evolutionary dynamics of a food web with recursive ... · evolutionary dynamics of a food web with...

in Artificial Life VIII, Standish, Abbass, Bedau (eds)(MIT Press) 2002. pp 207–215 1

Evolutionary Dynamics of a Food Web with Recursive Branching andExtinction

Hiroshi Ito1 and Takashi Ikegami2

Department of General Systems Sciences, The Graduate School of Arts and Sciences,University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902 Japan

[email protected]@sacral.c.u-tokyo.ac.jp

Abstract

A dynamic systems approach to evolutionary branch-ing and its development in an artificial food web ispresented. Predator-prey interaction among trophicspecies with two traits generates a variety of evolu-tionary branching patterns, depending on interactionstrengths and mutation rates. Studying branching pat-terns in a phenotypic space reveals three branching pat-terns: prey-bifurcation, predator-bifurcation, and net-bifurcation. In particular, a complex food web net-work emerges through net-bifurcation. A relationshipbetween biomass and the number of species is analysedby changing the interaction strength. We report thatthe branching and extinction rates of trophic speciesreaches a maximum when the number of species, butnot the entire biomass, reaches a maximum.

Introduction

The origin and evolution of species have been tested bymany mathematical models. Recent advances in mod-els for sympatric speciation (Dieckmann & Doebeli 1999;Higashi, Takimoto, & Yamamura 1999; Doebeli & Dieck-mann 2000; Kaneko & Yomo 2000) have shown thatvarious intra- and inter-species interactions split a sin-gle founding population into two di!erent populations,called “evolutionary branching”. Doebli and Dieckmannhave demonstrated that various ecological interactions(resource competition, mutualism and host-parasite re-lationships) can cause branching with the evolution ofassortative mating. Highashi’s sexual selection modelhas shown evolutionary branching only through evolu-tionary divergence of sexual preferences, while Kanekoand Yomo have shown that dynamic clustering causedby the internal degrees of freedom in each reproductiveunit drives the branching.

The main concern of these models is evolutionarybranching at each trophic level. Here we are moreconcerned with higher-order branchings, which generatepopulations at new trophic levels. In particular, we in-vestigate how the complexity of the ecosystem may bebuilt up from a bottom trophic species (e.g. plants) onlyby predator-prey interaction.

Mathematical studies on the evolution of a food webhave been mainly based on the replicator or Lotka-Volterra models. Yasutomi and Tokita (Tokita & Yasu-tomi 2002) have studied the evolution of replicator dy-namics by adding new columns and rows to the originalinteraction matrix. Jain and Krishna (Jain & Krishna2002) have shown the role of innovation and of keystonespecies in large extinctions using an evolutionary replica-tor model. The other approach can be found in Lindgrenand Mats Nordal’s (Lindgren & Nordahl 1993) study onthe evolution of a food web, using the model of the it-erated prisoner’s dilemma game. In this model, a ba-sic strategy (Tit for Tat) is used and new strategies areconstantly introduced through genetic algorithms. Theyanalysed what kind of food web emerges.

These previous approaches for evolutionary dynamicsof a food web invite new species from outside. Namely,these models cannot deal with the mechanism of creatingnew species from the intrinsic ecological dynamics thatmaintain the food web. In this paper, we investigatehow patterns of autonomous development and collapseof a food web with recursive branching and extinctionmay be classified and we show how ecological diversityand evolution are interrelated.

