Journal
CAN
of G e n e t i c s
A
SINGLE
Volume
RESOURCE
62
Number
SUPPORT
1
MANY
July
CONSUMER
SPECIES
1975
?
SULOCHANA
GADGIL AND MADHAV GADGIL
C e n t r e f o r T h e o r e t i c a l S t u d i e s , I n d i a n I n s t i t u t e of S c i e n c e
Bangalore
560 012
I N
T
R
O
D
U
C
T
I
O N
T h e n u m b e r of s p e c i e s w h i c h c a n b e p a c k e d w i t h i n a b i o l o g i c a l c o m m u n i t y c o n t i n u e s to h e o n e of t h e m o s t i n t r i g u i n g p r o b l e m s
in e c o l o g y
(Volterra
1931, G a u s e 1934, M a c A r t h u r
a n d L e v i n s 1964, L e v i n s 1968,
L e v i n 1970, M a y a n d M a c A r t h u r
1972, S t e w a r t a n d L e v i n 1973).
A very
g e n e r a l r e s u l t on t h i s p r o b l e m s t a t e s t h a t t h e n u m b e r of s p e c i e s in a c o m m u n i t y c a n n o t e x c e e d t h e n u m b e r of l i m i t i n g f a c t o r s ( L e v i n 1970).
Levin's,
as m o s t e a r l i e r m o d e l s a r e of t h e f o r m
1
x.
dx i
dt
= fi
(zI . . . . . . .
l
Zp)
=
p
~
j=l
ctij z j
+ %/i
(1)
H e r e x i is the density of the i2h species and zj is intensity of the jth
Iimitiff~-gfactor w h i c h in turn is a function of th~-population densities
x I ........ x n and the externally d e t e r m i n e d resources Yl ......... Ym' T h e
p
[ i m i t i n g ~ c t o r s determine the g r o w t h rates fl,
fnc~
9
Levin has s h o w n that (i) w h e n the n u m b e r !P~ of limiting factors required to
determine'the g r o w t h rates fi i s s m a l l e r than the total n u m b e r 'n' of the
Species and (ii) w h e n fi's a--~e a s s u m e d to be linear functions of t--he zj's ;
there will be n-p rel~'ionships b e t w e e n the population densities Xl, - Xn
'which, in general, depend on time and are of the f o r m
x%3 . . . . .
Xn n
Ke bt
(21
W h e n it is a s s u m e d ,
in a d d i t i o n , t h a t (iii) a l l t h e p o p u l a t i o n s a r e n o n - v a n i s h ing, a n d r e m a i n b o u n d e d f o r a l l t i m e , the. v a l u e of h in e q u a t i o n (2) h a s to
e q u a l z e r o a n d t h e e q u a t i o n b e c o m e s i n d e p e n d e n t of riffle.
In t h i s c a s e , e v e n
w h e n e q u i l i b r i u m p o i n t s e x i s t , t h e y w i l l not b e a s y r n p t o t i c ' a i l y s t a b l e a n d a s
s u c h t h e p o s s i b i l i t y of a S t a b l e c o - e x i s t e n c e
of n s p e c i e s is r u l e d out w h e n
(i), (ii) a n d ( i i i ) h o l d .
It is i m p o r t a n t to n o t e t h a t t h i s r e s u l t d e p e n d s c r u c i a l l y on t h e ! i n e a r i t y
a s s u m p t i o n a n d w i l l not in g e n e r a l , b e v a l i d f o r n o n l i n e a r s y s t e m s .
The non-
34
Species
Packing
linearities present in realistic m o d e l s of biological c o m m u n i t i e s limit severely
the applicability of this result in understanding the diversity of these c o l n m u nities.
F u r t h e r m o r e the a s s u m p t i o n of h o u n d e d n e s s of xi's , w h i c h is also
crucial to the proof is an additional a s s u m p t i o n w h i c h may--or m a y not be
r e a s o n a b l e for the s y s t e m s m o d e l l e d by (I). In fact, w e have presented a
m o d e l in this p a p e r in w h i c h it is not n e c e s s a r y to expect xi's to r e m a i n
b o u n d e d although the governing equation is of the. type (I).-The important question that remains is: under what conditions can a
single resottrce support m o r e than one species?
F r o m a biological view
point this will occur w h e n both of the following conditions are satisfied.
(i)
T h e r e l a t i v e e f f i c i e n c y of t w o c o m p e t i n g s p e c i e s i n u t i l i z i n g a r e s o u r c e
d e p e n d s u p o n t h e d e n s i t y of t h e r e s o u r c e .
