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Artif Life Robotics (2002) 6:78-81
9 ISAROB 2002
Takashi Shimada 9 Satoshi Yukawa 9 Nabuyasu Ito
Self-organization in an ecosystem
Received and accepted: September 27, 2002
A b s t r a c t A n ecosystem, especially a food web, is essentially characterized as a m a n y - b o d y system in which the
m e m b e r s interact with each o t h e r u n d e r the limitations of
the energy and resources. 1 W e introduce a coevolutional
p o p u l a t i o n dynamics m o d e l for food webs which contains
energy-conserving interactions, energy dissipation, and
rules for changing the degrees of f r e e d o m (extinction and
mutation). In this model, the diversity of the system increases spontaneously. T h e statistical p r o p e r t i e s of the system, such as the distribution of the life time of the species,
are also discussed.
K e y words Ecosystem 9P o p u l a t i o n dynamics 9F o o d webs 9
Evolution - E x t i n c t i o n . M u t a t i o n 9 Self-organization
1 Introduction
Since the L o t k a - V o l t e r r a equations were proposed, population dynamics has occupied an i m p o r t a n t position in the
study of ecosystems. A l t h o u g h t h e r e are many variations of
L o t k a - V o l t e r r a - t y p e models, these can be generalized into
the form 2'3
dx i
dt
- cixi + ~_,aijxixj
j
(1)
where x~, a~j, q represent the p o p u l a t i o n density of the i-th
species, the interaction coefficient b e t w e e n the i-th and
j-th species, and the intrinsic growth rate, respectively.
T h e analysis of this p o p u l a t i o n dynamics and its linearized
T. Shimada ( ~ ) 9S. Yukawa 9N. Ito
Department of Applied Physics, University of Tokyo, 7-3-1 Hongo,
Bunkyo-ku, Tokyo 113-8656, Japan
e-mail: [email protected]
This work was presented, in part, at the Sixth International Symposium
on Artificial Life and Robotics, Tokyo, Japan, January 15-17, 2001
m o d e l by R. May and others in the 1970s yielded a r e m a r k able harvest. .4 This is the fact that complex systems with
strong interactions are m e r e l y stable, which is the o p p o s i t e
of the belief at that time. H o w e v e r , it has also b e c o m e
k n o w n that the argument of M a y et al. is too strict to prohibit the existence of middle-sized systems with m o d e r a t e
interactions. F u r t h e r m o r e , the existence of exceptionally
large and strongly interacting systems is also not forbidden.
It is also found that such exceptionally large or strongly
interacting systems may b e constructed by the trial-andelimination scheme. 7-9
W e now focus on the food web. F o o d webs are the
trophic links in ecosystems. Therefore, when using p o p u l a tion dynamics models for ecosystems, one should t r e a t n o t
the p o p u l a t i o n but the resource or energy as a the dynamic
variables. A s m e n t i o n e d below, this brings a n o t h e r difficulty when trying to m a k e a m o d e l of a food web. A l t h o u g h
there are m a n y type of m o d e l s for food webs, 1~ t h e r e has
not b e e n a m o d e l in which the system is self-organized f r o m
the p o p u l a t i o n dynamics and trial-and-elimination.
2 Model
W e now start to construct a m o d e l of a food web from the
generalized L o t k a - V o l t e r r a equations (Eq. 1). To give the
system a chance to be stable, following the M a y et al. arguments, we set most of the interactions at 0. The average
n u m b e r of interactions p e r species is r e p r e s e n t e d as m. Since
interactions here are limited to p r e y - p r e d a t o r relations, we
also define the interaction coefficients as antisymmetric
(aq = -aji). N o t e that the p o p u l a t i o n xi actually refers to the
energy or resource in the biomass of the species. It is obvious
that the sum of the p o p u l a t i o n is conserved by p r e y - p r e d a t o r
interactions. W e then decide that there are single species of
plants which have a positive growth rate, and animals which
have a negative growth rate. T h e s e conditions c o r r e s p o n d to
the p r o d u c t i o n of energy in the plants and energy dissipation
by m e t a b o l i s m in animals. T h e trial-elimination scheme is
then realized as adding the rules of r a n d o m m u t a t i o n and
79
extinction. New species which have rn totally r a n d o m interactions come randomly. Species whose population becomes
very small become extinct and are eliminated from the
system.
