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A model of hierarchical ecosystems with migration
Bulletin of Mathematical Biology, Vol. 42, pp. 119-130 Pergamon Press Ltd. 1980. Printed in Great Britain © Society for Mathematical Biology
0007-4985/80/0101-0119 $02.00/0
A M O D E L OF H I E R A R C H I C A L ECOSYSTEMS WITH MIGRATION • HIRONORI HIRATA Department of Electronics, Chiba University, 1-33, Yayoi-cho, Chiba-shi, 260 Japan A model of ecosystems with migration is proposed from the viewpoint of flow. This model explains the following two points: (1) How the density-dependent terms in population dynamics arise as a consequence of migration. (2) How the ecosystem exhibits a hierarchy in energy per unit biomass.
1. Introduction. Ulanowicz pointed out the following important aspects in ecosystems (Ulanowicz, 1972): (1) How the density-dependent terms in population dynamics appear by deduction. (2) H o w the stable system exhibits a hierarchy in energy per unit biomass. Since the population of any species cannot continue to increase, the density-dependent terms, i.e. the saturation effect, is important in population dynamics. The more the population increase, the more organisms die, because according to the increase of the population in the area the competition intensifies, the environmental conditions become bad, etc. To escape these things, animals migrate to other a r e a s . We may expect that the ecosystem exhibits a hierarchy in energy per unit biomass. This tendency accords with the fact that animal carbohydrates and proteins have generally higher calorific values than those for plants. Although Ulanowicz tried to show the above two points, unfortunately, it was pointed out that his paper contained a physical mistake and his model could not explain them. A new model of mass and energy flow in ecosystems has been proposed (Hirata and Fukao, 1977). This new model without migration, in which the density effect was assumed, could explain 119
120
H I R O N O R I HIRATA
only point (2) above: how the stable system exhibits a hierarchy in energy per unit biomass. But further extension of this model to include migration has made it possible to explain both of the above points. The density effect is not assumed in this paper, however, the density-dependent term in population dynamics is shown to arise by deduction. In the following section, we first consider separately the fundamental process of migration and that of hierarchical foodweb. Thereafter, we combine them to create a model of the hierarchical ecosystems with migration.
2. A Model of the Fundamental Process of Migration.
We make the
following assumptions on migration: (1) We consider the migration of only animals and neglect that of plants. (2) Animals migrate only according to the ease of predation, i.e. they migrate to the area where they can easily prey on host species. We evaluate the degree of ease of predation in the area by the energy expended in predation, called "activity energy." Consider the migration of one species in one dimension. There are areas which are characterized by the different activity energy per unit biomass, as in Figure 1. Without any loss of generality, in this section we consider only
~~ m n - I ' n m nn'n+l +nl , F-.
/Zn+l
Figure 1. A model of the fundamental process of migration. - - - - - Activity energy
Mass;
migration and neglect the flows along the foodweb for simplicity. We assume that the flow streaming from the nth area to the n + l t h one is positive. The balance equations of mass and activity energy at the nth area are
dM. dt =m"-m'"-m"'"+l"
(1)
+c~,(M,), n=l,...,7. dHn=h,_l,,-h,,,+l
(2)
A M O D E L O F HIERARCHICAL ECOSYSTEMS W I T H M I G R A T I O N
121
M, is the mass (the population) in the nth area. H, is the activity energy required by all organisms in the nth area to capture their prey. m,_ a,, and h,_l, . are the mass flow from the n - l t h area to the nth one and the activity energy flow accompanying respectively, qS,(M,) is a motive force of migration. We assume the following relation between M , and H,.
H,=#,M,,
(3)
where #, is the activity energy per unit biomass. We add the following assumptions on migration: (3) Each area has the carrying c a p a c i t y / ~ r . (4)
#1 < # 2 < ' " < # ~ < # ~ + ~ = g d ,
(4)
where #a shows the limitation of the energy loss which can be expended on predation. The organisms die when # > # d , that is, when they expended more energy in predation and metabolism than they obtain from their prey. Condition (4) means that areas have been arranged in order of the ease of predation. Animals migrate gradually from area to area along this order. For example, animals in the nth area can migrate only to the n - l t h area and the n + l t h area. They cannot directly move to the other areas. Migration can be bidirectional and its direction is determined by the sign of the motive force ~b,(M,), i.e. it is determined by assumption (5). (5) When the capacity of the nth area is not filled (i.e. M , < 2 ~ , ) , the organisms migrate from the n + lth area to the nth one, i.e. they can move to a more favorable area. When the capacity of the nth area is filled (i.e., M , > M , ) , the weak organisms must migrate to the ,unfavorable area, i.e. the n + lth area. Therefore, the direction of migration is determined by the sign of.the difference ( M , - ) ~ , ) . Here the motive force of migration in equation (2) is expressed as: q~,(M,) = b,(#,+ 1 - # , ) ( M , - ) I 4 , ) ,
(51
where b, is a positive constant, because the difference ( M , - / ~ , ) (i.e. the difference between the population M , and the carrying c a p a c i t y / ~ , ) causes the migration, and the increase (or the decrease) in the activity energy per unit mass of ( M , - ) ~ r ) is proportional to the difference (g, + 1 - #,). The following relation is assumed: h,_ 1,, = #.m,_ 1 ,-
(6)
122
HIRONORI HIRATA
We can see below that this relation is reasonable. The following is derived from (1~(3), and (6): /~.+ lm.,.+ 1 - # . m . , . + l = q~.(M.).
