a mathematical model of a biological arms race with a dangerous

J. theor. Biol. (2002) 218, 55–70doi:10.1006/yjtbi.3057, available online at http://www.idealibrary.com on

A Mathematical Model of a Biological Arms Race with a Dangerous Prey

PaulWaltmanw , James Braselton*z and Lorraine Braseltonz

wDepartment of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, U.S.A.

and zDepartment of Mathematics and Computer Science, Georgia Southern University, P.O. Box 8093,Statesboro, GA 30460-8093, U.S.A.

(Received on 25 January 2002, Accepted in revised form on 15 April 2002)

In a recent paper, Brodie and Brodie provide a very detailed description of advances andcounter-measures among predator–prey communities with a poisonous prey that closelyparallel an arms race in modern society. In this work, we provide a mathematical model andsimulations that provide a theory as to how this might work. The model is built on a two-dimensional classical predator–prey model that is then adapted to account for the geneticsand random mating. The deterministic formulation for the genetics for the prey populationhas been developed and used in other contexts. Adapting the model to allow for geneticvariation in the predator is much more complicated. The model allows for the evolution ofthe poisonous prey and for the evolution of the resistant predator. The biological paradigm isthat of the poisonous newt and the garter snake which has been studied extensively althoughthe models are broad enough to cover other examples.

r 2002 Elsevier Science Ltd. All rights reserved.

1. Introduction

This paper develops a mathematical model of abiological arms race between a class of predatorsand a class of prey where the prey is dangerousto the predator. In their survey article, Brodie &Brodie (1999) describe predator–prey arms racesand draw interesting parallels with the morefamiliar arms races of modern society, usingas one example law enforcement and speedingmotorists. The biological arms race has alsoappeared in the popular media, PBS (2001). Themost interesting example described in Brodie &Brodie (1999) concerned the unusual predator–

*Corresponding author. Tel.: +1-912-681-0874; fax:+1-912-681-0654.E-mail address: [email protected]

(P. Waltman), [email protected] (J. Braselton),[email protected] (L. Braselton).

0022-5193/02/$35.00/0

prey relationship of the garter snake Thamnophissirtalis (predator) and the Oregon newt Taricha

granulosa (prey). The newt defends itself byproducing a toxin, tetrodotoxin (TTX). Thesnake is the only known predator of the Oregonnewt that has developed resistance to TTX. TheTTX need not cause the death of a garter snakedirectly: when a snake consumes a newt, it maybe immobilized by the TTX contained in thenewt’s skin for several hours. In this state, thesnake is susceptible to other predators and, if itcannot move, may not be able to thermoregulateproperly, and may die, Brodie & Brodie (1999).The newt–garter snake predator–prey rela-

tionship is a particular example of a biologicalarms race where the prey is dangerous to thepredator. The prey (newt) develops a defenseagainst the predator by becoming poisonousto the predator. The predator (garter snake)

r 2002 Elsevier Science Ltd. All rights reserved.

P. WALTMAN ET AL.56

develops a resistance to the prey’s toxicity. Theinteractions involving dangerous prey are differ-ent from other predator–prey relationships andresult in a co-evolutionary biological arms race.Although the snake–newt relationship is unu-sual, predator–prey arms races are not and havebeen observed in a variety of predator–preyrelationships. Brodie & Brodie (1999) and thereferences cited there provide many details andfield data. We use this newt–garter snakerelationship to guide the development of themathematical model using continuous modelsfrom population genetics and standard ecologi-cal models. Adaptation in the prey producesselection pressure on the predator. Our approachuses continuous models that incorporate bothgenetic and ecological considerations and allowa genotype of the prey to be lethal to somegenotypes of the predator. Although compli-cated, numerical simulations can be easilycarried out. These simulations are presented ina sequence of graphs. The model can also includethe more typical arms race: ‘‘the fox lineage mayevolve improved adaptations for catching rab-bits and the rabbit lineage improved adaptationsfor escaping’’ (quoted from Dawkins & Krebs,1979).For the prey’s growth, we use the logistic

equation, one of the building blocks of popula-tion ecology. A derivation and the fitting of agreat deal of biological data can be found inHutchinson (1978, Chapter 1). The equation isalso studied in standard elementary differentialequations course and appears in many texts, forexample in Abell & Braselton (2000). We add tologistic growth one of the standard prey capturefunctions. There is a great deal of literature onpredator–prey models: Freedman (1980) devotesa chapter to such Kolmogorov models and ourbeginning, ecological model is a special case ofthose considered there. We then seek to add thegenetics to the model in such a way that whencapture parameters are all equal, the modelreduces to the basic, well-established, predator–prey equations.A broad overview of predator–prey interac-

tions and co-evolution is given by Abrams(2000). Deterministic genetic models were devel-oped in a fundamental paper by Nagylaki &Crow (1974) and have been used by Beck (1982,

1984) in a model of cystic fibrosis and in a modelof co-evolution and by Beck et al. (1982, 1984) ininfectious disease models. There is also work,using the approach of Nagylaki and Crow formodels of growth with genotypic fertility differ-ences, Hadeler & Lieberman (1975), Butler et al.

