discrete time spatial models arising in genetics

8/8/2019 Discrete Time Spatial Models Arising in Genetics

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ELSEVIERDiscrete Time Spatia l Models Ar is ing in Genetics,Evolut ionary Game Theory , and Branching ProcessesJ . R AD CL IFF E AND L. RASSSchool o f M athematical Sciences, Queen Ma ry an d WesOqeld College,Universi ty of London, Mile End Road, L ond on E1 4N S, EnglandReceived 29 Decem ber 1995; revised 17 September 1996

A B S T R A C TA saddle point method is used to obta in the speed of f i rs t spread of newgenotypes in genetic models and of new strategies in game theoretic models. I t isalso used to obtain the spe ed of the forw ard tail o f the distribution of farthest spre adfor branching process models. The technique is applicable to a wide range o f models.They include multiple allele and sex-linked models in genetics, multistrategy andbimatrix evolutionary games, a nd multitype an d de m ograp hic branching processes.The speed of propagat ion has been obta ined for genet ics models ( in s imple casesonly) by Weinberger [1, 2] and Lui [3-7], using exact analytical methods. The exact

results were obtained only for two-allele, single-locus genetic models. The saddlepoin t meth od agrees in these very simple cases with the results obtaine d by using theexact analytic methods. Of course, i t can also be used in much more generalsituations far less tractable to exact analysis.The connection between genetic and game theoretic models is also brieflyconsidered, as is the extent to which the exact analytic methods yield results forsimple mod els in ga m e theory. Elsevier Science Inc., 1997

1. I N T R O D U C T I O NA s a d d l e p o i n t m e t h o d h a s b e e n u s e d t o o b t a i n t h e s p e e d o f

p r o p a g a t i o n f o r c e rt a in c o n t i n u o u s t i m e m o d e l s w h e n t h e s p a ti a l a s p e c tis d e s c r i b e d b y c o n t a c t d i s tr i b u ti o n s . T h e m a i n a r e a w h e r e t h e m e t h o dh a s b e e n a p p l i e d i s i n t h e m o d e l i n g o f e p id e m i c s . I t h a s a ls o b e e na p p l i e d t o c o n t a c t b r a n c h i n g p r o c e s s e s .

T h e r e s u l t fo r a s i m p l e o n e - t y p e S ~ I e p i d e m i c m o d e l o n a l in e w a so b t a i n e d b y D a n i e l s [8]. A n a p p l i c a t i o n t o t h e s p a t ia l b i r th p r o c e s s i sg i v e n i n R e f . 9. A r i g o r o u s a p p r o a c h t o t h e s a d d l e p o i n t m e t h o d is g i v e ni n R e f . 10, i n w h i c h t h e s p e e d o f p r o p a g a t i o n f o r a n n - t y p e S ~ I ~ Rm o d e l is d e r i v e d . T h e r e s u l ts w e r e l a t e r e x t e n d e d t o c o v e r N - d i m e n -s i o n a l n o n s y m m e t r i c c o n t a c t d i s t r i b u t i o n s w h e n t h e i n f e c t i o n m a t r i x i sr e d u c i b l e [ 1 1 ] .

M A T H E M A T I C A L B I O S C I EN C E S 140"101-129 (1997) Elsevier Science Inc., 1997655 Avenue o f the Americas, New York, NY 10010 0025-5564/97/$17.00P II S0025-5564(97)00154-X


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102 J. RA DC LIFFE AN D L. RAS SR i g o r o u s an a l y t i c a l t e ch n i q u es co n f i r m ed t h e sp eed s o f p r o p ag a t i o no b t a i n e d b y t h e s a d d l e p o i n t m e t h o d ; t h e s e r i g o r o u s t e c h n i q u e s w e r er e s t r i c t ed t o t h e ca se i n w h i ch t h e co n t ac t d i s t r i b u t i o n s a r e sy m m et r i c

[12-151.T h e s a d d l e p o i n t m e t h o d h a s a l s o b e e n a p p l i e d t o o t h e r m o d e l s f o rw h i ch an ex ac t an a l y s i s i s l e s s t r a c t ab l e . T h ese m o d e l s i n c l u d e ep i -d em i cs a l l o w i n g a r e t u r n t o t h e su scep t i b l e s t a t e an d b i r t h s i n t o t h esy s t em [1 6, 1 7 ] . T h e m e t h o d g i ve s th e sp eed o f f ir s t sp r ead o f t h ef o r w ar d f r o n t o f i n f ec t i o n . F o r s t o ch as t i c co n t ac t b r an ch i n g p r o ces sm o d e l s , t h e sp eed a t w h i ch t h e e x t r em e t a il o f th e d i s t ri b u t io n f u n c t i o no f f a r t h e s t sp r ead m o v es o u t i n an y sp ec i f i ed d i r ec t i o n is g i v en i n R e f .17.T h i s p ap e r co n s i d e r s d is c r e t e t im e m o d e l s , w h i ch a r e m o r e ap p r o p r i -

a t e f o r t h e m o d e l i n g o f g en e t ic s , ev o l u t io n a r y g am e t h eo r y , an d ce r t a i nb r an ch i n g p r o ces se s . A g a i n , co n t ac t d i s t ri b u t io n s a r e u sed t o m o d e l t h esp a ti a l a sp ec t . T h ey r ep r e s en t t h e d i s t an ce m o v e d b y i n d iv i d u a ls e i t h e ra t b i rt h o r a f t e r re a c h i n g m a t u r i t y p r i o r t o t h e p r o d u c t i o n o f t h e n e x tg en e r a t i o n . T h i s i s l i k e l y t o b e m o r e r ea l i s t i c t h an a l l o w i n g co n t i n u a ld i f fus ion dur ing the l i f e t ime o f an ind iv idua l .A s a d d l e p o i n t m e t h o d c a n b e u s e d f o r th e s e d i s c re t e ti m e m o d e l s .R esu l t s a r e o b t a i n ed f o r a g en e r a l m o d e l . T h i s is t h e n i n t e r p r e t ed i n t h eco n t ex t o f sp eci fi c m o d e l s u sed i n g en e t ic s , g am e t h eo r y , an d b r an ch i n gp r o ces se s, y ie l d i n g r e su lt s o n t h e sp ee d o f f ir s t sp r ead o f an a l le l e , ageno type , o r a s t r a t egy in an in i t i a l ly s t ab le popu la t ion , as wel l as thesp eed o f t h e f o r w ar d t a i l o f t h e d i s t r i b u t i o n o f f a r t h e s t sp r ead f o rb r an ch i n g p r o ces s m o d e l s . T h ese r e su lt s a r e sh o w n t o b e co n s i s t en t w i t ht h e r e su lt s f o r s i m p l e t w o - a ll e le , s in g l e- lo cu s g en e t i c m o d e l s co n s i d e r e dby Weinberger [1 , 2] and Lui [3 , 4 , 5 , 7] .T h e c o n n e c t i o n b e t w e e n g e n e t i c s a n d g a m e t h e o r e t i c m o d e l s is a ls ob r i e f l y co n s i d e r ed , a s i s t h e ex t en t t o w h i ch t h e ex ac t m e t h o d s y i e l dr e su lt s i n a s i m p l e g am e t h eo r y co n t ex t.2. T H E S A D D L E P O I N T M E T H O DF O R D I S C R E T E T I M E M O D E L S

C o n s i d e r a n o n s p a t ia l d i s c r e te t i m e m o d e l o f t h e f o r mX ! e + l ) . ~ - f i [ X ( r a ) ] ,

w h e r e X ~ m ) = { x ( r a ) } i is th e v a l u e o f th e i t h v a r ia b l e a t t h e m t h t i m ep o i n t . T h e v a r i ab l e s r ep r e sen t t h e p r o p o r t i o n s o f sp ec i f i c t y p es , w h e r et h e i n t e r p r e t a t i o n o f t y p e d e p e n d s u p o n t h e a p p l i c a t i o n . I n g e n e t i c s ,t y p es co r r e sp o n d t o sp ec if ic a ll e ll e s o r g en o t y p es ; w h e r ea s , i n ev o l u t i o n -a r y g am e t h eo r y , t y p es r e f e r t o i n d iv i d u a ls p l ay in g a g i v en s tr a teg y .


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DISCRJZE TIME SPATIAL MODELS 103A spatial aspect that may be continuous or discrete is introduced. In

the continuous-space models, individuals may be at any position r inRN. For discrete-space models, individuals may be at any site on theregular lattice Z N. Type j individu als at position r are allowed to moveto a new position s either at birth or upon maturity before giving rise tothe next generation. For spatial models therefore, xim(s) represents thevalue of xi at position s for generation m. Here xi(@+)(s) depends uponxjm)(r) and the contact distribution p(r) representing the vector dis-tance r moved, which may depend upon the type of individual.Consider a population containing n1 types, which is initially stablewith xi(s) = qi > 0 for i = 1,. ..,n,. Here vi = fi(n) for i = l,.. .,n, with{q}i = vi for i = 1,. . . , n, and {v}~=O for i=n,+l,...,n. New typesi=n,+l,..., n are introduced into a bounded region B of RN (or Z).Hence, if this is taken to form part of the zeroth generation, xi()(s) = ni>Ofor sGB and i=l,..., n,, and X))(S)= ni = 0 for s e B and i = n,+1 ,. . . , n. Then, for s far from region B, the approximate equationscorresponding to a wide range of models (examples of which are givenin Sections 3, 4, and 5) are given, for the continuous-space model, by

for i=n,+l,..., n, where the rij are nonnegative, with their values andthose of the pii depending on the particular application. For thediscrete-space model, the integral is replaced by a summation over ZN.For the continuous-space model, we consider the speed of spread ofthe forward front for type i in a specified direction, with directioncosines (Y.Take 5 to be small and positive, and define s(m) so that

/ xi(u) du = 5.u: au 2%(m)For the discrete-space model, we consider the speed of spread of theforward front for type i in the direction of one of the coordinates of the

lattice (which without loss of generality may be taken to be the firstcoordinate). This corresponds to using direction cosines (Y= (l,O, . . . ,O).For 6 small and positive, define s(m) to be the smallest integer suchthat C o tujl> s(mj~i(m)(~) G . Define t(m) to be the exact value of thesummation.In either case, the speed of first spread for type i individuals isc=lim ,,,Md/ml.


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104 J. RA DCLIFFE AND L. RASST h e va lue o f c f o r E q ua t i on ( 1) c a n be ob t a i ne d by u s ing t he s a dd l epoin t m e th od . Le t [e*


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DISCRETE TIM E SPATIAL MODELS 105F o r s i m p l i c it y , t h e d i s c u s s i o n i s c o n f i n e d t o t h e c a s e w h e r e (Y ij) i s

n o n r e d u c i b l e , s o A ( A ) i s n o n r e d u c i b l e . T h e e x t e n s i o n t o t h e r e d u c i b l ec a s e c a n b e o b t a i n e d b y u s in g t h e a p p r o a c h i n R e f . 1 1.