(7)
Replacing the second m.,.+l by m.,. to understand the energy balance equation (7) easily, the microfeature of balance in the nth area can be derived as illustrated in Figure 2. If the motive force of migration 4 . is
Area n
~
Areon+l
!
m
m
Figure 2.
Micro-feature of balance in the nth area, where m.,.= m.,.+ 1
positive, it changes the activity energy per unit mass of m.,.+l from #. to #.+1 and m.,.+l is positive (i.e. the mass flow steams from the nth area to the n + lth one). If ~b. is negative, these are vice versa. The following equation (8) is derived from ( 5 ) a n d (7), i.e. equations (1)(3), (5) and (6). m.,.+ t = b . ( M . -
M.).
(8)
Equation (8) shows that the a m o u n t of migration flow is proportional to the difference ( M . - M . ) . This agrees well with assumption (5). This model of fundamental process of migration satisfies all the above assumptions. 3. A Model of the Fundamental Process of Foodweb. With no loss of generality, in this section we consider only foodweb transfers and neglect migration for simplicity. This model is the same as Model A in the previous paper (Hirata and Fukao, 1977) except for the assumption of the density effect. We assumed the density effect in Model A without migration, i.e. the model contained the t e r m -qi(Mz) 2. In this paper, it is possible to explain h o w the density-dependent terms in population dynamics arise as a consequence of migration.
A MODEL OF HIERARCHICAL ECOSYSTEMS WITH MIGRATION
123
The f o o d w e b depicted in Figure 3 is considered. The trophic levels 0, 1,..., p constitute the p r i m a r y producers, the herbivores, the p r i m a r y a n d secondary carnivores, etc. M a s s circulates t h r o u g h the producers, the consumers a n d the decomposers. Solar energy is fixed at the first trophic level (i.e. producers), a n d some a m o u n t of energy is dissipated at each trophic level.
i
riMi ~
(
i_~i)mi-l,i ,
I sii i
\/
Deoth Lossof the moss due fo the respirotion k Excretion (COz) / Leavings
Fromlevel 0 (02)
~ /{ i"
l- - -9
I
0 All nutrient
elements
C,H,O,N,P,S,etc.
]
Figure 3. A model of the fundamental process of foodweb. - ---Energy
Mass;
This mass a n d energy equations at the ith (i :p O) trophic level are dMi dt
= ~ i m i - 1, i _ m i, i + 1 - - p i M i 4- (e,i - -
(9)
S i)Mi,
d E i = ~ i e i - 1, i _ e i, i+ 1 _ j, i p i M i + (a i _ r i ) M i , dt i=l,...,p
(10)
124
HIRONORI
HIRATA
where
ai=
o>O (i=1) (i@1)'
Pi, 8i, si, ri > 0, 0 < ~ i < 1, ~ 1 = 1 , m P'p+I=e p'p+I=O.
(11)
Mi is the biomass (the population) of the ith species, E~ is the energy of the ith species, and m ~'j and e ~'j are the mass and energy flows from the ith species to the jth species respectively. ~ is the utility efficiency of mass. The meaning of each term of equations (9) and (10) is as follows and is also illustrated in Figure 3. In (9), the first term is the digested part of mass flow mi-l"i; the second term, the loss of the mass flow due to predation of the i + lth species; the third term, the effect of death, etc., and the fourth term is the mass flow due to respiration etc. (For example, at the ith (i@1) trophic level--i.e, the c o n s u m e r s - - o x y g e n flows in and carbon dioxide flows out; whereas at the first level--i.e, the p r o d u c e r s - carbon dioxide flows in and oxygen flows out.) In (10), the first term is the energy flow accompanying the digested part of mass flow m~-l,~; the second term, the energy flow due to predation of the i + l t h species; the third term, the energy loss due to death, etc. The fourth term refers to the energy fixation (aM1) or the energy loss due to respiration (riMi). At the ground (0th) level the mass and energy equations are dMo dt
p
too, 1 + ~ ( l _ ~ l ) m t _ l , l l=2 p
+ ~ {p~M~+ (sz-e~)M,},
(12)
/=1
Eo = 0 .
