Research Article Information Transfer between Generations …downloads.hindawi.com/archive/2015/128980.pdf · 2019. 7. 31. · is modelled as a problem of communicating symbols across

Research ArticleInformation Transfer between Generations Linked toBiodiversity in Rock-Paper-Scissors Games

Ranjan Bose

Department of Electrical Engineering IIT Delhi Hauz Khas New Delhi 110016 India

Correspondence should be addressed to Ranjan Bose rboseeeiitdacin

Received 4 February 2015 Accepted 19 May 2015

Academic Editor Francois Guerold

Copyright copy 2015 Ranjan Bose This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Ecological processes such as reproduction mobility and interaction between species play important roles in the maintenanceof biodiversity Classically the cyclic dominance of species has been modelled using the nonhierarchical interactions amongcompeting species represented by the ldquoRock-Paper-Scissorsrdquo (RPS) game Here we propose a cascaded channelmodel for analyzingthe existence of biodiversity in the RPS game The transition between successive generations is modelled as communication ofinformation over a noisy communication channel The rate of transfer of information over successive generations is studied usingmutual information and it is found that ldquogreedyrdquo information transfer between successive generations may lead to conditions forextinction This generalized framework can be used to study biodiversity in any number of interacting species ecosystems withunequal rates for different species and also competitive networks

1 Introduction

Ecological processes such as reproduction mobility andinteraction between species have been shown to play impor-tant roles in the maintenance of biodiversity [1ndash5] Oneof the useful methods for investigating codevelopment ofpopulations is to use evolutionary game theory [6ndash8] In thiscontext the Rock-Paper-Scissors (RPS) game is a paradigmto describe species diversity [3 4] The cyclic dominance ofspecies is modelled using the nonhierarchical interactionsamong competing species represented by the ldquoRock-Paper-Scissorsrdquo (RPS) game in which rock crushes scissors scissorscut paper and paper wraps rock [9ndash16] Communities ofsubpopulations or species exhibiting such dynamics havebeen identified in several ecosystems [17] For example cyclicdominance of three male strategies has been reported inlizards [18]

It has been shown that cyclic dominance alone is notsufficient to preserve biodiversity When the interactionsbetween individuals are local spatially separated domainsare dominated by one species form leading to stable coex-istence [1ndash3] It is shown that beyond a critical value ofmobility biodiversity is lost [10] Researchers have shownthat spiral formation takes place without a conservation

law for the total density and in general fast diffusion candestroy species coexistence [11] The final survival proba-bilities characterizing different species in cyclic competitionhave also been reported [12] The interplay of evolutionarydynamics discreteness of the population and the nature ofthe interactions also influence the coexistence of strategies[13] Pattern formations due to cyclic dominance of threespecies operating near a bifurcation point have also beeninvestigated [14] The spreading of epidemic and its effecton biodiversity has been studied using the RPS game [15]The role of swarming in the context of biodiversity has alsobeen investigated [16] A recently published review paperon the cyclic dominance in evolutionary games covers indetail the current advances on RPS and related evolutionarygames In particular it focuses on pattern formation theimpact of mobility and the spontaneous emergence of cyclicdominance The authors also review mean-field and zero-dimensional RPS models and the application of the complexGinzburg-Landau equation and highlight the importance ofstatistical physics for the study of large-scale ecological sys-tems [19] Theoretical explorations of the relevance of spatialstructure in evolutionary games have also received a lot ofattention in the recent past [20ndash22] Realistic models beyond

Hindawi Publishing CorporationInternational Journal of BiodiversityVolume 2015 Article ID 128980 9 pageshttpdxdoiorg1011552015128980

2 International Journal of Biodiversity

well-mixed populations have been studied recently [2324] Researchers have also used mean-field theory to showthat group interactions at the mesoscopic scale have beentaken into account for the coexistence of a large number ofspecies [25] Recent work on mobility and velocity-enhancedcooperation of moving agents has further linked mobilityto biodiversity [26 27] The Ginzburg-Landau theory hasalso been applied successfully to study biodiversity usingmodels of spatial RPS games [28 29] Investigation of allianceformation among multiple number of interacting speciesshows that stable solutions are possible depending on theparameter values that determine invasion and mixing in themultispecies predator-prey systems [30]

Theprimarymotivation of this work is tomodel the infor-mation transfer between successive generations of evolutionand link it to biodiversity Further we wish to explore theeffect of the rate of information transfer on the existenceof biodiversity Yet another motivation is to study the non-hierarchical cyclic interactions between 119873 species using thisinformation-theoretic approach In this paper we providea novel approach using a cascaded communication chan-nel for analyzing biodiversity in the classic Lotka-Volterramodel Specifically we investigate the transitions from onegeneration to the next based on the transition probabilitymatrix between two consecutive generations The transitionbetween successive generations is modelled as communica-tion of information through a noisy communication channelWe derive explicit relation between the probabilities of thespecies in two consecutive generations in terms of the threebasic processes selection reproduction and mobility Thisgives us an elegant method to study the evolution of thespecies for long waiting times and ultimately leads to thecondition for the existence of biodiversity We also show howselection reproduction and mobility together influence theexistence of biodiversity

The paper is organized as follows Section 1 is the Intro-duction In Section 2 the systemmodel is discussed in detailIn Section 3 the transition from one generation to the nextis modelled as a problem of communicating symbols acrossa noisy channel The information transfer across generationsis linked to the existence of biodiversity in Section 4 Theproposed method is generalized for 119873 species in Section 5Results from lattice simulations are presented in Section 6Finally the paper is concluded in Section 7

2 System Model

We consider the cyclic Lotka-Volterra model where threestates 119860 119861 and 119862 cyclically dominate each other [31] Letthe time densities for the three species be 119886(119905) 119887(119905) and 119888(119905)respectively The deterministic rate equations describing thetime evolution of these three species can be written as

119886 = 119886 (119896119862119887 minus 119896119861119888)

= 119887 (119896119860119888 minus 119896119862119886)

119888 = 119888 (119896119861119886 minus 119896119860119887)

(1)

B AA B

C BB C

A CC A

E BB E

E CC E

E AA E

A E

B EB C

A B

C EC A

A E A A

B E B B

C E C C

Mobility (rate 120576p)

Selection (rate 120590p) Reproduction (rate 120583p)

Figure 1 The stochastic model The individuals of the three speciesare represented by119860 (blue) 119861 (yellow) and 119862 (red)The individualsinteract only with their immediate neighbors The selection repro-duction and mobility are modelled as the Poisson processes withrates 120590

119901 120583119901 and 120576

119901 respectively

As the spatial version of the Lotka-Volterra model weconsider three subpopulations or species (119860 119861 and 119862) themembers of which are distributed randomly over an 119872 times

119872 spatial lattice There are also empty spaces (119864) on thelattice which are essentially unoccupied For the possibleinteractions with the neighbours we consider a version of theRPS game namely a stochastic spatial variant of the model[4] The Rock Paper and Scissors are depicted by 119860 119861 and119862 respectively Figure 1 shows the basic processes of selec-tion reproduction and mobility modelled as the Poissonprocesses occurring at rates 120590

119901 120583119901 and 120576

119901 respectively The

individuals interact only with their immediate neighbors

3 Modeling Transition between Generations

Let us model the transition from generation 119894 to 119894 + 1 as aproblem of communicating four types of symbols (119860 119861 119862and 119864) across a noisy channel This channel occasionallymakes errors which result in one symbol being received as adifferent symbolThis is similar to a standard communicationchannel problem encountered in electrical engineering [32]This channel model is shown in Figure 2 In our case thesymbols are the lattice locations occupied by the individualsof the three species (or the empty locations) In the caseof transition from one generation to the next a site inthe lattice currently occupied by a particular species maybecome occupied by an individual from another species dueto mobility or fall vacant due to selection Alternatively anempty site may become occupied by a member from one ofthe species due to reproduction It is also possible that there isno change in the occupancy of a lattice site These transitionprobabilities are influenced by the selection rate reproductionrate and mobility

