell In thisway the dynamics becomes very sensitive to the initial population and how locally agglutinatedit is leading to the observed behavior for I le 4 For I ge 5 local clusterization must be reallyvery high for any state change Since we are not working with high initial densities for + suchsituation is not frequently met and we start to see a very weak correlation between p+(0) andthe average final c+ until the limit of I = 8 for which no correlations do exist

Figure 3 shows the average final population + versus the initial clustering c+(0) For lowinertia values I le 3 we have that the final p+ decreases fairly linearly with the initial c+ Atfirst sight it may seem a contradictory result since in principle by increasing c+(0) we couldexpect larger final + state populations However an important aspect to recall is that ourensemble englobes a relatively small range for the initial p+(0) 162ndash182 Therefore in this


5

012 014 016 018 020 022

180

200

220

240

260

280

300

Initial c+

Ave

rage

fina

lp +

I=8

I=7

I=6

I=5

I=4

I=3

I=2

I=1

I=0

Figure 3 Average final p+ as a function of the initial c+ for all values of inertia

012 014 016 018 020 02200

02

04

06

08

Initial c+

Ave

rage

fina

lc +

I=8

I=7

I=6

I=5

I=4

I=3

I=2

I=1

I=0

Figure 4 Average final c+ as a function of initial c+ for all values of inertia

case the higher c+(0)rsquos do not come from many small + clusters scattered off in the CA initialconfigurations Rather it is due to a small number of particular regions with relatively largepopulations of + (a result verified by a direct inspection of our original ensemble but not shownhere) To promote the proliferation of the + state from the CA evolution rules it is moreefficient to have many small + clusters randomly distributed (acting as nucleation centers) thanto have few localized regions rich in + state The latter can lead to a crowded region of +rsquos butnot allowing efficient migration to other locations of the CA lattice It explains the behavior ofthe I le 3 curves in Fig 3 On the other hand when 4 le I le 7 the final p+ presents a veryweak increasing with the initial c+ in a small but positive correlation This time only the fewregions with highly clusterized + can give rise to further creation of the + state Thus higherc+rsquos may lead to small increments in the final p+

In Figure 4 we see the curves of the average final versus the initial clustering of the state +Qualitatively these plots resemble those in Fig 3 although here with a much weaker dependenceof c+ on c+(0) For instance observe a very slightly negative (when I le 2) and a positive (when


6

Figure 5 The mean convergence time as a function of the inertia The same values seen forI = 0 and I = 1 is because I = 1 is still a too low inertia value to change such characteristictime

I ge 3) slope for the lines Note that typically the initial clustering (recalling our procedure ofaveraging over the initial p+(0)rsquos Sec 22) is not very critical to set the final clustering whenthe inertia is zero As previously mentioned p+(0) is much more important to define the finalclusterization values Indeed for 0 le I le 2 the average final c+ in Fig 4 is basically equalto c+(p+(0) = 172) in Figure 2 ie the value at the middle of the interval range for p+(0)Nevertheless for intermediate values of the inertia (3 le I le 5) we observe a positive dependencebetween the initial and final clusterizations

Finally we present in Figure 5 the mean (over the entire evolved ensemble) convergence timeas a function of I Of course when I = 8 the convergence time is zero since for this maximumresistance any initial configuration is already a steady one For the other values of I the meanconvergence time decreases for an increase in the inertia a result that should be expectedIndeed for higher Irsquos there are more resistance to changes so overall fewer number of changesresulting in a quicker relaxation to the stationary condition

4 DiscussionAs already mentioned in the Introduction the formation and evolution of a spatio-temporalpattern is a fundamental problem in different branches of science Although much progress in theunderstanding of such phenomena has been achieved [24] many important open questions remain[25] In particular in the context of biology it is still not quite clear which are the main factorsgenerating different structures in the organization and distribution of living systems [26] Forinstance competition among different species their ability to adapt to different environmentalconditions their diffusive skills [27] among others all play fundamental roles in the waymeta-populations grow giving rise to distinct biodiversity pattern in different ecosystems [28]Nevertheless the relative influence of each one of these mechanisms competition strengthfitness diffusiveness etc are not yet properly quantified [29]