F i r s t , c o n s i d e r t h e d o m i n a n t t e r m i n [ A (A )]m L ( )(A ) f o r A = 0 r e a lw i t h A* < 0 < A . L e t p ( 0 ) a n d E ( 0 ) b e t h e P e r r o n - F r o b e n i u s r o o t a n dc o r r e s p o n d i n g i d e m p o t e n t o f A ( 0 ) . T h e n

A ( 0 ) mL ( ) ( 0 ) = [ p ( O ) ] m E ( 0 ) L ( ) ( 0 ) .F r o m L e m m a s 1 a n d 4 i n R e f . 11 , 0 (A ) c a n b e d e f i n e d a s t h e

e i g e n v a l u e o f A (A ) w i t h l a r g e s t r e a l p a r t [ w i th c o r r e s p o n d i n g i d e m p o -t e n t E ( ) 0 ] f o r A i n a n o p e n n e i g h b o r h o o d o f 0 , f o r a n y A* < 0 < A . I nt h is n e i g h b o r h o o d , p ( A ) is a n a ly t i c a n d

e- ;~s(m){[A( A)] mE ()( A)} i =e-XS(m)+mlg[ ( ;O]{E(A) L()( A)}i.T a k e g ( A ) = - A s ( m ) + m a ( A ) , w h e r e a ( A ) = lo g [ p ( A )] . I n t h e f o l lo w -

i ng l e m m a , R e [ g ( A ) ] i s s h o w n t o h a v e a s a d d l e p o i n t o n t h e r e a l l i n e .T h e O ( m ) u s e d i n t h e i n t e g r a t i o n i s t a k e n a s t h is s a d d l e p o i n t .L E M M A 1

R e [ g ( A )] h a s a s a d d l e p o i n t o n t h e r e a l l in e a t A = 0 w h e r e 0 s a t i s f i e sthe re la t ion a ' ( O = [ s ( m ) / m ] .P r o o f C o n s i d e r t h e m a t r i x A (O + i y ) fo r y O . I t s ( i j ) t h en t ry i s

y i jP i y ( O + i y ) and I 'YijP ij( O + iy )[ ~ 0 f o r a l l A* < 0 < A . N o t e t h a t g(O)--- ,oo as0 1 ' A a n d as 0 J, A*. H en ce g ( O ) i s a c o n v e x f u n c t i o n f o r a ll r e a l 0 s u c ht h a t A* < 0 < A a n d h a s a m i n i m u m a t a p o i n t s u c h th a t g ' ( O ) = 0 ; t ha ti s , such tha t a ' ( O ) = [ s ( m ) / m ] . T h i s p o i n t i s t h e s a d d l e p o i n t o f g ( h ) .T h e r e f o r e 0 = O ( m ) i s t a k e n t o b e t h e r e a l v a l u e s a t i s f y i n g a ' ( O ) =[ s ( m ) / m ] . B e c a u s e l im m _ . ~ [ s ( m ) / m ] = c , O ( m ) w i l l a l so t end to a l imi t00 a s m --->% w ith a ' ( O o ) = c .I t i s a s s u m e d t h a t t h e r e i s a l i m i t c w i t h 00 > 0 a n d , f o r s i m p l i c it y ,t h a t A ( 0 0 ) h a s d i s t i n c t e i g e n v a l u e s . T h e n e x t s t e p i s t o f i n d a n a p p r o x i -m a t i o n t o ~ f o r r n l a r g e. T o d o t h i s, w e r e q u i r e t h e f o l l o w i n g t h e o r e m ,t h e p r o o f o f w h i c h i s g i v e n i n t h e A p p e n d i x .


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106 J. RADCLI FFE AND L. RASS

T H E O R EM 1Given any 8 * > 0 t here ex i st s an M ,0 < 8 < 8" , such tha t E , a n d 8 , w i t h 0 < ~ < I a n d

p 1O( m )e-g t tm) l ( 2zr _ 8 0 ( m ) + i y t O ( m ) + iy ] L( ) [O ( m ) [(m ) eg[O(m,+iy]_g[o(m,] lE + iY l} i d y< E m

f o r m > M .H e n c e , f o r a n y p o s it iv e 8 a n d fo r m s u f fi c ie n t ly l a rg e ,~'0( m ) e -St o(m)l= ~ { E [ O ( m ) ] L ( ) [ O (m ) ]} i

[ 8 O ( m ) eg[O(m)+iyl_g[o(m)ldyx j_ o N o w g[ O(m) + i y ] - g[ O(m)] = i yg '[O (m )] - ~ y2g"[ O (m )] fo r y smal l .A l s o g'[O(m)] = 0 fr o m L e m m a 1 . H e n c e~ O (m )e -gt(m>l= ~ -~ { E [ 0 (m ) ] L ( ) [O(m)]]iL~se-l /z '2'~a"t(m'ldy

{ E [ O ( m ) ] L ( ) [ O ( m ) ] } i 1~/2~ rd'[O (m )] vr-m "

T h e r e f o r e w e o b t a ins ( m ) a[O(m)]= 1 l o g ( m )m O ( m ) 2 0 ( m ) m

a n d h e n c ec = l i m s ( m ) = a ( 0 o )

m~ o m 00 '

w h e r e 00 i s s u c h th a t a ' (O o )= c .D e f i n e f ( ) 0 = [ a (A ) / A ] . N o t e t h a t f ' ( O o ) - - [ a '( 0 o ) - f (Oo)] /O o = O.A l s o

f "( O o ) = d ' ( O o ) - f ' ( O O ) o o = " - - ~ oO ' d ' ( O )T h e r e f o r e f ( O o ) is t h e m i n i m u m o f f () O .


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D I S C R E T E T I M E S P A T I A L M O D E L S 1 0 7H e n c e c = i n f , > o[a (A) /A] , p r o v i d e d t h a t t h i s i s p o s i t i v e . W h e n t h e

i n f i m u m o f f ( A ) i s n o t p o s i t i v e , t h e n s ( m ) / m d o e s n o t t e n d t o ap o s i t iv e l im i t a s m ~ ~ . T h i s i m p l i e s t h a t, i f y o u m o v e a t a n y p o s i t iv es p e e d i n t h e s p e c i f i e d d ir e c t i o n , y o u w il l e v e n t u a l l y c r o s s t h e f o r w a r df ro n t. T h e s p e e d o f t h e f r o n t is t h e r e f o r e t a k e n t o b e z e r o .T h u s t h e s p e e d o f p r o p a g a t i o n o b t a i n e d b y us in g t h e s a d d l e p o i n tm e t h o d i s

w he re a(A ) = log{ p[A(A)]} w ith {A( A)}0 = 7ijPij(A).C A S E 2 : D I S C R E T E S P A C E

I n v e r t in g t h e L a p l a c e t r a n s f o r m i n E q u a t i o n ( 2) , f o r a s u i t a b l e c h o i c eo f O ( m ) with 0 < O ( m ) < A, g ives

E xm)(u)a : { ' } 1 = v

" tr - -"~- 2 -~ L : [ (m )+'Y]V{L (m )[O ( m ) + & ] } i dY= 2 , / r_ ' I r l 7r e - [ O ( m ) + i y ] v { P m [ O ( m ) + i y ] L ( o ) [ o ( m ) + i Y ] ) i d Y "

H e n c e~ ( m ) = Y'~ x~m)(u )

u : {u}1 ~ s ( m )1 ~ e -[e(m)+iy]s(m) { p m [ o ( m ) + ~ ] L ( ) [ O ( m ) + ~ ] } i d Y "

A s i n C a s e 1 f o r c o n t i n u o u s s p a c e , A = O ( m ) is ta k e n t o b e t h e s a d d l ep o i n t o f g ( A ) = - A s ( m ) + m a ( A ) , w h e r e a ( A ) = l o g[ p ( A )] a n d O ( A ) i st h e e i g e n v a l u e o f A ( A ) w i t h l a r g e s t r e a l p a r t w i t h c o r r e s p o n d i n g i d e m -p o t e n t E ( A ). A g a i n , A ( A ) i s t a k e n t o h a v e d i s ti n c t e i g e n v a l u e s w h e nA = 0 0 = limm -~ O(m).

W e n o w s ta te t h e t h e o r e m e q u i v a l e n t t o T h e o r e m 1. T h e p r o o ff o l lo w s i n a n a l m o s t i d e n t ic a l m a n n e r , e x c e p t t h a t a = ~r a n d t h ec o m p o n e n t 11 i s n o t r e q u i r e d . T h e p r o o f is t h e r e f o r e o m i t t e d .


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108 J . RAD CLIFFE AND L. RASSTHEOREM 2

Given any 6" > 0 there exists an M , ~ , an d 6, wi th 0 < ~ < 1 a nd0 < 8 < 8" , such tha t[1 t ~ ~ - - - - - ~ .-] ,e g[O(m)+iy]-g[O(m)]e_O (m)]e_g tO(m )l;~(m)_ 2_~ f 8 r l _ ~ ,-O(m)l- 8 [1 - e - (m) - 'y ]

X {E[ O (m ) + ~1 L('[ O ( m ) + i y l } i d y I < ~:m1I

f o r m > M .P r o c e e d i n g i n a si m i la r m a n n e r t o S e c t i o n 2 , w e o b t a i n

l im m_.~o[s (m) /m ] = a(Oo) /O o an d hen ce c = max{0, inf~> 0[a (A)/A]} ,w he re a (A) = log{ o [A(A)]}. Th is con c ludes Ca se 2 .T h e r e s u l t o b t a i n e d f o r t h e s p e e d o f f ir s t s p r e a d i s t h e s a m e f o r b o t hc a s e s ( 1 a n d 2 ) a n d i s s u m m a r i z e d i n t h e f o l l o w i n g t h e o r e m .THEOREM 3

The speed o f propagation obtained by us ing the saddle poin t m etho d isc = m a x [ 0 ' i n f a ( - ~ ]x > 0

where a(3,) = log{ 0[A(A)]} with {A( A)} . = yijP~j(A).3. S E L E C T I O N - M I G R A T I O N M U L T I P L E A L L E L E

G E N E T I C M O D E L SG e n e t i c m o d e l s a r e d i s c u s s e d i n R e f . 1 8 . D i s c r e t e t i m e c o n t a c t

m o d e l s h a v e b e e n u s e d i n g e n e t ic s t o d e s c r i b e t h e s p a ti a l s p r e a d o f ag e n e w h e n t h e r e a r e t w o a l le l e s p r e s e n t . T h e n a l le l e ex t e n s i o n o f t h ism od e l i s d i scussed in Sec t ion 3 .1. A n exac t ana lys is o f the tw o-a l le lemode l i s g iven in Re fs . 1 -5 , 19 , and 20 .A s e l e c ti o n - m i g r a t i o n m o d e l w i t h s e x - li n k e d lo c u s , a g a in w h e n t h e r ea re tw o a l l ele s p re sen t , ha s be en ana lyzed b y Lu i [6, 7 ]. Th e re i s, aga in ,an n a l l e le ana logue , which i s cons ide red in Sec t ion 3 .2 .M o r e - g e n e r a l m o d e l s t h a t a l lo w t h e m i g ra t io n t o d e p e n d u p o n t h eg e n o t y p e c a n b e p o s t u l a t e d . S u c h m o d e l s a r e f o r m u l a t e d i n S e c t i o n s 3 .3and 3 .4 .T h e s a d d l e p o i n t m e t h o d i s a p p l i e d t o a l l t h e m o d e l s i n t h i s s e c t i o nt o g i v e t h e s p e e d o f f ir s t s p r e a d o f a n e w a l le l e ( g e n o t y p e ) i n apopu la t ion tha t i s in i t i a l ly s tab le . For the s imple ca se wi th two a l l e le s ,t h e r e s u lt s o b t a i n e d a r e e x a c t ly t h e r e s u lt s o b t a i n e d b y u s i n g th e e x a c ta n a ly t ic m e t h o d s o f t h e a u t h o r s r e f e r r e d t o a b o v e .