Modelling phenotype space anddynamics

In nature, individuals use inorganic resources and otherindividuals as foods. Individuals are also used by otherindividuals as resources. Quantifying those character-istics of individuals as a resource by a simple measure,we represent individuals as a resource distribution onthe measured space. Suppose that a single variable zfor the measure and each individual is characterized bytwo traits, which are measured by the factor z and in-terpreted as a resource trait (prey) and a utilizer trait(predator). Formally, they are labelled as r and u, re-spectively. With these labels, we define a phenotypicspace (Fig.1). A phenotype distribution function p(u, r)is defined on this space. A resource distribution R(z) anda resource utilization distribution U(z) are computed as

2 in Artificial Life VIII, Standish, Abbass, Bedau (eds) (MIT Press) 2002. pp 207–215

z: resource axis

u: utilizer trait

r: re

sour

ce tr

ait

resource space

phenotype space

R(z)

U(z)

L(z)

F(z)

Figure 1: A phenotype space (u, r) and a resource space.Light grey and dark grey circles denote phenotypic clus-ters in the phenotype space (below rectangle) with axesof utilizer trait u and resource trait r. In this case, thedark phenotype uses the light source (L(z)) and is eatenby the light grey phenotype. On the resource space(above), we express the situation in terms of the dis-tributions of U(z) and R(z).

follows:

R(z) =!

up(u, z)du + L(z), (1)

U(z) =!

rp(z, r)dr, (2)

where L(z) is a constant inward resource distributiongiven from without. Species at the lowest trophic levelin the focussed range of the food web use this re-source. Thus, the type of the inward resource (or-ganic/inorganic, plants/animals) depends on what rangeof food web we consider. Because we are concerned with

the total biological community here, the inward resourceL(z) corresponds to sunlight. We define L(z) as theGaussian distribution function:

L(z) = L0 ·1!

2!"Lexp[

"(z " µL)2

2"L2

], (3)

where L0 is the total resource amount. µL and "L denotethe position and the width of the resource, respectively.

An assumption here is that phenotypes whose r areequal to u of phenotypes will be eaten by those pheno-types. In other words, the phenotypes that have the traitu utilize the phenotypes whose r is equal to u. We nowdefine a “resource flow” at each point in the resourcespace (Fig. 1) as follows:

F (z) = " · U(z)R(z)1 + R(z)

M

(4)

where " determines the strength of predator-prey in-teraction, and M gives a maximum predation constantper individual when there are infinite amounts of food.This formula is a type II function response, which isknown to be a characteristic response of actual predatorpopulations to prey density.

It should be noted that, when R(z) is relatively muchsmaller than U(z), the resource flow is proportional tothe cross term U(z) · R(z). However, when R(z) be-comes larger, F(z) becomes proportional to M · U(z),because the maximum predation amount per individualis saturated to M . To keep the amount of the maxi-mum resource flow into the system constant irrespectiveof changes to the interaction strength ", we assume thatthe amount of the inward resource L0 is given by La/",where La is a constant positive value.

Each individual of phenotype (u, r) acquires a 1/U(u)rate of resource flow, assuming that it is equally dis-tributed. Similarly, each phenotype (u, r) is used by theothers at the rate 1/R(r). Concerning this point, wecompute the gain (g(u)) and loss (l(r)) per individual ofeach phenotype as,

g(u) = F (u)/U(u) (5)l(r) = F (r)/R(r) (6)

We finally obtain the following equation for the timeof evolution of the density function p(u, r) as,

#p(u, r)#t

= p(u, r) · (c · g(u) " l(r) " d) · (1 " p(u, r)K

)

+Du · #2p(u, r)#u2

+ Dr ·#2p(u, r)

#r2(7)

where c denotes the e#ciency of resource use, which inpractice is estimated at 0.1 in empirical studies (Pauly &


Christensen 1995). and K denotes the physical carrying-capacity of the habitat. The natural death rate is givenby the parameter d. The last two terms correspond tothe mutation flow in the phenotype space. The muta-tions generate new phenotypes in parents’ neighboursin the phenotype space. In practice, we run the aboveequation by discretizing the explicit Euler method. Toavoid unnecessary numerical underflow, we remove thephenotype whose population density is below a giventhreshold $ that is significantly small. We also assumethat the phenotype space has absorbing boundaries toavoid numerical divergences.

