The "Battle of the Sexes": A Genetic Model with Limit Cycle Behavior

Working Paper

THE " B m OF _THE

S-":

A GENJiTIC

MODEL

WITH

CYCLE

BEHAVIOR

J. M a y n a r d Smith Josef H o f b a u e r

J a n u a r y 1986 WP-86-12

International Institute for Applied Systems Analysis

A-2361 Laxenburg, Austria

NOT FOR QUOTATION WITHOUT THE PERMISSION OF THE AUTHORS

THE

" B A m OF

T H E

SEXE?':

A G E m I C MODEL

m H LIHlT CYCLE

BEHAVIOR

J. Maynard Smith Josep Hoflauer

January 1986 WP-86-12

Working Papers a r e i n t e r i m r e p o r t s o n w o r k of t h e I n t e r n a t i o n a l I n s t i t u r e f o r Applied S y s t e m s A n a l y s i s a n d h a v e r e c e i v e d o n l y l l m i t e d r e v i e w . Views or o p i n i o n s e x p r e s s e d h e r e i n d o n o t n e c e s s a r i l y r e p r e s e n t t h o s e of t h e I n s t i t u t e o r of its N a t i o n a l M e m b e r O r g a n i z a t i o n s .

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 L a x e n b u r g , A u s t r i a

Abstract

A two-locus genetic model, based on Dawkin's "sex war" game, with the fitness of the genotypes a t each locus depending on the gene frequencies a t the o t h e r , is shown t o give r i s e t o a stable limit c y c l e . The mathematical analysis involves averaging techniques and elliptic integrals.

Acknowledgement

We would like to t h a n k P r o f e s s o r K a r l Sigmund f o r h i s a d v i c e a n d encourage- ment during t h e p r e p a r a t i o n of t h i s p a p e r .

THE "BA-

OF

THE

SEXES":

A GENGTIC MODEL

WITH _LMIT

CYCLE BEHAVIOR

* **

J. M a y n a r d .%Smith a n d J o s q f H o f b a u e r

*

School of Biological S c i e n c e s , University of Sussex Falmer, Brighton. Sussex BN1 9QG. England

**

lnstitut f u r Mathematik, Universitat Wien Strudlhofgasse 4, A-1090 Wien. Austria

1. I n t ~ oduction

This n o t e d e s c r i b e s a simple two-locus g e n e t i c model t h a t gives r i s e t o a s t a b l e limit cycle. The o r i g i n of t h e model i s Dawkins' (1976) "sex w a r " game, a simple evolutionary game d e s c r i b i n g t h e p a r e n t a l investment conflict. In t h i s game males h a v e two possible s t r a t e g i e s , f a i t h f u l ( E l ) o r p h i l a n d e r e r i E 2 ) , and females may play t h e s t r a t e g i e s f a s t (F1) o r coy (F2). Assuming +15 as t h e r e w a r d f o r hav- ing a child ( t o both p a r e n t s ) . -20 as t h e total c o s t of raising t h e offspring, a n d -3 f o r t h e prolonged c o u r t s h i p coy females insist on. Dawkins obtained t h e following pay-off m a t r i c e s A f o r males and B f o r females:

5 3 1 5

This bimatrix game h a s a totally mlxed Nash equilibrium at (- -) and (- -) 8 ' 8 6 ' 6 ( s e e Dawkins (1976). Maynard Smith (1982). S c h u s t e r and Sigmund (1981)). I t is not a n evolutionarily s t a b l e state (ESS), however, s i n c e t h e second (stability) condi- tion i s not s a t i s f i e d . In f a c t , S e l t e n (1980) h a s shown in g e n e r a l t h a t in asymmetric conflicts, only p u r e s t r a t e g i e s c a n b e evolutionarily s t a b l e . Hence t h e s t a t i c , game t h e o r e t i c a p p r o a c h i s not v e r y s a t i s f a c t o r y f o r t h i s p a r t i c u l a r example, and i t seems r e a s o n a b l e t o i n t r o d u c e some dynamics.