Thus one species may be more
e f f i c i e n t in u t i l i z i n g a r e s o u r c e
at a low density while the other may be more
efficient at a high resource
d e n s i t y ( s e e e . g . , f i g . 1).
J
fO
W
:E
z
oi"
13.
w
z
LIGHT INTENSITY
Figure
i.
A diagramatic
representntlon
of t h e r a t e of n e t p h o t o s y n t h e s i s
a s a f u n c t i o n of H g h t i n t e n s i t y f o r a s u n - l o v i n g v e r s u s a
shade-tolerant
plant.
(ii)
T h e r e is a persistent t e m p o r a l or spatial heterogeneity in the density
of the resource.
Clearly, for a c o m m u n i t y in w h i c h two species with properties described in (t) are initially present and a t e m p o r a l or spatial variation in the
density of the r e s o u r c e is externally imposed, the final state will contain a
m i x e d population for a certain range of p a r a m e t e r s .
This is s h o w n by
i R o u g h g a r d e n (1971) for a genetic s y s t e m a n d by Stewart and L e v i n (1973)
for a c o m m u n i t y supported b y a t i m e - d e p e n d e n t resource.
It is important
to note that: co-existence is r e n d e r e d possible in such a m o d e l only w h e n the
spatial or t e m p o r a l variation of the r e s o u r c e is externally m a i n t a i n e d i.e.
only imthe p r e s e n c e of a favourable b o u n d a r y condition ir~posed by forces
and
S. Gadgil
M. Gadgil
35
outside the community.
Stewart, and Levin have shown that for their model
t h e n u m b e r s p e c i e s c a n n o t e x c e e d t h e n u m b e ~ : of r e s o u r c e s
when the resource
inputs are uniform in space and time.
N o w if i t c a n b e s h o w n t h a t f o r a r e s o u r c e
w h i c h is s u p p l i e d u n i f o r m l y
in space and time, the community
itself generates
the heterogeneity
in resource density which in turn supports the diversity,
then we have a mechanism to maintain the diversity which is not crucially dependent on the externally
imposed
boundary conditions.
Such a mechanism
will clearly add another dim e n s i o n t o t h e e x t e n t of d i v e r s i t y p o s s i b l e in a c o m m u n i t y .
The most famil i a r e x a m p l e of a s i t u a t i o n t n w h i c h s p a t i a l v a r i a t i o n is g e n e r a t e d
internally
within the biological community
i s t h e v e r t i c a l g r a d i e n t of l i g h t i n t e n s i t y i n
a plant community:
In this paper, we develop a mathematical
m o d e l of t h e
inter- and intra-species
interaction within a plant community
in the presence
of a u n i f o r m l i g h t i n t e n s i t y a t t h e t o p of t h e c a n o p y , a n d d e m o n s t r a t e
the
p o s s i b i l i t y of c o - e x i s t e n c e
of m o r e t h a n o n e s p e c i e s s u p p o r t e d b y a s i n g l e
resource
for this System.
PLANT
COMPETITION
W e u s e a n e x t e n s i o n of C o h e n ' s
( 1 9 7 1 ) m o d e l of g r o w t h a n d s e e d
Thus,
production in a plant with a limited growing period as our basic model.
defining,
W(t)
R
F(t)
T
The growth
=
=
=
=
rate
w e i g h t o f t h e v e g e t a t i v e t i s s u e s a t t i m e t_
net assimilation
rate per unit vegetative tissue
f r a c t i o n of n e t a s s i m i l a t i o n
d i v e r t e d to s e e d s a t t i m e
the total life-span
and the rate
--
dW
dt
=
R
dS
dt
=
Rrw
--
(1
F)
of s e e d p r o d u c t i o n
are
t
given by
(3)
W
(4)
F o r a d i s c r e t e m o d e l C o h e n ( 1 9 7 1 ) h a s s h o w n t h a t t h e O p t i m a l s e q u e n c e of
a l l o c a t i o n of t h e r e s o u r c e s
t o s e e d p r o d u c t i o n c o n s i s t s of n o a l l o c a t i o n u p t o
a transition
time
t and total allocation after t.
Thus,
the life history
comprises
a p h a s e of e x c l u s i v e v e g e t a t i v e g r o w t h f o l l o w e d b y a p h a s e of
exclusive reproductive
growth.
We will assume
r(t)
this for our model
=
o
t
~
=
i
t
~
also and take
~t
4.