The transition probabilities are based on averaged prob-abilities (densities) ignoring the spatial structure This is

International Journal of Biodiversity 3

E

C

A

B

E

C

A

B

Generation i Generation i + 1

(1 minus 3120576 minus 120590) (120576) (120576) (120590 + 120576)

(120576) (1 minus 3120576 minus 120590) (120576) (120590 + 120576)

(120576) (120576) (1 minus 3120576 minus 120590) (120590 + 120576)

(120583 + 120576) (120583 + 120576) (120583 + 120576) (1 minus 3120583 minus 3120576)120583 + 120576

120583 + 120576120583 + 120576

1 minus 3120576 minus 120590

120576120576

Figure 2 The channel model The lattice locations occupied by the individuals of the three species undergo transition from one generationto the next Instead of labelling all the arrows the transition probabilities have been listed towards the left of the species for clarity Note that120590 120583 120576 are the normalized rates

a mean-field approach assuming a well-mixed populationwith the limit of population size 119873 rarr infin The ideacan be extended by considering joint densities of pairs ofneighbours and to approximate higher order densities onecan consider products of lower order densities (so-called pairapproximation and moment closure) [33]

The channel transition probability matrix for a well-mixed population can be written as

A B C E

A

B

C

E120583 + 120576 120583 + 120576 120583 + 120576

120576

120576 120576

120576

120576 1205761 minus 120590 minus 3120576

1 minus 120590 minus 3120576

1 minus 120590 minus 3120576

120590 + 120576

120590 + 120576

120590 + 120576

1 minus 3120583 minus 3120576

=P(2)

Any element 119901119897119898

of the channel transition probability matrixrepresents the probability of transition of a particular siteoccupied by a member of species 119897 to the same site occupiedby a member of species 119898 in the next generation Here119897 119898 = 1 2 3 and 4 correspond to119860 119861119862 and119864 respectivelyFor example 119901

12represents the probability that the site

occupied by species 119860 will become occupied by species 119861 inthe subsequent generation This is possible only by mobilityand hence 119901

12= 120576 Similarly 119901

34= (120590 + 120576) represents the

probability that the site occupied by species119862will translate to119864 in the next generation This is possible either by selection(120590) or by mobility (120576) which are mutually exclusive eventsand hence the probabilities add up It is important to notethat the entries in the transition probability matrix (P) areprobabilities consisting of normalized 120590 120583 120576 such that eachrow adds up to unity In order to differentiate the rates havebeen denoted by 120590119901 120583119901 and 120576119901

From (2) the eigenvalues [34] of matrix P are found outto be 1205821 = 1 1205822 = 1205823 = 1 minus 4120576 minus 120590 and 1205824 = 1 minus 4120576 minus 3120583 minus 120590In the case of biodiversity 119875(119860) = 119875(119861) = 119875(119862) that is inthe process of cyclic dominance the occupation of the sitesin the lattice by the three species is equiprobable Let 119901 be theprobability of occurrence of species 119860 119861 or 119862 at a particularlattice site for generation 119894 and let 119902 be the probability ofoccurrence of species 119860 119861 or 119862 at a particular lattice site

for generation 119894 + 1 We can alternately write the transitionprobability matrix as P = VDVminus1 where V is the matrixof the eigenvectors of P and D is the diagonal matrix withthe diagonal elements as the eigenvalues [34] Thus P can bewritten in terms of the eigenvalues as

P =

[[[[[

[

120572 120573 120573 120574

120573 120572 120573 120574

120573 120573 120572 120574

120575 120575 120575 1 minus 3120575

]]]]]

]

(3)

where

120572 =1

(120590 + 3120583 + 4120576)((120583 + 120576) +

23(120590 + 3120583+ 4120576) 1205822

+13(120590 + 120576) 1205824)

120573 =1

(120590 + 3120583 + 4120576)((120583 + 120576) minus

13(120590 + 3120583+ 4120576) 1205822

+13(120590 + 120576) 1205824)

120574 =1

(120590 + 3120583 + 4120576)((120590 + 120576) minus (120590 + 120576) 1205824)

120575 =1

(120590 + 3120583 + 4120576)((120583 + 120576) minus (120583 + 120576) 1205824)

(4)

We note that the transition probability matrices given in(2) and (3) are equivalent Using (3) we can express theprobabilities of individual species For example for species119860 the next generation 119860 may result from 119860 119861 119862 or 119864as dictated by the transition probability matrix given in (3)Thus for species119860 the probability of the next generation canbe expressed as

119902 = 119901120572+119901120573+119901120573+ (1minus 3119901) 120575 (5)

Similar expressions can be derived for species 119861 and 119862Solving (5) we obtain

119902 = (120583 + 120576

120590 + 3120583 + 4120576) +(119901minus(

120583 + 120576

120590 + 3120583 + 4120576))1205824 (6)


Channel Channel Channel Channel

Gen 1 Gen i Gen i + 1 Gen n

middot middot middot

P P P P

(a)

Equivalent channel

Gen 1 Gen n

Pn

(b)

Figure 3The cascaded channel model (a)The transition from one generation to the next can bemodelled as communication of informationthrough a noisy channel Thus long waiting times can be modelled as a cascade of similar channels (b) The equivalent channel for 119899generations

Equation (6) gives and explicit relation between the probabil-ities of species 119901 in generation 119894 to the probabilities of species119902 in the consecutive generation 119894 + 1

4 Link to Biodiversity

So far we have considered two consecutive generations onlyWe now develop a mathematical framework for large waitingtimes that is for a large number of generations (119905 prop

1198722) As we let the generations evolve using the stochastic

spatial variant of the of the RPS game it is equivalentto communicating through a cascade of noisy channels asdepicted in Figure 3(a) The equivalent channel transitionprobability matrix for 119899 cascaded channels [32] is given byP119899 Since we can express P119899 = VD119899Vminus1 the eigenvalues ofP119899 are given simply by 120582119899

1 120582119899

2 120582119899

3 120582119899

4 where 120582

1 1205822 1205823 1205824

are the eigenvalues of P Thus 119899 cascaded channels canbe represented as one ldquoequivalent channelrdquo as shown inFigure 3(b) From (6) we can easily write the probabilities ofspecies 119860 119861 and 119862 after 119899 generations as

119902119899= (

120583 + 120576

120590 + 3120583 + 4120576) +(119901minus(

120583 + 120576

120590 + 3120583 + 4120576))120582119899

4 (7)

Since the matrix P is a probability transition matrix|1205824| lt |120582max| = 1 [22]Thus 1205821198994 rarr 0 Hence from (7) we canconclude that a well-mixed population must always result inbiodiversity

We now consider the real-life scenario when the popula-tion is notwell-mixed In fact clustering is routinely observedas the population evolves [10 14 31] In this case also theassumption of large population sizes exists that is size119873 rarr

infin Let us represent the resultant transition matrix for a non-well-mixed (clustered) scenario as

R119894= P+Q

119894 (8)

where P represents the transition matrix corresponding to awell-mixed population and Q119894 represents the effect of localclustering at generation 119894 We observe that the effect of theQ119894 matrix is to alter the transition probabilities (the branchlabels of Figure 2) Since the resultant matrix R119894 is also aprobability transitionmatrix with the individual rows addingup to unityQ