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D I S C R E T E T I M E S PA T IA L M O D E L S 109T h e s a d d le p o i n t m e t h o d c a n b e a p p l ie d i n a f a ir ly o b v io u s w a y t o

m o r e - c o m p l e x m o d e l s i n c lu d i n g se v e ra l l oc i. W e d o n o t p u r s u e t h isf u r t h e r , b e c a u s e t h e e x t e n s i o n i s c l e a r .3.1. A M U L T IP L E A L L E L E M O D E LT h e d y n a m i c s o f t h e m o d e l a r e d e s c r i b e d i n R e f . 1 f o r t w o a ll el e s. Ah a b i t a t t h a t h a s c o n s t a n t c a r r y i n g c a p a c i t y a t a l l p o i n t s o f R N o r a l lp o i n t s o f t h e N - d i m e n s i o n a l l a tt ic e Z N i s c o n s i d e r e d . C o n s i d e r t h ec o m p o s i t i o n o f th e a l le l e s f o r g e n e r a t i o n m a t p o s i ti o n s p r i o r t om a t i n g . L e t x ! m )(s ) b e t h e p r o p o r t i o n o f a ll e l e A i , f o r e a c h i = 1 . . . , n .R a n d o m m a t i n g oc c u r s a t e a c h p o s i ti o n s, t h e n u m b e r o f o ff s p ri n gb e i n g u n a f f e c t e d b y t h e g e n o t y p e . S u r v i v a l t o m a t u r i t y i s g o v e r n e d b yt h e f i tn e s s m a t r i x W a n d r e a p i n g o c c u r s t o r e d u c e t h e p o p u l a t i o n t o t h ec a r r y in g c a p a c i ty o f t h e h a b i t a t . M i g r a t i o n t h e n t a k e s p l a c e , w i t h p ( r )b e i n g t h e d e n s i t y f u n c t i o n ( p r o b a b i l it y ) in t h e c o n t i n u o u s ( d i s c re t e )s p a c e m o d e l c o r r e s p o n d i n g t o m i g r a t i o n b y a v e c t o r d i s t a n c e r i n R N( z N ) . A n e w g e n e r a t i o n i s t h e n p r o d u c e d a s b e f o r e , a n d t h e p r e c e d i n gg e n e r a t i o n d ie s. D e f i n e w ij = {W}ij.

T h e e q u a t i o n s d e s c r i b i n g t h e c o n t i n u o u s - s p a c e p r o c e s s a r e g i v e nb e l o w . E q u i v a l e n t e q u a t i o n s a r e o b t a i n e d f o r t h e d i s c r e t e - sp a c e p r o c e s s .

w h e r ex~m+ 1 ' (s ) = f n f i [x ( ~ ' ( r ) l P ( s - r ) d r ,

x ~ ) ( r ) { W x ( ~ ) ( r ) } if i t x ( m ) ( r ) ] = [ x ( m , ( r ) ] , W x ( m , ( r )

( 4 )

fo r i = 1 . . . . n . H e re [x('n ) (r )] i = x~m)(r) .C o n s i d e r t h e s p r e a d o f a s in g le n e w a l le l e w h e n i n t r o d u c e d i n t o a

b o u n d e d r e g i o n B o f th e h a b i t a t b y m u t a t i o n o r i m m i g r a ti o n , t h is b e in gt a k e n t o f o r m p a r t o f th e z e r o t h g e n e r a t i o n . P r i o r t o t h is o c c u r r e n c e ,t h e p o p u l a t i o n c o n t a i n e d n I a ll e le s i n s ta b l e e q u i l i b r iu m , t h e r e b e i n g ap r o p o r t i o n ~/i o f a l l e l e A i f o r i = 1 . . . . , n 1 a t a l l p o s i t i o n s s . H e r e7 / / = { w * - l l } i / ( I ' W * - l l ) > 0 f o r i = 1 . . . . n l , w h e r e W * is t he p a r t o ft h e f i t n e s s m a t r i x r e l a t i n g t o t h e f i r s t n 1 a l le l e s. N o t e t h a t , w h e n t h e r e isa s t a b l e i n t e r i o r e q u i l i b r i u m f o r t h e n 1 a l l e l e m o d e l , n e c e s s a r i l y i t isu n i q u e a n d W * is n o n s i n g u l a r w i t h W * - 11 > 0. L e t t h e n e w a l le l e b eA , , w he re n = n 1 + 1 . D ef in e r/n = 0 a nd l e t {7/} /= r/i fo r i < n r T h en t hez e r o t h g e n e r a t i o n h a s x ( )(s ) -= 7 / f o r s ~ B .

C o n s i d e r a p o s i t i o n s i n t h e f o r w a r d r e g i o n , f a r f r o m t h e r e g i o n Bw h e r e t h e n e w a l l e le A . w a s i n t r o d u c e d . T h e n x (m )(s) is s m a l l a n dx~m)(s) i s c l o se t o r/i fo r i = 1 . . . . n 1. In t he fo r w ard f ron t , E qu a t io n (4 )


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11 0f o r i = n m a y b e a p p r o x i m a t e d b y

J . RA D CL I F F E A N D L . RA S S

X ( D + 1 ) ( S ) = f R N p ( S - - r ) ( f n ( ~ ) + I )= l [ 0 { X } j ] x = [ x } m ) ( r ) - ~ j ] d r .(5 )

N o w ,

[ d fn (x ) ] = / 0{Wr/}n j n ,[ 0 { x } y J x ~ n [ 7 / ' W ~ / j = n .

H e n c e w e o b t a in t h e f o l lo w i n g a p p r o x i m a t e e q u a t io n s , w h i c h a r e v a li di n th i s f o r w a r d r e g i o n :

- f x(~m )(r){W r/}nx ~ m + I ) ( S ) - - J R N ~ 'W r / p ( s - r ) d r . ( 6 )

E q u a t i o n ( 6) is j u s t a s p e c i a l c a s e o f E q u a t i o n ( 1) w h e n a s i n g le n e wt y p e i s i n t r o d u c e d . T h e e q u a t i o n h a s n = n 1 + 1 , 3 ,nn = { W '0 }n /(r /'W ~ 7 )a n d p , n ( s ) = p ( s ) . I f w e c o n s i d e r t h e s p e e d o f f ir s t s p r e a d c i n a s p e ci fi cd i r e c ti o n f o r t h e n e w a ll el e A n , th e n , f r o m T h e o r e m 3 ,

c = m a x ( 0 , i n f l g [ P ( h ) ] + l g ( 3 , ) )X >0 A ' ( 7 )

w h e r e P ( A ) i s t h e L a p l a c e t r a n s f o r m o f t h e p r o j e c t io n o f t h e m i g r a ti o nd i s t ri b u t io n p ( s ) i n t h e s p e c i f i e d d i r e c t io n a n d 3' = { W ~ } n / ( ff W T / ). T h es a m e r e s u l t i s o b t a i n e d f o r t h e d i s c r e t e - s p a c e p r o c e s s .S u p p o s e t h a t t h e r e a r e s e v e r a l n e w a ll el es . L e t c i b e t h e s p e e d o fs p r e a d i f o n l y a s i n g le a l le l e A i i s i n t r o d u c e d i n t o t h e o r i g i n a l s y s t e m .I ts s p e e d o f s p r e a d is g iv e n b y E q u a t i o n ( 7) w i th 3' =

n l n l n 1Y'~jffilWijT~j/~.,kffilY'~]ffil"l'~kWjkT~j. N OW s u p p o s e t h a t w e t a k e n = n 1 + k a n di n t r o d u c e k n e w a ll e le s A n1 + 1 , .. -, A n i n t o t h e s y s t e m a t t h e s a m e t im e .L e t c = m a X ( n l + l) < r ~ n C r ; t h e n t h e a p p r o x i m a t io n i n t h e f o r w a r d fr o n tw i l l b e v a l i d o n l y f o r a l l e l e s A i w i t h c i = c ; t h a t i s , w i t h f a s t e s t s p e e d .T h i s c o r r e s p o n d s t o a l le l es A i , f r o m i = n I + 1 , . . . , n , f o r w h i c h Z , ~ L l w i j ~ l ji s a m a x i m u m . T h e v a l u e o f c is t h e s p e e d a t w h i c h _ a d i s t u r b a n c e t o t h ee q u i l i b r i u m i s f i r s t f e l t , t h i s d i s t u r b a n c e b e i n g c a u s e d b y t h e r e l a t i v e l yf i t t e s t n e w a l l e l e ( s ) .


11/29

DISCRETE TIME SPATIAL MODELS 111T h e c a s e n = 2 a n d n 1 = 1 w a s c o n s i d e r e d i n R e f s . 1 - 5 , 1 9, a n d 2 0 .E x a c t a n a l y t i c m e t h o d s w e r e u s e d , t h e p r o b l e m b e i n g s p l i t i n t o f o u rcases . T h e case w h en Wl~ = w2 = w22 i s exc lu de d, bec au se , in th i s c ase ,f x t " ) ( s ) d s - - f x ( )( s )d s f o r a l l m . T h e f i rs t c a se h a s W ll ~ W12 ~ W22 with

a t l e a s t o n e i n e q u a l i t y s t r i c t , s o t h e f i t n e s s o f t h e h e t e r o z y g o t e i si n t e r m e d i a t e b e t w e e n t h e h o m o z y g o t e fi tn e ss e s. T h e s e c o n d ( a n d t h i rd )c a s e s c o r r e s p o n d e d t o t h e h e t e r o z y g o t e f i t n e s s b e i n g g r e a t e r t h a n ( o rl e ss t h a n ) b o t h h o m o z y g o t e f i tn e s s e s . T h e f o u r t h c a s e h a s w22 ~ w12W l l , w i t h a t l e a s t o n e i n e q u a l i t y s t ri c t, s o t h e h e t e r o z y g o t e i s a g a i ni n t e r m e d i a t e , b u t t h e h o m o z y g o t e c o r r e s p o n d i n g t o al le l e A 1 is m o r e f i tt h a n t h e h o m o z y g o t e c o r r e s p o n d i n g t o t h e n e w a l le l e A 2.I n t h e f ir st c as e , th e s p e e d o f p r o p a g a t i o n w a s o b t a i n e d , p r o v i d e d W l la n d w 2 2 a r e n o t e q u a l a n d1 ( 8 )