T
(5)
36
Species
Packing
I n t e g r a t i o n of e q u a t i o n s (3) a n d (4), y ! e t d s
W(t)
=
W(o)
S(t)
=
R W(o)
e Rf
(T - [)
The total seed production
S(T)
is m a x i m i z e d w h e n
t
eR [
>_- [
d S(T)
-
0
dt
This implies
:
T
- I/R
iS(T) ] m a x i m u m
=
W(o) e R(T - I / R )
C o n s i d e r now two p l a n t s [ a n d j g r o w i n g s i d e b y s i d e and c o m p e t i n g
for light.
We a s s u m e that t h e i r h e i g h t s a r e p r o p o r t i o n a l to the w e i g h t s of
t h e i r v e g e t a t i v e t i s s u e s Wi a n d Wj.
Th'e l a r g e r p l a n t w i l l be u n a f f e c t e d by
the s m a l l e r one in t h i s c o m p e t i t t o n b u t the s m a l l e r one w i l l be s h a d e d b y the
l a r g e r p l a n t a n d its p h o t o s y n t h e s i s w i l l be d e p r e s s e d .
T h i s m a y be m o d e l l e d
b y a s s u m i n g the g r o w t h of the s m a l l e r p l a n t [ to be g i v e n b y
t
<
t-i
_
t i < t < T
where
m
~ O.
dWi
dt
=
R i Wi
(W i / W j )m
dS i
dt
:
R[ Wi
(W i / W j )m
(6)
T h e g r o w t h of the l a r g e r p l a n t w i l l be g i v e n as b e f o r e b y
dWj
t
&
[J
d'--7---
Rj Wj
dS.
t.<t<T
9
....
dt
~
=
R
W
J
J
Note t h a t the i n d e x m___d e t e r m i n e s the e x t e n t to w h i c h the p h o t o s y n t h e s i s is
d e p r e s s e d b y the s h a d i n g .
S i n c e this d e p r e s s i o n w i l l b e g r e a t e r in s u n l o v i n g p l a n t $ in c o m p a r i s o n w i t h the s h a d e t o l e r a n t p l a n t s we take m = l f o r
the f o r m e r and m ~.
l f o r the l a t t e r type of p l a n t s . T h i s d i s a d v a n t a g e
to the s u n - l o v i n g p l a n t s w i l l be c o u n t e r b a l a n c e d b y a h i g h e r r a t e of n e t p h o t o s y n t h e s i s w h e n e x p o s e d to f u l l s u n l i g h t . We t h e r e f o r e , a s s u m e that the
v a l u e of R f o r the s u n - l o v i n g p l a n t s is g r e a t e r t h a n t h a t f o r s h a d e - t o l e r a n t
plants.
S. G a d g i l
Competition
between
two s u n - l o v i n g
and
M. G a d g i l
37
plant s
Consider the c o m p e t i t i v e i n t e r a c t i o n s of two s u n - l o v i n g
plants
i. e
p l a n t s w{th m : l w h e n t h e y h a v e g e r m i n a t e d
simul[aneously
a n d h a v e th e
same biomass
W(o)
a t the i n i t i a l i n s t a n t .
L e t i_ c e a s e its v e g e t a t i v e
growth first, at t =[i and j_ cease its vegetative growth s o m e time later at
t_j where
[j
:
['t +
a
;
a
>
o
(7)
T h e two plants grow side by side upt0 the instant ti, neither shading
the other till t = [ i .
Beyond that, the seed production o--f l_ will be affected
by the shading by j_ whereas j_ will remain unaffected by the presence of [__
throughout 'its life history,
Hence using (6)
S i (T)
J W~ (tl) dt
W] (t)
:
W iZ (t i) d t
+ R
Wj (-tj)
ti
7
]
This is given by
-aR
Rti
S i (T)
:
W(o) e
Sj (T)
=
R W (o) e
[i +
R
e
+ a)
(ti
[(T - [i - a) R - I ]
(T
t i - a)~.
Define
Dij ([i, a)
=
S i (T)
-
Dij ( t-i ,
=
W(o) e R t i
sj (T)
Then
a)
['1-
e -aR
2R (T " ["i -
a)
sinh
(aR)]
(81
The function D i. represents the difference between the total seed-outputs
of the p l a n t s ---J[_ a n d j_ where the i_th p l a n t is a s s u m e d r s l o p its v e g e t a t i v e g r o w t h b e f o r e j_ . It is c l e a r f r o m (8) t h a t Di.. iS a m o n o t o n i c a l l y
i n c r e a s i n g f u n c t i o n of [" i , t h e t i m e at w h i c h t h e l i f f - h ! p l a n t s t a r t s s e e d
production.