119894must be a matrix with individual rows adding

up to zero After 119899 generations the effective channel canbe represented by Reff = prod

119899

119894=1R119894 where R119894is the resultant

transition matrix for generation 119894 The relation between the

probabilities for species119860 119861 and 119862 and the vacant locations119864 can be expressed as

[119901119860 119901119861119901119862119901119864]119899

= [119901119860 119901119861119901119862119901119864]1 Reff (9)

where [119901119860 119901119861119901119862119901119864]119899

represents the resulting probabili-ties for species119860 119861 and119862 and the vacant locations 119864 after 119899generations while [119901119860 119901

119861119901119862119901119864]1

represents the startingprobabilities Extinction implies that one of the species diesout that is the left hand side of (9) should be [0 119901

119861119901119862119901119864]119899

or [119901119860 0 119901119862119901119864]119899

or [119901119860 1199011198610 119901119864]119899

for arbitrary startingprobabilities [119901119860 119901

119861119901119862119901119864]1

This is possible if one of thefirst three columns ofReff is zero which gives us the necessarycondition for extinction (uniformity) |120582119877eff |min = 0Thus thecondition for the existence of biodiversity can be stated as

10038161003816100381610038161003816120582119877eff

10038161003816100381610038161003816min gt 0 (10)

Since Reff = prod119899

119894=1R119894 the condition for biodiversity can alsobe written as |120582

119877119894|min gt 0 for all 119894 This is one of our

central results An important contribution of this paper isto model the ldquotransitionsrdquo of cells like the ldquotransitionsrdquo ofsymbols over a noisy channel in communications This leadsto a matrix representation which eventually has an elegantsolution in terms of eigenvalues when a cascade of channelsis considered

5 Generalization to 119873 Species

The given theory can be easily extended to study the nonhier-archical cyclic interactions between 119873 species in general Inthis case the two consecutive generations of 119873 interactingspecies can be linked using an (119873 + 1) times (119873 + 1) transitionprobability matrix with eigenvalues given by 120582

1= 1 120582

2=

1205823= sdot sdot sdot = 120582

119873= 1 minus 120590 minus (119873 + 1)120576 and 120582

119873+1 = 1 minus 120590 minus 119873120583 minus(119873+ 1)120576 For a well-mixed population the expression for theprobabilities of119873 species after 119899 generations is calculated tobe

119902119899 = (120583 + 120576

120590 + 119873120583 + (119873 + 1) 120576)

+(119901minus(120583 + 120576

120590 + 119873120583 + (119873 + 1) 120576)) 120582119899

119873+1

(11)

We surmise from (11) that in case of biodiversity the latticearea occupied by each species on an average will be ((120583 +


0 100 200 300 400 5000

02

04

06

08

1

Generation number

120582i

1205822

1205823

1205824

(a)

0 100 200 300 400 5000

02

04

06

08

1

Generation number

120582i

1205822

1205823

1205824

(b)

Figure 4 Typical plot of the eigenvalues of resultant probability matrix R119894where 119894 represents the generation number 120582

2(red) 120582

3(green)

and 1205824(blue) (a) when biodiversity occurs and (b) when extinction happens

120576)(120590 + 119873120583 + (119873 + 1)120576))1198722 where1198722 is the total number oflattice sites

Another interesting aspect of cascaded channels is studyof mutual information between consecutive generations Adiscrete channel is a system with input alphabet 119883 outputalphabet 119884 and a probability transition matrix 119901(119910 | 119909) Theamount of information conveyed by the discrete channel isquantified by the mutual information between 119883 and 119884 andis defined as [32]

119868 (119883 119884) = 119867 (119884) minus119867 (119884 | 119883) (12)

where119867(119884) is the entropy of 119884 and119867(119884 | 119883) is the entropyof 119884 given119883 The entropy119867(119884) (in bits) is calculated using

119867(119884) = minus

4sum

119894=1119910119894log2119910119894 (13)

and in general 119867(119885) = 119867(1199111 1199112 ) = minussum119911119894log2119911119894

where 119885 is a discrete random variable The notion of mutualinformation is routinely used to determine the capacity of achannel in the area of communicationsHerewewish to studyhow the rate of transfer of information over successive gen-erations affects the possibility of coexistence among speciesIn our case for a 4 times 4 P matrix the expression for mutualinformation can be written as

119868 (119883 119884) =

4sum

119895=1

4sum

119894=1119875 (119910119895) 119875 (119909119894 | 119910119895) log

119875 (119909119894| 119910119895)

119875 (119909119894)

(14)

Solving for mutual information we obtain

119868 (119883 119884)

= 119867 (119884)

minus (3119901119867 (120572 120573 120573 120574) + (1minus 3119901)119867 (120575 120575 120575 1minus 3120575))

(15)

where 120572 120573 120574 and 120575 are given in (4)

6 Lattice Simulations

Extensive computer simulations were carried out to testthe condition for biodiversity as predicted by (10) Cyclicdominance is modeled using a stochastic lattice [2 4] Inour stochastic lattice simulations we have arranged the threespecies on a two-dimensional square lattice with periodicboundary conditions By periodic boundary conditions weimply that the sites on the left edge of the grid are viewed asadjacent to ones on the right edge and those on the bottomare viewed as adjacent to those on the top Every latticesite is occupied by an individual of species 119860 species 119861 orspecies119862 or left empty (119864) At each simulation step a randomindividual interacts with one of its eight nearest neighbours(corresponding to the move of the King in a chess game)Thechoice of the neighbour is also randomly determined (eachof the eight neighbours are equally likely) The interactionwith the neighbour reproduction or migration as well asthe corresponding waiting time is carried out accordingto a built-in function in MATLAB that generates randomvalues from a Poisson distribution In our simulations onegeneration is counted when every individual has reacted onan average once

Figure 4(a) shows a typical plot of the eigenvalues of R119894

1205822(red) 120582

3(green) and 120582

4(blue) when biodiversity occurs

Here the lattice size is 100 times 100 It is clear from the plot that120582119877119894|min = 1205824 gt 0 and hence biodiversity is observed In

Figure 4(b) we show the casewhen extinction happensHereas soon as 120582

4asymp 0 one of the species dies out

Simulations were also carried out for different values of120590119901 120583119901 and 120576

119901 and the system was tested for the existence

of biodiversity (or uniformity) after long waiting periodsThese operating points are plotted in Figure 5(a) We showthe results for different system sizes 50 times 50 lattice sites70 times 70 lattice sites and 100 times 100 lattice sites To obtainthe extinction probability the results are averaged over 100


0 02 04 06 08 10

02

04

06

08

1

50 times 50

70 times 70

100 times 100

Pex

t

|120582|min

(a) (b)

Figure 5The probability of extinction 119875ext (a) 119875ext for different system sizes 50 times 50 (red 998779) 70 times 70 (blue +) and 100 times 100 (black o) (b) Atypical snapshot from lattice simulations after a long waiting time for the case when biodiversity exists (three colors 119860 119861 and 119862 black dotsempty spaces 119864)

realizations Simulation results corroborate the theory Theprobability of extinction 119875ext tends to zero for the caseswhere |120582

119877eff|min gt 0 while |120582

119877119894|min = 0 leads to uniformity

We observe that as the system size increases the transitionbetween biodiversity and uniformity sharpens In order todetermine |120582

119877119894| the actual values of the normalized rates 120590 120583

and 120576were extracted by carrying out a frequency count of thenumber of lattice-site transitions from one generation to thenext and then averaging these values over 500 generationsFigure 5(b) shows a typical snapshot from lattice simulationsafter a long waiting time for the case when biodiversity exists