T h e s p e e d o f p r o p a g a t i o n w a s o b t a i n e d f o r a l l v a l u e s o f t h e f i t n e s sp a r a m e t e r s c o v e r e d b y t h e s e c o n d c a s e. I n t h e f o u r t h c a s e , a ll e le 2 d i eso u t u n i f o r m l y in R N f o r t he g ive n in i t i a l c ond i t i ons t ha t x ( 2 ) ( s ) = 0 f o rs e~ B f o r s o m e b o u n d e d r e g i o n B . F o r t h e t h i r d c a s e , n o e x p r e s si o n w a sg i v e n f o r t h e s p e e d o f p r o p a g a t i o n ; n o r w a s a n y w a y g i v e n o f o b t a i n i n gi t. H o w e v e r , n o t e t h a t a l l el e 2 d i e s o u t u n i f o r m l y i n R N i f, i n a d d i t i o n tothe c o nd i t i o n th a t x (2)(s) = 0 f o r s ~ B , i t is a l so t r u e t ha t

W l l - - W I2W l l d" W22 --2w12

f o r a l l s ~ B . T h e v a l u e s o f c o b t a i n e d i n t h e f i r s t a n d s e c o n d c a s e sw h e n n = 2 c o r r e s p o n d t o o u r g e n e r a l r e s u lt s f o r n a ll el es .3.2. A M U L T I P L E A L L E L E M O D E L W I T H S E X - L I N K E D L O C U SA t w o - a l l e le m o d e l w i t h s e x - l i n k e d l o c u s w a s c o n s i d e r e d b y L u i [ 6, 7 ].T h e d y n a m i c s a r e s i m i l a r t o t h a t o f S e c t i o n 3 . 1 e x c e p t t h a t t h e r e a r es e p a r a t e m a l e a n d f e m a l e g e n e t i c o u t p u t s a n d t h e m i g r a t i o n d i s t r i b u -t i o n s f o r m a l e s a n d f e m a l e s m a y d i f f e r. L e t y / ( ') ( s ) a n d z ~m )(s ) b e t h ep r o p o r t i o n s o f t h e a l le l e A i i n th e f e m a l e a n d m a l e o u t p u t s a t l o c a ti o ns i n g e n e r a t i o n m p r i o r t o m a t i n g . T h e m a l e i s h a p l o i d f o r t h e Xc h r o m o s o m e a n d t h e r e f o r e o n l y h a s n g e n e t ic t yp e s, t h e i t h b e i n g ,4 i .T h e f e m a l e i s d i p l o i d a n d s o h a s n ( n + 1 ) / 2 g e n o t y p e s A i A j f o r1 ~< i ~


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112 J . R ADC LIFFE AND L. RASSv i = {v}i a n d o f t h e f e m a l e A i A j g e n o t y p e is w i / = { W }i/. R e a p i n g o c c u r st o m a i n t a i n t h e d e n s i t i e s o f m a l e s a n d f e m a l e s . M i g r a t i o n n o w o c c u r s ,t h e d e n s i t y f u n c t io n s f o r t h e d i s ta n c e r m o v e d b e i n g p ( r ) f o r f e m a l e sa n d q ( r ) fo r m a le s . A n e w g e n e r a t i o n is t h e n p r o d u c e d a s b e f o r e a n dt h e p r e c e d i n g g e n e r a t i o n d i e s . T h e e q u a t i o n s f o r t h e c o n t i n u o u s - s p a c em o d e l a r e g iv e n b e l o w . T h e r e is a n e q u i v a l e n t d i s c r e t e -s p a c e m o d e l .

y ~ m ) ( r ) { W z ( m ) ( r ) } i +y~i n + ) ( s ) = z ~ m ) ( r ){ W y ( m ) ( r ) } i ' p ( s - r ) d r ,( 2 (y ( m ) (r ) ) ' W z ( m ) (r ) ) ( 9 )z~ - + 1) ( s ) = f R N v i Y ~ m ) ( r)[ v ,y ( m ) ( r) ] q ( s - r ) d r ,

f o r i = l , . . . , n .C o n s i d e r t h e s p r e a d o f a n a l le le A ~ w h e n i n t r o d u c e d i n t o a b o u n d e dr e g i o n B o f a h a b i t a t b y m u t a t i o n o r i m m i g r a t io n , t h is b e i n g t a k e n t of o r m p a r t o f th e z e r o t h g e n e r a t i o n . P r i o r t o t h is o c c u r r e n c e , th ep o p u l a t i o n c o n t a i n e d ( n - 1 ) a l le l es in st a b le e q u i l ib r i u m , t h e r e b e i n gp r o p o r t i o n s ~ i ( ~ i ) f o r f e m a l e s ( m a l e s ) o f a ll e le A i f o r i = 1 . . . , n - 1 a ta l l p o s i t i o n s s . H e r e

{ [ W * d / a g ( v * ) + d / a g ( v * ) W * ] - 11 } i7 / / = { I ' [ W * d i a g ( v * ) + d / a g ( v * ) W * 1 -1 1 } > 0 ,

{ [ W * + d i a g ( v * ) W * d i g g ( v * ) - l ] - l l ) i 0 ,~ ' i = { l ' [ W * + d i a g ( v * ) W * d i a g ( v * ) - l ] - l l }

( 1 0 )

f o r i = l . . . . . n - l , w h e r e W * a n d v * a r e t h e p a r t s o f t h e f it n es sm a t r i c e s r e l a t i n g t o t h e f i r s t ( n - 1 ) a l le l es . D e f i n e ~ , = 0 a n d ~ , = 0 .L e t { ~ }i = ~ i a n d { ~ } i = ~i- T h e n t h e z e r o t h g e n e r a t i o n h a s y ( ) ( s ) -a n d z ( ) (s ) = ~ f o r s ~ B .C o n s i d e r a p o s i t i o n s i n t h e f o r w a r d r e g i o n , f a r f r o m t h e r e g i o n Bw h e r e t h e n e w a l le l e A ~ w a s i n t r o d u c e d . T h e n y ~m )(s) a n d z~ m )(s) a r es m a l l , y /t m )( s) i s c l o s e t o ~ i , a n d z ! m ) ( s ) i s c l o s e t o ~ i f o r i = 1 . . . . n - 1 .F r o m E q u a t i o n s ( 9) , t h e f o ll ow i n g a p p r o x i m a t e e q u a t i o n s m a y b e o b -t a i n e d , w h i c h a r e v a l i d in t h i s f o r w a r d r e g i o n :

y~m + a)(S) = --jR y~ m )(r {W~"} ~ + z ~ m ) ( r) { W r /} ~ p ( s - r ) d r ,z ( m + l ) ( s ) - f v"Y~(~( r ) - " s - r "- J R u ( v r / ) q ( ) d r . ( 1 1 )


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DISCRETE TIME SPATIAL. MODELS 113Note that Equations (11) are just a special case of Equation (1) withn replaced by 2n, n, = 2n -2, and x$z? I(s) = y:)(s) and x$;)(s) =zim)(s). Also pi 2n_ (r) = p(r), pi 2n(rl = q(r) for i = (2~2 1),2n, and

YZ_l zn-1 ={wkI,/(2rlrWs), Yz:-l,zn =MI~J(29WS), Y2n,2n-1=u, / and Y~,~,, = 0. Hence we obtain the result that the speed offirst spread in a specific direction of allele A, among both males andfemales is c, where

log{y,,-,,,,-,[P(A)/21}+log[l+\/1+48Q than the homozygote for allele 1. Thespeed of propagation was obtained for all values of the parameters inthe superior case and with the restriction in the intermediate case withw22 > wll that (w12 - wI1) 2 wIl(w22 - w12)/2w12.Note that we could consider the spread of several new alleles. Thiscan be treated in a similar fashion to Section 3.1.3.3. A MULTIPLE ALLELE MODEL WITH MIGRATION DEPENDENT

UPON GENOTWEConsider dynamics similar to that of Section 3.1 but where the

density function for the migration depends upon the genotype. We nowconsider the proportion of genotype A,A, at position s in generation mprior to mating, which is denoted by 2x$)(s) for i # j and x~~~(s). Let{X()(S))~~ {Xcm))li x:7)(s). Note that the matrix X@)(s) s symmet-ric and that, at posrtron s in generation m , the proportion of allele A, isgiven by {X(m)(s)l}i. The density function for migration of genotypeAiA j by position vector r is pii( The equations describing the processare

1xi? + l)() = p)(s) RN

Wij{Xc(r)l},{X(r)l}jlfX(m)(r)~(m)(r)l pij(s-r)dr, (12)


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114 J . RA DC LIFFE AND L. RASSw h e r e

W i j { x ( m ) ( r ) 1 } i { X ' m ) ( r ) 1}yk(ra )(s ) = t. Z f R N l ' X ( m ) ( r ) $ r c x ( m ) ( r ) l P i j ( s - r ) d r "

JC o n s i d e r a s t a b l e p o p u l a t i o n w i t h a l l el e s A 1 . . . . A n - 1" L e t 2 , / i > 0,i ~ j , a n d 7 /ij > 0 , i = j b e t h e p r o p o r t i o n s o f g e n o t y p e A i A j a t a l lp o s i t io n s s ~ R N . A n e w a l le l e o c c u r s b y m u t a t i o n i n ( o r i m m i g r a t i o ni n t o ) a b o u n d e d r e g i o n B , s o t h e z e r o t h g e n e r a t i o n i s t a k e n t o h a v ex!)(s) = 7// j fo r i < n an d j < n an d s ~ B . A lso x~)(s) = 0 fo r i = n o rj = n a n d s ~ B . L e t 7 / / b e t h e p r o p o r t i o n o f a ll e le A i and let {7/*} = r/if o r i < n . T h e n r/* = W * - 1 1 / 1 ' W * - 11 > 0 . H e n c e

~ l i w i j w j { W * - 1 1 1 'W * - 1 } i j { W * } i j~ l i j = ~ 7* 'W * r/* = l ' W * - l lw h e r e W * is t h e p a r t o f t h e f i tn e s s m a t r i x re l a t in g t o t h e f i rs t ( n - 1 )a l le les .