Despite the expectation that the system grows to a rich
structure spontaneously, it turns out that this naive modification does not work. Such systems cannot yield a diverse
structure, and tend to collapse to a poor structure by mutation and extinction (Fig. 1). The m a i n reason for the failure
is a difficulty in deciding suitable values for the interaction
coefficients (aij). Since the interaction coefficients have dimensions of time per energy density, they must be distributed over a wide range. For example, the interaction
between cows and grass would be represented as far larger
than the one between eagles and rats. O n e solution is to
decide the proper range for the interaction coefficients according to the trophic levels. However, when we adopt this
scheme, we must arrange for the emergence of trophic
levels and a large variety of life-styles.
Here, we introduce another way, i.e., rescaling the interaction coefficient. In the L o t k a - V o l t e r r a equations, the rate
of preying per unit predator is a linear function of the
p o p u l a t i o n of the prey. W e modify this preying rate to a
function of the ratio of the populations of the prey (j) and
the predator (i) as
where the size of )~ corresponds to the saturation of the
preying rate. This preying rate (Eq. 2) yields the interaction
term
trophie level 3
;';". energyflow
t
r g y flow
~
Fig. 1. An example of the trivial star-like structure which is yielded by
simple Lotka Volterra-type interactions. Systems tend to collapse to
such structures or totally die out
Fig. 3. Time-development of the
number of species in the system.
The mutation rate is 0.1 per time
interval. The four lines represent
the total number of species, the
species in the second trophic level,
the species in the first trophic level,
and the species in the third trophic
level, in order from the top to the
bottom, respectively
trophic level 1
Fig. 2. An example of a self-organized food web with a rich structure.
Species are represented as circles. Each a r r o w represents a preying
interaction. There are 21 species and 3 tropic levels
180
160
140
9~ 120
100
80
Z
6O
40
20
0
0
50000
100000 150000 200000 25000.0 300000 350000 400000 450000 500000
Time
80
Fig. 4. Long-time behavior of the
system. The upper line represents a
system with weak interactions, and
the lower line represents a system
with strong interactions
1000
i
900
i
. . . . . .
i t,,,A
i-1
...................................................................................................................................................................................
800
700
i,,,,~!,
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"
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ff~',,J/!~,/%,~'
............
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........................ i ......................... i ......................... ~......................... i ......................... i ..................... :;~;f ................. ~'~'t .......................
........................ i ......................... i ......................... i ......................... i ...........
600
~
,,.';~'!s
......................... - .......................
500
400
........................
.........................
.............
.......................
i....................'.,t,~i,:~ .......................................................................
..................................................................................................
300
.........................
:d, i. + " ' l . ~.. ~,,,,:.~',,~ .d," + ,if ;i - !'.v~,'f
"
''r
~ '~i~i,
.-:
.........t ........... 'J~,r ........... :z,~............. ~+,~,~4~,,,,~,,r .............. kb,,,t,,~,~,.~ .................................................. ,'~,,,,,~ ........ ~ ................ - I
200
/
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,~'~"-"
100 L,,.~ :~;,~,,+,,~,~L/fL,,,:
0
[,;(i~ " +i
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0
500000
'"
~' "
'
i ..................... i
"',~'
+
. . . . . . . . . . .
le+06
"
............. [. . . . .
i
1.5e+06
"
~
d'
~,,b'4. t
++
/
3.5e+06
4e+06
+
........... i......
2e+06
,7','1
............................. !..............~Y,,.,,,-I
2.5e+06
3e+06
Time
Fig. 5. Distribution of the lifetime of species. The horizontal
axis is the number of mutations
which a species has endured before it becomes extinct. In the
large-number region, power-law
behavior is seen
le+07
. . . . . . . . .
