Chapter XVIII A Mathematical Model of Density Dependent Dispersive Motions

Biomathematics in 1980 L.M. Ricciardi and A.C. Scott (eds.) 0 North-Holland Publishing Company, 1982

CHAPTER X V I I I

A MATHEMATICAL MODEL OF U E N S I T Y DEPENDENT DISPERSIVE MOTIONS E i Teramoto

1. I n t r o d u c t i o n D i s p e r s i v e b e h a v i o u r o f l i v i n g o r g a n i s m s has been s t u d i e d b y many a u t h o r s m o s t l y based on t h e u s u a l d i f f u s i o n e q u a t i o n . The a p p l i c a t i o n o f t h e u s u a l d i f f u s i o n e q u a t i o n seems t o be adequate f o r p a s s i v e d i s p e r s i o n s such as t h e d i s p e r s i o n o f s p o r e s i n t h e a i r o r p h y t o p l a n k t o n s i n t h e sea w a t e r . However t h e a c t i v e d i s p e r s i v e m o t i o n s o f a n i m a l s show t h e v a r i e t y o f b e h a v i o u r p a t t e r n s depending upon t h e s p e c i e s c h a r a c t e r i s t i c s and a p p l i c a b i l i t y o f t h e u s u a l d i f f u s i o n e q u a t i o n i s c o m p l e t e l y i m p o s s i b l e even i n t h e c a s e o f a p p a r e n t l y s i m p l e d i s p e r s i v e motions o f small organisms. R e c e n t l y Okubo ( 1 9 8 0 ) p u b l i s h e d an e x c e l l e n t book i n w h i c h h e gave c a r e f u l d i s c u s s i o n s on a w i d e r a n g e o f d i f f u s i o n p r o b l e m s i n b i o l o g y . Okubo p r o p o s e d t h r e e t y p e s o f p a r t i a l d i f f e r e n t i a l e q u a t i o n s i n which t h e s t a t e dependent d i f f u s i v e motions a r e taken i n t o account. L e t us c o n s i d e r a one d i m e n s i o n a l random walk model w i t h t h e p r o b a b i l i t y o f a jump t o r i g h t h , t o l e f t p and r e m a i n i n g &('A+ !+ K = 1). Here t h e s e p r o b a b i l i t i e s a r e assumed t o b e dependent on t h e s t a t e a t a p o s i t i o n between n e i g h b o r i n g p o i n t s . F o r example, t h e p r o b a b i l i t y o f a jump f r o m t h e p o i n t x - 1 t o x d u r i n g t h e t i m e i n t e r v a l ? i s g i v e n b y a f u n c t i o n 'A(x - a l , t ) , where a i s a p a r a m e t e r w i t h v a l u e O a < 1. S i m i l a r l y t h e probab i l i t y o f a jump x + 1 t o x i s g i v e n b y a f u n c t i o n p ( x + a l , t ) .

<

Then we have ')\{x+(l-a)l,tj

+ p{x-(l-a)l,tj

+ &(x,t) = 1

(1)

and t h e m a s t e r e q u a t i o n f o r t h e p r o b a b i l i t y d e n s i t y p ( x , t ) p(x,t+t)

= \(x-al,t)p(x-1,t)

+ k(x,t)p(x,t)

+p(x+al,t)p(x+l,t)

(2)

We expand t h e s e e q u a t i o n s i n t e r m s o f i n f i n i t e s i m a l q u a n t i t i e s 1 and 2; and t a k e I n t h i s l i m i t , i t can b e shown t h a t x - 1 can b e e x p r e s s e d as t h e l i m i t l,T+O .~Qs(x, t ) where E approaches z e r o i n t h e same o r d e r w i t h 1. Then assuming t h e e x istence o f the l i m i t s

.

- lim

1$40

1 6 / ~ =uO

,

2

1 /2?=

liin 1 ,,Po

we o b t a i n

where

u ( x , t ) = u0@x,t)

,

D(x,t)

=

245

D

Do

246

E. TERAMOTO

Due t o t h e v a l u e o f a , Okubo c l a s s i f i e d t h r e e t y p e s o f d i f f u s i o n e q u a t i o n s .