Practically, when the number of predators is muchlarger than that of prey, the consumption rate may beproportional to the number of prey irrespective of thenumber of predators. In order to compensate for thisfact, we can also use a modified version for the resourceflow as,

F̃ (z) = " · U(z)R(z)U(z) + R(z)

M

(8)

With this formulation, F̃ (z) approaches to MU(z) forR(z)/U(z) # $ and to R(z) for R(z)/U(z) # 0. Wemainly report below the simulation results of the firstmodel. However, we also argue for a di!erence betweenthe first and second model by comparing the main resultwith preliminary results of the second model. Insteadof parameter value c (i.e. an e#ciency of resource util-ity), we chose other parameters arbitrarily. However, thebasic scenario presented below (three type of branchingpatterns, etc.) is not sensitive to the parameter values.

Simulation ResultsAn initial food web is prepared with a single phenotypeutilizing an inward resource (L(z)), that corresponds to,for example, plants (Fig.2 (a)). Within the wide rangeof parameter values, several isolated phenotype clustersare generated.

These clusters correspond to “trophic species” whichare functional groups of taxa consisting of species sharingthe same predators and prey in a food web. Here, we callthese phenotypic clusters “trophic species.” To countthese species, we define a base line level for the densityfunction %(> $) to identify each connected cluster in thephenotype space. The number of species does not changesignificantly unless we take base line values that are toolow or too high. Below, we report the details of thisbranching phenomenon and analyse it with regard to theinteraction strengths and biomass.

Branching patternWe observe evolutionary branching of trophic species ina wide range of parameter values. These bifurcation pat-terns are classified into three types in terms of biologicalfunctions, by referring to Figure 2 (a–i).

(a) (b) (c)

Light u: using trait

r: re

sour

ce tr

ait

a

b c d

a

bc

de

f

gh

i

(e)

(h)

(d)

(g)

(f)

(i)

1.0

1.00

0

Figure 2: Evolutionary branching of trophic species issimulated. Darker colours indicate higher densities ofthe phenotype. Contours with solid lines correspond tothe base line abundance truncated by %. Dashed lineson the u axis indicate the organic resource distribu-tions (R(z)-L(z)). Model parameters: M = 10.0, " =13.0, Du = 1.2 · 10!7, Dr = 0.6 · 10!7, c = 0.1, d =1.0, K = 1.0 · 104, $ = 1.0 · 10!9, % = 5.0 · 10!2, L0 =15/", µL = 0, "L = 0.08, $t = 5.0 · 10!4

• Type u: predator-branching

In Figure 2(a) # (d), the initial trophic speciesbranches to produce new trophic species. The r-traitof the initial trophic species, expressed as a distribu-tion function (with a broken line) on the u-axis, pro-vides another potential resource to the initial trophicspecies. Thus, a part of the trophic species evolves toexploit the new resource by branching its u-trait.

This predation forces “prey” species to evolve an r-trait to avoid being predated and “predator” speciesalso evolve a u-trait to pursue the “prey” species,which gives rise to a evolutionary “arms race” (Fig.2 (c) # (d)). Whenever we start from a single trophicspecies, the primary branching is always this type.

• Type r: prey-branching

Instead of escaping from predators, a trophic speciesas prey sometimes branches its r-trait. In Figure 2(e),


a

b

c d

Light

a

bc

d

Light

e

f i

h

g

(a)

(b)

Figure 3: Extracted food-web topology from Fig. 2. (a)and (b) correspond to (g) and (i), respectively. Eachletter denotes a trophic species. Arrows indicate theresource flows among the trophic species.

the two phenotypes simultaneously branch their r-characteristics, developing four new phenotypes.

• Type n: net-branching

Simultaneous branching of both r- and u-traits. Aclear example can be found in Figure 2(g). Some othercases are found when two predators exploit a singleprey from both sides or when two predator-prey sys-tems collide with each other. By this branching, thestructure of the entire food web diverges abruptly andbecomes complicated. The complexity of the networkis well observed by abstracting the relational networktopology as in Figure 3. A topology associated withFigure 2(f) is depicted in Figure 3(a). This will evolveinto the network-Figure 3(b)-of Figure 2(i).