I'
1
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le+06
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10000
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100
[
10
1 110
100
1000
10000
100000
N u m b e r of Mutations Before Extinction
1L
aijx i
?~
Xy
(3)
A l t h o u g h k c a n also b e v a r i e d like a o, i n this a r t i c l e w e s e t a
u n i f o r m )~ f o r simplicity. I n t h i s case, w e s h o u l d l i m i t t h e
u n i f o r m v a l u e ~. i n e (0,1/2) t o k e e p t h e p r e y i n g r a t e c o n v e x
under certain transformation of variables. The metabolism
r a t e ci is also u n i f o r m , a n d is c h o s e n t o 1 e x c e p t f o r t h e
p l a n t s . T h e n t h e t i m e - e v o l u t i o n e q u a t i o n s in o u r m o d e l a r e
easily w r i t t e n as
Plants
dx',
dt
=
Ox,(1
-
x,)
+
~.aUx~lx} -x
j
Animals
-d~ti
= -- X,9 + 2
aq < 0
aijx~i xl]-)' + Z
aijx 1-~
i Xy
aii > 0
(a+ = - a , / ~ (-1,1t, ~ ~ (0,1/21 a > 0)
(4)
81
where G denotes the growth rate of the plants. The condition aq ~ ( - 1 , 1 ) comes from the requirement that not all the
animals can survive when the total population of the prey is
smaller than the population of the animals. This model has
additional rules to change the degrees of freedom of the
system, as described below.
2.1 Additional rules
-
-
If the energy of species i (xi) becomes 0 or less,
that species is extinct and the i-th degree of freedom is
eliminated. In addition to this rule, an instant extinction
rule is applied to a species which is completely isolated
from others.
Mutation
(invasion).
A new species comes into the system
randomly at any time. The initial energy of the newcomer
is chosen to be very small (10 8). The number of interactions is decided randomly in the range of (1, 2m), where
m is the average number of interactions per species. The
amplitudes of the interactions ( a q ) are also decided randomly from a certain distribution P, which is defined in
(0,1).
when the average number of interactions (m) and the
strength of interactions (decided by P) are chosen to be
smaller than the value proposed by May et al. (Fig. 4). In
cases where interactions are strong, the system diversity
goes on fluctuating, and the power spectrum of the number
of species shows the 1/f distribution. In the latter case, the
distribution of the life-time of the species shows a powerlaw tail (Fig. 5).
Extinction.
It is worth stressing that we do not require a threshold for
extinction. In Lotka-Volterra-type models, populations of
unmatched species decrease exponentially, and therefore a
threshold is necessary to cause the extinction. 3'9 In contrast,
ai~x~ ~x; type interactions cause the algebraic decay of the
unmatched species, so the orbits of the unmatched species
decrease to 0 in finite time. The irreversibility of the extinction is represented naturally as the breaking of the uniqueness of the solution at xi - 0.
3 Results
We now investigate the model described above by a numerical simulation. The parameters in this simulation are
chosen as G = 50 and m = 5. A t the starting time there are
only plants in the system. This time the system diversity
grows gradually by mutations (invasions), although most of
the newcomers fail to survive (Fig. 2). To get the structural
information, we classify the species by trophic level.
Trophic level is defined as the minimum distance to the
plants. In other words, the trophic level of the species who
eat the plants is one, and the trophic level of the species who
do not eat plants and prey on the species whose trophic
level is one is two, and so on. In Fig. 3, we can see the
development of the food web at trophic levels.
We now consider the long-time behavior of the system.
The number of the species grows to an infinite number
4 Summary
We have introduced a simple model of a food web. In this
simple model, the system self-organizes to a rich structure
by modifying the interactions as we have demonstrated.
Systems with weak interactions grow to an infinite size.
Systems with relatively strong interactions continue growing and shrinking.
In the present model, we have two essential features:
p r e d a t o r - p r e y interactions, and the directionality of the
energy flow, which comes from the introduction of an energy source (plants) and energy dissipation. We have seen
that the structure is self-organized by these two factors, as
same as in real food webs.
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