I t t u r n s o u t , however, t h a t t h e s t a n d a r d continuous time game dynamics, as studied e.g. by Zeeman (1980), may b e generalized in (at l e a s t ) two d i f f e r e n t ways t o asymmetric conflicts: t h e one, see S c h u s t e r and Sigmund (1981), gives r i s e t o c o n s e r v a t i v e oscillations, t h e o t h e r , see Maynard Smith (1982), Appendix J, makes

the equilibrium asymptotically stable. With discrete time, however, the stationary solution would always be unstable (Eshel and Akin, 1983).

The f i r s t type of dynamics is also relevant f o r the analysis o f our new model in 5 2: With z i

,

y j denoting the relative frequencies o f the strategies Ei,Fj respec- tively and the basic assumption

z i

- ⁼ growth rate o f Ei = average payoff f o r Ei

zi

-average payoff f o r males = ( A y ) i - g t A y .

Schuster and Sigmund (1981) derlve the equations

f o r z = z l = 1-z2.

Y

= y 1 =

l y 2 .

They show that

1s a constant o f motion, so that all solutions oscillate around the Nash equilibrium.

which also represents the tlme-average o f any orbit. (Note t h e strong analogy t o the Lotka-Volterra predator-prey equations.)

This model has one main drawback, however: I t neglects the genetic struc- ture. The derivation ( 2 ) assumes implicitly that individuals reproduce asexually.

There is thus a need f o r a genetic model whlch takes into account the sexual in- teraction between males and females. One such diploid model was studied by Bomze e t al. (1983), again using continuous time and leading t o similar conservative oscil- lations. As this lacks structural stability too, a more natural model would be t o consider a diploid model with separated generations.

We consider two loci, with two alleles at each locus. The A-locus, with alleles A and

a ,

regulates male behavior: A + strategy E l

(=

faithful), a +E2

(=

phi- landerer), and i s not expressed Ln females. More precisely, we assume that homoz- ygotes

AA

and

aa

play the pure strategies E l and E2 respectively, and that the heterozygote

A a

plays the mixed strategy -(El 1 + E 2 ) . In a similar way the B-locus

2

determines female behavior:

B +

F1 = f a s t ,

b

+F2 = coy. Let

pA,qA

be the f r e -

quencies o f allele A in adult males and females respectively, and pg,qg b e t h e corresponding frequencies o f allele B .

Assuming linkage equilibrium, t h e frequencies o f AA

.

^{Aa ,}

^aa

in t h e next gen- eration a r e P A ~ A s P A ( 1 - q ~ ) + qsubA ( 1 - p A ) = P A + QA

-

2pAqA, (1 *A ) ( I - q A ) respectively. The marginal frequency o f gene A , which also measures t h e proba- bility o f strategy E l among males in t h e new generation, is then given by

1 1

pAqA

+

-(pA +qA -2pAqA) = ?(pA +qA). Similar results hold f o r t h e B-locus.

2

In order t o simplify t h e mathematical analysis, we replace Dawkins' original pay-off matrix ( 1 ) b y t h e following more symmetric one (this does not change t h e situation qualitatively):

1+L 1+K, 1-L

I

^{with 0}

^<

^{K L}

^<

^1.

+K, 1-L 1-K. 1+L

This gives t h e following fitness values f o r males, depending on t h e frequency o f f e - males:

with similar results f o r females

Hence writing p r A , e t c .

for

t h e allelic frequencies in the adults o f t h e new generation,

1

QA

PA

^{+qA -2pA )}

q k

= - _-

_{PA +qA}

1 2 , etc.

The mean fitness o f males at t h e A-locus, wA

,

is given by

This leads finally t o the equations

P'g

=

-(Pg+qB) 1 2

Since O H , L <I this is a well-defined transformation on t h e state space [0,114 Obviously pA

=

qA

=

pg

=

qg

= y

I is a stationary solution o f ( 7 ) with two zero eigenvalues (corresponding t o a v e r y quick convergence t o a sex ratio 1 : l ) and a pair o f complex conjugate eigenvalues with real part +1 (due t o t h e cyclic struc- t u r e o f Dawkins' game) which make this solution unstable.