In our previous simple model of CA we have analyzed some general factors influencing pattern


7

formation all them associated to realistic situations in biology [22] In fact as a methodologywe have considered an ensemble of initial configurations and then performed averages to identifygeneral trends instead to focus on particular initial situations Of course such protocol maynot be appropriated to model punctual instances like the purple loosestrife invasion in NorthAmerica [30] Nevertheless by properly delimiting the range of values of relevant parameters(like the ones assumed for p0(0) p+(0) and c+(0) see Sec 22) a such statistical approachmay be useful in pointing to typical global behavior of specific ecosystems [31] For instance weverified that as long as the inertia (intrinsic resistance) of the CA elements are zero the initialpopulation of the majority state is more important to define the final majority population than apossible spatial aggregation-clusterization of such state at t = 0 However this is true providedthe other active competing state (in our case minus) has the same degree of clusterization We shouldmention that for the largest considered initial population of + 182 the initial population ofminus is 141 and thus in average cminus(0) is not too radically different from c+(0) So our presentanalysis would be for example more appropriate for homogeneous biomes

Figure 6 Typical succession of generations of a CA until reaching the steady configurationThe parameters are given in the main text and the colors black grey and white representrespectively the states + minus and 0

A new factor introduced here is the inertia quantifying the intrinsic resistance of the cellelements to change their actual states In a biological context such quantity could be associatedto a fitness of a given species [29] allowing it to resist to other invasive species From our resultsone realizes that already very low inertia can change the final pattern of a evolutive process or


8

even the relative influence of the other parameters for so Indeed depending on the I valuethe relevance of c+(0) (the initial clusterizaiton) increases This becoming clear from the curvesin Figs 3 and 4 Moreover I can accelerate the process of reaching a final steady state for abiological pattern (see Fig 5) Although it should be expected for larger Irsquos this also holdstrue for small and intermediate resistance values

Actually the above previous observations bring about another interesting point whethera weaker less aggressive (in terms of competitiveness) species can survive in an environmentwith more aggressive species For instance models using coupled logistic maps (see eg [29])points that in fact this is possible provided the neutral (weaker) species has minimal survivalskills However such type of models generally does not include spatial factors very importantto determine the geographic distribution of meta-populations With the present CA eitheraspects can be considered as exemplified in Figure 6 which displays for different Irsquos a typicalautomaton evolving until it reaches a stationary configuration In the example the initialclustering and population are c+(0) = 0169864 and p+ = 172 with a convergence time τ forI = 0 equal to 9 iterates In Figure 7 we show the resulting number of elements in the 0 (neutral)state at the steady situation For I = 0 none survives But already for the lowest value of I = 1such ldquoweakerrdquo species can resist to the competitive pressure to change their states (which ina biological view could mean the death of such species and the establishment of the strongerspecies in that same location) Thus the zero state can manage to survive resisting to rdquoattacksrdquoin particular spatial spots of the landscape

0 2 4 6 80

50

100

150

Value of Inertia

Ave

rage

fina

lp 0

Figure 7 For the automaton of Fig6 the final population of state 0 as a function of the inertia

If now we come back to Fig 6 we observe a very interesting phenomenon in biology associatedto ecotones [15 32] Ecotones are transition zones between distinct geographic regions of differentbiomes In many instances it is not clear how groups of species coming from the differentbiomes can coexist in this transition region Also how the even less fitted animals and plants(eg due to a less developed response to the area particular climate condition) can survive inan ecotone [33] whereas they eventually would perish in other more homogeneous environmentsif competing with the same stronger species [34] In Fig 6 we see that the neutral state 0with no colonization skills will go extinct if I = 0 However even for small values of I it can