1)

ci = 1 / 2

Neutral type ( F i c k ' s law)

se= -A(up) at bx 2)

a = 0

3)

c1 =

+ -I(&) z b

(4)

Attractive type

1

Repulsive type

8=

,

&up)

L

+ hy(Dp) bx

I n t h e case o f a t t r a c t i v e type, cesses o f d i s p l a c e m e n t i n w h i c h dependent on t h e r e c o g n i t i o n o f t h e o t h e r hand, i n t h e c a s e o f d i s p l a c e m e n t depends on o n l y t h e

t h e dispersion consists o f t h e elementary prot h e p r o b a b i l i t y o f displacement i s e s s e n t i a l l y t h e s t a t e a t t h e end p o i n t o f d i s p l a c e m e n t . On r e p u l s i v e dispersion, p r o b a b i l i t y o f elementary state at the i n i t i a l position.

I n t h e f o l l o w i n g s e c t i o n s we s h a l l p r e s e n t an example i n w h i c h t h e e x p e r i m e n t a l d a t a o f d i s p e r s a l b e h a v i o u r o f some i n s e c t s c a n b e s u c c e s s f u l l y d e s c r i b e d b y a d e n s i t y dependent d i f f u s i o n e q u a t i o n o f t h e r e p u l s i v e t y p e . 2. E v i o u r a l c h a r a c t e r o f a n t l i o n s The e x p e r i m e n t s o f d i s p e r s a l b e h a v i o u r o f a n t l i o n s was done b y M o r i s i t a (1952) u s i n g a box, j u s t one h a l f o f w h i c h i s f i l l e d w i t h f i n e sand and o t h e r h a l f i s f i l l e d w i t h c o a r s e sand. The a n t l i o n s were p u t one b y one on t h e b o r d e r l i n e o f t h e t w o a r e a s and t h e number o f i n d i v i d u a l s s e t t l e d i n each sand a r e a was counted a f t e r t h e i r p i t formation. I t was f o u n d t h a t when t h e p o p u l a t i o n d e n s i t y i s r e l a t i v e l y low, t h e a n t l i o n s have s t r o n g t e n d e n c y t o p r e f e r f i n e sand t o c o a r s e sand f o r p i t f o r m a t i o n , b u t t h i s t e n d e n c y g r a d u a l l y f a l l s w i t h i n c r e a s i n g d e n s i t y u n t i l an a l m o s t e q u a l number o f i n d i v i d u a l s s e t t l e t h e i r r e s i d e n c e s i n b o t h sands. T h i s f a c t r e f l e c t s t h e e f f e c t o f r e p u l s i v e i n t e r f e r e n c e among t h e i n d i v i d u a l s. I n o r d e r t o e x p l a i n t h i s e x p e r i m e n t a l r e s u l t q u a n t i t a t i v e l y , M o r i s i t a assumed t h a t t h e p r o b a b i l i t y o f s e t t l e m e n t o f an i n d i v i d u a l i n f i n e sand a r e A ( o r c o a r s e sand a r e a B ) i s 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 d e g r e e o f " u n f a v o r a b l e n e s s " o f t h a t h a b i t a t , and t h e degree o f u n f a v o r a b l e n e s s i s g i v e n b y t h e sum o f t h e q u a n t i t y EA ( o r E ) named " e n v i r o n m e n t a l d e n s i t y " , w h i c h i s a measure o f t h e i n t r i n s i c u n f a v o r a b l g n e s s o f t h e h a b i t a t i t s e l f , and t h e number o f i n d i v i d u a l s nA ( o r n B ) a l r e a d y s e t t l e d i n t h e f i n e sand ( o r c o a r s e s a n d ) a r e a . Then t h e p r o b a b i l i t i e s t h a t an i n d i v i d u a l s e t t l e s t h e r e s i d e n c e and f o r m s a p i t i n f i n e sand a r e a A o r c o a r s e sand a r e a B when n A and nB i n d i v i d u a l s have a l r e a d y s e t t l e d i n area