On t h e other hand, using t h e method

of

"average Ljapunov functions", developed by Hofbauer and Sigmund (1984) and Hutson and Moran (1982), one can analyze the local behavior near t h e boundary o f t h e state space. showing that this boundary is repelling, and system ( 7 ) exhibits "permanent coexistence'' (see Ap- pendix A). The attractors o f ('7) thus have t o lie in t h e interior

o f

[0.114, and since the only stationary solutlon t h e r e is unstable, this attractor has t o be a more com- plicated set.

Numerical simulations show that this attractor is a 'limit cycle", i.e. a closed invariant attracting curve surrounding t h e stationary solution (see Fig. 1 ) . In t h e next section we will give a mathematical proof o f this observation

for

small values

of

L

,K

(this is t h e biological most relevant case) and determine t h e explicit equa- tion

of

this limit cycle in t h e limit K

=

L +O. This is done by averaging techniques, as, f o r example, when treating Van der Pol's equation as a perturbation

of

t h e har- monic oscillator.

PA

FIGURE

1. The limit c y c l e .

3. The Limit Cycle

We begin with a l i n e a r c h a n g e of v a r i a b l e s which t r a n s f o r m s t h e s t a t i o n a r y solution to t h e origin:

Then ( z , y , u . v ) ~ [ - l . + l ] ~

.

F u r t h e r m o r e w e assume K

=

L to make t h e analysis simpler and set c = K / 4 . Then (7) i s transformed into

NOW

I

^{u r}

1, I

V , 1s c and so u and v will b e of o r d e r 0 (c) f o r c+O, unlformly o n 1 + 4 c S

t h e whole s t a t e - s p a c e , in a l l subsequent g e n e r a t i o n s . (This c o r r e s p o n d s t o a rough equllibration of t h e s e x - r a t i o . ) Thus t h e four-dimensional system (9) r e d u c e s f o r small r t o a two-dimensional one:

z f = z

+ r f ( z , y )

+

r 2 ? ( z , y )

+

0 ( r 3 ) y'

=

y

+

e g ( z , y )

+

r 2 f ( z . v ) + o ( r 3 ) with

f ( z . y )

=

a / ( l - 2 2 ) f

-

( 2 . y )

=

-4zy2 ( 1

- z 2 )

f ( z , y )

= 4 z 2

y ( l

-

y 2 ) . (11)

g ( z . v )

= z ( 1 - y 2 )

F o r c -r 0 w e o b t a i n , as a f i r s t - o r d e r approximation, t h e d i f f e r e n t i a l equation

z

= f ( z , y ) = a / ( l - 2 2 )

ai

= g ( z , y ) = z ( l

- v

2 ). (12)

These are

-

u p t o a t r a n s l a t i o n of t h e Nash solution t o t h e o r i g i n

-

just t h e Schuster-Sigmund equations ( 3 ) f o r Dawkins' game, see a l s o Maynard Smith (1982).Appendix J .

These equations are i n t e g r a b l e with

as a c o n s t a n t of motion. T h e r e f o r e w e c h a n g e t o "canonical c o o r d i n a t e s " (R,cp) such t h a t (12) a p p e a r s in t h e normal form

with z

=

~ e ~ ^P, o r equivalently

~ 2 0 ,& = u ( R 2 )

The canonical a n g l e v a r i a b l e cp a n d t h e a n g l e velocity o ( R 2 ) are e v a l u a t e d in Ap- pendix B. They c a n b e e x p r e s s e d only in terms of elliptic i n t e g r a l s ,

Now we s h a l l e x p r e s s o u r d i f f e r e n c e equation (10) in terms of t h e canonical v a r i a b l e s I

=

R~ and cp:

Now t h e l i n e a r terms in t vanish s i n c e (14) implies

i =

I,f +Iyg

=

0

.