9

survive in small ldquocoloniesrdquo exactly at the borders of two regions dominated by the states + andminus Of course we do not expect the present model to describe all the complexity found in suchecosystems Nevertheless if we interpret the states 0 and plusmn as meta-species the CA with inertiamay be helpful in revealing some important aspects and key mechanisms underlying ecotonesbiodiversity

5 Final Remarks and ConclusionIn this contribution we have considered a CA model with three states one being neutral (passive)in terms of dynamical pressure to change the other states Furthermore we have associated aninner variable to each cell the inertia representing an intrinsic resistance to change the cellactual state under the action of the CA dynamical rule

By calculating averages properties of final states (after reaching the steady condition) of anensemble of initial configurations for the CA we have analyzed how different initial featuresnamely population aggregation of states and the inertia strength can influence the formationand evolution of spatio-temporal pattern in the system Moreover based on our findings wehave presented a general discussion on how patterns of biodiversity could emerge from scenariosof competition-fitness mechanisms We have done so assuming initial random spatial distributionfor the species (the states) but also considering a certain degree of clusterization of the speciesinvolved in the process

A particularly interesting result has been that the proposed CA naturally yields as a possiblesolution configurations in which the neutral (or weaker in some evolutive sense) state cansurvive provided its inertia is not null Indeed the 0 state can establish itself in small colonieslocated at the borders (transition zones or ecotones) of regions full of the stronger plusmn states Suchphenomenon which does occur in real situations is still poorly understood So the present canbe faced as null (or even a first approximation) model to quantify such rich and complex kindof ecosystem

Finally this simple and exploratory implementation of our CA with inertia of course cannotexplain realistic spatio-temporal pattern originated from say species competition Neverthelessit may be faced as a first step towards the incorporation of a very important feature intrinsicresistance (or fitness) into the process Surely further studies are necessary for a betterunderstanding of the problem in terms of the present model For instance to consider broaderranges for the initial parameters to use clusterization measures which can distinguish betweenlocal and global state aggregations to allow temporal as well as spatial variations for the inertiaparameter to study the influence of the CA size etc All such extensions would help to addressimportant open problems in biodiversity pattern specially in ecotones [33] Presently thementioned possibilities are being investigated and hopefully will be reported in the due course

AcknowledgmentsM G E da Luz is in great debt to M C M Marques for very valuable discussions about patternformation in biological systems specially about ecotones We thank CNPq and CAPES forresearch grants and F Araucaria CNPq and CT-Infra (FINEP) for financial support

References[1] von Neumann J 1966 Theory of Self-Reproducing Automata (Champaign-IL University of Illinois Press)[2] Gardner M 1971 Scientific American 224 112 Gardner M 1985 Wheels Life and Other Mathematical

Amusements (New York-NY W H Freeman amp Company)[3] Zuze K 1970 Calculating Space (Cambridge-MA MIT University Press)[4] Wolfram S 2002 Cellular Automata and Complexity Collected Papers (Boulder-CO Westview Press)[5] Rendell P 2002 in Collision-Based Computing Adamatzky A (ed) (London Springer) pp 513-539[6] Schiff J L 2008 Cellular Automata a Discrete View of the World (New York-NY Wiley-Interscience)


10

[7] Ermentrout G B and E-Keshet L 1993 J Theor Biolg 160 97 Lin Y Mynett A and Chen Q 2009 inAdvances in Water Resources and Hydraulic Engineering - Vol II Zhang C and Tang H (eds) (BerlinSpringer) pp 624-629

[8] Deutsch A and Dormann S 2005 Cellular Automaton Modeling of Biological Pattern Formation (Boston-MABirkhauser)

[9] Boccara N 2004 Modeling Complex Systems (New York-NY Springer-Verlag)[10] Meinhardt H 2009 The Algorithmic Beauty of Sea Shells - 4th Edition (New York-NY Springer)[11] Miramontes O 1995 Complexity 1 56 Miramontes O and de Souza O 2008 J Insec Sci 8 22[12] Lin A L Mann B A Torres-Oviedo G Lincoln B Kas J and Swinney H L 2004 Biophys J 87 75[13] Moldovan S and Goldenberg J 2004 Technol Forecast Soc Change 71 425 Zhang T Xuan H and Gao