(1981), Hadeler & Glas (1983). So (1986) andJosic (1997). Discrete models with fertilitydifferences are considered in Doebeli (1997)and Doebeli & de Jong (1998). Freedman &Waltman (1978). So & Freedman (1986), Freed-man et al. (1987) and So (1990) use continuousformulation in a model of predator–prey systemswhere only the genetics of the prey is modeled. Adiscussion of the general topic of an arms racecan be found in Dawkins & Krebs (1979) wheremany examples are mentioned as well as Epstein(1997). The predator–prey dynamics by theirclassification is ‘‘asymmetric’’ or ‘‘attack-defense’’ type and, of course, inter-specific. Thefact that the prey is dangerousFthe prey cankill the predatorFdifferentiates our model ofan arms race from those involving mimicry orphysical improvements of the prey or thepredator, like those referenced above.Although, introducing genetics into the logis-

tic equations is fairly straightforward, incorpor-ating them into the predator systems is moredifficult because one of the basic assumptionof predator–prey systems is that growth comesfrom prey capture, not just the quantity ofpredators. We believe that the model of predatorgrowth through prey capture, when the capturerates differ, is new.

2. The Basic Model

We begin with a standard predator–preyequation of Kolmogorov type

x0 ¼ ax 1�x

K

� ��

mxy

a þ x

y0 ¼ ymx

a þ x� s

� �: ð1Þ

The basic working assumption is that when thegenetics, introduced below, are not relevant tothe predator–prey interactions, then the systemwith genetics should reduce to eqn (1). Equation(1) often occurs with other parameters, for

10 20 30 40 50 t

0.2

0.4

0.6

0.8

x

y

0 .10. 20. 30. 40. 5x

0.1

0.2

0.3

0.4

0.5

y

(a)(b)

Fig. 1. (a) Predator–prey time course: parameters as above with m ¼ 2:5; a ¼ 0:37; s ¼ 1:1; K ¼ 1; a ¼ 1:2: (b) Phase-plane plot corresponding to (a).

MATHEMATICAL MODEL OF A BIOLOGICAL ARMS RACE 57

example, a conversion constant to convertcaptured prey to predator biomass or an extrarate constant in front of the predator equation.These may be scaled out and we assume thatscaling has been done. One could also scale theparameter K out of the system but we choose toretain it. The prey capture term is of Monod type(also called a Hollings Type II response) which isusually justified as allowing for prey-handlingtime. m reflects the difficulty of prey capture, s isthe death rate of the predator in the absence ofprey, and a and K are discussed below whenconsidering the logistic equation. Figure 1(a)shows the time course of the predator–preysystem. We have selected the parameters to bein the oscillatory range for this system. Thecorresponding phase plane plot is given inFig. 1(b).We begin with the growth of the prey without

a predator which, is assumed in eqn (1) to followa basic logistic equation,

x0 ¼ ax 1�x

K

� �: ð2Þ

K is called the ‘‘carrying capacity’’ and reflectsthe level to which the prey will grow if thereare no predators. a reflects the rate at whichthe prey approaches the carrying capacity.The prey are divided into three classes represent-ing three genotypes and random mating isassumed. The usual classification of a one locus,two allele problem is labeled AA; Aa; aa;corresponding to two choices of an allele atone location. There is a deterministic formula-tion of the evolution of the genotypes, Nagylaki& Crow (1974), Butler et al. (1981) which takes

the form

x01 ¼ a

1

xx1 þ

x2

2

� �2�aK

x1x;

x02 ¼ 2a

1

xx1 þ

x2

2

� �x3 þ

x2

2

� ��

aK

x2x;

x03 ¼ a

1

xx3 þ

x2

2

� �2�aK

x3x ð3Þ

with initial conditions x1ð0Þ ¼ x10; x2ð0Þ ¼ x20;x3ð0Þ ¼ x30:Built into this format is the interpretation of

the logistic equation that growth is representedby the linear term (ax) and (natural) death by thequadratic term (x2=K) although other interpre-tations are possible (and will not affect the workhere). The principal result of Freedman &Waltman (1978) for eqn (3) is that the threegenotypes evolve (have limits) in the ratio ðx1 :x2 : x3Þ ¼ ðc2 : 2c : 1Þ where

c ¼x10 þ 1

2x20

x30 þ 12x20

:

This reflects the basic Hardy–Weinberg principlefor random mating in an asymptotic form.Adding the three equations in eqn (3) with xðtÞ ¼x1ðtÞ þ x2ðtÞ þ x3ðtÞ recovers the logistic equa-tion (2), that is, the total grows logistically.In Fig. 2(a), we illustrate the evolution of the

genotypes. The total (x) follows a typical logisticequation. We take the parameters (arbitrarily) tobe a ¼ 1:2; and K ¼ 1: As to be expected, growthoccurs according to Hardy–Weinberg propor-tions without selection. Selection is illustrated by

t5 10 15 20

t

0.2

0.4

0.6

0.8

1 x

AaAA

aa

5 10 15 20

0.2

0.4

0.6

0.8

1

1.2 x

Aa

AA

aa(a) (b)

Fig. 2. (a) The evolution of three genotypes following logistic growth. The initial conditions are x1ð0Þ ¼ 0:06; x2ð0Þ ¼0:1; x3ð0Þ ¼ 0:02: (b) The evolution of three genotypes where x1 has an increase in its growth rate from 1.2 to 1.31. The

10 20 30 40 50t

0.025

0.05

0.075

0.1

0.125

0.15

0.175

0.2Prey

aa

AaAA

10 20 30 40 50t

0.2

0.4

0.6

0.8

Genotypic Proportions

aaAa

AA

(a) (b)

Fig. 3. (a) The prey population broken into three genotypes with x1ð0Þ ¼ 0:02; x2ð0Þ ¼ 0:002; x3ð0Þ ¼ 0:001; and yð0Þ ¼0:1: (b) The genotypic frequencies corresponding to (a).