A and B r e s p e c t i v e l y a r e g i v e n b y

DENSITY DEPENDENT DISPERSIVE MOTIONS

247

A n a l y z i n g t h i s Markov p r o c e s s , t h e average number o f i n d i v i d u a l s f o u n d i n t h e f i n e sand area, a f t e r n i n d i v i d u a l s a r e i n t r o d u c e d i n t o t h e box, can b e o b t a i n e d as

1/2. as n i n c r e a s e s . The e x o e r i m e n t a l r e s u l t s c o u l d be q u a n t i t a t i v e l y e x p l a i n e d b y t h i s f o r m u l a v e r y successfully. I t i s seen t h a t t h i s r a t i o aooroaches a v a l u e

3. G e n e r a l i z a t i o n

Now we c a n g e n e r a l z e M o r i s i t a ' s i d e a b y g i v i n g an e t h o l o g i c a l i n t e r p r e t a t i o n t o t h e d i s p e r s a l movement o f i n d i v i d u a l a n t l i o n ( S h i g e s a d a , Kawasaki and Teramoto 1979). Here we assume t h a t t h e movement o f an i n d i v i d u a l i s a t t r i b u t e d t o t h e following three forces: i ) t h e d i s p e r s i v e f o r c e w h i c h i s a s s o c i a t e d w i t h random movement o f i n d i v i d u a l s , i i ) t h e p o p u l a t i o n p r e s s u r e due t o m u t u a l i n t e r f e r e n c e s between i n d i v i d u a l s and i i i ) t h e a t t r a c t i v e f o r c e toward t h e favorable environment. The p o p u l a t i o n p r e s s u r e may b e t a k e n i n t o a c c o u n t b y c o n s i d e r i n g t h a t t h e d i s p e r s i v e f o r c e i s enhanced b y t h e r e p u l s i v e i n t e r f e r e n c e w i t h t h e i n c r e a s e o f population density, Then we can assume t h e i s o t r o p i c d i s p e r s i v e f o r c e , w h i c h i s t h e sum o f t h e f o r c e s i ) and ii),t o b e g i v e n b y a d i s p e r s i o n c o e f f i c i e n t e x p r e s s e d as

a

+

Bn(x)

where n ( x ) i s t h e p o p u l a t i o n d e n s i t y a t t h e p o s i t i o n x. As f o r t h e measure o f " f a v o r a b l e n e s s " , we can i n t r o d u c e a q u a n t i t y - U ( x ) , where t h e n e g a t i v e s i g n i s a t t a c h e d s o as t o r e g a r d t h e f u n c t i o n U ( x ) as an e n v i r o n m e n t a l p o t e n t i a l f u n c t i o n . Thus t h e mean v e l o c i t y o f t h e movement caused b y t h e f a v o r a b l e n e s s o f h a b i t a t i s assumed t o b e p r o p o r t i o n a l t o t h e f o r c e p r o d u c e d b y t h e p o t e n t i a l f u n c t i o n U ( x ) , t h a t i s -grad U(x). As a p r e p a r a t o r y s t e p t o t h e d e r i v a t i o n o f a n o n l i n e a r d i f f u s i o n e q u a t i o n , I n s t e a d o f i n t r o d u c i n g t h e organisms we s h a l l c o n s i d e r t h e f o l l o w i n g e x p e r i m e n t . one b y one, i f we p u t a l l n i n d i v i d u a l s s i m u l t a n e o u s l y i n t h e box, t h e y move around i n t h e box u n d e r t h e i n f l u e n c e o f m u t u a l i n t e r f e r e n c e s . Then we c a n assume t h a t t h e t r a n s f e r p r o b a b i l i t y r a t e f r o m B t o A i s g i v e n b y a+pnB+k/2 and s i m i l a r l y from A t o B b y a+BnA-k/2, UB-UA

where k i s p r o p o r t i o n a l t o t h e p o t e n t i a l d i f f e r e n c e

and t h e r e l a t i o n n A + n B = n h o l d s t h o u g h o u t t h e p r o c e s s .