(13)
The first term of (12) is the mass flow absorbed by the first species, and the other terms are the feedback from all the upper levels due to leavings, excretion, death, etc. Here the following relations of the proportionality are assumed. E i = ~iMi,
(14)
e i - l ' i = , ~ i _ l m i - l'i
(15)
A MODEL
OF HIERARCHICAL ECOSYSTEMS WITH MIGRATION
125
Model of Hierarchical Ecosystems with Migration. We consider a model of the global closed ecosystems with migration. W e c o m b i n e the model for m i g r a t i o n in Section 2 a n d t h a t for f o o d w e b in Section 3 as in Figure 4. 4. A
i+I,n
li %' ,h~,' m'o.,,°
..;.°~,
n-l,n
en, n+l
h;-l,n
ilk,n,I i-l,i
l,i
mn
,h°i - l , i
i-l,n
Figure 4. A model of hierarchical ecosystems with migration. - - - - - - - Energy; - - - - Activity energy
Mass ;
T h e b a l a n c e e q u a t i o n of mass, e n e r g y a n d activity e n e r g y (the p a r t of energy) at the ith (i@0) t r o p h i c level in the nth area are dMi , dt
,
=~imin-l'i
--
i,i+l _ p i M i , n
ml, 1
+ (ei - s i ) M i , .
i
i
+ r a n - l , . - mn, n + 1,
i=l,...,p,n=l,...,7 dE, ,
_ ¢iei ~ l,i
_ei, i+l
(16)
- 2ipiM,,,
i -I- (6i -- r i ) m i , n + ein - 1, n -- e,, ,+ x, i=l,...,p,
n = 1,..., 7
(17)
HIRONORI HIRATA
126
d~tt,, = {ihi - ,,i_ h~,,+ 1 _ #~.p,M,,. + #i,(e,-
i si)M,,, + 4)i,,(Mi,,) + hi.- 1,n -- h,,, + 1, i=l,...,p, n = l , . . . , y (18)
where ~ b i , . ( M , , , ) = b ~ , . (i # . + l - #i, ) (Mi , , - / ~ i , . )
(i•1),
~bl,,(Mj..) = 0 (a>O
(i=1) (i#1)'
ai=~o
Pi, 8i, Si, ri, hi, n, JVli,n > O,
0<¢i<1,~1=1, mO,O+ x = ep,O+, = h p, p+, = 0 , 1
1
1
1
1
1
I/?ln_ l , n ~ l q l n , , + l = e n _ l , n = e n , n + 1 ~ h n _ l , n = h n , n+ 1 = 0 .
(19)
Mi.,, Ei, . and Hi, n a r e the mass, the energy and the activity energy of the ith species in the nth area respectively. The superscript and subscript of m, e, h, etc. indicate the species and the area respectively. All symbols have the same meaning as in Sections 2 and 3. We assume that the species of all areas has the same properties except those for migration, i.e. 2i, ei, etc. are i dependent on only i, and #'.- is dependent on both i and n. m~,r+ 1 = msi , o in (16), ey, i r+ x _- e i,o in (17), and i _ i,o i,o means the increase of r hr, r+ 1 - h ~ i n ( 1 8 ) . m r the death rate of the ith species due to the excessive energy loss over #~, i.e. this means that the organisms expend more energy in predation than the energy obtained from the prey captured. At the ground (0th) level, the balance equations are dMo,n
dt
=-m.
0,1
P
+ E [(1-~,)mt,,-l"+P,Mt.,,+(st-e,)M,,,,], 1=1
n=l,...,y Eo,,=Ho,,=0.
(20) (21)
Here the following relations of the proportionality are assumed as (3), (6), (14) and (15):
Ei., = 2~Mi_,,
(22)
A M O D E L O F HIERARCHICAL ECOSYSTEMS W I T H M I G R A T I O N Hi, n =
# .i M i , . ,
(23)
i-- l , i i en - - ~ i - 1 Din
h.
i=
1,i
i
i
#.m.
127
1,i
(24)
i-- l , i
(25)
i
(26)
en-
1 , n = " ~ i m n - 1,n~
hi_
1, n = # n m n
i i-
l ,n •
(27)
Equation (22) holds for i = 1,..., p, n = 1,..., 7 and (23)-(27) hold for i = 2,..., p, n=l,...,7. The conservation of the total biomass can be easily derived from (16) and (20): P
Y
Z
Z M,,,=K
(28)
i=On=l
where K is a constant. The mass flow Di in (16) contains m a n y direct or indirect interactions implicitly. Although we cannot see the behavior of the population only from (16), the help of (17) and (18) makes it possible. The following equation (29) is derived from (16), (17), (22), (24) and (26). mn
i - 1, i = a i I"V l"i, n ,
(29)
where i(Si-- r i i(2i_2i_l )
a i-
al -
(all i#1),
~ - r l - ~q (~i - s l ) 21
(30) (31)
In the same way, the following equation (32) is derived from (16), (18), (23), (25) and (27). i n+ 1 Din,
=
bi , n(Mi
,n
-- M-- i , , ) - ki,,Di,i , i + l
where
'
#.+i -F.'