B 2005 in 2005 International Conference on Service System and Services Management - Proceedings ofICSSSMrsquo05 - Vol 2 Chen J (ed) (Los Alamitos-CA IEEE Computer Society) Zupan N 2007 TechnolForecast Soc Change 75 798

[14] Young T P 2000 Biological Conservation 2 73[15] Zang Y and Malanson G P 2006 Geograph Analys 38 271 Favier C Chave J Fabing A Schwartz D and

Dubois M A 2004 Ecolog Model 171 85[16] Ferreira D A C Noguera S P Carneiro Filho A and Soares-Filho B 2008 Ciencia Hoje 42(248) 26 (in

Portuguese)[17] Platkowski T 2002 Physica A 311 291[18] Alonso-Sanz A and Martn M 2004 Chaos Solitons and Fractals 21 809[19] Eppsteina M J Beverb J D and Molofskyc J 2006 Ecolog Model 197 133[20] Wallet F and Dussert C 1998 Europhys Lett 42 493[21] Lahiri T Chakrabarti A and Dasgupta A K 1998 J Struct Biolog 123 179[22] Schneider M F 2001 J Appl Ecolog 38 720 Jenerette G D and Wu J G 2001 Landscape Ecolog 16 611

Kupfer J A and Runkle J R 2003 Oikos 101 135 Malanson G P Wang Q and Kupfer J A 2007 EcologModel 202 397

[23] Druzhinin O A and Mikhailov A S 1989 Radiophys Quant Elect 32 334 Kometer K Zandler G and Vogl P1992 Semicond Sci Technol 7 B559 Dzwinel W 2006 in Cellular Automata Lecture Notes in ComputerScience - 4173 Yacoubi S E Chopard B and Bandini S (eds) (Berlin Springer-Verlag) pp 657-666Wissner-Gross A D 2008 J Cellul Autom 3 27

[24] Mocenni C Facchini A and Vicino A 2010 Proc Nat Acad Sci 107 8097[25] Grigoriev D and Vakulenko S 2006 Ann Pure Appl Logic 141 412[26] Bolliger J Lischke H and Green D G 2005 Ecol Complexity 2 107[27] Santos M C Raposo E P Viswanathan G M and da Luz M G E 2004 Europhys Lett 67 734 Faustino C

L da Silva L R da Luz M G E Raposo E P and Viswanathan G M 2007 Europhys Lett 77 30002[28] Ormerod S J and Watkinson A R 2000 J Appl Ecol 37 1[29] Nowak M A 2006 Evolutionary Dynamics (Cambridge-MA Harvard University Press)[30] Welk E 2004 Ecol Model 179 551[31] Gower S T Kucharik C J and Norman J M 1999 Remote Sens Environ 70 29[32] Peters D P C (2002) Ecol Model 152 5[33] Ngai J T and Jefferies R L 2004 J Ecol 92 1001 Chaneton E J Mazia C N and Kitzberger T 2010 J Ecol

98 488[34] Freeman D C Wang H Sanderson S and McArthur E D 1999 Evol Ecol Res 1 487 Scarano F R 2009

Biol Conserv 142 1201


11

Cellular automata with inertia species competition

spatial patterns and survival in ecotones

K Kramer M Koehler and M G E da Luzlowast

Departamento de Fısica Universidade Federal do Parana CP 19044 81531-980 Curitiba-PRBrazil