I n t h i s case

settlement o f t h e residence i s not apparently considered. C l e a r l y t h i s elementary p r o c e s s i s a t y p i c a l example o f r e p u l s i v e t y p e d i s c u s s e d i n t h e p r e v i o u s s e c t i o n . The t e m p o r a l change o f t h e number o f i n d i v i d u a l s i n each a r e a c a n b e r e p r e s e n t e d b y t h e equations

The s o l u t i o n o f t h e s e e q u a t i o n s c a n b e e a s i l y o b t a i n e d a n a l y t i c a l l y d i s t r i b u t i o n a l w a y s approaches a s t a t i o n a r y d i s t r i b u t i o n g i v e n b y

'A n

=

(a+k/2)/2@ + (n-1)/2 a / @+ ( n - 1 )

and t h e

248

E. TERAMOTO

I t w i l l b e seen t h a t t h e e x p r e s s i o n ( 1 0 ) becomes e q u i v a l e n t t o M o r i s i t a ' s r e s u l t ( 8 ) b y e q u a t i n g t h e e n v i r o n m e n t a l d e n s i t i e s as

EA = ( a

-

,

k/2)/213

EB = ( a + k/2)/2B

As t h e n e x t s t e p , we c o n s i d e r a l i n e a r a r r a y o f many h a b i t a t s w h i c h have d i f f e r e n t e n v i r o n m e n t a l c o n d i t i o n s , f o r i n s t a n c e d i f f e r e n t p a r t i c l e s i z e s o f sand f o r ant l i o n s . Then, i n j u s t t h e saine way as t h e e q u a t i o n s ( 9 ) f o r t h e case o f t w o h a b i t a t s , we c a n o b t a i n a s e t o f e q u a t i o n s e a c h o f w h i c h d e s c r i b e s t h e t e m p o r a l change o f t h e number o f i n d i v i d u a l s ni i n t h e i t h h a b i t a t . I n order t o d e r i v e t h e space c o n t i n u o u s model, we can t a k e t h e c o n t i n u o u s l i m i t o f t h e equat i o n s f o r t h i s d i s c r e t e a r r a y model i n a s t r a i g h t f o r w a r d way. Then i t can b e shown t h a t t h e p o p u l a t i o n f l o w a t t h e p o s i t i o n x i s g i v e n b y

3 =

-

grad

l(a +

Bn(x,t))n(x,t)x

-

n(x,t)grad U(x)

(11)

where n ( x , t ) i s t h e p o p u l a t i o n d e n s i t y a t t h e p o s i t i o n x and t i m e t . The f i r s t and second t e r m s o f t h e e q u a t i o n ( 1 1 ) c o r r e s p o n d t o t h e d e n s i t y dependent d i s p e r s i o n and t h e f l o w due t o t h e e n v i r o n m e n t a l p o t e n t i a l r e s p e c t i v e l y . Using t h i s f l o w f u n c t i o n , we can e x p r e s s t h e e q u a t i o n f o r t h e change o f d e n s i t y d i s t r i b u t i o n by the c o n t i n u i t y equation An(x,t) at

=

-

div

J

T h i s i s j u s t t h e e q u a t i o n o f r e p u l s i v e t y p e ( 7 ) , w i t h u = - g r a d U ( x ) and D = c1 + B n ( x , t ) . I f we c o n s i d e r a one d i m e n s i o n a l box w i t h b o u n d a r i e s a t w h i c h J = 0, a s t a t i o n a r y d i s t r i b u t i o n n * ( x ) i s g i v e n as a s o l u t i o n o f t h e e q u a t i o n J = 0, t h a t i s

( a+2Bn *

dn* t - dU n*=O )x dx

and b y i n t e g r a t i n g t h s e q u a t i o n we have r

where n ( 0 ) and U(0) a r e t h e v a l u e a t t h e o r i g i n o f t h e c o - o r d i n a t e chosen i n t h e h a b i t a t a r e a .

arbitrarily

F u r t h e r m o r e i t can b e p r o v e d t h a t t h e s t a t i o n a r y s o l u t i o n ( 1 4 ) i s g l o b a l l y s t a b l e , b y showing t h a t t h e f u n c t i o n