(32)
128
HIRONORI
HIRATA i
i
PT+t =#d"
(34)
rain,n+ 1 = bi, n(Mi,, -- 1~i,,) -- qi+ 1,nMi+ 1,n,
(35)
From (29) and (32)
where qi+ 1,n = ki, nai+ 1" From (16), (20), (29) and (35), we obtain the population equations: dMo, n & - ~ ciMz,,, n = l , . . . , 7 - - 1 dt 1=1 ~vact ~0,
P
7
dt-
P
2 ( c , + b , , , ) M t , , - 2 b,,,M,,,, l=1
/=1
dMl,n dt = e l M t , , - a 2 M 2 , , ,
n=l,...,7
n dmi, dt = ° q M i ' n - f l i + l ' n M i + l ' n - q i + l ' n
+
~
k=n-i
(36)
(37)
(3s)
1Mi+l'n-1
(--1)n-g+l{bi, k(mi,k--mi,k)} i=2,...,p,n=1,...,7
(39)
where ~zl=alq-gl--sl--Pl,
ei = ~iai + e i - s i - Pi, n = 2,..., p fli,,=ai-qi,,, i=2,...,p, n=1,...,7 ct = - a l + s t - e l
+Pt,
q = (1--~l)al+pl+sl--el, / = 2 , . . . , p qi, o=bi, o=O, i = 2 , . . . , p
(40)
We can see that the components of the coefficients of the population equations (36)-(39) are all energy parameters except for a~ and s¢. The equilibrium of (36)-(39) implies the point of balance between foodweb and migration. The value of the equilibrium depends deeply on the relation of dimensions between the total carrying capacity and the total mass K. PROPOSITION 1. I f the biomass M c , is non-negative for all i and n and each level satisfies the "energy condensing condition", i.e. 2i(si-ei)>r~ for
A M O D E L O F H I E R A R C H I C A L ECOSYSTEMS W I T H M I G R A T I O N
129
all 2 G i G p and ¢ - r 1 >21(a 1 - s l ) , then a necessary and sufficient condition for the main mass flows m.i--1.i to stream f r o m the lower level to the upper one is
(41)
)up > )~o_ t > . . - >,Z 1
The proof of Proposition 1 is obvious from (30) and (31). Proposition 1 shows how the ecosystem exhibits a hierarchy in energy per unit biomass, i.e. the important aspect (2) pointed out by Ulanowicz. The condition (41) means that the specific energies should increase as one ascends the trophic ladder. The mass flow contains m a n y direct or indirect interactions implicitly. We can see that mass flow contains the density-dependent terms as shown in the following Proposition 2. PROPOSITION 2.
The term dependent on M 2,,. appears in
i
ran, n+ 1 a s
mi
Oi,M 2.+"
n=l,
-,7
(42)
~'"-K
1-
n=l,...,7.
(43)
where
,n > 0
Each remainder part of (42), which is shown by the symbol "- .... , does not have any terms dependent on M 2 This is proved in the Appendix. Proposition 2 yields dMi,, _ dt
Oi,,M2, + - ' ' , n = l , . . . , y .
(44)
Let y
Mi = ~ M i , , ,
(45)
n=l
then dMi dt-
O~"~M~'7+
(46)
Equation (44) for n=#? means that the saturation of the ith species in the nth area occurs and some weak organisms move to the other areas. Equation (46) shows the density effect, i.e. the death rate of the ith species
130
H I R O N O R I HIRATA
increase due to the increase of the population of the ith species. Therefore Proposition 2, i.e. equation (46), shows how the density-dependent terms in population dynamics arise as a consequence of migration, i.e. the important aspect (1) pointed out by Ulanowicz. 5. Conclusion. Considering the spatial distribution of the population has
made it possible to explain both how the density-dependent terms in population dynamics appear by deduction and how the ecosystem exhibits a hierarchy in energy per unit biomass. The author wishes to express his thanks to Dr. R. E. Ulanowicz, one of the reviewers, for his valuable comments and suggestions in the reviewing process.
APPENDIX (Proof of Proposition 2) Using equations (28) and (35), we can derive m,,+i
1 = b i . , ( M i. ,-37/Ii . . , ) - q. i + - b i , n Mi, n K
qi+ l.,, M
l
nMi+ l n
Mj, t 1
p
r
-bi'"(l-~)
(A1)
Here let ,,, =-K= \ - ~ " - - j ,
(A2)
then m,,+i
1 =Oi,,M~, + ' " ,
(A3)
and we can easily see that 0~,, is positive.
LITERATURE Hirata, H. and T. Fukao. 1977. "A Model of Mass and Energy Flow in Ecosystems." M a t h . Biosci., 33, 321-334. Ulanowicz, R. E. 1972. "Mass and Energy Flow in Closed Ecosystems." J. T h e o r . Biol., 34, 239-253.