D ef in e ~Tn = 0 , a nd le t r / b e th e ve c to r w i th {~7}i= r / i . N o t e t h a tn - - 1r/i = E j = I ~ i j , T h e n , i n t h e f o r w a r d f r o n t , E q u a t i o n ( 1 2) f o r i < n m a y b ea p p r o x i m a t e d b yn - - 1

x ! 7 + 1 ) ( s ) = E q ~ i l f _ l , , q i t ( s - r ) x ~ m ) ( r ) d r , ( 1 3 )l= 1 "1w h e r e ~ b i t = w i n r l i / r f W r l a n d q i t ( r ) = P i n ( r ) . H e r e b o t h q ~ i t a n d q i l ( r )d e p e n d o n l y o n i a n d n o t o n I. E q u a t i o n ( 13 ) i s i d e n t ic a l w i t h E q u a t i o n( 1 ) i f w e t a k e x i + ( m ) l ( S )= x!m)(s) fo r i = 1 , . . . , ( n - - 1), rep lac e Y u b y ~bil an dP i l b y q i l , a n d l e t n 1 = 1 .I n t h i s c a se , A (A ), a s d e f i n e d i n S e c t i o n 2 , h a s i d e n t i c a l c o l u m n s . T h em a x i m u m e i g e n v al u e is t h e r e f o r e e q u a l t o t h e s u m o f th e e n t r ie s o f t h ec o m m o n c o l u m n v ec to r . H e n c e ,

n - 1 Win~ip [ A ( A ) ] = ~ ] P i n ( A ) w - - T W " = ' y P ( A ) ,i = I

w h e r e 3, = { W ~ } n / w ' W v / a n d P ( A ) = Y ' . ~ - ~ a i P i n ( A ) , w h e r e a i =W i n T l / { W T } } . N o t e t h a t P ( A ) i s t h e L a p l a c e t r a n s f o r m o f t h e p r o j e c t i o ni n t h e s p e c i f i e d d i r e c t i o n o f a d e n s i t y p ( r ) , w h i c h i s a w e i g h t e d f u n c t i o no f m i g r a t i o n d e n s i t i e s ; t h a t is, p ( r ) - n - 1- Z i = i i P i n ( r ) . H e n c e t h e s p e e d o ff i r s t s p r e a d c a n b e g i v e n i n t h e s i m p l e f o r m

c = m a x{ O , in f l g [ P ( A ) ] + l g ( ' ) )x > 0 A "


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DISCRETE TIME SPATIAL MO DELS 115N ot e that , i f w e le t x(~m)(s)b e t h e p r o p o r t i o n o f a l le l e A n ( a t p o s i t i o n

s an d i n g en e r a t i o n m ) , t h en b y su m m i n g o v e r i i n E q u a t i o n ( 13 ), w eo b t a i n

x (m + l ) ( s ) = f R N T X ~ m ) ( r ) p ( s _ r ) d r .H en ce t h e sp eed o f sp r ead o f al le l e A n can b e d e r i v ed f r o m t h i s s in g l ee q u a t i o n a n d is t h e c o m m o n s p e e d o f s p re a d o f t h e g e n o ty p e s , c , g iv e nabove .3.4. A M U L T IP L E A L L E L E M O D E L W I T H S E X -L I N K E D L O C U S A N DM I G R A T I O N D E P E N D E N T U P O N G EN O T Y PE

T h e d y n am i cs o f t h is m o d e l a r e s i m i l a r to S ec t i o n 3.3 , ex cep t t h a t t h em a l e a n d f e m a l e g e n e t i c o u t p u t s a r e s e p a r a t e a n d m a y h a v e d i f f e r e n tmigra t ion d i s t r ibu t ions . As in Sec t ion 3 .2 , the male i s hap lo id fo r the Xc h r o m o s o m e a n d t h e f e m a l e is d ip lo id . C o n s i d e r th e p r o p o r t i o n o f t h ed i f f e r en t g en o t y p es a t p o s it io n s i n g en e r a t i o n m f o r e ach sex p r i o r t om a t i n g . F o r t h e f em a l e s , l e t 2 y ! 7 )( s ) b e t h e p r o p o r t i o n o f g en o t y p eA i A j w h en i ~ j an d l e t y ~m ) (s ) b e t h e p r o p o r t i o n o f g en o t y p e A i A i.A l so l e t z~m ) (s ) b e t h e p r o p o r t i o n o f g en o t y p e A i f o r t h e m a l e s . T h em i g r a t i o n d en s i t y an d f i t n e s s f o r f em a l e s o f g en o t y p e A i A / a r e p i j ( r )an d w i j . T h e co r r e sp o n d i n g m i g r a t i o n d en s i t y an d f it n e s s f o r m a l e s o fg e n o t y p e A i a r e q i ( r ) an d v i . L e t {Y(m) ( s ) } ij = y ~ ) ( s ) . A l so l e t { z ( m ) ( $ ) } i= z!m) (s ). Th e e qua t ions descr ib ing the p rocess a rey ~ 7 + l ) ( s ) = 1 w i j [ { y ( m ) ( r ) l } i { z ( m ) ( r ) } j + { Y ( m ) ( r ) l } y { z ( m ) ( r ) } i ]

k(mS(s) f : 2 [ ze~) ( r) ] 'W [y(m ) ( r ) ] 1 p i y ( s - r ) d r , ( 1 4 )

z~m+ O(S ) : 1 , " v i { y ( m ) ( r ) l } ib(m)(s) JR N [ v 'y ( m ) ( r )l ] q i ( s - r ) d r ,fo r j < i and i = 1 . . . . n , wh ere

k(m) ( s ) = ~ . E f R N w i y [ { Y ( m ) ( r ) l } i tz ( m ) ( r ) } j + { Y ( m ) ( r ) l } j t z ( m ) ( r ) } i ] j 2 [ z ( m ) ( r ) ] ' W [ y ( m ) ( r ) ] l

p i j ( s - r ) d r ,f v i { Y ( m ) ( r ) l } ib ( m ) ( s ) = t ~ . J n N [ v ' y ( m ) ( r ) l ] q i ( s - r ) d r "


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1 1 6 J . R A D C L I F F E A N D L . R A S SC o n s i d e r a s t a b le p o p u l a t i o n w i t h a ll e le s A 1 . . . . A n - 1 w i t h 2 r /q > 0 ,

i :~ j , a n d ~ /i: > 0 , i = j b e i n g t h e p r o p o r t i o n o f f e m a l e s o f g e n o t y p eA ~ A i a t a l l p o s i t io n s s ~ R N. T h e p r o p o r t i o n o f m a l e s o f g e n o t y p e A ~ a ta ll p o s i t i o n s s i n R N is ~ i. A n e w a l le l e o c c u r s b y m u t a t i o n i n ( o ri m m i g r a ti o n in t o ) a b o u n d e d r e g i o n B , s o th e z e r o t h g e n e r a t i o n i s t a k e nto ha ve y! ) (s ) = ~Tij an d z~)(s ) = ~i fo r i < n an d j < n an d s ~ B. A lsoyi


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DISCRETE TIME SPATIAL MODELSw h e r e

117

~/11 = 2 ~ , , W r I , P~l( r) = ~ W i n ~ ii = l ( - - ~ n ' P i n ( F ) ' Y12 = 2b r,W ~,w i n ( n } i , . v .P~2( r ) = ~ - ~ ' P i , t r ) , Y2, = v'-~'i=1

a n dp ~ l ( r ) = q n ( r ) .

H e n c ec = m a x ( O , i n f l o g [ p ( A ) ]}A>O A '

w h e r e

1 [ yl,P ~,l(A ) + V/y21P~'12 A ) +4Y 21 ylZ P~'z(A) P~'I(A) ],gw i t h P i ~ ( A ) t h e L ap l ace t r an s f o r m o f t h e p r o j ec t i o n o f p ~ . ( r ) i n t h espec i f i ed d i r ec t ion .4. E V O L U T I O N A R Y G A M E T H E O R Y M O D E L S

I n t h is s ec t i o n , w e co n s i d e r t w o m o d e l s i n ev o l u t io n a r y g am e t h eo r y .F o r a g en e r a l d i s cu s s io n o f ev o l u t i o n a r y g am e t h eo r y , s ee R e f s . 2 1 - 2 3 .S p a t ia l m o d e l s h av e b een co n s i d e r ed in R e f s . 24 an d 2 5, w i t h d i ff u s i o nt e r m s b e i n g u sed t o m o d e l t h e sp a t i a l a sp ec t . I n co n t r a s t , w e u secon tac t d i s t r ibu t ions to descr ibe the spa t i a l sp read .T h e f ir s t m o d e l co n s i s ts o f a s in g le p o p u l a t i o n i n w h i ch i n d iv i d u a lscan p l ay n p o s s i b l e s t r a t eg i e s . T h i s t u r n s o u t t o b e an a l o g o u s t o t h em o d e l in S ec t i o n 3.1 ex cep t t h a t t h e m a t r ix W i s n o w a p ay o f f m a t r i xt h a t is n o t i n g en e r a l sy m m et r i c . T h e seco n d m o d e l is t h a t o f a b i m a t r ixg a m e . T h e r e su l ts o f th e s a d d l e p o i n t m e t h o d a r e a p p l ie d t o g i ve t h esp eed o f sp r ead o f n ew s t r a t eg i e s i n p o p u l a t i o n s t h a t w e r e i n eq u i l i b -r i u m .T h e c o n n e c t i o n b e t w e e n t h e f i r s t m o d e l a n d t h e g e n e t i c m o d e l o fSec t ion 3 .1 i s b r i e f ly d i scussed , as i s the ex ten t to which the exac tan a l y t ic m e t h o d s o f W e i n b e r g e r [1 , 2 ] an d L u i [3 ] y i e ld r e su lt s f o r t h et w o - s tr a te g y m o d e l i n g a m e t h e o r y .


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118 J . RAD CLIFFE AND L. RASS4.1. A S IN G L E - P O P U L A T I O N E V O L U T I O N A R Y G A M E T H E O R Y M O D E L

A s e t u p a n a l o g o u s t o t h e m o d e l o f S e c t i o n 3.1 is c o n s i d e r e d . E a c hind iv idua l in the popu la t ion p lays one o f n s t r a teg ie s , w i th x !m)( s ) nowd e n o t i n g t h e p r o p o r t i o n i n t h e m t h g e n e r a t i o n a t p o s i t i o n s t h a t p l a y ss t ra t e g y i . T h e s u r v iv a l t o m a t u r i t y n o w d e p e n d s o n a p a y o f f m a t r ix A ,w h i c h "is n o t i n g e n e r a l s y m m e t ri c . T h e m o d e l i s d e s c r i b e d b y t h ee q u a t i o n s

x } m ) ( r ) [ { A x ( m ) ( r ) } i + k ]x } m + l ) ( s ) = J R N [ X ( m ) ( r ) ] , [ A x ( m ) ( r ) + k ] P ( s - r ) d r , ( 1 5 )

f o r i = 1 , . . . , n .H e r e k i s a la r g e p o si t iv e c o n s t a n t r e p r e s e n t i n g t h e c o m m o n b a c k -g r o u n d f i tn e s s. T h i s i s a s p a t ia l v e r s i o n o f t h e m o d e l d e s c r i b e d o n p a g e

133 o f Re f . 23 . Le t W i j ~ - { W } i j = {A}ij + k . P ro v ide d k > m a x ( - {A}ij)t h e n w ij > 0 fo r a l l i and j . When W i s subs t i tu ted in to Equa t ions (15) ,t h e y b e c o m e i d e n t i c a l w i t h E q u a t i o n s ( 4 ) . N o t e t h a t , f o r t h e m o d e ldesc r ibed in Sec t ion 3 .1 , the f i tne s s ma t r ix W i s symmet r ic . For thep r e s e n t m o d e l , h o w e v e r , t h e m a t r i x W i s n o t i n g e n e r a l s y m m e t r i c .

C o n s i d e r t h e s p r e a d o f a n e w s t r a t e g y S n. T h i s i s i n t r o d u c e d i n t o apo pu la t ion in s tab le equ i l ib r ium, wi th s t r a teg ie s S 1 . . . , Snl be ing p layedby p rop or t io ns ~ . . . . . */nl o f the pop u la t ion , r e spec t ive ly , a t a l l po in t s so f R N . T h e expre s s ion fo r 7 / / i s iden t ic a l w i th tha t ob ta ined fo r thec o r r e s p o n d i n g g e n e t i c m o d e l o f S e c t i o n 3.1 . T h e s p e e d o f f ir s t s p r e a d co f n e w s t r a t e g y Sn i s g i v e n b y E q u a t i o n ( 7 ) . R e s u l t s f o r s e v e r a l n e ws t r at e g i e s a r e i d e n ti c al t o t h o s e o b t a i n e d f o r g e n e s i n S e c t i o n 3 .1 .