F o r t h e r2

-terms w e obtain from ( 1 3 )

I, = 2 z ( 1 y 2 ) , I,, = 2 ( 1 y 2 ) , IZY = - 4 Z y ; . . Therefore

I'

=

I + E ~ F ( Z , ~ ) + O ( & ~ )

wlth

F ( z ,Y

=

7 1 , +&?IY

++u~I,,,

1 +Zf91xy + 9 2 1 y y )

=

4 z y 2 ( 1 - z 2 ) 2 z ( l y 2 ) - 4 z 2 y ( l - Y 2 ) 2 y ( 1 2 2 )

+

Y 2 ( 1 - z 2 ) 2 ( 1 q 2 ) + ~ ( 1 - z 2 ) ( 1 y 2 ) 4 z y + ~ ~ ( l y ~ ) ~ ( l - z ~ )

=

(l-z2)(ly2)[-16z2y2+y2(1~2)+422y2+z2(1--y2)]

=

( 1 - ~ ~ ) ( z ~ + y ~ - 1 4 z ~ y ~ )

Similarly, using

b =

v , f + v y g

=

o ( I ) , w e obtain

fp'

=

p ( z ' , y ' )

=

v ( z + & f + , . . . , y + c g + .

. .

⁾

= v ( z

,Y ) + t f

v ,

^{+ ~ g f p y} + 0 ( c 2 )

=

f p + ~ o ( R ~ ) + o ( e ~ ) . S o w e end up wlth

I ,

=

I + E ~ F ( I , v ) + o ( E ~ )

The next step i s to average out the pdependence in the c2-term o f ( 1 8 ) by the An- satz P

=

I + & h ( 1 , ~ ) . Then

N o w , following the usual averaging procedure, compare e . g . Arnold ( 1 9 8 3 ) , p. 147

f f , we s e p a r a t e F into i t s mean and i t s oscillating p a r t ,

with

I n s e r t i n g t h i s into (19) w e may c h o o s e h in such a way t h a t

2 n

1 Oh

(Since h i s a 2~ -periodic function in (p, t h e mean of i t s d e r i v a t i v e ,

-J

^{-d 9.}

27T

av

equals z e r o . Since t h i s will not in g e n e r a l b e t h e c a s e f o r t h e mean of F ( I , v ) we c a n a v e r a g e o u t only t h e oscillating p a r t F(I,rp).

S o with o u r new v a r i a b l e P Instead of I we have simplified (18) to

The a v e r a g e d function G ( P ) i s computed in Appendix B (in t e r m s of e l l i p t i c In- t e g r a l s again), w h e r e w e a l s o p r o v e t h e e x i s t e n c e of a number Po s u c h t h a t G(0)

=

G(1)

=

G ( P o )

=

0

G ( P ) > O f o r O < P < P o G ( P ) < O f o r Po < P < 1 Thus t h e g r a p h of G i s similar to t h a t shown in Fig. 2.

If w e neglect 0-terms in (22). t h e n P =Po is obviously t h e equation of a n in- v a r i a n t globally a t t r a c t i n g c i r c l e . That t h i s c i r c l e also p e r s i s t s u n d e r t h e 0- p e r t u r b a t i o n s f o r small c

>

0 , c a n be shown e.g. by t h e technique developed by Iooss (1979) f o r proving t h e Hopf b i f u r c a t i o n theorem f o r maps.

FIGURE 2. The g r a p h of t h e function G ( P ) .

In p a r t i c u l a r i t i s found t h a t for small ^E

>

0 ( 1 0 ) h a s a n a t t r a c t i n g limit c y c l e of t h e form

S o o n e p a r t i c u l a r p e r i o d i c o r b i t from t h e Hamiltonian system ( 1 2 ) , i.e. t h e Schuster-Sigmund dynamics f o r Dawkins' game,

R~ = z2+y2-z2y2

= P O , s e r v e s as t h e limiting case E + O f o r t h e limit c y c l e s of o u r more r e f i n e d g e n e t i c model of t h e ' b a t t l e of t h e sexes". The constant Po which determines t h e position of t h i s c y c l e i s t h e z e r o of t h e function G ( P ) and i s computed in Appendix B.