P. WALTMAN ET AL.58

increasing the value of a in its first occurrencein the equation for x1: Keeping all of the otherparameters the same and replacing the value a ¼1:2 by 1:31 in its first occurrence in the equationfor x1 yields Fig. 2(b). Clearly, the AA genotypeis replacing the others in the mix. This isexpected and shown here to illustrate that thecontinuous version of the evolution of genotypesmatches the discrete one.We now add a predator to the system that

preys equally on each of the three genotypes. Aswith all simple models we also assume that thepredator feeds exclusively on this prey. Whilethat is not realistic, to assume otherwise eitherrequires that one know the other prey andadd them to the model or to assume that thepredator also has a logistic growth term inaddition to the prey, which does not allow one toseparate out the effects of this particular prey.One hopes that the simple model captures theessence of the effect even if it is not totallyrealistic in modeling the natural situation. Theequations become

x01 ¼ a

1

xx1 þ

x2

2

� �2�aK

x1x �x1

x

mxy

a þ x;

x02 ¼ 2a

1

xx1 þ

x2

2

� �x3 þ

x2

2

� ��

aK

x2x �x2

x

mxy

a þ x;

x03 ¼ a

1

xx3 þ

x2

2

� �2�

aK

x3x �x3

x

mxy

a þ x;

y0 ¼ ymx

a þ x� s

� �: ð4Þ

Again, adding the prey equations produces astandard predator–prey system (1).In Fig. 3(a), the prey is broken in three

genotypes, showing that each oscillates asexpected. The initial conditions are x1ð0Þ ¼0:02; x2ð0Þ ¼ 0:002; x3ð0Þ ¼ 0:001; and yð0Þ ¼0:1; the system is prejudiced in favor of AA:Since the predominant prey is AA we have

broken the graph at the top in order to show theothers because aa has such small numbers it doesnot show clearly in the graph. Figure 3(b), whichis uninteresting in this context but will beimportant in the discussion that follows, plotsthe evolution of the relative proportions of thethree genotypes. After a slight adjustment atthe beginning, the frequencies are constant.The figures illustrate that breaking the prey


population into three genotypes preserves theexpected predator–prey behavior. In what fol-lows we will disturb this basic predator–preyrelations to formulate the models of the armsrace.

3. Elusive and Poisonous Prey

In this section, we let the prey develop adefense against the predator. Defenses areevolutionary traits that can be physical (faster,quicker turning, etc.), passive (camouflage),offensive (poisonous) or a combination of theseand give the prey a survival advantage. We alsoconsider the possibility that a poisonous preyis able to alter the predator capture rate. Weassume that the trait is genetic and by conven-tion we let the ‘‘special’’ prey be of the aa type,denoted by x3ðtÞ: Complete dominance of A isassumed so that neither AA nor Aa are elusive ordangerous (produce a toxin). We illustrate howthese different strategies affect the evolution ofthe genotypes. We first relabel the m parameterin eqn (4) to be m1; m2; m3; respectively, toproduce the system

x01 ¼ a

1

xx1 þ

x2

2

� �2�

aK

x1x �x1

x

m1xy

a þ x;

x02 ¼ 2a

1

xx1 þ

x2

2

� �x3 þ

x2

2

� ��

aK

x2x �x2

x

m2xy

a þ x;

x03 ¼ a

1

xx3 þ

x2

2

� �2�aK

x3x �x3

x

m3xy

a þ x;

y0 ¼ ym1x1 þ m2x2 þ m3x3

a þ x� s

� �: ð5Þ

100 200 300 400t

0.1

0.2

0.3

0.4

y Predator

100 200

0.10.20.30.40.50.60.7

Pre

AA

Aa

(a) (b)

Fig. 4. Evolution of the (a) predator and (b) prey with anthe relative frequencies of an elusive prey.

We take m1 ¼ m2 ¼ 2:5 as in the previouscomputations but reduce the capture rate forthe aa genotype (x3) by 10% to 2:25%: Thismeans that x3 is more difficult or less desirable tocapture corresponding to a genetic trait as notedabove. For example, x3 may be faster or quickerat turning when being pursued making it moredifficult to capture; x3 may exhibit coloring ora marking that causes the prey y to find itundesirable; x3 may taste bad so that when ycaptures it, y ‘‘spits it’’ out leaving it unharmed,which has been observed in some newt–snakeinteractions; x3 may secrete a chemical thatcauses it to smell bad to y and so on. We plot theevolution of the predator and the three preygenotypes in separate graphs. To make thisevolution more dramatic we also plot theevolution of the relative frequencies in Fig. 4(c).The elusive prey has become the establishedprey; contrast Fig. 4(c) with Fig. 3(b). Thepredator survives but at a lower level.A more serious capture avoidance can lead to

the extinction of the predator. If instead ofreducing the capture rate by 10%; the improve-ment in the prey’s ability to avoid capturereduces the capture rate to 50%; then thepredator becomes extinct. The prey again isdominated by aa although the route is notsmooth as in the previous case. Figures 5(a) and(b) show the predator and prey time courses andFig. 5(c) shows the evolution of the relativefrequencies.Of course, a 50% improvement represents a

drastic step. Although we do not consider thecase here, one might want to model theimprovement as a separate process, allowinggradual improvement in avoiding capture. Thiscould probably be done with a multi-locus modelwhich would introduce considerable complexity.