Furthermore,
Dij ( [i
=
T - 1/R
Dij ( [ i
=
T
-
'
- a) = 1 - e
a ) = l - e
-aR
aR
<
>
0
0
(9)
T h i s i m p l i e s t h a t the s e e d p r o d u c t i o n of t h e i 'th p l a n t is l e s s t h a n t h a t of the
j t h p l a n t w h e n the j t h s t a r t s s e e d p r o d u c t i o n a t
T- t/R.
T h i s is o n l y to
b e e x p e c t e d s i n c e t h e s e e d p r o d u c t i o n of the j t h p l a n t w i l l be the m a x i m u m
38
Species Packing
p o s s i b l e in t h i s c a s e .
On the o t h e r h a n d , w h e n t h e i th p l a n t s t o p s its v e g e tative, g r o w t h at T - a, the _~th p l a n t s p e n d s its e n t i r e I i f e - s p a n T_ in v e g e t a t i v e g r o w t h a n d h e n c e the s e e d p r o d u c t i o n of the i th p l a n t is c l e a r l y g r e a t e r
t h a n tha~ of j.
In b e t w e e n t h e s e two e x t r e m e s is a p o i n t t i so t h a t when~'i=t~[
the s e e d p r o d u c t i o n of the _[th p l a n t e q u a l s that of j t h p l a n t - S i n c e we e x p e c t b o t h the c o m p e t i t o r s to h a v e e v o l u t i o n a r y f l e x i b i l i t y , e a c h of t h e m w i l l c h o o s e
a l i f e h i s t o r y w h i c h w i l l i m p l y a m i n i m u m d i s a d v a n t a g e in t e r m s of t h e i r s e e d
p r o d u c t i o n r e l a t i v e to the c o m p e t i t o r .
T h u s t_ w i l I t r y to m a x i m i s e
Si (T) - Sj (T) and _~ wilt t r y to m a x i m i z e
S_j(T) - S i (T).
The outcome
of t h i s c o m p e t i t i o n m a y t h e n be e x p e c t e d to be
S[ (T)
:
Sj (T
(I0)
Now (10) is t r i v i a l l y s a t i s f i e d wheri we take
ii
=
tj
(ll)
T h e o t h e r c o n t o u r a l o n g w h i c h (10) h o l d s is g t v e n f r o m
(8) as
-aR
T
-
ti
- a
=
[
-
e
(lZ)
R (e +aR- e - a R )
T
St>Sj
ij
~
/
T - - ~ T- 1
i'x~ st~ sl
st,sj
T--~R
\f--st=sJ
T 0.#4
i t --~
F i g u r e Z.
S e e d p r o d u c t i o n till the e n d of g r o w t h p e r i o d S(T}, f o r two
c o m p e t i t o r s i a n d ] a s a f u n c t i o n of the t i m e of t r a n s i t i o n
[ r o r n v e g e t a t i v e g r o w t h to s e e d p r o d u c t i o n .
S. G a d g i l
and
M. G a d g i l
39
In F i g u r e
Z, b o t h (1l) a n d ( [ Z ) ' a r e d r a w n , i n a ( t i , ~ j ) p l a n e .
Note
t h a t the i n t e r s e c t i o n of ( l l ) a n d (IZ) o c c u r s at "~i w h i c h c a n be r e a d i l y
o b t a i n e d f r o m (lZ) b y t a k i n g t h e l i m i t a s _a t e n d s to z e r o .
A
ti
Lt
a -~0
:
-
I
2R
I/ZR
=
iT
- a + 0 (a 2) ]
Hence
i
=
T
-
(13)
]
T h e point (13) is a saddle point and it can be seen that. the strategy of.minimizing the disadvantage in seed production will lead to the choice of the value
(13) for _ti and t_j by the following argument.
If the ith plant chooses_t i to
be _~i then S i (T)>. Sj (T) whereas for any other value of t_i, the _ith plant
can choose a--tj which will m a k e its seed output greater than that of [ i.e.
Si (T) <
Sj (T). T h e argument is synanaetrical[y true for j and the plant j
will again choose_t] to be equal to 2j'
Thus the evolutionary outcome will
be that both the plants will start seed production at the instant given by ([3)
and will achieve the s a m e seed output and hence a relative fitness equal to one.
It is seen that the competition between these plants has delayed the time of
transition f r o m vegetative to seed production f r o m T - I/R for the noncompeting plants to T - I/ZR.