W e h a v e d e s c r i b e d limit cycle b e h a v i o r in a simple g e n e t i c model, with two al- l e l e s at e a c h of two loci, with sex-dependence and f i t n e s s e s depending on t h e f r e - quencies of t h e o t h e r s e x . The model i s based o n Dawkins' (1976) 'battle of t h e sexes". F o r small E , measuring t h e intensity of f r e q u e n c y dependent selection, w e h a v e p r o v e d t h e e x i s t e n c e and stability of a limit cycle. The amplitude of t h i s limit c y c l e i s r a t h e r insensitive to c h a n g e s in t h e s e l e c t i o n intensity, while t h e "period"

i s approximately i n v e r s e l y p r o p o r t i o n a l t o t h e selection intensity. This i s in con- trast t o limit cycle behavior constructed by s t a n d a r d Hopf bifurcation techniques where t h e amplitude i s small and t h e p e r i o d i s approximately c o n s t a n t .

In t h e limit e -. 0 , t h i s limit c y c l e tends t o one p a r t i c u l a r c y c l e of t h e simple Hamiltonian system identified by S c h u s t e r a n d Sigmund (1981) as t h e f i r s t dynamic model f o r Dawkins' game. In p a r t i c u l a r , at least for e -. 0 , t h e time a v e r a g e o v e r t h e limlt c y c l e r e p r e s e n t s t h e i n t e r i o r s t a t i o n a r y solution p r e d i c t e d by game t h e o r y . Thus t h i s solution, though unstable both dynamically a n d evolutionarily, still h a s some biological r e l e v a n c e . This shows o n c e more t h a t d e s p i t e t h e i r sim- plicity such haploid models a l r e a d y c a r r y t h e b a s i c ingredients.

Numerical simulations show t h a t t h i s limit c y c l e b e h a v i o r i s r a t h e r r o b u s t . I t d o e s not depend o n t h e assumption t h a t f i t n e s s e s a r e additive, or t h a t t h e selection intensity i s t h e same in t h e two s e x e s (i.e. K

=

L ). Limit c y c l e b e h a v i o r i s s t i l l ob- s e r v e d if t h e r e i s some d i r e c t i o n a l , non-frequency-dependent selectlon. provided t h a t t h i s d i r e c t i o n a l s e l e c t i o n i s l e s s intense t h a n t h e frequency-dependent selec- tio;i. Also, t h e conclusion d o e s not depend on t h e assumption t h a t t h e g e n e s are sex-limited in t h e i r e f f e c t s . A similar limit c y c l e a r i s e s if t h e f i t n e s s e s of t h e genotypes at t h e A locus, in both s e x e s , depend additively o n t h e f r e q u e n c i e s of t h e a l l e l e s at t h e B locus, a n d vice-versa.

All t h e s e o b s e r v a t i o n s could b e p r o v e d using t h e same method of a v e r a g i n g . We h a v e confined o u r s e l v e s to t h e simplest possible c a s e , however, in o r d e r t o k e e p t h e mathematical analysis t r a c t a b l e . In m o r e g e n e r a l models dlfficulties could a r i s e s i n c e t h e n i c e e l l i p t i c i n t e g r a l s h a v e t o b e r e p l a c e d by more g e n e r a l i n t e g r a l s where formulas l i k e (B5-7) are probably not available.

That limit c y c l e b e h a v i o r i s possible in r a t h e r simple g e n e t i c systems h a s b e e n shown in t h e l a s t f e w y e a r s . The pioneering work in t h i s d i r e c t i o n i s d u e to Akin (1979). O t h e r p a p e r s dealing with t h i s t o p i c are Akin (1982, 1983). Hastings (1981), H o f b a u e r a n d Iooss (1984). H o f b a u e r (1984, 1985). Hunt (1982). Koth a n d Kemler (19851, S e l g r a d e and Namkoong (1984).

Appendix A: Behaviour Near the Boundary

F i r s t i t i s e a s y t o c h e c k t h a t o n e of , qi , pi ,

qi

is z e r o if and only if o n e of t h e f o u r a l l e l e s A , a , B , b i s missing in both s e x e s . In t h e new v a r i a b l e s (8) :

z =

ⁱ 1 or y

=

ⁱ 1. Thus t h e set

which i s a union of f o u r s q u a r e s , i s t h e maximal I n v a r i a n t s u b s e t of t h e boundary of t h e state s p a c e : S t a r t i n g on t h e boundary outside M , t h e n e x t g e n e r a t i o n a n d a l l subsequent o n e s will b e in t h e i n t e r i o r of [0,114. i.e. a l l f o u r a l l e l e s will b e p r e s e n t in both s e x e s . Thus in o r d e r t o show t h a t t h e boundary i s a repelling s e t , i t i s suf- f i c i e n t to p r o v e t h a t M i s repelling.