300 400t

y

aa

100 200 300 400t

0.2

0.4

0.6

0.8

1Genotypic Proportions

aa

Aa

AA

(c)

elusive prey and m1 ¼ m2 ¼ 2:5; m3 ¼ 2:25: (c) Evolution of

20 40 60 80 100t

0.1

0.2

0.3

0.4

y Predator

20 40 60 80 100t

0.0250.05

0.0750.1

0.1250.15

0.1750.2

Prey

aa

AAAa

20 40 60 80 100t

0.2

0.4

0.6

0.8


aa

AA

Aa

(a) (b) (c)

Fig. 6. (a) Evolution of the (a) predator and (b) prey with a poisonous prey with low initial density. (c) Evolution of therelative frequencies of a poisonous prey with low initial density.

20 40 60 80 100t

0.1

0.2

0.3

0.4

y Predator

20 40 60 80 100t

0.2

0.4

0.6

0.8

1Prey

aa

A A

Aa

20 40 60 80 100t

0.2

0.4

0.6

0.8


aa

A A

Aa

(a) (b) (c)

Fig. 5. Evolution of the (a) predator and (b) prey with an elusive prey (aa) and with m1 ¼ m2 ¼ 2:5; m3 ¼ 1:125: (c)Evolution of the relative frequencies of an elusive prey.

P. WALTMAN ET AL.60

We now turn to another improvement in thedevelopment of the prey, a poisonous genotype.The snake and newt system discussed in theIntroduction is the prime example of such asystem. We again assume that the poisonousprey is represented by the aa genotype and thatits consumption is fatal to the predator. Asnoted in the Introduction, this is an extremeassumption because most newts may only renderthe snake immobile for a while and the snake issubject to other forces while in this state. A‘‘correction’’ factor could be entered in theremoval term for the predator but the value ofsuch a correction factor seems unlikely to beknown. The equations for a poisonous prey takethe form

x01 ¼ a

1

xx1 þ

x2

2

� �2�

aK

x1x �x1

x

mxy

a þ x;

x02 ¼ 2a

1

xx1 þ

x2

2

� �x3 þ

x2

2

� �

�aK

x2x �x2

x

mxy

a þ x;

x03 ¼ a

1

xx3 þ

x2

2

� �2�

aK

x3x �x3

x

mxy

a þ x;

y0 ¼ ymðx1 þ x2Þ

a þ x� s

� ��

mx3y

a þ x: ð6Þ

The Monod term, formerly reflecting the addedgrowth of the predator by capturing x3; now nolonger does so and, in addition, the capture of x3contributes to the death rate of the predator. Weuse the same parameters as before with initialconditions x1ð0Þ ¼ 0:02; x2ð0Þ ¼ 0:002; x3ð0Þ ¼0:001; which represents a rare, poisonous prey.We plot the predator evolution, prey evolution,and the evolution of the relative frequenciesin Fig. 6. Although the predator is diminishedslightly, almost nothing changes from theoriginal model (4). From the standpoint ofthe predator, this is an acceptable ecosystem.The poisonous prey is providing a type of‘‘group defense’’ with little effect.However, if the initial density of the poisonous

prey is high, the results are disastrous for thepredator. The same three plots follow in Fig. 7except that now the initial conditions are x1ð0Þ ¼0:02; x2ð0Þ ¼ 0:002; and x3ð0Þ ¼ 0:03:These figures illustrate what could happen if

the density of the poisonous prey becomes high.However, if the genetic event occurs from arandom mutation that makes the prey lethal tothe predator but does not give the prey a survivaladvantage, Fig. 6 shows that the poisonousprey will not achieve high enough densities toeliminate the predator.Now suppose the poisonous prey has a slight

advantage, like those described earlier, thatmakes it less susceptible to being captured. Themodel is adjusted to take this into consideration

5 10 15 20t

0.2

0.4

0.6

0.8

1y Predator

5 10 15 20t

0.2

0.4

0.6

0.8

1Prey

aa

AA

Aa

5 10 15 20t

0.10.20.30.40.5


aa

AA

Aa

(a) (b) (c)

Fig. 7. (a) Evolution of the (a) predator and (b) prey where aa corresponds to the poisonous prey with high initialdensity. (c) Evolution of the relative frequencies of a poisonous prey with high initial density.

50 100 150 200t

0.2

0.4

0.6

0.8

1y Predator

50 100 150 200t

0.2

0.4

0.6

0.8

1Prey

aa

AA

Aa

50 100 150 200t

0.2

0.4

0.6

0.8


aa

AA

Aa

(a) (b) (c)

Fig. 8. Evolution of the (a) predator and (b) prey with a poisonous prey low initial density and a slight advantage incapture avoidance. aa corresponds to the poisonous prey with low initial density and a slight advantage in captureavoidance. (c) Evolution of the relative frequencies of a poisonous prey with low initial density and a slight advantage incapture avoidance.