E x a c t m e t h o d s t o o b t a i n t h e s p e e d o f p r o p a g a t io n f o r th e g e n e t i cm o d e l o f S e c t i o n 3 .1 h a v e b e e n u s e d f o r t h e c a s e n = 2 a n d a r ed i scussed b r ie f ly in tha t s ec t ion . Equ iva len t r e su l t s us ing exac t me th od sm a y a l s o b e o b t a i n e d f o r t h e g a m e t h e o r y m o d e l i n t h e g e n e r a ln o n s y m m e t r i c c a s e .B e c a u s e n = 2 , w e n e e d c o n s i d e r o n ly th e s e c o n d e q u a t i o n o f E q u a -t ions (4) w i th W no nsy m m etr ie , be ca us e x[m)(s ) = 1 - - x~2m)(s). This m ayb e w r i t t e n i n t h e f o r m

( 1 6 )w h e r e

w~ x 2 + w21x(1 - x)g ( x ) - - w 2 2x 2 + ( w l 2 + w l)x(1- x ) + w 1(1- x )


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D I SCR E T E T I M E SPA T IA L MO D E L S 1 19C o n s i d e r t h e s o l u t i o n s t o g ( x ) = x fo r 0 ~< x ~< 1. In all ca se s, x = 0

a n d x = 1 a r e s o l u t i o n s . T h e r e i s a n a d d i t i o n a l s o l u t i o n x * , w h e r e

W 2 1 - - W l l )x * = [ ( w 2 , - w H ) + ( w 1 2 - w 2 )]

D e f i n e .~ = x * i f 0 < x * < 1 ; o t h e r w i s e d e f i n e $ = 1 .F o r t h e g e n e t i c s m o d e l w i t h n = 2 , x~2m )(s) s a t is f i es E q u a t i o n ( 1 6 ) a n dg ( x ) i s d e f i n e d a s a b o v e b u t h a s w 12 = w 2 1. W e i n b e r g e r [1 , 2 ] a n d L u i [ 3]

p r o v e d r e s u lt s f o r th e a s y m p t o t i c s p e e d o f p r o p a g a t i o n f o r t h e g e n e t i cm o d e l i n t e rm s o f a g e n e r a l f u n c t i o n g ( x ) t h a t s a t i s f i e d c e r t a i n c o n d i -t i o n s . A s u f f i c i e n t s e t o f c o n d i t i o n s i s a s f o l l o w s :

(1 ) g ( x ) h a s c o n t i n u o u s b o u n d e d d e r i v a t i v e s f o r x ~ [ 0 , 1 ] .( 2 ) g ( 0 ) = 0 , g ( 1 ) = 1 , a n d g ( $ ) = $ .(3 ) g ( x ) > x f o r x ~ ( 0 , $ ) a n d g ( x ) < x f o r x ~ ( $ , 1 ).(4 ) g ( x ) / x i s a n o n i n c r e a s i n g f u n c t i o n o f x f o r x ~ ( 0, $ ) .(5 ) 0 < g' (x ) ~ g '(O) fo r x ~ [0 , :~] .( 6 ) T h e r e e x is t s a D > 0 s u c h t h a t g'(O)(x - D x 2 ) < g (x )


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120 J. RA DCL IFFE AND L. RASSB . I n C ase 4 , a s i n t h e g en e t i c m o d e l , S t r a teg y 2 w ill d i e o u t u n i f o r m l yin R N . I n C ase 3 , S t r a t eg y 2 w i l l a l so d i e o u t an d n o t sp r ead i n an ydire ct ion w he n, in ad di t ion, x~) (s) < ~ for a l l s ~ B.W h e n n = 2, th e s p e e d o f fi rs t s p re a d o b t a i n e d b y t h e s a d d l e p o in tm e t h o d f o r t h e e v o l u ti o n a r y g a m e t h e o r y m o d e l a g r e e s w i th t h e r e su l tsf o r t h e s p e e d o f p r o p a g a t io n o b t a i n e d b y e x a c t m e t h o d s .4.2. A B I M A T R I X G A M E

C o n s i d e r t w o d i f f e r en t p o p u l a t i o n s t h a t a r e i n co n f l ic t . I n d i v id u a l s inP o p u l a t i o n 1 can p l ay s tr a t eg ie s S ~ . . . . . S t , an d i n d iv i d u a ls in P o p u l a t i o n2 can p lay s t r a t eg ies T 1 , . . . , T ~ . Le t x !m) (s ) and y~m)( s ) de no te thep r o p o r t i o n s i n P o p u l a t i o n s 1 an d 2 i n t h e m t h g en e r a t i o n a t p o s i t i o n stha t p lay s t r a t eg ies i and j , r espec t ive ly . The su rv iva l to matur i ty inP o p u l a t io n s 1 a n d 2 n o w d e p e n d s o n t w o p a y o f f m a t r ic e s , A a n d B ,w h i c h a r e n o t i n g e n e r a l s y m m e t ri c . T h e m o d e l i s d e s c r i b e d b y t h eeq u a t i o n s

. x ~ m ) ( r ) [ { A y ( m ) ( I ) } i + k ~ ]x ~ m + 1 ' ( S ) = J R N [ x ' m ' ( r ) ] t t A y ( m ' ( r ) J r k l ] P l ( s - r ) d r '

. y ( m ) ( r ) [ { B x ( m ) ( r ) } j J r k 2 ]y ( m + 1)(S = JR s [Y(m)(r) ]' [Bx(m )(r ) + k2] p 2 ( s _ r ) d r ( 1 7 )

fo r i = 1 . . . . . t a n d j = 1 , . . . , n . H e r e k 1 a n d k 2 a r e l a rg e p o s it iv ec o n s t a n ts r e p r e s e n t i n g t h e c o m m o n b a c k g r o u n d f i tn e s s o f i n d iv id u a ls i nPo pu lati on s 1 an d 2, respe ctive ly. Le t {V}ij = {A}i/+ kl an d {W}i. = {B}ij+ k 2 . T h e n p r o v id e d k 1 > m a x ( - { A } i ) a n d k 2 > m a x ( - { B } i j ) , ( V } i j > 0an d {W}ij > 0 fo r a l l i a nd j . E qu a t ion s (17) can be w r i t t en as

r x ~ m ) ( r ) { V y ( m ) ( r ) } ix ! m + I ) ( S ) = J R N ~ P l ( s - r ) d r '

r y ( m ) ( r ) { W X ( m ) ( r ) } jy ( m + l ) ( s ) = J R N ~ P 2 ( s _ r ) d r ' ( 1 8 )f o r i = l . .. . . t a n d j - - 1 . .. .. n .In i ti a lly , P opu la t ion 1 p lays s t r a t eg ies S 1 . . . . S5 a n d P o p u l a t i o n 2p lays s t r a t eg ies T 1 , . . . , T n c T h e p o p u l a t i o n s a r e m eq u i l i b r i u m , w i t hp r o p o r t i o n s {0" } - - 0 * p l ay in g s t r a teg y S i ( i - -1 . . . . . t 1) an d {~/*}j- -~7p lay ing s t r a t egy T / ( j = 1 , . . . , n 1 ) a t a ll p o in t s s , .where~ 1 ' 0 " = 1 a n d1 '7" = 1 . T he n 0* and ~j* sa t is fy 0* = g i ( O * , ~ * ) and ~j* = h j ( O * , ~ * ) ,w h e r e g i (O* , 7 1" ) = ( O * { V * ~ * } i / O * ' V * 7 1 * ) a n d h i ( O * , 7 1 ") =


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DISCRETE TIME SPATIAL MODELS 121( 7 / p { W * 0 * } j / r/ * 'W * 0 * ) . H e r e iV * }0. = {A*}ij + k 1 a n d {W *}0 = { B * }0. +k 2 , w h e r e A * a n d B * a r e t h e p a y o f f m a t r ic e s f o r P o p u l a t i o n s 1 a n d 2c o r r e s p o n d i n g t o t h e f i r st t I a n d n I s t r a t e g i e s , r e s p e c t iv e l y . T h u s{ 0 " } i = { V * - 1 1 } i / / ( l ' V * - 1 1 ) a n d { ,/* }j = {W *- 1 1 } / . / ( r W * - 1 1 ) .

L e t t = t 1 + 1 a n d n = n I + 1 . S u p p o s e t h a t P o p u l a t i o n 1 i n t r o d u c e s an e w s t r a t e g y S t , b u t P o p u l a t i o n 2 c o n t i n u e s t o p l a y t h e e x i s ti n g s t r a t e -g i e s . D e f i n e { 0 } = 0* fo r i = 1 . . . t 1 an d { 0 } = 0 . A lso le t { ,/}i = r /* fo ri = 1 . . . . n 1 a n d {7/}~ = 0 .

T h e a p p r o x i m a t e e q u a t i o n o b t a i n e d f r o m E q u a t i o n s ( 18 ) i n t h ef o r w a r d f r o n t is

x ~ m + 1 ) ( $ ) = ~ f t t f R N P l ( r ) x ~ m ) ( s _ r ) d r , ( 1 9 )w he re 3~, = {V~/} / ( 0 ' V~/) .

T h e s p e e d o f f ir s t s p r e a d o f s t r a te g y S t i n a s p e c i fi c d i r e c t i o n i s

c l = m a x ( O , i n f l o g [ P l ( A ) ] + l o g ( ' Y " ) )~ > 0 A 'w h e r e P I (A ) is th e L a p l a c e t r a n s f o r m o f th e p r o j e c ti o n o f th e m i g r a t i o nd i s t r ib u t i o n p l ( s ) i n t h e s p e c i f ie d d i r e c t io n .S i m i la r ly , s u p p o s e t h a t P o p u l a t i o n 2 i n t r o d u c e s a n e w s t r a t e g y T~,w h i l e P o p u l a t i o n 1 c o n t i n u e s p l a y i n g s t ra t e g i e s S 1 . . . . S t1 . T h e a p p r o x i-m a t e e q u a t i o n o b t a i n e d f r o m E q u a t i o n s ( 18 ) i n th e f o r w a r d f ro n t is

y ~ m + 1)(s ) = ~ n , fR n p 2 ( r ) Y ~ n m ) ( s _ r ) d r , ( 2 0 )

w h e r e ~n~ = { W 0 } n / (1 7 'W 0 ) . T h e s p e e d o f f i r s t s p r e a d o f s t r a t e g y Tn in as p e c i f ic d i r e c t i o n i s

w h e r e P 2( A ) is th e L a p l a c e t r a n s f o r m o f th e p r o j e c t io n o f t h e m i g r a t io nd i s t r i b u t i o n p 2 ( s ) i n t h e s p e c i f i e d d i r e c t i o n .