This implies a decline in the total seed output
to
W(o) e
RT-
I/Z
2
C o m p e t i t i o n in a c o m m u n i t y of s u n - l o v i n g and S h a d e - t o l e r a n t p l a n t s
C o n s i d e r n e x t the c o m p e t i t i o n bet:ween a s u n - l o v i n g p l a n t , d e n o t e d
b y the s u b s c r i p t L and a s h a d e - t o l e r a n t Plant, d e n o t e d b y the s u b s c r i p t s.
Then
-mL
=
RL
>
1
ms
Rs
<
L:
(14)
W e further a s s u m e that the shade-tolerant plants, b e i n g less affected b y
shading are not involved in a race for light a s d[sc:ussed for the sun-loving
Plants, and stop:vegetative growth uniformly at T . I/Rs~ W e also a s s u m e
W L (0) =
Ws (o) and the time Of germination to he: the same.
Let!i ] and
S[--](T) be the $ and the seed output 0f: ith plant w h e n competing for light
with the jth plant, then w e have -:
fsL =
iss
tLs
T-I/R
:
iLL =
T
-- T .! l : / R s
L'
- I / ZR L
40
Species Packing
Note t h a t (14) implies
iLs
:sL
>
R s (T - I/Rs)
T (
Sss
SLL
= T-I/R s
)
R s W
(o) e
T I/"
RL(T
L~ R L W(o) e
=
dt
(R s T-1 )
=
W
(o) e
- I/2RL)
(R LT-I/Z)
dt
=
W(o)e
T - 1/ZR
T
L/
SLs
RL
:
(T-I/R
(R L T - I )
W(o) e
L )
R L W(o) e
dt
=
T- 1/R
T-1/RJ-I R sWs~+11
(is) dt
SsL
=
+
IT
m
W L (t)
T'I/Rs
~n+ l
RsWs
m
WL (iL
-1/R
L
F r o m the above expressions we get by using assumption (14)
(R L
SLs
=
Rs) T
e
>
1
Sss
T
S
SS
T-I/R
--
SsL
/
q
Rs
dt
s)
>
T.
W (T.I/R)
Rs [ s
s
-1/R
WL (t)
S
since
Thus
Ws / WL
~
1
S
Ls
>
S
ss
>
S
sL
m
]
dt
_
(is) dt
S. Gadgil
and M. Gadgil
41
T h e r e is no c l e a r cut inequality b e t w e e n S L L and Ss.L
=
S
sL
2be
Rs/R
L
+ m-(m+l) RL T+
S LL
Defining
I/Z
- l/m
(l
(1-e - m ( l / b
-1)
[ l - b (1 - e - m R L T - m / b ) ]
0.9
0.7
t
m
0.5
SSL> 1
/
0.~
0.1
SL L
I
,t"
0.91
F i g u r e 3,
I
I
!
0.93
0'95
Rs/R L
0'97
'
I
0'99
B e h a v i o u r of the r a d o
S s L / S L L a s a function of R s / R L
and m.
R L i s a s s u m e d to be 2 , 0, a n d T to be 1. 0.
In figure 3, S s L / S L L
is plotted as a function of R s / R L and m___. As
expected, this ratio increases as the ratio of the net assimilation rates R s / R L
increases and decreases as rn, which m e a s u r e s the detrimental effect of the
larger plant on the smaller, increases.
It is seen that there is a range of
p a r a m e t e r values for which S s L > S L L .
F o r this particular range of p a r a meters we have:
SL s >
Ss s
>
SsL
>
SLL
(15)
A n inspection of this inequality i m m e d i a t e l Y reveals that under these
Conditions sun-loving and Shade-tolerant plants would co-exist because of the
frequency dependence of their seed production rates, which m a y be equated to
population growth rates. If a population largely comprises of _L plants, then
the relevant seed production rates are S s L and S L L .
Since S s L > SLL, the
proportion of s's will increase in such a populatib--n-7-. O n t h e other hand, if
a population w e r e to be largely m a d e up of s plants, the relevant seed
production rates are Sss and S L s 9 Since S L s -> Sss , L's would increase,
in proportion in such a--'p-opulati-o-~. In either case then the minority type will
tend to increase. W e would then expect a co-existence of the two competing
types in such a system.
Species
42
Packing
POPULATION
DYNAMICS
This co-existence
m a y b e s h o w n f o r m a l l y in a m o d e l of a m i x e d
p o p u l a t i o n of L a n d
s plants.
W e a s s u m e _L and s to d i f f e r in S e e d p r o d u c t i o n , but be equivalent in mortality rates. Consider a m i x e d population
of such plants to be held at s o m e constant density through density dependent
mortality which does not distinguish between s and L_ plants.
We may
study such a situation by using a m o d e l without expliclt[y introducing the
density induced mortality.