Because each trophic species acts both as prey andpredator for other trophic species at the same time, thegeneric mechanism of the evolutionary branching is com-plicated. The network evolves from a directed graph(Fig. 3 (a)) to a rhizome structure (Fig. 3 (b)). To seethis whole process, the total branching tree is depictedin three-dimensional space (Fig. 4). Successive branch-ings occur along the time axis. A type-n branching onlyoccurs at the later stages.

The degree of complexity a network can attain de-pends on, for example, the interaction strength. Theevolution is not a homogeneous process, but accompa-nies various dynamics, We study these features below.

r: resource traitu: using trait

: Type u: Type r

: Type n

branching type

40

30

20

10

0

time

step

(×10

)4

Figure 4: Phylogenetic tree extracted from the simu-lation with the same conditions as in Figure 2. Eachline indicates an historical trace of each trophic species,where the line width expresses the population size of thetrophic species at that time. The time slice of this phy-logenetic tree gives a phenotype distribution on a phe-notype space (u, r). Each branching type is denoted bya di!erent line style.

Dynamics of evolutionary branching

This evolutionary branching is associated with varioustemporal dynamics related to the numbers of trophicspecies and the total biomass. The critical control pa-rameter is the strength of interaction in the equation(7). The dynamic changes from periodic to chaotic byincreasing its strength. In the strong interaction region,the large number of species is sustained via intermittentchaotic dynamics.

In the relatively lower strength regions of Figures 5 (a)and (b), divergences and extinctions of biomass occurrepeatedly. In these regions, the system repeats type-rand -u branchings, but seldom shows type-n branching.


(a)

(c)

(d)

(b) 0 100 200

-10

0

10

20species

log(biomass)

num

ber o

f spe

cies

-10

0

10

20

num

ber o

f spe

cies

0

20

40

num

ber o

f spe

cies

-10

0

10

20

num

ber o

f spe

cies

specieslog(biomass)

specieslog(biomass)

specieslog(biomass)

time step (×10 )4

0 100 200time step (×10 )4

0 100 200time step (×10 )4

0 100 200time step (×10 )4

(e) (f)

Figure 5: Time evolution of the number of trophicspecies and the biomass at di!erent strengths of interac-tion ("). (a):" = 10,(b):" = 11,(c):" = 20,(d):" = 100.Other parameters used have the same values as in Figure2 except that %L0 = 22/". (e) is a typical phenotype dis-tribution of (a) and (b), and (f) is a typical phenotypedistribution of (c).

Finally, several large clusters of phenotype spread theregion, arriving at massive extinction. As a result, theentire food web resets and it starts again from the bot-tom trophic species that use the inward resource (L(z)).The same scenario; an excess reproduction of a particu-lar trophic species, triggers excess reproduction and itsdistribution is repeated. The associated two-dimensionalphenotype space is depicted in Figure 5(e), where fourtrophic species are expanding without branching. Thus,no food web increases in complexity.

On the other hand, the larger interaction strengthevolves into a regular lattice on the two-dimensional phe-notype space (Figure 5(f)). The excess reproductionof trophic species is suppressed by forming a laterally-inhibiting network. The organized food web continuesexpanding outwards. At the outside of the food web,sub-networks based on the bottom trophic species be-come extinct. At the same time, new networks based onthe new bottom trophic species are created in the insideof the food web. Because the speed of this extinction andexpansion of the network is stabilized, the total biomassand the numbers of all trophic species are maintaineddynamically. This picture is well demonstrated in Fig-ure 5(c). It is interesting to note that the system has amuch larger number of species in Figure 5(c) rather thanin (a) and (b): even the entire biomass is suppressed inthe case of (c). However, a big extinction is aperiodicallyinduced in Figure 5(c). When the interaction strengthis much larger, the amplitude of biomass changes is sup-pressed and shows more stable oscillations. Instead, thefood web at this stage cannot sustain a large number ofspecies because of the shortage of biomass.