The proposed mathematical framework can be conve-niently used to study systems with unequal reaction rates byappropriatelymodifying the transitionmatrixP Suppose thethree species have different normalized rates for selectionreproduction and mobility given by 120590119894 120583119894 120576119894 119894 = 1 2 and3 Then the corresponding transition probability matrix forunequal normalized rates can be written as

P119880

=

[[[[[[[[

[

1 minus 1205903 minus 31205761 1205761 1205761 1205903 + 1205761

1205762 1 minus 1205901 minus 31205762 1205762 1205901 + 1205762

1205763 1205763 1 minus 1205902 minus 31205763 1205902 + 1205763

1205831 + 1205761 1205832 + 1205762 1205833 + 1205763 1 minus3sum

119894=1(120588119894 + 120576119894)

]]]]]]]]

]

(16)

The eigenvalues of this generalized transition probabilitymatrix can be used to study the effects of the different param-eters on biodiversity As an illustrative example supposewe wish to investigate the effect of unequal (normalized)reproduction rates 1205831 1205832 and 1205833 for the three species (120590 and 120576are the same for all three species) From (12) we can calculatethe eigenvalues as 120582

1= 1 120582

2= 1205823= 1 minus 120590 minus 4120576 and

1205824= 1 minus 120590 minus (120583

1+ 1205832+ 1205833) minus 4120576

Another interesting observation from nature is the factthat resource competitors can benefit one another throughcontainment of shared competitors [35 36] When several

factors determine the outcome of competition the inter-acting species can be modeled as a competitive networkas shown in Figure 6(a) Here species are shown as nodesand arrows connect the competitive inferior to the superiorcompetitor Researchers have shown that intransitivity incompetitive networks can maintain diversity [3 37] Theproposed cascaded channel model can be easily adapted toinclude different numbers of competitive relationships in acompetitive network as illustrated in Figure 6(a) Figure 6(b)gives the generic channel model for 119903 species and 119904 compet-itive relationships which can be represented as a transitionprobability matrix and analyzed for the existence of biodiver-sity as discussed earlier

Figure 7 shows the plot of mutual information versusgeneration number for typical cases where coexistence isobserved (cases (a) and (b)) and typical cases where extinc-tion occurs (cases (c) and (d)) These simulations are forlattice size 100 times 100 In these experiments for differentsets of values for 120590 120583 120576 we determine whether the latticesimulations lead to biodiversity or uniformity It is interestingto observe that 119868(119883 119884) between successive generations isfairly stable for the cases when biodiversity occurs whileit shows an increasing trend for the cases when uniformityoccurs The trend is depicted by the dotted line in thefigure It appears that ldquogreedyrdquo information transfer betweensuccessive generations may lead to conditions for extinction

7 Conclusion

We have developed a framework using cascaded channelmodel for analyzing the existence of biodiversity in nonhier-archical interactions among119873 competing speciesThismath-ematical framework gives a clear insight into how the threeparameters reproduction rate selection rate and mobilityinfluence biodiversity To give a biological interpretationwe can draw a parallel to colicinogenic strains of E coligrowing on a Petri dish [3]These strains can produce a toxinand the ldquoantidoterdquo which determine the selection rate (120590119901)


S1

Sr

S4 S3

S2

(a)

E E

S1S1

Sr Sr

S3 S3

S2 S2

(1 minus r120576 minus s120590) (120576) (120576) (s120590 + 120576)

(120576) (1 minus r120576 minus s120590) (120576) (s120590 + 120576)

(120576) (120576) (1 minus r120576 minus s120590) (s120590 + 120576)

(s120583 + 120576) (s120583 + 120576) (s120583 + 120576) (1 minus rs120583 minus r120576)

(b)

Figure 6 (a) Competitive network with 119903 species (1198781 1198782 119878

119903) 119903 = 5 and 119904 = 2 competitive relationships (b) Generic channel model with

119903 species and 119904 competitive relationships Note that it is a fully connected channel model however not all connections are shown for the sakeof clarity

0 100 200 300 400 500 60004

06

08

1

12

14

Generation number

I(XY

)

(a)

0 100 200 300 400 500 60002

04

06

08

1

12

Generation number

I(XY

)

(b)

0 100 200 300 400 5000

02

04

06

08

1

Generation number

I(XY

)

(c)

0 50 100 150 200 250 3000

02

04

06

08

1

Generation number

I(XY

)

(d)

Figure 7 Mutual information 119868(119883 119884) between each consecutive generation for all generations Cases (a) and (b) represent the typicalscenarios which lead to biodiversity while (c) and (d) represent the cases when extinction occurs Simulations are for lattice size 100 times 100


However both the toxin and the ldquoantidoterdquo are generated byusing resources and hence at the cost of their reproductionrate (120583

119901) The mobility of E coli can be increased by using

supersoft agar On the other hand this mobility can belowered by increasing the agar concentrationThis ismodeledby our mobility parameter 120576

119901

The significant contributions of this paper are as follows

(1) The transition between successive generations ismodelled as communication of information througha noisy communication channel This informationtheoreticmodel is a paradigm shift and is not linked toany of the previously reported models For exampleusing this model one can understand the interde-pendence on reproduction rate selection rate andmobility together on the existence of biodiversity

(2) The cascaded channel model exploits a very uniquecharacteristic ofmatrices the ability to express squarematrices in terms of their eigenvalues This approachpermits a cascade of channels to be modelled as aproduct of several matrices This approach leads toan elegant solution in terms of the power of theeigenvalues

(3) The versatility of this approach allows it to beextended to studymore than three interacting speciesspecies with ecosystems with unequal rates for differ-ent species and competitive networks with differentcompetitive relationships

(4) The rate of transfer of information over successivegenerations is studied using mutual information andit is found that ldquogreedyrdquo information transfer betweensuccessive generations may lead to conditions forextinction

(5) Explicit dependence of the probability of populationdistribution on the transition probabilities linked tothe reproduction rate selection rate and mobility isderived in this paper The condition for biodiversityis derived which is corroborated by simulations

(6) This generalized mathematical framework can beused to study biodiversity in any number of inter-acting species ecosystems with unequal rates fordifferent species and also competitive networks

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] R Durrett and S Levin ldquoAllelopathy in spatially distributedpopulationsrdquo Journal of Theoretical Biology vol 185 no 2 pp165ndash171 1997

[2] R Durrett and S Levin ldquoSpatial aspects of interspecific compe-titionrdquo Theoretical Population Biology vol 53 no 1 pp 30ndash431998

[3] B Kerr M A Riley M W Feldman and B J M BohannanldquoLocal dispersal promotes biodiversity in a real-life game of

rock-paper-scissorsrdquo Nature vol 418 no 6894 pp 171ndash1742002

[4] RMMay andW J Leonard ldquoNonlinear aspects of competitionbetween three speciesrdquo SIAM Journal on Applied Mathematicsvol 29 no 2 pp 243ndash253 1975

[5] R V Sole and J Bascompte Self-Organization in ComplexEcosystems Princeton University Press Princeton NJ USA2006

[6] J von Neumann and O Morgenstern Theory of Games andEconomic Behavior Princeton University Press Princeton NJUSA 1944

[7] J M Smith Evolution and the Theory of Games CambridgeUniversity Press 1982

[8] J W Weibull Evolutionary Game Theory The MIT PressCambridge Mass USA 2002

[9] D Neal Introduction to Population Biology Cambridge Univer-sity Press Cambridge UK 2004

[10] T Reichenbach M Mobilia and E Frey ldquoMobility promotesand jeopardizes biodiversity in rockndashpaperndashscissors gamesrdquoNature vol 448 no 7157 pp 1046ndash1049 2007