0007-4985/80/0101-0119 $02.00/0
A M O D E L OF H I E R A R C H I C A L ECOSYSTEMS WITH MIGRATION • HIRONORI HIRATA Department of Electronics, Chiba University, 1-33, Yayoi-cho, Chiba-shi, 260 Japan A model of ecosystems with migration is proposed from the viewpoint of flow. This model explains the following two points: (1) How the density-dependent terms in population dynamics arise as a consequence of migration. (2) How the ecosystem exhibits a hierarchy in energy per unit biomass.
1. Introduction. Ulanowicz pointed out the following important aspects in ecosystems (Ulanowicz, 1972): (1) How the density-dependent terms in population dynamics appear by deduction. (2) H o w the stable system exhibits a hierarchy in energy per unit biomass. Since the population of any species cannot continue to increase, the density-dependent terms, i.e. the saturation effect, is important in population dynamics. The more the population increase, the more organisms die, because according to the increase of the population in the area the competition intensifies, the environmental conditions become bad, etc. To escape these things, animals migrate to other a r e a s . We may expect that the ecosystem exhibits a hierarchy in energy per unit biomass. This tendency accords with the fact that animal carbohydrates and proteins have generally higher calorific values than those for plants. Although Ulanowicz tried to show the above two points, unfortunately, it was pointed out that his paper contained a physical mistake and his model could not explain them. A new model of mass and energy flow in ecosystems has been proposed (Hirata and Fukao, 1977). This new model without migration, in which the density effect was assumed, could explain 119
120
H I R O N O R I HIRATA
only point (2) above: how the stable system exhibits a hierarchy in energy per unit biomass. But further extension of this model to include migration has made it possible to explain both of the above points. The density effect is not assumed in this paper, however, the density-dependent term in population dynamics is shown to arise by deduction. In the following section, we first consider separately the fundamental process of migration and that of hierarchical foodweb. Thereafter, we combine them to create a model of the hierarchical ecosystems with migration.
2. A Model of the Fundamental Process of Migration.
We make the
following assumptions on migration: (1) We consider the migration of only animals and neglect that of plants. (2) Animals migrate only according to the ease of predation, i.e. they migrate to the area where they can easily prey on host species. We evaluate the degree of ease of predation in the area by the energy expended in predation, called "activity energy." Consider the migration of one species in one dimension. There are areas which are characterized by the different activity energy per unit biomass, as in Figure 1. Without any loss of generality, in this section we consider only
~~ m n - I ' n m nn'n+l +nl , F-.
/Zn+l
Figure 1. A model of the fundamental process of migration. - - - - - Activity energy
Mass;
migration and neglect the flows along the foodweb for simplicity. We assume that the flow streaming from the nth area to the n + l t h one is positive. The balance equations of mass and activity energy at the nth area are
dM. dt =m"-m'"-m"'"+l"
(1)
+c~,(M,), n=l,...,7. dHn=h,_l,,-h,,,+l
(2)
A M O D E L O F HIERARCHICAL ECOSYSTEMS W I T H M I G R A T I O N
121
M, is the mass (the population) in the nth area. H, is the activity energy required by all organisms in the nth area to capture their prey. m,_ a,, and h,_l, . are the mass flow from the n - l t h area to the nth one and the activity energy flow accompanying respectively, qS,(M,) is a motive force of migration. We assume the following relation between M , and H,.
H,=#,M,,
(3)
where #, is the activity energy per unit biomass. We add the following assumptions on migration: (3) Each area has the carrying c a p a c i t y / ~ r . (4)
#1 < # 2 < ' " < # ~ < # ~ + ~ = g d ,
(4)
where #a shows the limitation of the energy loss which can be expended on predation. The organisms die when # > # d , that is, when they expended more energy in predation and metabolism than they obtain from their prey. Condition (4) means that areas have been arranged in order of the ease of predation. Animals migrate gradually from area to area along this order. For example, animals in the nth area can migrate only to the n - l t h area and the n + l t h area. They cannot directly move to the other areas. Migration can be bidirectional and its direction is determined by the sign of the motive force ~b,(M,), i.e. it is determined by assumption (5). (5) When the capacity of the nth area is not filled (i.e. M , < 2 ~ , ) , the organisms migrate from the n + lth area to the nth one, i.e. they can move to a more favorable area. When the capacity of the nth area is filled (i.e., M , > M , ) , the weak organisms must migrate to the ,unfavorable area, i.e. the n + lth area. Therefore, the direction of migration is determined by the sign of.the difference ( M , - ) ~ , ) . Here the motive force of migration in equation (2) is expressed as: q~,(M,) = b,(#,+ 1 - # , ) ( M , - ) I 4 , ) ,
(51
where b, is a positive constant, because the difference ( M , - / ~ , ) (i.e. the difference between the population M , and the carrying c a p a c i t y / ~ , ) causes the migration, and the increase (or the decrease) in the activity energy per unit mass of ( M , - ) ~ r ) is proportional to the difference (g, + 1 - #,). The following relation is assumed: h,_ 1,, = #.m,_ 1 ,-
(6)
122
HIRONORI HIRATA
We can see below that this relation is reasonable. The following is derived from (1~(3), and (6): /~.+ lm.,.+ 1 - # . m . , . + l = q~.(M.).