and becomes

x01 ¼ a

1

xx1 þ

x2

2

� �2�aK

x1x �x1

x

m1xy

a þ x;

x02 ¼ 2a

1

xx1 þ

x2

2

� �x3 þ

x2

2

� �

�aK

x2x �x2

x

m2xy

a þ x;

x03 ¼ a

1

xx3 þ

x2

2

� �2�aK

x3x �x3

x

m3xy

a þ x;

y0 ¼ ym1x1 þ m2x2

a þ x� s

� ��

m3x3y

a þ x: ð7Þ

We take m1 ¼ m2 ¼ 2:5 and m3 ¼ 2:25 andinitial conditions x1ð0Þ ¼ 0:02; x2ð0Þ ¼ 0:002;and x3ð0Þ ¼ 0:001: This reflects genetic changeby giving the poisonous prey a slight advantagein capture avoidance but also a very low initialsize, as would be expected after an advantageousmutation has occurred. Figure 8 shows anopening salvo in the biological arms race: eventhough x3 has a very low initial size itsadvantage in avoiding the predator allows x3

to dominate, causing extinction of the predator.(cf. Fig. 8 with Fig. 6).If the poisonous prey becomes established at a

high level it eliminates the predator. If it has anadvantage in avoiding capture (or detection),it first out-competes its rivals because of theadvantage of a lower capture rate, and thusmoves from an extremely low level to asignificant level, eliminating the predator.Snakes have been observed ‘‘spitting out’’ thenewt (Brodie, private comm., 2001) which wouldgive it a slightly diminished capture rate, i.e.,m3omaxfm1;m2g:If the predator does not respond with a genetic

alteration, it will become extinct. Hence, one hasthe next step in the arms race.

4. The Arms Race

We now let the predator, y; evolve withimmunity to the dangerous prey, x: (We havein mind the example of the poisonous newt andthe garter snake discussed in the Introductionwhere the garter snake acquires resistance to thetoxin produced by the newt, but the model willhave wider applicability.) At the key locus wedenote the genotype for the predator as BB; Bb;and bb and label the concentrations of each byy1; y2; and y3; respectively. We assume that bb is

P. WALTMAN ET AL.62

the resistant genotype. The redistribution of thegenotypes due to random mating is much moredelicate than that of the prey discussed pre-viously. In the case of the model of the prey, thegrowth rate is constant (a) so the increase in preydepends on the numbers in each class, but in thecase of the model of the predator, growthfollows from prey capture.Since the model is somewhat complicated, we

develop it in stages in hopes of achieving greaterclarity for the form of the final model. If x

denotes the concentration of the prey (we ignoregenotypes at first) and y; the predator, theincrease in the concentration of the predator isdriven by the Monod term

mxy

a þ x:

We seek to incorporate the distribution ofgenotypes using this term. As long as there areno genotypic differences affecting predator–preyreactions, a basic hypothesis is that one must beable to recombine the genotypes into the basicpredator–prey system which we have assumedfrom the beginning to be of the form

x0 ¼ ax 1�x

K

� ��

mxy

a þ x;

y0 ¼ ymx

a þ x� s

� �: ð8Þ

We assume that the prey captured by yi is thefraction of the total catch that yi represents inthe population:

yi

y

mxy

a þ x:

(The y terms cancel.) The basic assumption ofpredator–prey models is that the captured preytranslates to growth of the predator. For a singleprey, x; this leads to the equations

x0 ¼ ax 1�x

K

� ��

mxðy1 þ y2 þ y3Þa þ x

;

y01 ¼

mx

ða þ xÞðy1 þ y2 þ y3Þy1 þ

y2

2

� �2�sy1;

y02 ¼ 2mx

ða þ xÞðy1 þ y2 þ y3Þ

y1 þy2

2

� �y3 þ

y2

2

� �� sy2;

y03 ¼mx

ða þ xÞðy1 þ y2 þ y3Þy3 þ

y2

2

� �2�sy3: ð9Þ

If one adds the last three equations in the system,then using y ¼ y1 þ y2 þ y3 reproduces the basicpredator–prey equations (8).We now rewrite the system with the full prey

genotypes, borrowing from the prey equationsdeveloped in the previous section. The modeltakes the form (where we are expansive in thenotation to illustrate the effect of incorporatingthe genotypes for both predator and prey)

x01 ¼

ax1 þ x2 þ x3

x1 þx2

2

� �2�ax1ðx1 þ x2 þ x3Þ

K

�mx1ðy1 þ y2 þ y3Þa þ x1 þ x2 þ x3

;

x02 ¼

2ax1 þ x2 þ x3

x1 þx2

2

� �x3 þ

x2

2

� �

�ax2ðx1 þ x2 þ x3Þ

K�

mx2ðy1 þ y2 þ y3Þa þ x1 þ x2 þ x3

;

x03 ¼

ax1 þ x2 þ x3

x3 þx2

2

� �2�ax3ðx1 þ x2 þ x3Þ

K

�mx3ðy1 þ y2 þ y3Þa þ x1 þ x2 þ x3

;

y01 ¼mðx1 þ x2 þ x3Þ

ða þ x1 þ x2 þ x3Þðy1 þ y2 þ y3Þ

y1 þy2

2

� �2�sy1;

y02 ¼ 2

mðx1 þ x2 þ x3Þða þ x1 þ x2 þ x3Þðy1 þ y2 þ y3Þ

y1 þy2

2

� �y3 þ

y2

2

� �� sy2;


y03 ¼mðx1 þ x2 þ x3Þ

ða þ x1 þ x2 þ x3Þðy1 þ y2 þ y3Þ

y3 þy2

2

� �2�sy3:

Again, if one adds the first three equations andthe last three equations, the basic predator–preyequation (8) is reproduced. At this point themodel with three genotypes for each has notchanged. However, suppose that each prey has adifferent capture rate which we denote by m1;m2; and m3; respectively. Then we produce a newmodel for predator–prey interactions. Note theassumption that the capture rate is dependent onthe x genotype and not on the y genotype, thatis, all y genotypes capture a given x genotypeequally. To conserve notation, we return tousing x ¼ x1 þ x2 þ x3 and y ¼ y1 þ y2 þ y3 forthe respective sum of genotypes if there is nomultiplication by an mi: For brevity, we let