Whether a system shows periodic, chaotic or other dy-namics is mostly determined by the parameters. Practi-cally, the interaction strength " determines the kinds ofbranchings. The maximum predation level M has asim-ilar e!ect to ".

In the next section, we study how the entire dynam-ics depend on interaction strengths and other ecologicalparameters.

Interaction Strengths and Mutation RatesBy changing parameter values, we measured the longtime averages of the number of species, and the specia-tion and extinction rates.

1)Interaction Strength ("): (Fig. 6)A larger strength of interaction leads to a smaller

biomass without temporal oscillation. Because it is di#-cult to sustain species, the number of species becomessmaller. On the other hand, the smaller interactionstrength delays the increase in the number of predators.Therefore, this can produce a large biomass but with anunstable temporal oscillation that easily leads to extinc-tion. The number of species is suppressed.

With a middle strength (around 20), the average num-ber of species attains a maximum value having the same


0 2 4 6 8

10 12 14 16 18

10 100 20 40

speciesspeciationextinction

num

ber o

f spe

cies

Ω: strength of interaction

Figure 6: A relation between the strength of interaction(") and the numbers of trophic species. An average num-ber of species, extinction and evolutionary branchingthrough a long time-step (6 ·105) is computed. The rela-tive amounts of extinction and branching almost coincideover this wide range. Model parameters: L0 = 22/",and the others are used with the same values as in Fig-ure 2.

temporal variation as the biomass. When the number ofspecies attains a maximum value, the extinction and spe-ciation rates also become maximum, as is seen in Figure6. Note that those two rates have almost equal averagevalues. This reflects that the food web temporally col-lapses to the initial state so that it reorganizes the entirenetwork, or the food web with balanced speciation andextinction.

2) Mutation rates (Fig.7)When the mutation rates of two traits (Du, Dr) are

set at the same, the number of species gets larger. Ingeneral, when Dr is less than Du, a system holds morespecies than in the opposite situation. A similar ar-gument can be found given by Ikegami and Kaneko(T.Ikegami & K.Kaneko 1990). The entire biomass (Fig.7(b)) becomes larger when the absolute values of Du andDr are smaller and the asymmetry is larger.

A relation between biomass and speciesdiversityA large biomass does not necessarily imply a maximalnumber of species. In Figure 8, the relationship be-tween the number of species and the biomass is depicted.The number of species forms a convex function of thebiomass. This convex form is observed over a wide rangeof parameter values (e.g. mutation rates and the inter-action strength). Figure 8(a) is computed from Figure5(d), which shows a sharp peak around 0.1. Here thebiomass is well distributed over the trophic species andthe hole in the network caused by sub-extinction will

10-8 10-7 10-610-8

10-7

10-6

0

80

Du

Dr

10-8 10-7 10-610-8

10-7

10-6

0

50

Du

Dr

(a) (b)

Figure 7: The e!ect of mutation rates on the numbersof trophic species (a) and of biomass (b). Horizontaland vertical axes are mutation rates of the utilizer traitu and resource trait r, respectively. Model parameters:" = 13.0, L0 = 20/", and the others are used at thesame values as in Figure 2.

soon be compensated for by the new network.This convex shape is also observed in di!erent param-

eter changes such as mutation rate (Fig. 8(a)) and in-teraction strength.

A case of F̃ = UR/(U + R/M)Instead of using F, we analysed a model equation withF̃ . In this case too, type-r, -u and -n were observed. Bythese branchings, development and sudden extinctionsof trophic species occur (Fig.9 (a)). As a result, themutation rate produces the same e!ect on the averagenumber of trophic species and biomass.