(7)
Replacing the second m.,.+l by m.,. to understand the energy balance equation (7) easily, the microfeature of balance in the nth area can be derived as illustrated in Figure 2. If the motive force of migration 4 . is
Area n
~
Areon+l
!
m
m
Figure 2.
Micro-feature of balance in the nth area, where m.,.= m.,.+ 1
positive, it changes the activity energy per unit mass of m.,.+l from #. to #.+1 and m.,.+l is positive (i.e. the mass flow steams from the nth area to the n + lth one). If ~b. is negative, these are vice versa. The following equation (8) is derived from ( 5 ) a n d (7), i.e. equations (1)(3), (5) and (6). m.,.+ t = b . ( M . -
M.).
(8)
Equation (8) shows that the a m o u n t of migration flow is proportional to the difference ( M . - M . ) . This agrees well with assumption (5). This model of fundamental process of migration satisfies all the above assumptions. 3. A Model of the Fundamental Process of Foodweb. With no loss of generality, in this section we consider only foodweb transfers and neglect migration for simplicity. This model is the same as Model A in the previous paper (Hirata and Fukao, 1977) except for the assumption of the density effect. We assumed the density effect in Model A without migration, i.e. the model contained the t e r m -qi(Mz) 2. In this paper, it is possible to explain h o w the density-dependent terms in population dynamics arise as a consequence of migration.
A MODEL OF HIERARCHICAL ECOSYSTEMS WITH MIGRATION
123
The f o o d w e b depicted in Figure 3 is considered. The trophic levels 0, 1,..., p constitute the p r i m a r y producers, the herbivores, the p r i m a r y a n d secondary carnivores, etc. M a s s circulates t h r o u g h the producers, the consumers a n d the decomposers. Solar energy is fixed at the first trophic level (i.e. producers), a n d some a m o u n t of energy is dissipated at each trophic level.
i
riMi ~
(
i_~i)mi-l,i ,
I sii i
\/
Deoth Lossof the moss due fo the respirotion k Excretion (COz) / Leavings
Fromlevel 0 (02)
~ /{ i"
l- - -9
I
0 All nutrient
elements
C,H,O,N,P,S,etc.
]
Figure 3. A model of the fundamental process of foodweb. - ---Energy
Mass;
This mass a n d energy equations at the ith (i :p O) trophic level are dMi dt
= ~ i m i - 1, i _ m i, i + 1 - - p i M i 4- (e,i - -
(9)
S i)Mi,
d E i = ~ i e i - 1, i _ e i, i+ 1 _ j, i p i M i + (a i _ r i ) M i , dt i=l,...,p
(10)
124
HIRONORI
HIRATA
where
ai=
o>O (i=1) (i@1)'
Pi, 8i, si, ri > 0, 0 < ~ i < 1, ~ 1 = 1 , m P'p+I=e p'p+I=O.
(11)
Mi is the biomass (the population) of the ith species, E~ is the energy of the ith species, and m ~'j and e ~'j are the mass and energy flows from the ith species to the jth species respectively. ~ is the utility efficiency of mass. The meaning of each term of equations (9) and (10) is as follows and is also illustrated in Figure 3. In (9), the first term is the digested part of mass flow mi-l"i; the second term, the loss of the mass flow due to predation of the i + lth species; the third term, the effect of death, etc., and the fourth term is the mass flow due to respiration etc. (For example, at the ith (i@1) trophic level--i.e, the c o n s u m e r s - - o x y g e n flows in and carbon dioxide flows out; whereas at the first level--i.e, the p r o d u c e r s - carbon dioxide flows in and oxygen flows out.) In (10), the first term is the energy flow accompanying the digested part of mass flow m~-l,~; the second term, the energy flow due to predation of the i + l t h species; the third term, the energy loss due to death, etc. The fourth term refers to the energy fixation (aM1) or the energy loss due to respiration (riMi). At the ground (0th) level the mass and energy equations are dMo dt
p
too, 1 + ~ ( l _ ~ l ) m t _ l , l l=2 p
+ ~ {p~M~+ (sz-e~)M,},
(12)
/=1
Eo = 0 .