Tðx1; x2;x3Þ ¼ m1x1 þ m2x2 þ m3x3:

The model takes the form

x01 ¼

ax

x1 þx2

2

� �2�ax1x

K�

m1x1y

a þ x;

x02 ¼

2ax

x1 þx2

2

� �x3 þ

x2

2

� ��

ax2x

K�

m2x2y

a þ x;

x03 ¼

ax

x3 þx2

2

� �2�ax3x

K�

m3x3y

a þ x;

y01 ¼T x1;x2; x3ð Þða þ xÞy

y1 þy2

2

� �2�sy1;

y02 ¼ 2

T x1;x2; x3ð Þða þ xÞy

y1 þy2

2

� �y3 þ

y2

2

� �� sy2;

y03 ¼

T x1;x2;x3ð Þða þ xÞy

y3 þy2

2

� �2�sy3:

The variables in this model do not add toreproduce eqn (8) unless m1 ¼ m2 ¼ m3: How-ever, eqn (10) could be used to model the‘‘elusive’’ prey case discussed previously.We now turn to the main development, a

model for the poisonous prey and the resistantpredator. We remind the reader that we take thepoisonous prey to be the aa genotype (denotedby x3) and the resistant predator to be the bb

genotype (denoted by y3). The consumption ofx3 by y1 or y2 does not lead to added growth (sothe term must be subtracted from the precedingmodel) and does lead to increased death (so aterm must be added to the intrinsic death rate).However, the consumption of x3 by y3 does leadto increased growth. Thus, the distribution ofgenotypes will not be as convenient as thatexpressed in eqn (9). For y1; the growth termbecomes [corresponds to eqn (9)]

Tðx1; x2; 0Þy1a þ x

þ1

2

Tðx1;x2; 0Þy2a þ x

� �2

Tðx1; x2;x3Þya þ x

¼Tðx1; x2; 0Þ

2

ða þ xÞTðx1; x2;x3Þyy1 þ

y2

2

� �2:

Similarly, the term for y2 becomes

Tðx1; x2; 0Þy1a þ x

þ1

2


� �Tðx1; x2;x3Þy3

a þ xþ1

2

T x1; x2; 0ð Þy2a þ x

� �

Tðx1;x2; x3Þya þ x

¼ðTðx1; x2; 0Þy1 þ

1

2Tðx1; x2; 0Þy2ÞðTðx1; x2;x3Þy3 þ

1

2Tðx1;x2; 0Þy2Þ

Tðx1; x2;x3Þyða þ xÞ

while that of y3 becomes

Tðx1; x2;x3Þy3a þ x

þ1

2


� �2

Tðx1; x2; x3Þy3a þ x

¼ðTðx1;x2;x3Þy3 þ

1

2Tðx1;x2; 0Þy2Þ

2

Tðx1; x2;x3Þða þ xÞy

P. WALTMAN ET AL.64

We now incorporate these ideas into the modelof a biological arms race:

x01 ¼

ax

x1 þx2

2

� �2�ax1x

K�

m1x1y

a þ x;

x02 ¼

2ax

x1 þx2

2

� �x3 þ

x2

2

� ��

ax2x

K�

m1x2y

a þ x;

x03 ¼

ax

x3 þx2

2

� �2�ax3x

K�

m3x3y

a þ x;

y01 ¼

Tðx1; x2; 0Þ2

ða þ xÞTðx1; x2; x3Þyy1 þ

y2

2

� �2

�m3x3y1

a þ x� sy1;

y02 ¼ 2Tðx1; x2; 0Þy1 þ

1

2Tðx1;x2; 0Þy2

Tðx1;x2; x3Þy

Tðx1;x2; x3Þy3 þ

1

2Tðx1;x2; 0Þy2

a þ x

�m3x3y2

a þ x� sy2;

y03 ¼ðTðx1; x2;x3Þy3 þ

1

2Tðx1; x2; 0Þy2Þ

2

Tðx1; x2;x3Þða þ xÞy� sy3:

ð11Þ

Figure 9 shows a typical arms race. Theparameters have been selected to show an

1000 2000

0.10.20.30.40.50.6

Preda

y2

1000 2000

0.10.20.30.40.50.6

Pre

x2

Fig. 9. Co-evolution of predator and prey using parameteand initial conditions x1ð0Þ ¼ 0:6; x2ð0Þ ¼ 0:02; x3ð0Þ ¼ 0; y1ð0

oscillatory case and the initial conditionsreflect zero poisonous prey and resistant preda-tors but a very small number of heterozygotescarrying one copy of the respective alleles. Aneven lower number of heterozygotes (reflecting arandom perturbation) would present the sameresult but with a longer time-scale. An intuitiveexplanation begins with the fact that x1 and y1dominate the initial configuration that would bein a oscillatory regime, if they were the onlyorganisms present. Gradually, because of thelower capture rate, x3; the poisonous prey, out-competes x1 and x2 and lowers the predatorpressure by increasing the death rate of y1 and y2:This allows for the emergence of y3; the resistantpredator. Finally, y3 and x3 coexist in anoscillatory regime. This result is more dramati-cally portrayed in Fig. 10 which plots total preyx ¼ x1 þ x2 þ x3 and total predators, y ¼ y1 þy2 þ y3 against time in the middle part of theevolution.The reader is reminded that x and y are sums

of components of a system of differentialequations and do not satisfy a two-dimensionalsystem as Fig. 10 might suggest. However, if oneaccepts that the functions x1ðtÞ; x2ðtÞ; y1ðtÞ; andy2ðtÞ all tend to zero as t tends to infinity, asthe computations suggest, then eqn (11) is anasymptotically autonomous system with limitingequations of the form (8) with m ¼ m3 and theother parameters as specified. Of course, the