[11] M Peltomaki andM Alava ldquoThree- and four-state rock-paper-scissors games with diffusionrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 78 no 3 Article ID031906 7 pages 2008

[12] M Berr T Reichenbach M Schottenloher and E Frey ldquoZero-one survival behavior of cyclically competing speciesrdquo PhysicalReview Letters vol 102 no 4 Article ID 048102 2009

[13] J C Claussen and A Traulsen ldquoCyclic dominance and biodi-versity in well-mixed populationsrdquo Physical Review Letters vol100 no 5 Article ID 058104 2008

[14] T Reichenbach and E Frey ldquoInstability of spatial patterns andits ambiguous impact on species diversityrdquo Physical ReviewLetters vol 101 no 5 Article ID 058102 2008

[15] W-X Wang Y-C Lai and C Grebogi ldquoEffect of epidemicspreading on species coexistence in spatial rock-paper-scissorsgamesrdquo Physical Review E vol 81 no 4 Article ID 046113 2010

[16] R Bose ldquoEffect of swarming on biodiversity in non-symmetricrock-paper-scissor gamerdquo IET Systems Biology vol 4 no 3 pp177ndash184 2010

[17] J B C Jackson andL Buss ldquoAllelopathy and spatial competitionamong coral reef invertebratesrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 72 pp5160ndash5163 1975

[18] B Sinervo andCM Lively ldquoThe rockndashscissorsndashpaper game andthe evolution of alternativemale strategiesrdquoNature vol 380 no6571 pp 240ndash243 1996

[19] A Szolnoki M Mobilia L Jiang B Szczesny A M Rucklidgeand M Perc ldquoCyclic dominance in evolutionary games areviewrdquo Journal of the Royal Society Interface vol 11 no 100Article ID 20140735 2014

[20] M Perc and A Szolnoki ldquoCoevolutionary games a minireviewrdquo BioSystems vol 99 no 2 pp 109ndash125 2010

[21] E Frey ldquoEvolutionary game theory theoretical concepts andapplications to microbial communitiesrdquo Physica A vol 389 no20 pp 4265ndash4298 2010

[22] M Perc J Gomez-Gardenes A Szolnoki L M Florıa andY Moreno ldquoEvolutionary dynamics of group interactions onstructured populations a reviewrdquo Journal of The Royal SocietyInterface vol 10 no 80 Article ID 20120997 2013


[23] P Bednarik K Fehl and D Semmann ldquoCosts for switch-ing partners reduce network dynamics but not cooperativebehaviourrdquo Proceedings of the Royal Society B Biological Sci-ences vol 281 no 1792 Article ID 20141661 2014

[24] G S van Doorn T Riebli and M Taborsky ldquoCoaction versusreciprocity in continuous-time models of cooperationrdquo Journalof Theoretical Biology vol 356 pp 1ndash10 2014

[25] H Cheng N Yao Z Huang J Park Y Do and Y Lai ldquoMeso-scopic interactions and species coexistence in evolutionarygame dynamics of cyclic competitionsrdquo Scientific Reports vol4 article 7486 2014

[26] A Cardillo S Meloni J Gomez-Gardenes and Y MorenoldquoVelocity-enhanced cooperation of moving agents playing pub-lic goods gamesrdquo Physical Review E vol 85 Article ID 0671012012

[27] M H Vainstein and J J Arenzon ldquoSpatial social dilemmasdilution mobility and grouping effects with imitation dynam-icsrdquo Physica A vol 394 pp 145ndash157 2014

[28] I S Aranson and L Kramer ldquoThe world of the complexGinzburg-Landau equationrdquoReviews ofModern Physics vol 74no 1 pp 99ndash143 2002

[29] B Szczesny M Mobilia and A Rucklidge ldquoCharacterizationof spiraling patterns in spatial rock-paper-scissors gamesrdquoPhysical Review E vol 90 Article ID 032704 2014

[30] G Szabo A Szolnoki and I Borsos ldquoSelf-organizing patternsmaintained by competing associations in a six-species predator-prey modelrdquo Physical Review E vol 77 no 4 Article ID 0419192008

[31] T Reichenbach M Mobilia and E Frey ldquoSelf-organization ofmobile populations in cyclic competitionrdquo Journal ofTheoreticalBiology vol 254 no 2 pp 368ndash383 2008

[32] R Bose Information Theory Coding and Cryptography TataMcGraw-Hill Noida India 2002

[33] G Szabo and G Fath ldquoEvolutionary games on graphsrdquo PhysicsReports vol 446 no 4ndash6 pp 97ndash216 2007

[34] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press Cambridge UK 2012

[35] L Stone and A Roberts ldquoConditions for a species to gainadvantage from the presence of competitorsrdquo Ecology vol 72no 6 pp 1964ndash1972 1991

[36] S Allesina and J M Levine ldquoA competitive network theoryof species diversityrdquo Proceedings of the National Academy ofSciences of theUnited States of America vol 108 no 14 pp 5638ndash5642 2011

[37] R A Laird and B S Schamp ldquoCompetitive intransitivitypromotes species coexistencerdquo The American Naturalist vol168 no 2 pp 182ndash193 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Anatomy Research International

PeptidesInternational Journal of


Hindawi Publishing Corporation httpwwwhindawicom

International Journal of

Volume 2014

Zoology


Molecular Biology International

GenomicsInternational Journal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


BioinformaticsAdvances in

Marine BiologyJournal of



Signal TransductionJournal of


BioMed Research International

Evolutionary BiologyInternational Journal of



Biochemistry Research International

ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Genetics Research International


Advances in

Virolog y

Hindawi Publishing Corporationhttpwwwhindawicom

Nucleic AcidsJournal of

Volume 2014

Stem CellsInternational



Enzyme Research



Microbiology


well-mixed populations have been studied recently [2324] Researchers have also used mean-field theory to showthat group interactions at the mesoscopic scale have beentaken into account for the coexistence of a large number ofspecies [25] Recent work on mobility and velocity-enhancedcooperation of moving agents has further linked mobilityto biodiversity [26 27] The Ginzburg-Landau theory hasalso been applied successfully to study biodiversity usingmodels of spatial RPS games [28 29] Investigation of allianceformation among multiple number of interacting speciesshows that stable solutions are possible depending on theparameter values that determine invasion and mixing in themultispecies predator-prey systems [30]

Theprimarymotivation of this work is tomodel the infor-mation transfer between successive generations of evolutionand link it to biodiversity Further we wish to explore theeffect of the rate of information transfer on the existenceof biodiversity Yet another motivation is to study the non-hierarchical cyclic interactions between 119873 species using thisinformation-theoretic approach In this paper we providea novel approach using a cascaded communication chan-nel for analyzing biodiversity in the classic Lotka-Volterramodel Specifically we investigate the transitions from onegeneration to the next based on the transition probabilitymatrix between two consecutive generations The transitionbetween successive generations is modelled as communica-tion of information through a noisy communication channelWe derive explicit relation between the probabilities of thespecies in two consecutive generations in terms of the threebasic processes selection reproduction and mobility Thisgives us an elegant method to study the evolution of thespecies for long waiting times and ultimately leads to thecondition for the existence of biodiversity We also show howselection reproduction and mobility together influence theexistence of biodiversity

The paper is organized as follows Section 1 is the Intro-duction In Section 2 the systemmodel is discussed in detailIn Section 3 the transition from one generation to the nextis modelled as a problem of communicating symbols acrossa noisy channel The information transfer across generationsis linked to the existence of biodiversity in Section 4 Theproposed method is generalized for 119873 species in Section 5Results from lattice simulations are presented in Section 6Finally the paper is concluded in Section 7