E-mail lowastluzfisicaufprbr

Abstract We consider a two-dimensional CA model with three possible states for the systemindividual cells 0 and plusmn As for the dynamical rules only plusmn can exert pressure to change thecells actual states In this way the 0 state is neutral and in some sense competitively weakerthan the other two states We further assume an inner property the inertia which is an intrinsicresistance to changes in the system We evolve an ensemble of initial configurations for the CAuntil reaching steady states By calculating averages over some relevant quantities for the finalstationary configurations we discuss how certain features of the problem namely initial statespopulation and degree of aggregation as well as the values of inertia can determine the differentcharacteristics of the spatio-temporal pattern created by the CA evolution We finally discusshow our findings may be relevant in the understanding of structures formation due to speciescompetition in biology specially in the transition regions between different biomes the so calledecotones

1 IntroductionDuring the 1940rsquos John von Neumann with the help of Stanislaw Ulam introduced the idea ofcellular automata (CA) [1] The initial motivation was to mimic replication ie reproductionof living organisms by means of a simple mathematical construction the CA Such type ofsystems were then forgot until the 1970rsquos when John H Conway created the famous Game ofLife [2] again a CA which tries to simulate fundamental biological processes life and deathfrom very straightforward rules Around the same time Konrad Zuse [3] probably becomes thefirst person to propose CArsquos as useful tools to model different natural phenomena Since thenit has been an increasing interest in CArsquos specially after the detailed classification of the 1Dand 2D cases made by Stephen Wolfram [4] in the 1980rsquos

Given their simplicity and generality (even being possible realizations for universal Turingmachines [5]) nowadays CArsquos are considered in very broad contexts Indeed CArsquos are ableto reproduce the complex behavior seen in many systems in physics geology social sciencespsychology etc [6] But an area in which CArsquos find most applications (perhaps because theabove mentioned historical initial developments) is still biology [7] specially when related tospatio-temporal pattern formation [8] as eg in the study of spread of epidemics [9] the shapeof sea shells [10] and the organization of social insects [11]

In very general terms spatio-temporal structures often arise from driving forces acting onan extended system formed by spatially distributed cell (or sub-system) elements The cellsevolve and interact with each other under the influence of such driving forces Thus the time


ccopy 2010 IOP Publishing Ltd 1









2




sm(n) (1)





sk(n+ 1) =


(2)




c(n) =1

p(n)

sumk

[c]k(n)

cs(n) =1

ps(n)

sumk

[cs]k(n) (3)




3





165 170 175 180150

200

250

300

350

Initial p+

Ave

rage

fina

lp +

I=8

I=7

I=6

I=5

I=4

I=3

I=2

I=1

I=0




4


165 170 175 180

02

04

06

08

Initial p+

Ave

rage

fina

lc +

I=8

I=7

I=6

I=5

I=4

I=3

I=2

I=1

I=0






5

012 014 016 018 020 022

180

200

220

240

260

280

300

Initial c+

Ave

rage

fina

lp +

I=8

I=7

I=6

I=5

I=4

I=3

I=2

I=1

I=0


012 014 016 018 020 02200

02

04

06

08

Initial c+

Ave

rage

fina

lc +

I=8

I=7

I=6

I=5

I=4

I=3

I=2

I=1

I=0





6







7





8



0 2 4 6 80

50

100

150

Value of Inertia

Ave

rage

fina

lp 0




9










10















11









2




sm(n) (1)





sk(n+ 1) =


(2)




c(n) =1

p(n)

sumk

[c]k(n)

cs(n) =1

ps(n)

sumk

[cs]k(n) (3)




3





165 170 175 180150

200

250

300

350

Initial p+

Ave

rage

fina

lp +

I=8

I=7

I=6

I=5

I=4

I=3

I=2

I=1

I=0




4


165 170 175 180

02

04

06

08

Initial p+

Ave

rage

fina

lc +

I=8

I=7

I=6

I=5

I=4

I=3

I=2

I=1

I=0






5

012 014 016 018 020 022

180

200

220

240

260

280

300

Initial c+

Ave

rage

fina

lp +

I=8

I=7

I=6

I=5

I=4

I=3

I=2

I=1

I=0


012 014 016 018 020 02200

02

04

06

08

Initial c+

Ave

rage

fina

lc +

I=8

I=7

I=6

I=5

I=4

I=3

I=2

I=1

I=0





6







7





8



0 2 4 6 80

50

100

150

Value of Inertia

Ave

rage

fina

lp 0




9










10















11




sm(n) (1)





sk(n+ 1) =


(2)




c(n) =1

p(n)

sumk

[c]k(n)

cs(n) =1

ps(n)

sumk

[cs]k(n) (3)