W e then have
dNs
dt
rs
and r L
_
dNL
-dt
-
rs Ms ,
would be functions of
=
Sss, SsL, SLL,
rLN L
SLs
and Ns, N L.
N s
Ps
-
(16)
NL
PL
N s + NL'
-
Ns + NL
W e do not expect the total population to increase exponentially as implied by (16).
But since we a s s u m e that the total population is limited by density dependent
mortality which does not distinguish between the two categories, and w e are
mainly interested in the possibility of co-existence, it is sufficient to consider
(16) and investigate under what conditions the frequencies Ps and PL
remain
between zero and one. Since w e a s s u m e that the population-'-aensity-~s held at
a constant level, the incidence of neighbour types would be proportional to
their frequency.
In that case:
rs
=
S ss Ps
+ SsL PL
rL
=
S L s Ps + S L L
PL
O u r complete m o d e l then is :
dN
dt
s
-Ns
dNL
dt
:
[Sss
NL
[SLs
Ps + S s L PL ]
Ps + SLLPL
(17)
]
Hence
dPs
dt
-
1
dNs
Ns
N s +N L
dt
(N s + N L ) 2
d(N s + N L )
dt
S. G a d g i l
and
M. G a d g i l
43
i. e.
dP s
d--t
Ps(l-Ps). [ S s L - S L L + ( S s s + S L L - SsL -SLs)P s]
=
(18)
dPL
dp s
dt
dt
(19)
In t h i s f o r m a l i s m ,
e q u i l i b r i u m i m p l i e s a s t e a d y f r e q u e n c y a n d b o t h t y p e s of
plants will co-exist provided that there exists a steady non-zero frequency
l e s s t h a n o n e f o r e a c h of t h e m .
N o t e t h a t f r o m (18), w e g e t
]x
dP s
=
dt
0
at
Ps
= o,
1, Ps
where
SLI ~ S s L
/%
Ps
=
(20)
SLESsL + Sss-SLs
Both species will co-exist if
0 <
A
ps<
(zz)
1
Co-existence is possible under two sets of conditions
or
Sss >
SLs
and
SLL > SsL
(22)
Sss <
SLs
and
SLL < SsL
(23)
Stability
We h a v e s h o w n a b o v e t h a t t h e r e a r e t h r e e p o i n t s of e q u i l i b r i u m ;
two
corresponding
to p o p u l a t i o n s c o n s i s t i n g of o n e t y p e a l o n e a n d t h e l a s t o n e /Ps
corresponding
to a m i x e d p o p u l a t i o n .
W e i n v e s t i g a t e n o w t h e s t a b i l i t y of
a l l t h e s e p o i n t s in o r d e r to d e t e r m i n e t h e c o n d i t i o n s u n d e r w h i c h a m i x e d p o p u l a t i o n w i l l r e s u l t f r o m an a r b i t r a r y
initial frequency.
Definu
M
=
Sss
Ps
= Ps
+
SLL
- SsL
- SLs
(24)
+
~"
44
From
Species
(18) and (20)
dPs
.
dt
Equation
MPs(1
.
.
.
Ps ) (Ps
ps )
(24) g i v e s
d(-
:
I_
(-
d~_~
dt
d'-t-
Using
Packing
MPs
(1
-
p s ) (-
+ 0 ( (-2 )
(21), we g e t
>
<
0
0
if M
if IV[
>
,~
0
0
T h u s the p e r t u r b a t i o n _~ will g r o w if M > 0 a n d d e c a y if N i < O
implying
that the equilibrium will be stable for the latter case.
Note that this implies
that the equilibirurn is stable w h e n (23) holds and unstable w h e n (22) holds.
Similarly it can be s h o w n that the points Ps = 0, 1 are stable w h e n .MM >
0
and unstable when
To sum
(l)
~s = 1
I-lence
of one
M
<
O.
up:
A
Sss > SLs
and S L L > S L s
i m p l i e s t h a t the p o i n t s
Ps = 0 a n d
a r e s t a b l e , w h i l e the i n t e r m e d i a t e p o i n t , 0 < ~ s < 1 is uns-t-Kble.
in s u c h a s y s t e m a m i x e d p o p u l a t i o n w o u l d e v o l v e t o w a r d s the e x t i n c t i o n
o r t h e o t h e r s p e c i e s d e p e n d i n g on the iiaitia[ c o n d i t i o n s (fig 4a).