3000 4000 5000t

tor

y1

y3

3000 4000 5000t

y

x1

x3

r values m1 ¼ m2 ¼ 2:5; m3 ¼ 2:45; a ¼ 0:37; s ¼ 1:1; a ¼ 1:2Þ ¼ 0:6; y2ð0Þ ¼ 0:02; and y3ð0Þ ¼ 0:01:

200 400 600t

0.10.20.30.40.50.6

Predator

y1

y2

y3

200 400 600t

0.2

0.4

0.6

0.8Prey

x1 x2 x3

Fig. 11. Co-evolution of predator and prey using parameter values m1 ¼ m2 ¼ 2:5; m3 ¼ 2; a ¼ 0:37; s ¼ 1:1; a ¼ 1:2and initial conditions x1ð0Þ ¼ 0:6; x2ð0Þ ¼ 0:02; x3ð0Þ ¼ 0; y1ð0Þ ¼ 0:6; y2ð0Þ ¼ 0:02; and y3ð0Þ ¼ 0:

3250

3500

3750

4000

4250

t

0.2

0.3

0.4

0.5

x

0.2

0.25

0.3

y

t

0

Fig. 10. Co-evolution of the total predator population, y ¼ y1 þ y2 þ y3; and the total prey population, x ¼ x1 þ x2 þx3; using the same parameter values and initial conditions as in Fig. 9.


predator–prey relationship need not be oscilla-tory and Figs 11 and 12 show a similar evolutionbut with the parameters chosen so that thelimiting system is in a steady state.If the parameter m3 is lowered even

farther, it is possible for the prey to causeextinction of the predator as illustrated inFig. 13. Essentially, x3 out-competes x1 and x2but y3 cannot exist on x3 alone and thus becomesextinct. The levels are so low that they do notreally show on the graph; however, the prey sumis tending to K ¼ 1 so no predators will bepresent.

An important question remains: shouldthe capture rate for x3 by y3 be the same as thatby the others? It is possible to alter the modelto allow the capture rate to be dependent onboth the prey and the predator genotypes byreplacing mi by mij: This is a major increasein complexity and, in addition, it is unlikely thatsuch parameters could be realistically determined.However, to answer the question as towhether the dominant conclusion is due to thelow capture rate, we make a final alterationto the model to allow the capture of x3 (only) tobe different for y3: To avoid an unnecessarily

0

200

400

600

t

0

0.250.5

0.751

x

0

0.2

0.4

0.6

y0

Fig. 12. Co-evolution of predator y ¼ y1 þ y2 þ y3 and prey x ¼ x1 þ x2 þ x3 using the same parameter values andinitial conditions as in Fig. 11.

100 200 300t

0.10.20.30.40.50.6

Predator

y1

y2

100 200 300t

0.10.20.30.40.50.6

Prey

x1

x2

x3

Fig. 13. Co-evolution of predator and prey using parameter values m1 ¼ m2 ¼ 2:5; m3 ¼ 1:7; a ¼ 0:37; s ¼ 1:1; a ¼ 1:2and initial conditions x1ð0Þ ¼ 0:6; x2ð0Þ ¼ 0:02; x3ð0Þ ¼ 0; y1ð0Þ ¼ 0:6; y2ð0Þ ¼ 0:02; and y3ð0Þ ¼ 0:

P. WALTMAN ET AL.66

complex model, we allow just two capturerates: m is the capture rate of x1 and x2 by allpredators and of x3 by y3: We retain the capturerate notation m3 for the capture of x3 by y1and y2: The total capture of prey by predators isgiven by

mxy3 þ ðy1 þ y2ÞTðx1; x2;x3Þa þ x

;

where

Tðx1;x2;x3Þ ¼ mx1 þ mx2 þ m3x3:

We incorporate this into model (11) to obtain

x01 ¼

ax

x1 þx2

2

� �2�ax1x

K�

mx1y

a þ x;

x02 ¼

2ax

x1 þx2

2

� �x3 þ

x2

2

� ��

a x2x

K�

mx2y

a þ x;

x03¼

ax

x3 þx2

2

� �2�ax3x

K�

x3ðm3ðy1 þ y2Þ þ my3Þa þ x

;

y01 ¼m2ðx1 þ x2Þ

2

ða þ xÞðmxy3 þ ðy1 þ y2ÞTðx1; x2;x3ÞÞ

1000 2000 3000 4000 5000t

0.10.20.30.40.50.6

Predator

y1

y2y3

1000 2000 3000 4000 5000t

0.10.20.30.40.50.6

Prey

x1

x2 x3

Fig. 14. Co-evolution of predator and prey using parameter values m ¼ 2:5; m3 ¼ 2:45; a ¼ 0:37; s ¼ 1:1; a ¼ 1:2 andinitial conditions x1ð0Þ ¼ 0:6; x2ð0Þ ¼ 0:02; x3ð0Þ ¼ 0; y1ð0Þ ¼ 0:6; y2ð0Þ ¼ 0:02; and y3ð0Þ ¼ 0:01:

3000

3500

4000

4500

t

0.2

0.3

0.40.5

x

0.15

0.2

0.25

0.3

y

0

0

Fig. 15. Co-evolution of predator y ¼ y1 þ y2 þ y3 and prey x ¼ x1 þ x2 þ x3 using the same parameter values andinitial conditions as in Fig. 14.