However, we could not observe a dynamic stable stateas in Figure 5(c). Instead, a nearly fixed phenotype dis-tribution lasts for a long time in the region of Du > Dr

(Fig.9 (b)). It seems that many di!erent organizationscan be possible in this stable phase.

Some oscillating phases that hold stable phenotypicdistribution also appear in Figures 9(c) and (d). Thesephases tend to collapse by oscillationary divergence. Af-ter the extinction, a new food web structure becomesreorganized.

DiscussionsIn this model, we have studied the evolution of “trophicspecies”, which are functional groups of taxa consist-ing of species sharing the same predators and prey ina food web. The concept of trophic species is some-times criticized but is widely accepted, and structuralfood web studies can reduce biases in the data by usingit (Williams & Martinez 2000).

We distinguish two di!erent time scales in the presentsimulations. A longer time scale corresponds to an evo-lutionary food web scale, which is determined by succes-sive large extinctions. A shorter time scale corresponds


10 -4 10 -2 10 +0 10 +2 0

20

40

biomass

num

ber o

f spe

cies

0.01 1.0 100.0 0

20

40

60

biomass

num

ber o

f spe

cies

(a)

(b)

Figure 8: A relationship between biomass and the num-ber of trophic species. (a) generated from Fig. 5(d), and(b) from Fig. 6.

to a within-food web developmental time scale, wherenew trophic species are synthesized on the local portionsof the phenotypic space. The term evolvability usuallyimplies inheritance of genetic potential to produce newcharacteristics over a longer time scale. The term eco-logical diversity means to develop a food web network ina shorter time scale. An advantage of our model is thatevolvability and ecological diversity are studied in thesame model. Both evolvability and diversity require acertain degree of instability in the system. Evolvabilityneeds to destabilize the established food web to intro-duce new trophic species and thus introduce a new formof a food web into the system. Under a certain rangeof interaction strengths with the condition of mutationrates, µu < µr, large diversity is developed. Thus, preda-tors should have larger mutation rates than prey to pro-duce diversity. A similar condition has been reported forhost-parasitoid systems (T.Ikegami & K.Kaneko 1990).In the wide parameter range of our model, a system au-tonomously breaks down its own food web structure tosynthesize a new food web. In other words, instabilitythat generates a local species branching event will finallylead to a large scale instability that resets the entire foodweb.

Evolutionary branching of the trophic species in thepresent model is categorized into three types. Type-uis branching as a competitor (or predator), which corre-sponds to previous models for evolutionary branchingthrough niche-shift for new resources by intra-speciescompetition (Doebeli 1996; M. Kawata 2002). On the

fixedoscillating

(a)

u: using trait r: resource trait 0

200

400

600

time

step

(×2.

5·10

)2

(b) (c) (d)

u: using trait

r: re

sour

ce tr

ait

0.0 0.5 1.00.0

0.5

1.0

0.0 0.5 1.00.0

0.5

1.0

Figure 9: Evolutionary dynamics with F̃ . (a) Time evo-lution of distributions of traits u(left) and r(right). Aphenotype distribution in a stable phase (b), and inthat of the oscillating phase ((c) & (d)). Model pa-rameters: M = 10.0, " = 10.0, Du = 8.45 · 10!8, Dr =9.0·10!8, c = 0.1, d = 1.0, K = 1.0·104, $ = 1.0·10!9, % =1.0 ·10!2, L0 = 300/", µL = 0, "L = 0.08, $t = 4.0 ·10!3

other hand, type-r and type-n are branchings producedby predator-prey interaction, that is inter-species com-petition. Previous models for evolutionary branchingby predator-prey interaction have shown that, at onetime, a single pair of predator-prey systems splits intotwo pairs (four populations) or a single prey splits intotwo preys exploited by the same predator (three popu-lations) (Savill & Hogeweg 1998; Doebeli & Dieckmann2000). Type-n can generate more than two species at atime, because a single pair of predator-prey system splitse!ectively into six trophic species (two preys and fourpredators) at one time. This type of speciation might bepossible in real populations, which have both resourcecompetition and intra-guild predation.