-

H = \@nlog(n/n*)

a(n-n*)

+ B(n-n*)*?dx

> 0

J

i s a Lyapunov f u n c t i o n o f t h e e q u a t i o n ( 9 ) . the function H i s given by

*

dt =

-

)(J2/n)dx

<

A c t u a l l y t h e t i m e d e r i v a t i v e of

0

As an example, we can have t h e r e s u l t shown i n F i g u r e . 1 w h i c h shows t h e s t a t i o n a r y d i s t r u b u t i o n when t h e e n v i r o n m e n t a l p o t e n t i a l f u n c t i o n i s g i v e n b y 2 U=1.5(x-1) We c a n see a g a i n t h e c h a r a c t e r i s t i c f e a t u r e o f M o r i s t a ' s e x p e r i m e n t a l r e s u l t t h a t t h e d i s t r i b u t i o n becomes f l a t t e r w i t h i n c r e a s i n g t h e t o t a l number o f i n d i v i d u a l s N o r t h e p o p u l a t i o n p r e s s u r e B .

.

DENSITY DEPENDENT DlSPERSl VE MOTIONS

c

6.0

249

(b)

N=6

0

F i g u r e 1.

.

2 a) E n v i r o n m e n t a l p o t e n t i a l f u n c t i o n U ( x ) = 1.5(x-l) b) Stationary population densities. N i s t h e t o t a l number o f ndividuals.

4. A p p l i c a t i o n s P l a u s i b i l i t y o f t h e p r e s e n t f o r m u l a t i o n h a s been i n s p e c t e d b y S h i g e s a d a (1980) b y a p p l y i n g t h e n o n l i n e a r d i f f u s i o n equation (12) t o o t h e r experimental data o f t h e d i s p e r s i v e motions o f i n s e c t s s t u d i e d also by M o r i s i t a f o r ant l i o n s ( 1 9 5 4 ) , b y Watanabe, U t i d a and Y o s i d a f o r a z u k i bean w e e v i l s ( 1 9 5 2 ) and b y C l a r k f o r grass hoppers (1962). I n t h e s e e x p e r i m e n t s a number o f i n d i v i d u a l s a r e p l a c e d a t some p o s i t i o n i n t h e e x p e r i m e n t a l f i e l d w i t h u n i f o r r n e v i r o n m e n t a l c o n d i t i o n s . The r e l e a s e d i n d i v i d u a l s m i g r a t e r a n d o m l y i n t h e f i e l d and we can o b s e r v e t h e s p r e a d o f d i s t r i b u t i o n q u a n t i t a t i v e l y as t h e f u n c t i o n o f t i m e , b y c o u n t i n g t h e number o f i n d i v i d u a l s found i n s i d e o f a c i r c l e o f r a d i u s r centered a t t h e r e l e a s i n g p o i n t , o r b y evaluating t h e variance o f t h e d i s t r i b u t i o n o f t h e whole population. I n order t o analyse t h e experimental d a t a f o r these q u a n t i t i e s , Shigesada a p p l i e d t h e equation (12) t o t h e two dimensional d i s p e r s i v e motions o f t h e i n s e c t s i n t h e uniform field. S i n c e i t i s known f r o m t h e e x p e r i m e n t a l d a t a t h a t (Y i s u s u a l l y r e l a t i v e l y s m a l l compared w i t h B n ( x , t ) , she assumed u=O i n h e r a n a l y s e s . I n t h i s case t h e a n a l y t i c a l s o l u t i o n c a n be o b t a i n e d and t h e r a d i a l d i s t r i b u t i o n as a f u n c t i o n o f t i m e was s t u d i e d c o m p a r i n g t h e e x p e r i m e n t a l d a t a w i t h v e r y s u c c e s s f u l l y agreement. From t h e r e s u l t s o f t h e s e i n v e s t i g a t i o n s we c a n p r o b a b l y c o n c l u d e t h a t , a t l e a s t f o r some s p e c i e s o f s m a l l i n s e c t s , c h a r a c t e r i s t i c b e h a v i o u r o f d i s p e r s i o n can b e mathematically described b y t h e d i f f u s i o n equation o f repulsive type given b y e q u a t i o n s ( 1 1 ) and ( 1 2 ) . A n o t h e r a p p l i c a t i o n c o n c e r n s a more e c o l o g i c a l problem. Our p r o b l e m i s t o examine t h e p o s s i b i l i t y t h a t t h e s p a t i a l s e g r e g a t i o n o f h a b i t a t w h i c h r e s u l t s f r o m t h e m u t u a l i n t e r f e r e n c e s and t h e h e t e r o g e n e i t y o f t h e e n v i r o n m e n t h a s an e f f e c t t o r e l a x t h e i n t e r s p e c i e s c o m p e t i t i o n and s t a b i l i z e t h e c o e x i s t e n c e o f t w o competitive species.