F o r t h i s w e f i r s t need t h e behavior on M itself. S i n c e z'Sz iff y W , i t i s e a s y t o see t h a t on t h e p a r t w h e r e y =-I,

z

c o n v e r g e s monotonically, i n c r e a s i n g from -1 t o 1. Where

z

= I , y g o e s f r o m -1 t o 1 ; where y =1, z g o e s from 1 back to -1, a n d finally w h e r e z=-1, y d e c r e a s e s f r o m 1 to -1, a n d t h e c y c l e i s closed.

Thus t h e o

-

limit set of M c o n s i s t s only of t h e f o u r fixed polnts w h e r e jz l = l y = l ( a n d u = v = O ) .

Now c o n s i d e r t h e function

Then P ² 0 holds, with P = O on M . F u r t h e r m o r e

->

P' d

>

0 on t h e whole state P

s p a c e a n d

P '

a t t h e f o u r boundary fixed points in M . Thus

- ^>

^I o n t h e whole o-limlt set of M , P

a n d P i s a n " a v e r a g e Lyapunov function" f o r M . According to Hutson a n d Moran (1982), M i s a r e p e l l e r .

Appendix B: Elliptic Integrals in the Battle of the Sexes

We start with t h e normal form (14)

of t h e Schuster-Sigmund d i f f e r e n t i a l equation (12) f o r t h e b a t t l e of t h e s e x e s . Since, by (13), w e have

R2

=

z 2 + y 2 ~ 2 y 2 , (82)

t h e s t a t e s p a c e [ ( z , y ) cIK2 :

1

z

1 <

11 f o r (12) t r a n s f o r m s into f ( R , p ) E R ^{X R /} 2 n Z : 0 S R

<

11. In o r d e r t o d e t e r m i n e t h e canonical angle p and t h e angle velocity o(R2) w e treat p as a function of

z

and

R:v =

p ( z , R ) . Then

it =

rp,z + p R ~ implies

Eliminating y from (B2).

I t is well-known t h a t t h i s leads t o a n elliptic i n t e g r a l . In p a r t i c u l a r w e obtain

Now r e c a l l Legendre's formulas f o r complete elliptic i n t e g r a l s ( s e e e.g. G r o e b n e r and H o f r e i t e r (1950), p . 39):

where (a),

=

a ( a - 1 ) . .

.

(a--n + l ) . These s e r i e s c o n v e r g e f o r

kl <

1. (B4) and (B5) imply

The n e x t problem i s t o evaluate t h e a v e r a g e d function (20):

Since w e d o not have a n explicit expression for rp, w e have to t r a n s f o r m t h e in- t e g r a l . Using (82) and splitting t h e c i r c l e into 4 equal p a r t s w e obtain

S o again w e h a v e t o calculate a n elliptic i n t e g r a l . From (16) w e obtain

So (B10) t o g e t h e r with (BE) gives a n "explicit" formula f o r G:

G (R2)

=

( 1 -R2)(14~'-26

+

^{26 %).}

K(R) (812)

Inserting t h e s e r i e s expansions (B5), (B6) into ( B l l ) w e obtain after some cal- culation:

Obviously a l l t e r m (up t o t h e f i r s t two) of t h e sum have a negative coefficient.

S i n c e , f u r t h e r m o r e , G (R2)(1-R2)

"IR,l =

-12

<

0 from (BIZ), t h e function G ( P ) h a s a unique z e r o Po in t h e i n t e r v a l ( 0 , l ) and i s t h e r e f o r e of t h e d e s i r e d form (23).

The numerical value of Po c a n b e found approximateIy from a t a b l e of elIiptic func- tions: Po 0.46.

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