Based on our model simulations, we suggest the fol-lowing insights into the relationships between 1) interac-tion strength and stability, and 2) biomass and species-diversity.

1) Interaction Strength vs. StabilityEvolution of trophic species in our model may corre-

spond to a travelling wave phenomenon in the phenotypespace. The stability of the trophic species is sustained


when there exists a time lag between the growth of preyand that of predators, as it generates a traveling wavesof prey chased by predators. Thus, each species main-tains its population evolving directionally. This time lagincreases when the interaction strength is weak (a small").

However, a weak interaction rate tends to destabilizethe whole food web and results in relatively small diver-sity. By contrast, there are studies reporting that weakinteraction has a stabilizing e!ect to maintain high di-versity by dumping the population dynamics of the foodweb (McCann 2000; Neutel, Heesterbeek, & de Ruiter2002). This paradox may arise because we are concernedwith dynamic stability through the directional evolutionof each trophic species. On the other hand, these empir-ical studies do not concern evolutionary dynamics, butpopulation dynamics.

2) Biomass vs. Species-DiversityAnother important result is that the number of trophic

species becomes maximum at a medium biomass. Weargue that this dependency on the biomass can be ex-plained as follows. When the amount of biomass is low,the system simply cannot a!ord to have any higher orderniches, and the number of species may be suppressed. Onthe other hand, the system may develop a large biomassonly when plant species are prospering in the absence ofpredators. Thus, the number of trophic species is keptlower. Only the mid range of biomass corresponds to arich and complex food web network. There are manyempirical studies investigating the relationships betweenbiomass or productivity and species diversity. Mostof them (Abramsky & Z 1984; Tilman & Pacala 1993;Leibold 1999; Ritchie & Ol! 1999; Waide et al. 1999;Dodson, Arnott, & Cottingham 2000; Gaston 2000) re-port the same tendency as our results here: that di-versity peaks at intermediate productivity or biomass.However these empirical researches were focused on arestricted range of trophic levels in the whole food web;on the other hand, we investigated the relationship ata higher level between the whole biomass of a food weband the numbers of trophic groups, each of which maycontain several species. This interesting concordance be-yond scale di!erence in a food web might imply a univer-sal restriction on the relationship between biomass anddiversity.

A criticism of the present model is that to represent aresource by a single dimension is an extreme simplifica-tion. Indeed, our simplification from actual multidimen-sional niche-space should be reducing some aspects ofreal evolutionary dynamics. For example, in our model,when two pairs of arms-racing predators and preys areapproaching each other in the phenotype space they al-ways collide. On the other hand, they may not collideif there is enough distance between the pairs in the di-rection of another niche-axis. Thus, an extension of the

present model to take multidimensional traits into ac-count may be necessary. However, as food webs are as-sembled in the same way in our model, assigning eachspecies on one dimensional niche-axis helps explain manycharacteristics of real complex food webs (Williams &Martinez 2000) and may provide a certain assurancethat the evolutionary dynamics of food webs are alsodescribed su#ciently in our simple rule.

To make the present model closer to real food webs,the introduction of mating mechanisms is important.This provides a certain viscosity with populations bygene-flow between phenotypes, which will converge eachmating group as a “species” under ecological defini-tion and will provide more realistic evolutionary dy-namics of a food web consisting of those species. An-other extension will be to improve a new form of re-source flow F (z). As we have described, the new formof F̃ (z) certainly provides more rich evolutionary path-ways. It is also interesting to note a connection be-tween the present model and the approaches based on thereplicator equations (K.Hashimoto & T.Ikegami 2001;T.Ikegami & K.Hashimoto 2002). For example, the no-tion of keystone species should also be examined in thepresent model.

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