2 System Model

We consider the cyclic Lotka-Volterra model where threestates 119860 119861 and 119862 cyclically dominate each other [31] Letthe time densities for the three species be 119886(119905) 119887(119905) and 119888(119905)respectively The deterministic rate equations describing thetime evolution of these three species can be written as

119886 = 119886 (119896119862119887 minus 119896119861119888)

= 119887 (119896119860119888 minus 119896119862119886)

119888 = 119888 (119896119861119886 minus 119896119860119887)

(1)

B AA B

C BB C

A CC A

E BB E

E CC E

E AA E

A E

B EB C

A B

C EC A

A E A A

B E B B

C E C C

Mobility (rate 120576p)

Selection (rate 120590p) Reproduction (rate 120583p)

Figure 1 The stochastic model The individuals of the three speciesare represented by119860 (blue) 119861 (yellow) and 119862 (red)The individualsinteract only with their immediate neighbors The selection repro-duction and mobility are modelled as the Poisson processes withrates 120590

119901 120583119901 and 120576

119901 respectively

As the spatial version of the Lotka-Volterra model weconsider three subpopulations or species (119860 119861 and 119862) themembers of which are distributed randomly over an 119872 times

119872 spatial lattice There are also empty spaces (119864) on thelattice which are essentially unoccupied For the possibleinteractions with the neighbours we consider a version of theRPS game namely a stochastic spatial variant of the model[4] The Rock Paper and Scissors are depicted by 119860 119861 and119862 respectively Figure 1 shows the basic processes of selec-tion reproduction and mobility modelled as the Poissonprocesses occurring at rates 120590

119901 120583119901 and 120576

119901 respectively The

individuals interact only with their immediate neighbors

3 Modeling Transition between Generations

Let us model the transition from generation 119894 to 119894 + 1 as aproblem of communicating four types of symbols (119860 119861 119862and 119864) across a noisy channel This channel occasionallymakes errors which result in one symbol being received as adifferent symbolThis is similar to a standard communicationchannel problem encountered in electrical engineering [32]This channel model is shown in Figure 2 In our case thesymbols are the lattice locations occupied by the individualsof the three species (or the empty locations) In the caseof transition from one generation to the next a site inthe lattice currently occupied by a particular species maybecome occupied by an individual from another species dueto mobility or fall vacant due to selection Alternatively anempty site may become occupied by a member from one ofthe species due to reproduction It is also possible that there isno change in the occupancy of a lattice site These transitionprobabilities are influenced by the selection rate reproductionrate and mobility

The transition probabilities are based on averaged prob-abilities (densities) ignoring the spatial structure This is


E

C

A

B

E

C

A

B


(1 minus 3120576 minus 120590) (120576) (120576) (120590 + 120576)

(120576) (1 minus 3120576 minus 120590) (120576) (120590 + 120576)

(120576) (120576) (1 minus 3120576 minus 120590) (120590 + 120576)

(120583 + 120576) (120583 + 120576) (120583 + 120576) (1 minus 3120583 minus 3120576)120583 + 120576

120583 + 120576120583 + 120576

1 minus 3120576 minus 120590

120576120576




A B C E

A

B

C

E120583 + 120576 120583 + 120576 120583 + 120576

120576

120576 120576

120576

120576 1205761 minus 120590 minus 3120576

1 minus 120590 minus 3120576

1 minus 120590 minus 3120576

120590 + 120576

120590 + 120576

120590 + 120576

1 minus 3120583 minus 3120576

=P(2)

Any element 119901119897119898




12= 120576 Similarly 119901

34= (120590 + 120576) represents the




P =

[[[[[

[

120572 120573 120573 120574

120573 120572 120573 120574

120573 120573 120572 120574

120575 120575 120575 1 minus 3120575

]]]]]

]

(3)

where

120572 =1

(120590 + 3120583 + 4120576)((120583 + 120576) +

23(120590 + 3120583+ 4120576) 1205822

+13(120590 + 120576) 1205824)

120573 =1

(120590 + 3120583 + 4120576)((120583 + 120576) minus

13(120590 + 3120583+ 4120576) 1205822

+13(120590 + 120576) 1205824)

120574 =1

(120590 + 3120583 + 4120576)((120590 + 120576) minus (120590 + 120576) 1205824)

120575 =1

(120590 + 3120583 + 4120576)((120583 + 120576) minus (120583 + 120576) 1205824)

(4)


119902 = 119901120572+119901120573+119901120573+ (1minus 3119901) 120575 (5)


119902 = (120583 + 120576

120590 + 3120583 + 4120576) +(119901minus(

120583 + 120576

120590 + 3120583 + 4120576))1205824 (6)





P P P P

(a)

Equivalent channel

Gen 1 Gen n

Pn

(b)







1 120582119899

2 120582119899

3 120582119899

4 where 120582

1 1205822 1205823 1205824


119902119899= (

120583 + 120576

120590 + 3120583 + 4120576) +(119901minus(

120583 + 120576

120590 + 3120583 + 4120576))120582119899

4 (7)




R119894= P+Q

119894 (8)




119899




[119901119860 119901119861119901119862119901119864]119899

= [119901119860 119901119861119901119862119901119864]1 Reff (9)

where [119901119860 119901119861119901119862119901119864]119899


119861119901119862119901119864]1


119861119901119862119901119864]119899

or [119901119860 0 119901119862119901119864]119899

or [119901119860 1199011198610 119901119864]119899


119861119901119862119901119864]1


10038161003816100381610038161003816120582119877eff

10038161003816100381610038161003816min gt 0 (10)







1= 1 120582

2=

1205823= sdot sdot sdot = 120582

119873= 1 minus 120590 minus (119873 + 1)120576 and 120582


119902119899 = (120583 + 120576

120590 + 119873120583 + (119873 + 1) 120576)

+(119901minus(120583 + 120576

120590 + 119873120583 + (119873 + 1) 120576)) 120582119899

119873+1

(11)



0 100 200 300 400 5000

02

04

06

08

1

Generation number

120582i

1205822

1205823

1205824

(a)

0 100 200 300 400 5000

02

04

06

08

1

Generation number

120582i

1205822

1205823

1205824

(b)


2(red) 120582

3(green)




119868 (119883 119884) = 119867 (119884) minus119867 (119884 | 119883) (12)


119867(119884) = minus

4sum

119894=1119910119894log2119910119894 (13)



119868 (119883 119884) =

4sum

119895=1

4sum

119894=1119875 (119910119895) 119875 (119909119894 | 119910119895) log

119875 (119909119894| 119910119895)

119875 (119909119894)

(14)


119868 (119883 119884)

= 119867 (119884)


(15)





1205822(red) 120582

3(green) and 120582









0 02 04 06 08 10

02

04

06

08

1

50 times 50

70 times 70

100 times 100

Pex

t

|120582|min

(a) (b)









P119880

=

[[[[[[[[

[

1 minus 1205903 minus 31205761 1205761 1205761 1205903 + 1205761

1205762 1 minus 1205901 minus 31205762 1205762 1205901 + 1205762

1205763 1205763 1 minus 1205902 minus 31205763 1205902 + 1205763

1205831 + 1205761 1205832 + 1205762 1205833 + 1205763 1 minus3sum

119894=1(120588119894 + 120576119894)

]]]]]]]]

]

(16)


1= 1 120582

2= 1205823= 1 minus 120590 minus 4120576 and

1205824= 1 minus 120590 minus (120583

1+ 1205832+ 1205833) minus 4120576




7 Conclusion



S1

Sr

S4 S3

S2

(a)