(13)
The first term of (12) is the mass flow absorbed by the first species, and the other terms are the feedback from all the upper levels due to leavings, excretion, death, etc. Here the following relations of the proportionality are assumed. E i = ~iMi,
(14)
e i - l ' i = , ~ i _ l m i - l'i
(15)
A MODEL
OF HIERARCHICAL ECOSYSTEMS WITH MIGRATION
125
Model of Hierarchical Ecosystems with Migration. We consider a model of the global closed ecosystems with migration. W e c o m b i n e the model for m i g r a t i o n in Section 2 a n d t h a t for f o o d w e b in Section 3 as in Figure 4. 4. A
i+I,n
li %' ,h~,' m'o.,,°
..;.°~,
n-l,n
en, n+l
h;-l,n
ilk,n,I i-l,i
l,i
mn
,h°i - l , i
i-l,n
Figure 4. A model of hierarchical ecosystems with migration. - - - - - - - Energy; - - - - Activity energy
Mass ;
T h e b a l a n c e e q u a t i o n of mass, e n e r g y a n d activity e n e r g y (the p a r t of energy) at the ith (i@0) t r o p h i c level in the nth area are dMi , dt
,
=~imin-l'i
--
i,i+l _ p i M i , n
ml, 1
+ (ei - s i ) M i , .
i
i
+ r a n - l , . - mn, n + 1,
i=l,...,p,n=l,...,7 dE, ,
_ ¢iei ~ l,i
_ei, i+l
(16)
- 2ipiM,,,
i -I- (6i -- r i ) m i , n + ein - 1, n -- e,, ,+ x, i=l,...,p,
n = 1,..., 7
(17)
HIRONORI HIRATA
126
d~tt,, = {ihi - ,,i_ h~,,+ 1 _ #~.p,M,,. + #i,(e,-
i si)M,,, + 4)i,,(Mi,,) + hi.- 1,n -- h,,, + 1, i=l,...,p, n = l , . . . , y (18)
where ~ b i , . ( M , , , ) = b ~ , . (i # . + l - #i, ) (Mi , , - / ~ i , . )
(i•1),
~bl,,(Mj..) = 0 (a>O
(i=1) (i#1)'
ai=~o
Pi, 8i, Si, ri, hi, n, JVli,n > O,
0<¢i<1,~1=1, mO,O+ x = ep,O+, = h p, p+, = 0 , 1
1
1
1
1
1
I/?ln_ l , n ~ l q l n , , + l = e n _ l , n = e n , n + 1 ~ h n _ l , n = h n , n+ 1 = 0 .
(19)
Mi.,, Ei, . and Hi, n a r e the mass, the energy and the activity energy of the ith species in the nth area respectively. The superscript and subscript of m, e, h, etc. indicate the species and the area respectively. All symbols have the same meaning as in Sections 2 and 3. We assume that the species of all areas has the same properties except those for migration, i.e. 2i, ei, etc. are i dependent on only i, and #'.- is dependent on both i and n. m~,r+ 1 = msi , o in (16), ey, i r+ x _- e i,o in (17), and i _ i,o i,o means the increase of r hr, r+ 1 - h ~ i n ( 1 8 ) . m r the death rate of the ith species due to the excessive energy loss over #~, i.e. this means that the organisms expend more energy in predation than the energy obtained from the prey captured. At the ground (0th) level, the balance equations are dMo,n
dt
=-m.
0,1
P
+ E [(1-~,)mt,,-l"+P,Mt.,,+(st-e,)M,,,,], 1=1
n=l,...,y Eo,,=Ho,,=0.
(20) (21)
Here the following relations of the proportionality are assumed as (3), (6), (14) and (15):
Ei., = 2~Mi_,,
(22)
A M O D E L O F HIERARCHICAL ECOSYSTEMS W I T H M I G R A T I O N Hi, n =
# .i M i , . ,
(23)
i-- l , i i en - - ~ i - 1 Din
h.
i=
1,i
i
i
#.m.
127
1,i
(24)
i-- l , i
(25)
i
(26)
en-
1 , n = " ~ i m n - 1,n~
hi_
1, n = # n m n
i i-
l ,n •
(27)
Equation (22) holds for i = 1,..., p, n = 1,..., 7 and (23)-(27) hold for i = 2,..., p, n=l,...,7. The conservation of the total biomass can be easily derived from (16) and (20): P
Y
Z
Z M,,,=K
(28)
i=On=l
where K is a constant. The mass flow Di in (16) contains m a n y direct or indirect interactions implicitly. Although we cannot see the behavior of the population only from (16), the help of (17) and (18) makes it possible. The following equation (29) is derived from (16), (17), (22), (24) and (26). mn
i - 1, i = a i I"V l"i, n ,
(29)
where i(Si-- r i i(2i_2i_l )
a i-
al -
(all i#1),
~ - r l - ~q (~i - s l ) 21
(30) (31)
In the same way, the following equation (32) is derived from (16), (18), (23), (25) and (27). i n+ 1 Din,
=
bi , n(Mi
,n
-- M-- i , , ) - ki,,Di,i , i + l
where
'
#.+i -F.'