(~)
X>O
1
1
X<O
t
Ru
\
\
1
1
PS - ~
Figure 4.
ps--~
P h a s e d i a g r a m to i l l u s t r a t e the p o s s i b l e e q u i l i b r i a f o r the
~ y s t e m of e q u a t i o n s (18) and (19}. T h e t h r e e p o s s i b l e
points of e q u t t i b r l a a t wkich d p s / d t z d P L / d t = 0 a r e
a r e Ps = l, P--L = i o r s o m e - - ~ - - 8= bL w h e r e 0 C ~ < i .
(~} when M > 0, the f i r s t two a r e s t a b l e and M is uns t a b l e and (b) when M < 0 the f i r s t two a r e u n s t a b l e
and ~
is s t a b l e .
S. G a d g t l
and
M. G a d g i l
45
(2)
S s s ~,. S L s
and
SLL < SsL
i m p l i e s t h a t the p o i n t s k s : 0 o r
/Ps = 1 a r e p o i n t s of u n s t a b l e e q u i l i b r i a , w h i l e the i n t e r m e d i a t e ~ b r i u m
~ 0 ~
s < 1 is s t a b l e . S u c h a s y s t e m w o u l d p e r m i t the p e r m a n e n t c o e x i s t e n c - e of the two c o m p e t i n g s p e c i e s (fig 4b).
It m a y be m e n t i o n e d t h a t f o r t h i s s i t u a t i o n the p o p u l a t i o n of s p e c i e s L__
is r e l a t e d to t h a t of s p e c i e s s t h r o u g h a r e l a t i o n s i m i l a r to (2) v i z
---A
e+Bt
=
NL
A
kNL/t=O
S ss - S s L
SLs
However,
;
B
- SLL
= S Ls SsL
SLs
- Sss S L L
SLL
h e r e n o n z e r o v a l u e s of B a r e p e r m i t t e d
b e c a u s e the c o n d i t i o n of
The
i m p o s i t i o n of an a d d i t i o n a l d e n s i t y d e p e n d e n t m o r t a l i t y w h i c h d o e s not d i s c e r n
b e t w e e n the two c o m p e t i t o r s w i l l k e e p the p o p u l a t i o n s of t h e s e two s p e c i e s
b o u n d e d but w i l l not a l t e r the f r e q u e n c y of the f i n a l s t a t e and h e n c e w i l l a l s o
l e a d to a m i x e d p o p u l a t i o n f o r t h i s c a s e .
b o u n d e d n e s s of i n d i v i d u a l p o p u l a t i o n s at l a r g e t i m e is not i m p o s e d .
O u r m o d e l for the competitive interactions between sun-loving and
shade-tolerant plants generated the inequality (15) for certain values of
R fs and m ;
SLs ) Sss > SsL > SLL
This is obviously compatible with the set of inequalities (23), and such a
s y s t e m would pernnit the p e r m a n e n t co-existence of two species limited by a
single resource, which is being continually supplied at a constant level. This
co-existence is rendered possible by the spatial heterogeneity in the density
of that resource generated by the action of the plant c o m m u n i t y itself.
S u c h internally induced spatial heterogeneties are particularly likely
to occur in the case of sessile o r g a n i s m s such as plants, but need not be
restricted to them. F o r example, territorial animals m a y crop their food
plants m o r e intensely near the centre of the territory, and less intensely
towards periphery. Other animals using these as shelter m a y then diversify,
a species needing denser cover chosing the periphery and a species requiring
[hinner cover occurring towards the centre of the territory.
SPECIES
PACKING
T h e p r o b l e m of packing of species at different levels of resource
densities is analogous to.that of packing of species along a resource continuum
46
Species
Packing
a n a l y s e d by M a y and M a c A r t h u r (1972).
T h e s e a u t h o r s c o n s i d e r the p r o b l e m of p a c k i n g of s p e c i e s s p e c i a l i s e d , f o r e x a c n p l e , to t a k e p r e y of d i f f e r e n t
sizes.
T h e a h i i i t y of the s p e c f e s to u t i l i z e p r e y of v a r i o u s s i z e s m a y be r e p r e s e n t e d b y a u t i l i z a t i o n f u n c t i o n a l o n g t h e p r e y - S i z e , a x i s (fig 5a). T h e y
a s s u m e t h a t t h e s e u t i l i z a t i o n f u n c t i o n s a r e G a u s s i a n , and h a v e i d e n t i c a l a r e a s
u n d e r the c u r v e .
T h e y s h o w ~hat U n d e r t h e s e c o n d i t i o n s t h e s p e c i e s m a y be
p a c k e d in a v a r i a b l e e n v i r o n m e n t w i t h the p e a k s of u t i l i z a t i o n f u n c t i o n s
s e p a r a t e d b y a d i s t a n c e r o u g h l y e q u a l l i n g one s t a n d a r d d e v i a t i o n .