E E

S1S1

Sr Sr

S3 S3

S2 S2

(1 minus r120576 minus s120590) (120576) (120576) (s120590 + 120576)

(120576) (1 minus r120576 minus s120590) (120576) (s120590 + 120576)

(120576) (120576) (1 minus r120576 minus s120590) (s120590 + 120576)


(b)




0 100 200 300 400 500 60004

06

08

1

12

14

Generation number

I(XY

)

(a)

0 100 200 300 400 500 60002

04

06

08

1

12

Generation number

I(XY

)

(b)

0 100 200 300 400 5000

02

04

06

08

1

Generation number

I(XY

)

(c)

0 50 100 150 200 250 3000

02

04

06

08

1

Generation number

I(XY

)

(d)






119901










References















































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


E

C

A

B

E

C

A

B


(1 minus 3120576 minus 120590) (120576) (120576) (120590 + 120576)

(120576) (1 minus 3120576 minus 120590) (120576) (120590 + 120576)

(120576) (120576) (1 minus 3120576 minus 120590) (120590 + 120576)

(120583 + 120576) (120583 + 120576) (120583 + 120576) (1 minus 3120583 minus 3120576)120583 + 120576

120583 + 120576120583 + 120576

1 minus 3120576 minus 120590

120576120576




A B C E

A

B

C

E120583 + 120576 120583 + 120576 120583 + 120576

120576

120576 120576

120576

120576 1205761 minus 120590 minus 3120576

1 minus 120590 minus 3120576

1 minus 120590 minus 3120576

120590 + 120576

120590 + 120576

120590 + 120576

1 minus 3120583 minus 3120576

=P(2)

Any element 119901119897119898




12= 120576 Similarly 119901

34= (120590 + 120576) represents the




P =

[[[[[

[

120572 120573 120573 120574

120573 120572 120573 120574

120573 120573 120572 120574

120575 120575 120575 1 minus 3120575

]]]]]

]

(3)

where

120572 =1

(120590 + 3120583 + 4120576)((120583 + 120576) +

23(120590 + 3120583+ 4120576) 1205822

+13(120590 + 120576) 1205824)

120573 =1

(120590 + 3120583 + 4120576)((120583 + 120576) minus

13(120590 + 3120583+ 4120576) 1205822

+13(120590 + 120576) 1205824)

120574 =1

(120590 + 3120583 + 4120576)((120590 + 120576) minus (120590 + 120576) 1205824)

120575 =1

(120590 + 3120583 + 4120576)((120583 + 120576) minus (120583 + 120576) 1205824)

(4)


119902 = 119901120572+119901120573+119901120573+ (1minus 3119901) 120575 (5)


119902 = (120583 + 120576

120590 + 3120583 + 4120576) +(119901minus(

120583 + 120576

120590 + 3120583 + 4120576))1205824 (6)





P P P P

(a)

Equivalent channel

Gen 1 Gen n

Pn

(b)







1 120582119899

2 120582119899

3 120582119899

4 where 120582

1 1205822 1205823 1205824


119902119899= (

120583 + 120576

120590 + 3120583 + 4120576) +(119901minus(

120583 + 120576

120590 + 3120583 + 4120576))120582119899

4 (7)




R119894= P+Q

119894 (8)




119899




[119901119860 119901119861119901119862119901119864]119899

= [119901119860 119901119861119901119862119901119864]1 Reff (9)

where [119901119860 119901119861119901119862119901119864]119899


119861119901119862119901119864]1


119861119901119862119901119864]119899

or [119901119860 0 119901119862119901119864]119899

or [119901119860 1199011198610 119901119864]119899


119861119901119862119901119864]1


10038161003816100381610038161003816120582119877eff

10038161003816100381610038161003816min gt 0 (10)







1= 1 120582

2=

1205823= sdot sdot sdot = 120582

119873= 1 minus 120590 minus (119873 + 1)120576 and 120582


119902119899 = (120583 + 120576

120590 + 119873120583 + (119873 + 1) 120576)

+(119901minus(120583 + 120576

120590 + 119873120583 + (119873 + 1) 120576)) 120582119899

119873+1

(11)



0 100 200 300 400 5000

02

04

06

08

1

Generation number

120582i

1205822

1205823

1205824

(a)

0 100 200 300 400 5000

02

04

06

08

1

Generation number

120582i

1205822

1205823

1205824

(b)


2(red) 120582

3(green)




119868 (119883 119884) = 119867 (119884) minus119867 (119884 | 119883) (12)


119867(119884) = minus

4sum

119894=1119910119894log2119910119894 (13)



119868 (119883 119884) =

4sum

119895=1

4sum

119894=1119875 (119910119895) 119875 (119909119894 | 119910119895) log

119875 (119909119894| 119910119895)

119875 (119909119894)

(14)


119868 (119883 119884)

= 119867 (119884)


(15)





1205822(red) 120582

3(green) and 120582









0 02 04 06 08 10

02

04

06

08

1

50 times 50

70 times 70

100 times 100

Pex

t

|120582|min

(a) (b)









P119880

=

[[[[[[[[

[

1 minus 1205903 minus 31205761 1205761 1205761 1205903 + 1205761

1205762 1 minus 1205901 minus 31205762 1205762 1205901 + 1205762

1205763 1205763 1 minus 1205902 minus 31205763 1205902 + 1205763

1205831 + 1205761 1205832 + 1205762 1205833 + 1205763 1 minus3sum

119894=1(120588119894 + 120576119894)

]]]]]]]]

]

(16)


1= 1 120582

2= 1205823= 1 minus 120590 minus 4120576 and

1205824= 1 minus 120590 minus (120583

1+ 1205832+ 1205833) minus 4120576




7 Conclusion



S1

Sr

S4 S3

S2

(a)

E E

S1S1

Sr Sr

S3 S3

S2 S2

(1 minus r120576 minus s120590) (120576) (120576) (s120590 + 120576)

(120576) (1 minus r120576 minus s120590) (120576) (s120590 + 120576)

(120576) (120576) (1 minus r120576 minus s120590) (s120590 + 120576)


(b)




0 100 200 300 400 500 60004

06

08

1

12

14

Generation number

I(XY

)

(a)

0 100 200 300 400 500 60002

04

06

08

1

12

Generation number

I(XY

)

(b)

0 100 200 300 400 5000

02

04

06

08

1

Generation number

I(XY

)

(c)

0 50 100 150 200 250 3000

02

04

06

08

1

Generation number

I(XY

)

(d)






119901










References















































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology





P P P P

(a)

Equivalent channel

Gen 1 Gen n

Pn

(b)







1 120582119899

2 120582119899

3 120582119899

4 where 120582

1 1205822 1205823 1205824


119902119899= (

120583 + 120576

120590 + 3120583 + 4120576) +(119901minus(

120583 + 120576

120590 + 3120583 + 4120576))120582119899

4 (7)




R119894= P+Q

119894 (8)




119899




[119901119860 119901119861119901119862119901119864]119899

= [119901119860 119901119861119901119862119901119864]1 Reff (9)

where [119901119860 119901119861119901119862119901119864]119899


119861119901119862119901119864]1


119861119901119862119901119864]119899

or [119901119860 0 119901119862119901119864]119899

or [119901119860 1199011198610 119901119864]119899


119861119901119862119901119864]1


10038161003816100381610038161003816120582119877eff

10038161003816100381610038161003816min gt 0 (10)







1= 1 120582

2=

1205823= sdot sdot sdot = 120582

119873= 1 minus 120590 minus (119873 + 1)120576 and 120582


119902119899 = (120583 + 120576

120590 + 119873120583 + (119873 + 1) 120576)