(32)
128
HIRONORI
HIRATA i
i
PT+t =#d"
(34)
rain,n+ 1 = bi, n(Mi,, -- 1~i,,) -- qi+ 1,nMi+ 1,n,
(35)
From (29) and (32)
where qi+ 1,n = ki, nai+ 1" From (16), (20), (29) and (35), we obtain the population equations: dMo, n & - ~ ciMz,,, n = l , . . . , 7 - - 1 dt 1=1 ~vact ~0,
P
7
dt-
P
2 ( c , + b , , , ) M t , , - 2 b,,,M,,,, l=1
/=1
dMl,n dt = e l M t , , - a 2 M 2 , , ,
n=l,...,7
n dmi, dt = ° q M i ' n - f l i + l ' n M i + l ' n - q i + l ' n
+
~
k=n-i
(36)
(37)
(3s)
1Mi+l'n-1
(--1)n-g+l{bi, k(mi,k--mi,k)} i=2,...,p,n=1,...,7
(39)
where ~zl=alq-gl--sl--Pl,
ei = ~iai + e i - s i - Pi, n = 2,..., p fli,,=ai-qi,,, i=2,...,p, n=1,...,7 ct = - a l + s t - e l
+Pt,
q = (1--~l)al+pl+sl--el, / = 2 , . . . , p qi, o=bi, o=O, i = 2 , . . . , p
(40)
We can see that the components of the coefficients of the population equations (36)-(39) are all energy parameters except for a~ and s¢. The equilibrium of (36)-(39) implies the point of balance between foodweb and migration. The value of the equilibrium depends deeply on the relation of dimensions between the total carrying capacity and the total mass K. PROPOSITION 1. I f the biomass M c , is non-negative for all i and n and each level satisfies the "energy condensing condition", i.e. 2i(si-ei)>r~ for
A M O D E L O F H I E R A R C H I C A L ECOSYSTEMS W I T H M I G R A T I O N
129
all 2 G i G p and ¢ - r 1 >21(a 1 - s l ) , then a necessary and sufficient condition for the main mass flows m.i--1.i to stream f r o m the lower level to the upper one is
(41)
)up > )~o_ t > . . - >,Z 1
The proof of Proposition 1 is obvious from (30) and (31). Proposition 1 shows how the ecosystem exhibits a hierarchy in energy per unit biomass, i.e. the important aspect (2) pointed out by Ulanowicz. The condition (41) means that the specific energies should increase as one ascends the trophic ladder. The mass flow contains m a n y direct or indirect interactions implicitly. We can see that mass flow contains the density-dependent terms as shown in the following Proposition 2. PROPOSITION 2.
The term dependent on M 2,,. appears in
i
ran, n+ 1 a s
mi
Oi,M 2.+"
n=l,
-,7
(42)
~'"-K
1-
n=l,...,7.
(43)
where
,n > 0
Each remainder part of (42), which is shown by the symbol "- .... , does not have any terms dependent on M 2 This is proved in the Appendix. Proposition 2 yields dMi,, _ dt
Oi,,M2, + - ' ' , n = l , . . . , y .
(44)
Let y
Mi = ~ M i , , ,
(45)
n=l
then dMi dt-
O~"~M~'7+
(46)
Equation (44) for n=#? means that the saturation of the ith species in the nth area occurs and some weak organisms move to the other areas. Equation (46) shows the density effect, i.e. the death rate of the ith species
130
H I R O N O R I HIRATA
increase due to the increase of the population of the ith species. Therefore Proposition 2, i.e. equation (46), shows how the density-dependent terms in population dynamics arise as a consequence of migration, i.e. the important aspect (1) pointed out by Ulanowicz. 5. Conclusion. Considering the spatial distribution of the population has
made it possible to explain both how the density-dependent terms in population dynamics appear by deduction and how the ecosystem exhibits a hierarchy in energy per unit biomass. The author wishes to express his thanks to Dr. R. E. Ulanowicz, one of the reviewers, for his valuable comments and suggestions in the reviewing process.
APPENDIX (Proof of Proposition 2) Using equations (28) and (35), we can derive m,,+i
1 = b i . , ( M i. ,-37/Ii . . , ) - q. i + - b i , n Mi, n K
qi+ l.,, M
l
nMi+ l n
Mj, t 1
p
r
-bi'"(l-~)
(A1)
Here let ,,, =-K= \ - ~ " - - j ,
(A2)
then m,,+i
1 =Oi,,M~, + ' " ,
(A3)
and we can easily see that 0~,, is positive.
LITERATURE Hirata, H. and T. Fukao. 1977. "A Model of Mass and Energy Flow in Ecosystems." M a t h . Biosci., 33, 321-334. Ulanowicz, R. E. 1972. "Mass and Energy Flow in Closed Ecosystems." J. T h e o r . Biol., 34, 239-253.