N o w c o n s i d e r t h e c a s e w h e n P o p u l a t i o n s 1 a n d 2 e a c h i n t r o d u c e an e w s t r a t e g y , S t a n d T , r e s p e c t i v e l y . A s i n S e c t i o n 3 . 1 , E q u a t i o n ( 1 9 )w i l l b e v a l i d o n l y i f c I > / c 2 , i n w h i c h c a s e t h e s p e e d o f f i r s t s p r e a d o fs t r a t e g y S t i s c 1. E q u a t i o n ( 2 0 ) i s s i m i l a r l y v a l i d o n l y i f c 2 > / C l, i n w h i c hc a s e t h e s p e e d o f f ir s t s p r e a d o f s t r a t e g y Tn i s c 2 . T h e v a l u e o f


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122 J . RAD CLIFF E AND L. RASSc = m a x ( c1 , c 2 ) g i v es th e s p e e d a t w h i c h a d i s t u r b a n c e t o t h e e q u i l i b r i u mi s f i r s t f e l t , t h i s d i s t u r b a n c e b e i n g c a u s e d b y t h e r e l a t i v e l y m o r e a d v a n -t a g e o u s s t ra t eg y .

T h e i n t r o d u c t i o n o f se v e r al n e w s t r a t e g ie s m a y b e t r e a t e d i n a s im i l a rf a s h i o n .5 . B R A N C H I N G P R O C E S S M O D E L S

T h e s a d d l e p o i n t m e t h o d c a n b e a p p l ie d t o a w i d e r a n g e o f d is c re t et i m e b r a n c h i n g p r o c e s s m o d e l s . T w o p r o c e s s e s a r e g i v e n i n th i s s e c t io na s il lu s t ra t io n s . T h e f ir s t i s a m u l t i ty p e G a l t o n - W a t s o n p r o c e s s a n d t h es e c o n d i s a d e m o g r a p h i c b r a n c h i n g p r o c e ss . A d i s c u s s io n o f m u l t i t y p ea n d d e m o g r a p h i c b r a n c h i n g p r o c e s s e s is g i v e n i n R e f . 2 6.

I n b r a n c h i n g p r o c e s s m o d e l s , a s y s t e m o f n o n l i n e a r i n t e g r a l e q u a -t i o n s i s o b t a i n e d f o r t h e p r o b a b i l i t y t h a t t h e f a r t h e s t s p r e a d i n a s p e c i f i cd i r e c t i o n a t d i s c r e t e t i m e m i s a t l e a s t s , g i v e n t h a t , a t t i m e z e r o , t h e r eis o n e i n d iv i d u a l i n th e p o p u l a t i o n w h o i s at t h e o r i g i n a n d is o f i t h t y p e( o r a g e o r a c o m b i n a t i o n o f b o t h ) f o r a ll p o ss i b le t y p e s i . T h i s i sd e n o t e d b y x ! m ) ( s ) . H e r e x ! ) ( s ) = 0 f or s t> 0 , an d x ~ ) ( s ) = 1 for s < 0 .

B e c a u s e x ~ ) ( s ) i s n o t z e r o o u t s i d e a b o u n d e d r e g i o n o f R , t h eb r a n c h i n g m o d e l s d o n o t s a ti s fy t h e c o n d i t i o n s i m p o s e d i n S e c t i o n 2 top r o v e T h e o r e m s 1 a n d 2 . N o t e h o w e v e r t h a t t h e a c tu a l c o n d i t io n u s e di n t h e p r o o f s o f t h e s e t h e o r e m s r e q u i r e s t h a t D ) ( 0 ) h a s e n t r i e s t h a tex i s t fo r R e ( O ) > 0 a n d a r e u n i f o r m l y b o u n d e d f o r 01 ~< R e ( O ) ~< 02 f o rany 0 < 01 < 02 < 0o.F o r t h e c as e N - - 1 , t o e n s u r e t h a t th e s e c o n d i t io n s h o l d w h e ne s t a b li s h in g t h e s p e e d o f s p r e a d i n t h e p o s i t iv e d i r e c ti o n , w e n e e d o n l yr e q u i r e t h a t e a c h x ~ ) ( s ) i s b o u n d e d a n d t h a t x ! ) ( s ) = 0 fo r s > t A fo re a c h i a n d s o m e f i n i t e A .

T h i s l a t t e r c o n d i t i o n i s m e t f o r t h e b r a n c h i n g p r o c e s s m o d e l s ; h e n c et h e r e s u l t s o f S e c t i o n 2 m a y s t i l l b e a p p l i e d .5.1. A M U L T I T Y P E G A L T O N - W A T S O N P R O C E S S

T h e r e a r e n t y p es . E a c h i n d i v id u a l l iv e s f o r o n e d i s c r e te t i m ei n t e rv a l , g iv i n g b i r t h a t t h a t t i m e t o o f f s p r i n g , w i t h a l l i n d i v i d u a l sc o n t r i b u t i n g o f f s p r i n g t o t h e p r o c e s s i n d e p e n d e n t l y . T h e p r o b a b i l i t yt h a t t h e r e a r e t l . . . . , t n o f f s p r in g o f ty p e s 1 , . . . , n o f a t y p e i p a r e n t i sg i ( t I . . . . . t n) . T h e d e n s i t y f u n c t i o n f o r t h e d i s t a n c e r i n a s p e c i f i e dd i r e c t io n o f a ll th e o f f s p r i n g f r o m t h e p a r e n t o f t y p e i i s p i ( r ) . L e t U mb e t h e p o s i t i o n o f t h e i n d i v i d u a l f a r t h e s t f r o m 0 i n th e r u t h g e n e r a t i o ni n t h e g i v e n d i r e c t i o n , a n d l e t x } m ) ( s ) = P [U rn > s g i v e n t h a t t h e r e i so n e t y p e i i n d i v i d u a l a t p o s i t i o n 0 in t h e 0 t h g e n e r a t i o n ] .

B y c o n d i t i o n i n g o n w h a t o c c u r s a t t h e e n d o f t h e f i r s t d i s c r e te t i m e


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D I S C R E T E T I M E S P A T I A L M O D E L S 123i n te r v a l, w e o b t a i n t h e f o l l o w i n g s y s t e m o f e q u a t i o n s d e s c r ib i n g t h i sm o d e l :. (m + f R t ~ [ 1 x ~ m ) ( r )] ' P i ( S r ) d r ,i 1 )(S ) . . . . ~ . ~ g i ( t l , . . . , t n ) 1 - - -

tnf o r i = 1 , . . . , n . N o t e t h a t x ~ ) ( s ) = 1 i f s < 0 a n d x ~) = 0 i f s > 0 .

I n t h e f o r w a r d t a il o f t h e d i s t ri b u t i o n o f f a r t h e s t s p r e a d , t h e s ee q u a t io n s m a y b e a p p r o x i m a t e d b y :

x } m + l ) ( s ) = f g ~ t 1 " '" ~ . ~ g i( tl . . . . . t n ) [ ~ t j x } m ) ( r ) ] p i ( s - r ) d rtn [ j = l 1n= E l z i j f x } m ) ( r ) p i ( s - - r ) d r ,

j ~ l Rw h e r e /z q is t h e e x p e c t e d n u m b e r o f o f f sp r in g o f t y p e j f r o m a ty p e ip a r e n t .

L e t { A(A )}q = / z i j P / ( A ) , w h e r e P / (A ) is t h e L a p l a c e t r a n s f o r m o f p i ( r ) .T h e n t h e s p e e d o f s p r e a d o f th e f o r w a r d ta il , c , is g i v en b y T h e o r e m 3w i t h t h i s m a t r i x A (A ) . A s i m p l i fi c a ti o n o c c u r s w h e n P i (A ) is t h e s a m e f o ra ll i w i th c o m m o n v a l u e P ( A ) . I n t h is c a s e , c i s g i v e n b y E q u a t i o n ( 7 )w i t h ~, = p ( / . 0 , w h e r e { ~ } i j = [tl'ij"5.2. A D E M O G R A P H I C B R A N C H I N G P R O C E S S M O D E L

C o n s i d e r a n a g e - d e p e n d e n t b r a n c h i n g p r o c e s s m o d e l , w i t h x i ( s ) n o wm e a s u r i n g t h e p r o b a b i l i t y t h a t t h e f a r t h e s t s p r e a d a t t i m e m i n as p e c i f i e d d i r e c t i o n i s a t l e a s t s , g i v e n t h a t t h e r e w a s o n e i n d i v i d u a li n it ia l ly o f a g e i a t p o s i t i o n 0 . A n i n d i v i d u a l o f a g e i h a s a p r o b a b i l it y / 3 ~o f d y i n g b e f o r e r e a c h i n g a g e i + 1 w i t h /3k = 1 , s o a n i n d i v i d u a l c a n n o tl iv e f o r k + 1 y e a r s o r m o r e . I f t h e i n d i v i d u a l s u r v i v e s t o a g e i + 1, t h e nt h e p r o b a b i l i t y o f t o f f s p r i n g i s g i ( t ) a n d , c o n d i t i o n a l o n t , t h e d e n s i t yf u n c t i o n o f t h e p o s i t i o n s r 1 . . . . r o f t h e s e o f f s p r in g i n t h e s p e c i f ie dd i r e c t i o n r e l a t iv e t o t h e p a r e n t is p i ( r 1 . . . . r t ) , w i t h c o m m o n m a r g i n a ld e n s i t y f u n c t i o n p i ( r ) . A g a i n , u s i n g a c o n d i t i o n i n g a r g u m e n t , w e o b t a i nt h e f o l lo w i n g e q u a t i o n s d e s c r i b in g t h e p r o c e s s :

x ! m + l ) ( S ) = ( 1 - - ~ i ) ~ - , g i ( t )t

(m )x 1 - [ 1 - f R , E [ 1 - - x ( m ) ( s - r j ) l P i ( r l ' " " r t ) d r l ' " " d r t '

f o r i = 1 , . . . , k . N o t e t h a t x ~ "~ )l( S) -= O .


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124 J . RAD CLIFFE AND L. RASSI n t h e f o r w a r d t a i l o f t h e d i s tr i b u t io n o f f a r th e s t s p r e a d , t h e s e

e q u a t io n s m a y b e a p p r o x i m a t ed b y

X } m + 1 ) ( $ ) = ( 1 - / 3 i ) [ X ~ + ~ ( S ) q " I,L i f x ( o m ) ( r ) p i ( s - - r ) d r ] ,f o r i = 1 . . . . k , w h e r e /z/ i s t h e e x p e c t e d n u m b e r o f o f f s p r i n g o f a nind iv idua l o f age ( i + 1 ). Le t P / (A) be t he Lap l ace t r ans form of p i ( r ) .D ef in e {A( A)}il = (1 - /3 i_ 1)/zi- 1P /- I(A) fo r i = 1 . . . . k + 1, {A( A)}i, + 1 =( 1 - / 3 / - 1 ) f o r i = 1 . . . . , k , a n d { A (A )}ij = 0 f o r j ~ : l o r ( i + 1 ) . T h e n t h es p e e d o f s p r e a d o f t h e f o r w a r d t a il, c , is g iv e n b y T h e o r e m 3 w i t h t h ismat r ix A(A).6 . C O N C L U S I O N

T h e s a d d l e p o i n t m e t h o d i s a p o w e r f u l t o o l . I t c a n b e a p p l i e d t o aw i d e r a n g e o f b io l o g ic a l m o d e l s i n w h i c h t i m e a n d s p a c e m a y b ed i s c r e t e o r c o n t i n u o u s . I t h a s a l r e a d y b e e n u s e d f o r t h e a n a l y s i s o fc o n t i n u o u s t i m e m o d e l s o f e p id e m i c s . In t h i s p a p e r , w e h a v e d e r i v e d t h es a d d l e p o i n t a p p r o x i m a t i o n f o r d i s c r e t e t i m e m o d e l s . F o r o u r r e s u l t s t ob e a p p l i c a b l e t o s u c h d i s c r e t e t i m e m o d e l s , t h e a p p r o x i m a t e e q u a t i o n sin t he fo rward f ron t mus t be l i nea r , a s i n Equa t ion (1 ) , wi th nonnega t i vec o e f f i c i e n t s y / j . S o m e c a r e i s n e e d e d w h e n t h e s e e q u a t i o n s i n t h eforward f ron t y i e ld a r educ ib l e sys t em, when t he o r ig ina l sys t em ofequa t i ons i s nonreduc ib l e . In such ca se s , t he equa t i ons a re va l i d on ly i nt h e f o r w a r d f r o n t f o r a n e w t yp e , w h e r e n o c o n n e c t e d n e w t y p e c anspread a t a f a s t e r speed .