+(119901minus(120583 + 120576

120590 + 119873120583 + (119873 + 1) 120576)) 120582119899

119873+1

(11)



0 100 200 300 400 5000

02

04

06

08

1

Generation number

120582i

1205822

1205823

1205824

(a)

0 100 200 300 400 5000

02

04

06

08

1

Generation number

120582i

1205822

1205823

1205824

(b)


2(red) 120582

3(green)




119868 (119883 119884) = 119867 (119884) minus119867 (119884 | 119883) (12)


119867(119884) = minus

4sum

119894=1119910119894log2119910119894 (13)



119868 (119883 119884) =

4sum

119895=1

4sum

119894=1119875 (119910119895) 119875 (119909119894 | 119910119895) log

119875 (119909119894| 119910119895)

119875 (119909119894)

(14)


119868 (119883 119884)

= 119867 (119884)


(15)





1205822(red) 120582

3(green) and 120582









0 02 04 06 08 10

02

04

06

08

1

50 times 50

70 times 70

100 times 100

Pex

t

|120582|min

(a) (b)









P119880

=

[[[[[[[[

[

1 minus 1205903 minus 31205761 1205761 1205761 1205903 + 1205761

1205762 1 minus 1205901 minus 31205762 1205762 1205901 + 1205762

1205763 1205763 1 minus 1205902 minus 31205763 1205902 + 1205763

1205831 + 1205761 1205832 + 1205762 1205833 + 1205763 1 minus3sum

119894=1(120588119894 + 120576119894)

]]]]]]]]

]

(16)


1= 1 120582

2= 1205823= 1 minus 120590 minus 4120576 and

1205824= 1 minus 120590 minus (120583

1+ 1205832+ 1205833) minus 4120576




7 Conclusion



S1

Sr

S4 S3

S2

(a)

E E

S1S1

Sr Sr

S3 S3

S2 S2

(1 minus r120576 minus s120590) (120576) (120576) (s120590 + 120576)

(120576) (1 minus r120576 minus s120590) (120576) (s120590 + 120576)

(120576) (120576) (1 minus r120576 minus s120590) (s120590 + 120576)


(b)




0 100 200 300 400 500 60004

06

08

1

12

14

Generation number

I(XY

)

(a)

0 100 200 300 400 500 60002

04

06

08

1

12

Generation number

I(XY

)

(b)

0 100 200 300 400 5000

02

04

06

08

1

Generation number

I(XY

)

(c)

0 50 100 150 200 250 3000

02

04

06

08

1

Generation number

I(XY

)

(d)






119901










References















































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


0 100 200 300 400 5000

02

04

06

08

1

Generation number

120582i

1205822

1205823

1205824

(a)

0 100 200 300 400 5000

02

04

06

08

1

Generation number

120582i

1205822

1205823

1205824

(b)


2(red) 120582

3(green)




119868 (119883 119884) = 119867 (119884) minus119867 (119884 | 119883) (12)


119867(119884) = minus

4sum

119894=1119910119894log2119910119894 (13)



119868 (119883 119884) =

4sum

119895=1

4sum

119894=1119875 (119910119895) 119875 (119909119894 | 119910119895) log

119875 (119909119894| 119910119895)

119875 (119909119894)

(14)


119868 (119883 119884)

= 119867 (119884)


(15)





1205822(red) 120582

3(green) and 120582









0 02 04 06 08 10

02

04

06

08

1

50 times 50

70 times 70

100 times 100

Pex

t

|120582|min

(a) (b)









P119880

=

[[[[[[[[

[

1 minus 1205903 minus 31205761 1205761 1205761 1205903 + 1205761

1205762 1 minus 1205901 minus 31205762 1205762 1205901 + 1205762

1205763 1205763 1 minus 1205902 minus 31205763 1205902 + 1205763

1205831 + 1205761 1205832 + 1205762 1205833 + 1205763 1 minus3sum

119894=1(120588119894 + 120576119894)

]]]]]]]]

]

(16)


1= 1 120582

2= 1205823= 1 minus 120590 minus 4120576 and

1205824= 1 minus 120590 minus (120583

1+ 1205832+ 1205833) minus 4120576




7 Conclusion



S1

Sr

S4 S3

S2

(a)

E E

S1S1

Sr Sr

S3 S3

S2 S2

(1 minus r120576 minus s120590) (120576) (120576) (s120590 + 120576)

(120576) (1 minus r120576 minus s120590) (120576) (s120590 + 120576)

(120576) (120576) (1 minus r120576 minus s120590) (s120590 + 120576)


(b)




0 100 200 300 400 500 60004

06

08

1

12

14

Generation number

I(XY

)

(a)

0 100 200 300 400 500 60002

04

06

08

1

12

Generation number

I(XY

)

(b)

0 100 200 300 400 5000

02

04

06

08

1

Generation number

I(XY

)

(c)

0 50 100 150 200 250 3000

02

04

06

08

1

Generation number

I(XY

)

(d)






119901










References















































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


0 02 04 06 08 10

02

04

06

08

1

50 times 50

70 times 70

100 times 100

Pex

t

|120582|min

(a) (b)









P119880

=

[[[[[[[[

[

1 minus 1205903 minus 31205761 1205761 1205761 1205903 + 1205761

1205762 1 minus 1205901 minus 31205762 1205762 1205901 + 1205762

1205763 1205763 1 minus 1205902 minus 31205763 1205902 + 1205763

1205831 + 1205761 1205832 + 1205762 1205833 + 1205763 1 minus3sum

119894=1(120588119894 + 120576119894)

]]]]]]]]

]

(16)


1= 1 120582

2= 1205823= 1 minus 120590 minus 4120576 and

1205824= 1 minus 120590 minus (120583

1+ 1205832+ 1205833) minus 4120576




7 Conclusion



S1

Sr

S4 S3

S2

(a)

E E

S1S1

Sr Sr

S3 S3

S2 S2

(1 minus r120576 minus s120590) (120576) (120576) (s120590 + 120576)

(120576) (1 minus r120576 minus s120590) (120576) (s120590 + 120576)

(120576) (120576) (1 minus r120576 minus s120590) (s120590 + 120576)


(b)




0 100 200 300 400 500 60004

06

08

1

12

14

Generation number

I(XY

)

(a)

0 100 200 300 400 500 60002

04

06

08

1

12

Generation number

I(XY

)

(b)

0 100 200 300 400 5000

02

04

06

08

1

Generation number

I(XY

)

(c)

0 50 100 150 200 250 3000

02

04

06

08

1

Generation number

I(XY

)

(d)






119901










References















































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


S1

Sr

S4 S3

S2

(a)

E E

S1S1

Sr Sr

S3 S3

S2 S2

(1 minus r120576 minus s120590) (120576) (120576) (s120590 + 120576)

(120576) (1 minus r120576 minus s120590) (120576) (s120590 + 120576)

(120576) (120576) (1 minus r120576 minus s120590) (s120590 + 120576)


(b)




0 100 200 300 400 500 60004

06

08

1

12

14

Generation number

I(XY

)

(a)

0 100 200 300 400 500 60002

04

06

08

1

12

Generation number

I(XY

)

(b)

0 100 200 300 400 5000

02

04

06

08

1

Generation number

I(XY

)

(c)

0 50 100 150 200 250 3000

02

04

06

08

1

Generation number

I(XY

)

(d)






119901










References















































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology





119901










References















































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology
























Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology








Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology

Research Article Information Transfer between Generations …downloads.hindawi.com/archive/2015/128980.pdf · 2019. 7. 31. · is modelled as a problem of communicating symbols across

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