3





165 170 175 180150

200

250

300

350

Initial p+

Ave

rage

fina

lp +

I=8

I=7

I=6

I=5

I=4

I=3

I=2

I=1

I=0




4


165 170 175 180

02

04

06

08

Initial p+

Ave

rage

fina

lc +

I=8

I=7

I=6

I=5

I=4

I=3

I=2

I=1

I=0






5

012 014 016 018 020 022

180

200

220

240

260

280

300

Initial c+

Ave

rage

fina

lp +

I=8

I=7

I=6

I=5

I=4

I=3

I=2

I=1

I=0


012 014 016 018 020 02200

02

04

06

08

Initial c+

Ave

rage

fina

lc +

I=8

I=7

I=6

I=5

I=4

I=3

I=2

I=1

I=0





6







7





8



0 2 4 6 80

50

100

150

Value of Inertia

Ave

rage

fina

lp 0




9










10















11





165 170 175 180150

200

250

300

350

Initial p+

Ave

rage

fina

lp +

I=8

I=7

I=6

I=5

I=4

I=3

I=2

I=1

I=0




4


165 170 175 180

02

04

06

08

Initial p+

Ave

rage

fina

lc +

I=8

I=7

I=6

I=5

I=4

I=3

I=2

I=1

I=0






5

012 014 016 018 020 022

180

200

220

240

260

280

300

Initial c+

Ave

rage

fina

lp +

I=8

I=7

I=6

I=5

I=4

I=3

I=2

I=1

I=0


012 014 016 018 020 02200

02

04

06

08

Initial c+

Ave

rage

fina

lc +

I=8

I=7

I=6

I=5

I=4

I=3

I=2

I=1

I=0





6







7





8



0 2 4 6 80

50

100

150

Value of Inertia

Ave

rage

fina

lp 0




9










10















11


165 170 175 180

02

04

06

08

Initial p+

Ave

rage

fina

lc +

I=8

I=7

I=6

I=5

I=4

I=3

I=2

I=1

I=0






5

012 014 016 018 020 022

180

200

220

240

260

280

300

Initial c+

Ave

rage

fina

lp +

I=8

I=7

I=6

I=5

I=4

I=3

I=2

I=1

I=0


012 014 016 018 020 02200

02

04

06

08

Initial c+

Ave

rage

fina

lc +

I=8

I=7

I=6

I=5

I=4

I=3

I=2

I=1

I=0





6







7





8



0 2 4 6 80

50

100

150

Value of Inertia

Ave

rage

fina

lp 0




9










10















11

012 014 016 018 020 022

180

200

220

240

260

280

300

Initial c+

Ave

rage

fina

lp +

I=8

I=7

I=6

I=5

I=4

I=3

I=2

I=1

I=0


012 014 016 018 020 02200

02

04

06

08

Initial c+

Ave

rage

fina

lc +

I=8

I=7

I=6

I=5

I=4

I=3

I=2

I=1

I=0





6







7





8



0 2 4 6 80

50

100

150

Value of Inertia

Ave

rage

fina

lp 0




9










10















11







7





8



0 2 4 6 80

50

100

150

Value of Inertia

Ave

rage

fina

lp 0




9










10















11





8



0 2 4 6 80

50

100

150

Value of Inertia

Ave

rage

fina

lp 0




9










10















11



0 2 4 6 80

50

100

150

Value of Inertia

Ave

rage

fina

lp 0




9










10















11










10















11















11

cellular automata with inertia: species competition - iopscience

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