E. TERAMOTO

250

I n o r d e r t o s t u d y t h i s problem, l e t u s c o n s i d e r f i r s t l y s p a t i a l l y u n i f o r m p o p u l a t i o n s o f two c o m p e t i t i v e species. The V o l t e r r a t y p e e q u a t i o n s f o r t h i s system a r e w r i t t e n as

where E l and

pll

and pz2

€2 a r e t h e i n t r i n s i c g r o w t h r a t e s o f t h e p o p u l a t i o n s o f t w o s p e c i e s , a r e t h e c o e f f i c i e n t s o f i n t r a s p e c i f i c c o m p e t i t i o n s and pI2

and pZ1

are those o f i n t e r s p e c i f i c competitions. I t i s w e l l known t h a t t h e c o e x i s t e n c e o f t h e s e t w o s p e c i e s becomes p o s s i b l e o n l y when t h e c o n d i t i o n s Pll/tl

> bl/%

and

P 221% > p 12 l E 1

a r e s a t i s f i e d , o t h e r w i s e o n l y one o f t h e s p e c i e s can s u r v i v e and a n o t h e r s p e c i e s i s l e d t o e x t i n c t i o n (Gause's c o m p e t i t i v e e x c l u s i o n p r i n c i p l e ) . Here l e t u s c o n s i d e r t h a t t h e s e t w o a n i m a l s p e c i e s h a v e a l m o s t t h e same f a v o r a b l e n e s s f o r t h e e n v i r o n m e n t and a r e m i g r a t i n g under t h e i n f l u e n c e o f t h e p o p u l a t i o n p r e s s u r e due t o i n t r a - and i n t e r - s p e c i f i c i n t e r f e r e n c e s . Then t h e population flows are given by

Combining t h e n o n l i n e a r d i s p e r s i v e movement w i t h p o p u l a t i o n g r o w t h , we h a v e t h e equations

4 bt

=-A

b x J1

' b - b & t n2 - - 5 J2

+

+

-

pllnl

-

p12nz)nl

(CZ -

Pz1n1

-

@zznz)nz

(61

(17)

T h i s e q u a t i o n (.17) i s a c t u a l l y d i f f i c u l t t o d e a l w i t h a n a l y t i c a l l y . However, we can show b y computer c a l c u l a t i o n t h a t t h e s p a t i a l s e g r e g a t i o n caused b y t h e nonlinear flows (16) a c t u a l l y s t a b i l i z e s t h e coexistence o f two c o m p e t i t i v e s p e c i e s . F o r t h e n u m e r i c a l c a l c u l a t i o n s , we assumed t h e c o n d i t i o n s

pz14

>

1-111/€1

Under t h e s e c o n d i t i o n s , c r i t i c a l s t a t e (€l/pll,O) s i t y nl

and

p12/%

>

P22/%

t h e s p a t i a l l y u n i f o r m system (15) leads t o t h e s t a b l e when i t s t a r t s f r o m a s t a t e i n w h i c h t h e p o p u l a t i o n den-

i s r e l a t i v e l y l a r g e r t h a n nz, c o n v e r s e l y t h e system approaches t h e s t a t e

(0,E2/p22)i f i t s t a r t s f r o m a s t a t e i n w h i c h n2 i s r e l a t i v e l y l a r g e r t h a n nl.