M a n y c o n t a c t m o d e l s t h a t f i t i n t o t h e f o r m a t d e s c r i b e d c a n b ec o n s i d e r e d . W e h a v e s h o w n t h e w i d e r an g e o f a p p l ic a b il it y o f t h em e t h o d b y a p p ly i n g it t o s e v e r a l c o n t a c t m o d e l s i n g e n e ti c s, e v o l u t io n -a r y g a m e s , a n d b r a n c h i n g p r o c e s s e s . M a n y v a r ia t io n s a n d m o r e - c o m p l e xm o d e l s c a n b e w r i t t e n f o r w h i c h t h e m e t h o d i s s t i l l a p p l i c a b l e a n d f o rw h i c h t h e s p e e d o f fi rs t s p r e a d c a n b e o b t a i n e d b y t h e s a d d l e p o i n tm e t h o d d e s c r i b e d i n t h i s p a p e r .A P P E N D I X

P r o o f o f T h e o r em 1 . T a k e M 0 su c h t h a t 1 0 ( m ) - 001 < ( 0 0 / 2 ) f o rm > M 0 . Then , fo r m > M 0, wr i t e1 [ O ( m ) + i ~ O ( m )

O ( m ) e -g [ (m ) ] ~ = ~ " O ( m ) - i~ ) t{ [ A ( A ) ] m L( ) ( A ) } i d A = I 1 d - I 2 -I- 1 3 d - I 4 ,

- - e - A s ( m ) - g [ O ( m ) ]


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D I S C R E T E T I M E S P A T IA L M O D E L Sw h e r e

125

N o w

1I I = 2 - ~ f l > a J l y O ( . m ) e _ i y s ( m ) 1O ( m ) + i y { p [ O ( m ) ] }m { { A [ 0 ( m ) + / y ] } 'n L( )[ O (m) + iY]} idY ,

O ( m ) 112 = 2-~ fs< [yl 0 t h e r e e x is tsa n a * s u c h t h a t y i / k ij ( y ) . < < d f o r l y l I> a * . T a k e . p =minx~tOo/2,kOol p (x ). T h e c o n t i n u i t y a n d p o s i t i v i t y o f p ( x ) i m p l y t h a tp > O .


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126T h e n f o r o~ > / a * ,

J . R A D C L I FF E A N D L . R A S S

E y # k i y ( Y ) { ( d l l ) I L ' @l l l < 2_~ f lyI p - m , m-i> a jh n l , ] m 1

= E ~ ' ~ YiYfly > ? k i t ( Y ) d y .jT a k e d = p / 2 ( n - n 1) a n d f in d th e c o r re s p o n d i n g a * . T h e n

h1Ill < E -~ p "Yiy2-m lyl> a k U ( Y ) d yJf o r c~ >~ c~*.

N o w c h o o s e a > a * s u c h t h a t

fly 1rp> ,~ i j ( y ) dy < 3hY.Ty#f o r a l l i , j .H e n c e 1 1 mI l 1 < ~ ( ~ ) f o r t h is ot a n d f o r m > ~ M o.

N e x t c o n s i d e r 1 3. B e c a u s e t h e e i g e n v a l u e s o f A ( 0 0 ) a r e d i s ti n c t, f r o mD i e u d o n n 6 [ 2 7 ] , p a g e 2 4 8 , t h e r e e x i s t s a n e i g h b o r h o o d IA - 0 0 l < 80 f o rw h i c h A ( A ) h a s d i s t in c t e i g e n v a l u e s /.~ I(A ) . . . , / ~ n_ n l( A ) w i t h p ( A ) - -/ Z l( A ), w h e r e R e[ p ( A ) ] > I /z ,( A ) l f o r s t> 2 . A l s o / ~ s ( A ) , a n d t h e e n t r i e s o ft h e c o r r e s p o n d i n g i d e m p o t e n t E , (A ) , a re c o n t i n u o u s f u n c t i o n s o f A f o ra l l s a n d IA - 0 0 l < 80 .L e t E ( A ) = E l (A ) . In t h e n e i g h b o r h o o d o f 0 , d e f i n e

mK ( ; O = ( 1 } ( X )]

F r o m t h e c o n t i n u i t y o f t h e i d e m p o t e n t s a n d t h e e n t r i e s o f L /2 . Ina d d i t i o n , I ~ s ( A ) l < R e[ p ( A ) ] < p[Re(A)] a n d h e n c e I / z~ ( A ) / o [ R e ( A ) ]I < 1f o r s > i 2 . B e c a u s e m a x , ~ 2 1 / z s ( A ) / p [ R e ( A ) ] l ~ m a x ~ l /~ s ( 0 o ) l P ( 0 o ) < 1a s A ~ 0 0, t h e r e e x is ts a p o s i t i v e 8 1 < 80 a n d y < l s u c h t h a tI / z ~ ( A ) / p [ R e ( A ) ] I ~< y f o r s > / 2 a n d IA - 0 0 l < 61. H e n c e IK ( A ) I ~< y m ( n- n 1 - 1 ) D f o r a ny m ~ M o a n d I A - 0 1 < B1


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D I S C R E T E T I M E S P A T IA L M O D E L S 127C h o o s e M I > _ . M o s u c h t h a t I O ( m ) - O o l < < . ( 8 1 / v r 2 ) . T h e n 1 1 3 1 ~ / M 1.

F i n a ll y , f o r t h e a a n d 8 p r e v i o u s l y o b t a i n e d , c o n s i d e r t h e i n t e g r a l 1 2.D e f i n e A * j ( x ) = su p 8 ,~ lyl ,~ ~ I A o ( x + iy )l. W h e n Y o = 0 , t h e n A o ( x + iy )- - 0 a n d h e n c e A * j ( x ) = 0 I f Yij ~ : 0 , t h en A * . (x ) < A o ( x ) I n t h i s c a s e ,j

l im x_~ o o [ A * j ( x ) / A o ( x ) ] = [ A * j( O o ) /A i j( O o ) ] < 1. H e n c e t h e r e e x i st s ap o s i t i v e /3 < 1 a n d a n M 2 t> M 1 s u c h t h a t { A ' ~ j [ O ( m ) ] / A o [ O ( m ) ] } < / 3f o r m > / M 2 a n d a ll i , j s u ch t h a t 3 'o ~ 0. H e n c e A * [ 0 ( m ) ] < / 3 A [ O ( m ) ]fo r a l l m i> M 2 .

N o w , b e c a u s e M 2 > / M o , w e h a v e ~ {L ( )[0 (m )+ /Y ]} jl < h f o r a l l j a n da ll m > / M o. T h e r e f o r e

2 h a rlI2l ~< - - ~ - p [ 0 ( m ) ] - m {(A * [ 0 ( r e ) l ) m } ih a p [ O ( m ) ] - m / 3 m { ( A [ O ( m ) ] ) m l } i

f o r m >~ M 2 .B u t A [ 0 ( m ) ] h a s d i s t in c t e i g e n v a l u e s f o r m 1> M 1 a n d h e n c e f o rm >i M 2 , a n d t h e i d e m p o t e n t E s [ 0 ( m ) ] t e n d s t o E ~ ( 0 ) a s m -- ,o o f o r a l l s ,

s o t h e e n t r i e s a r e b o u n d e d . T h e r e f o r en ~ .~ n 1 ~ l ~ s [ O ( m ) ] mP[O(m)]-m{(A[O(m)])ml}i< = 1 p [O (m ) ] { I E s [ 0 ( m ) ] l } in - - n 1< { I E s [ 0 ( m ) ] l l } iS = I

< Ff o r s o m e co n s t a n t F .H e n c e 1 1 2 1 ~< ( h o t F / ' n ' ) f l m f o r m I> M 2 . T a k e ~b = ( 1 + / 3 ) .

1 mT h e n J I2 1 < ~ b p r o v i d e d m t> M 3 , w h e r e M 3 = m a x [ M 2 ,l o g ( 3 h a F / ' t r ) / l o g ( ~b /3 ) ].

T h e t h e o r e m t h e n f o l lo w s b y t a k i n g = m a x ( , y , ~b) a n d M = M 3 .R E F E R E N C E S

1 H .F . Weinberger , Asymptotic behavior of a mo del in popula t ion genet ics . InNon linear P artialDifferentialEqu ations and A pplications.J . Chadam , Ed. , LectureN otes in M athematics, Vol. 648, Springer, New Y ork, 1978, pp. 47-98 .2 H . F . W einberger, Lo ng-t im e beh avio r of a class of biological models. S/AM ZMath. Anal . 13:353-396 (1982).


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128 J . R A D C L I F F E A N D L . R A S S3 R . Lu i , A n o n l in e a r in te g r a l o p e r a to r a r i sin g f r o m a mo d e l in p o p u la t io n

g e n e t i cs , I : m o n o to n e in i t ia l d a ta . SIA M J . Math. Ana l . 13(6) :913-937 (1982) .4 R . L u i , A n o n l i n e a r i n t e g ra l o p e r a t o r a ri s in g f r o m a m o d e l in p o p u l a t i o ng e n e t i cs , I I: i n i ti a l d a ta w i th c o mp a c t s u p p o r t . S /A M J . Math. Anal . 13(6) :938-953

(1982).5 R . L u i , Ex i s t e n c e a n d s t a b i l i ty o f tr a v e l l in g wa v e s o lu t io n s o f a n o n l in e a rin te g r a l o p e r a to r . J . Math. Biol. 16:199-220 (1983) .6 R . Lu i , B io lo g ica l g r o wth a n d s p r e a d m o d e le d b y s y s tems o f r e c u r s io n s , I:m a t h e m a t i c a l t h e o ry . Math. Biosci. 93:269 -295 (1989) .7 R . Lu i , B io lo g ic a l g r o wth a n d s p r e a d m o d e le d b y s y s tems o f r e c u r s io n s , I I :

b io lo g ic a l th e o r y . Math. Biosci. 93:297 -312 (1989).8 H . E . D a n i e l s , T h e d e t e r m i n i s t i c s p r e a d o f a s im p l e e p i d e m i c . I n Perspectives in

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discrete time spatial models arising in genetics

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