For our
AVAILABILITY.%
......
>~
<
(b}
PREY SlZE
AVAILABILITY
>-
L
_J
_
~.4_ION
~
N
RESOURCE DENSITY
Figure
5.
Utils
function,a, or the efficiency with which a resourc,: m a y be u s e d (a) along a resout'cc c o n t i n u u m :
of prc~y size and (b) at d i f f ~ r ~ n t r ~ s o u ~ c e (lensit[es.~[ t
T h e ~Ipp(:rn~ost c u r v e represents l]I(~ tot;ll c o n c e n t r a tion of r e s o u r c e s at (a} dlffer~nt p r e y sizes or (b) a~
various resc)urc~ d~inslt[~8.
m o d e l , w e m a y consider analogous utilization functions along a r e s o u r c e density axis. T h e s e could not he G a u s s i a n but w o u l d increase m o n o t o n i c a l l y to
an a s y m p t o t e as in the case of the photosynthetic r e s p o n s e to light intensity
(fig i). In this case, the r e s o u r c e density at w h i c h the species is m o r e efficient than all of its c o m p e t i n g species w o u l d c o r r e s p o n d to the p e a k of the
utilization functions of the r e s o u r c e c o n t i n u u m case (fig 5a and b). T h e
p r 0 b [ e m of h o w closely the species could be p a c k e d in this case d e s e r v e s to
be investigated.
S
U M
M
A
R
Y
T h e conclusion that the n u m b e r of species co-existing within a biological c o m m u n i t y cannot e x c e e d the n u m b e r of limiting factors is not Valid
if w e a s s u m e that (i) the relative efficiency of two competingispecies in uti~[izing a r e s o u r c e is not independent of the r e s o u r c e densityL but one spelcies
S. G a d g i l
and
M. G a d g i l
47
m a y be m o r e e f f i c i e n t at a l o w e r ' d e n s i t y a n d l e s s e f f i c i e n t at a h i g h e r d e n s i t y
and (ii) t h e r e is a s p a t i a l o r t e m p o r a l h e t e r o g e n e i t y in t h e d e n s i t y of the
resource.
T h i s s p a t i a l o r t e m p o = a l h e t e r o g e n e i t y d o e s n o t h a v e to be f u r n i s h e d by f a c t o r s e x t e r n a l to th e b i o l o g i c a l c o m m u n i t y , b u t m a y be g e n e r a t e d
w i t h i n the b i o l o g i c a l c o m m u n i t y i t s e l f as in the c a s e of a v e r t i c a l g r a d i e n t of
light in a plant community.
This possibility of a stable co-existence of
m o r e than one species in a c o m m u n i t y limited by a single resource, even
when. the resource is being supplied uniformly in space and time, is formally
demonstrated.
R E F E R E N C E S
COHEN,
D. (1971). M a x i m i z i n g final yield w h e n growth is limited
by time or by limiting resources.
Journal of Theoretical
Biology, 33: 299-307.
GAUSE,
G F.
(1934).
The struggle for existence.
W i l k i n s , B a l t i m o r e , 163p.
Williams
&
L E V I N , S A. (1970). C o m m u n i t y e q u i l i b r i a and s t a b i l i t y , and an
e x t e n s i o n of the 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 .
A m e r Nat ,
104: 413-423.
4
LEVINS,
R. (1968). Evolution in changing environments.
Princeton University Press, Princeton, N J. 130p.
MACARTHUR,
R H and L E V I N S , R. (1964). Competition, habitat
selection, and character displacement in a patchy environment.
P r o c Nat A c a d Sci, U S, 51: 1207-1210.
MAY,
R R and M A C A R T H U R ,
R H. (1972). N i c h e o v e r l a p as a
function of environmental variability. P r o c N a t A c a d Sci,
U S, 69: II09-II13.
R O U G H G A R D E N , J. ( 1 9 7 l ) .
E c o l o g y , 52: 4 5 3 - 4 6 8 .
Density dependent natural Selection.
STEWART,
F N and L E V I N , B R. (1973). Partitioning of resources and the o u t c o m e of interspecific competition" a
m o d e l and s o m e general considerations. A m e r
Nat ,
107: 1 7 1 - 1 9 8 .
VOLTERRA,
V. (1931). L e c o n s sur la theorie m a t h e m a t i q u e de la
lutte pour la vie. Gauthier - Vtllars. Paris, 214p.