9960 9970 9980 9990 10000t

0.05

0.1

0.15

0.2

0.25

0.3

0.35Predator and Prey

x1

x2

x3

y3

Fig. 16. Figure 14 for 9950ptp10 000:


y1 þy2

2

� �2�

m3x3y1

a þ x� sy1;

y02 ¼ 2

mðx1 þ x2Þðy1 þ1

2y2Þ

mxy3 þ ðy1 þ y2ÞTðx1; x2;x3Þ

ðmxy3 þ1

2mðx1 þ x2Þy2

a þ x�

m3x3y2

a þ x� sy2Þ;

y03 ¼ðmxy3 þ

1

2mðx1 þ x2Þy2Þ

2

ðmxy3 þ ðy1 þ y2ÞT x1;x2; x3ð ÞÞða þ xÞ� sy3:

ð12Þ

100 200 300 400 500t

0.10.20.30.40.50.6

Predator

y1

y2

y3

100 200 300 400 500t

0.10.20.30.40.50.6

Prey

x1

x2

x3

Fig. 17. Co-evolution of predator and prey using parameter values m ¼ 2:1; m3 ¼ 1:9; a ¼ 0:37; s ¼ 1:1; a ¼ 1:2 andinitial conditions x1ð0Þ ¼ 0:6; x2ð0Þ ¼ 0:02; x3ð0Þ ¼ 0; y1ð0Þ ¼ 0:6; y2ð0Þ ¼ 0:02; and y3ð0Þ ¼ 0:01:

0

200

400

600

800

1000

t

00.2

0.40.6

0.8

x

0.2

0.4

0.6

0.8

1

y

0

0

0

Fig. 18. Co-evolution of the total prey and total predator using parameter values as in Fig. 17.

P. WALTMAN ET AL.68

We repeat the first two simulations above foreqn (12) with the same parameters and initialconditions; the results are shown in Figs 14and 15.We next plot a short time period to illustrate

the periodic nature of the final outcome inFig. 16.In this case, all three prey genotypes survive

but only the resistant predator, y3; survives. Thearms race ends as it had begun but with onlythe resistant predator surviving. This is to beexpected since now y3 feeds equally on all prey

genotypes, so x3 cannot eliminate its competitorsalthough it does eliminate the non-resistantpredators. However, it does increase its relativefrequency during the time that y1 and y2dominate the mix. Thus its final proportion ismuch higher than if the same problem wassimulated with only the resistant predatorpresent (and the same prey initial conditions).The choice between eqns (11) and (12) could bedecided by observable data. If a territory couldbe found where the resistant predator dominatesbut the non-poisonous prey survives in quantity,


then eqn (12) is supported. Figure 15 shows thethree-dimensional plot of total prey and totalpredators against time.The result need not be oscillatory as Figs 17

and 18 illustrate.

5. Conclusion

We have provided a model of a biologicalarms race motivated by a predator–prey systemwhere the prey develops the ability to produce atoxin against the predator and then the predatorresponds with resistance to the toxin. We have asa biological model that of the newt–garter-snakerelationship studied by Brodie and Brodie (1999).Our model and simulations seem to capture mostof the points discussed there. The principalmodeling difficulty was to expand the determi-nistic genetic modeling to the predator dynamicswhere growth depends on prey capture.One thinks of the genetic change as occurring

by a random mutation that we model by taking avery small initial condition in the differentialequations. The simulations seem to show that thepoisonous prey cannot become established inlarge enough numbers to affect the system with-out having an advantage. Then we assumed thatthe special prey has an advantage with respect toprey capture. This is also observed in the newt–snake system: the poisonous prey sometimesescapes the predator’s grasp alive. The size ofthe advantage determines how rapidly the systemevolves but any advantage will lead to establish-ment. Once the poisonous prey is established insufficient quantities, the non-resistant predatorsface extinction (although this is an artifact of ourassumption that it lives only on this prey) and theresistant predator does not have this added deathrate and thus thrives. The arms race ends as itbegan except with slightly altered players. Thenext step in the arms race requires a newmutationFperhaps an altered poison.The model suffers the usual deficiencies of

predator–prey models in that it assumes thepredator lives exclusively on the prey. It alsopresumes that the change in the genetic trait is atone locus whereas major alterations usuallyreflect multiple loci. Additional loci can beincluded in the model at a considerable increasein complexity but well within the reach of

modern computers. In the model of So (1990)two loci are considered and the number ofequations increase from three to nine. Addi-tional alleles can also be included. However, theassumptions here are no worse than thoseusually associated with such systems. Geneticimprovement often comes at a cost, usuallyreflected in a lower reproduction rate. For thepoisonous newt this seems not to be the case (orit is negligible). That is not the case with thegarter snake as Brodie & Brodie (1999) showthat the resistant snake has a lower sprintvelocity that would be modeled in our systemby a change in capture rate. We feel that this canbe incorporated into a more general model,alluded to in the main text, by making thecapture rate dependent on both predator andprey genotypesFintroducing mij instead of mi:We hope to do this in a later study.

All of the differential equations were solved with,and all of the figures created with, Mathematica 4,Wolfram Research, Inc., 1999. (see Wolfram, 1999).The authors wish to thank Edmund D. Brodie III forcommunicating his insights regarding predator–preyarms races and to thank Martha Abell for assistancewith the graphics.

The research of Paul Waltman was supported byNational Science Foundation Grant, DMS-9801622.

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a mathematical model of a biological arms race with a dangerous

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