In

any way e i t h e r o f t h e s p e c i e s i s d e s t i n e d t o r u i n . A r e s u l t o f o u r computer c a l c u l a t i o n o f e q u a t i o n s ( 1 7 ) i s g i v e n i n F i g u r e 2 w h i c h shows t h e s t a t i o n a r y d i s t r i b u t i o n s n,(x,-) and n,(x,-) which are e s t a b l i s h e d

a f t e r a s u f f i c i e n t l y l o n g t i m e s t a r t i n g f r o m an i n i t i a l l y u n i f o r m d i s t r i b u t i o n . F u r t h e r computer s i m u l a t i o n s f o r t h i s system w i t h t h e p a r a m e t e r v a l u e s o f F i g u r e 2 have shown t h a t t h i s n o n - u n i f o r m s t a t i o n a r y d i s t r i b u t i o n i s l o c a l l y s t a b l e f o r s m a l l p e r t u r b a t i o n s o f p o p u l a t i o n d e n s i t i e s and i s e s t a b l i s h e d s t a r t i n g f r o m a w i d e v a r i e t y o f i n i t i a l c o n d i t i o n s . However t h e r e r e m a i n s a p o s s i b i l i t y t h a t t h i s

251

DENSITY DEPENDENT DlSPE RSI VE MOTIONS

F i g u r e 2. Population d e n s i t i e s o f two s i m i l a r competing species. 2 U,(X) = 1 . 5 ( ~ - 1 ) ; a1 = a2 = 1; Bll = 8 2 2 = B12 = 0;

El

=

e2

= 6;

pll = p Z 2 = 14; pZ1 = p12 = 2.8.

d i s t r u b u t i o n s , nl(x,O)

and n,(x,O).

stationary distributions,

n,(x,-)

U,(x)

-

421 = 10;

(a) I n i t i a l

(b) Finally attained and n,(x,-).

p a t t e r n c a n n o t b e r e a c h e d b y s t a r t i n g f r o m some s p e c i a l t y p e o f i n i t i a l d s t r i b u t i o n . It s h o u l d b e n o t e d t h a t t h e e s t a b l i s h m e n t o f such a s t a b l e s e g r e g a t on o f d i s t r u b u t i o n a l p a t t e r n depends on t h e p a r a m e t e r v a l u e s . F o r i n s t a n c e , t h e s mulat i o n has shown t h a t such a p a t t e r n becomes u n s t a b l e w h e n C 1 = c2 = 18, pll =

p22 =

8.4 ( j u s t t h r e e t i m e s t h e v a l u e s used i n F i g u r e 2 w i t h r e m a i n i n g p a r a m e t e r v a l u e s and t h e i n i t i a l u n i f o r m d i s t r i b u t i o n unchanged). Thus i t h a s been c l a r i f i e d t h a t i f we t a k e i n t o a c c o u n t t h e e n v i r o n m e n t a l h e t e r o g e n e i t y and t h e n o n l i n e a r d i s p e r s i v e f o r c e s , t h e c o e x i s t e n c e o f t w o s i m i l a r and c o m p e t i n g s p e c i e s can b e r e a l i z e d a t l e a s t u n d e r some c o n d i t i o n s .

E. TERAMOTO

252

REFERENCES

111

0.P C l a r k , An a n a l y s i s o f d i s p e r s a l and movement i n P h a u l a c r i d i u m v i t t a t u m , Aust. J . Zool. 182 ( 1 9 6 2 ) .

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M. M o r i s i t a , H a b i t a t p e r f e r e n c e and e v a l u a t i o n o f e n v i r o n m e n t o f a n i m a l : e x p e r i m e n t a l s t u d i e s on t h e p o p u l a t i o n d e n s i t y o f an a n t - l i o n , G l e n u r o i d e s j a p o n i c u 2 M ' L (1) P h y s i o l . and E c o l . 5, 1 ( 1 9 5 2 ) .

lo,

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[7J

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2,

Biol

. 9, 85

(1980).

(L),

1,