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Predation, competition and environmental variables: Some mathematical models
J. theor. Biol(l970)
27, 175-195
Predation, Competition and EnvironmentalVariables: SomeMathematical Models VERNA LOUISE ENGSTROM-HEG Department of Mathematics, State University College, Oneonta, New York, U.S.A. (Received 20 February 1969, and in revisedform 9 September 1969) Mathematical models adapted to computer programming were prepared to simulate various ecological interactions. The models were tested on an Alwac III-E computer, using various hypothetical population parameters. Single-species populations in a constant environment became stabilized in number of individuals and age-composition. Prey-predator interactions resulted in damped oscillations in both populations. However, when the effect of climate was introduced as a random variable, the oscillations were no longer damped. Simulations of two species competing on the same trophic level resulted in the extinction of one species or the other. Interactions among three species had various outcomes, depending on the trophic relationships among the species. The two- and three-species models depart from the classical approach in that the age composition of each species is taken into account. The three-species model can be expanded to handle larger numbers of species, and can be modified to handle reproductive cycles of various lengths and to approximate instantaneous mortality rates. If desired, the model can be programmed to solve for unknowns other than the population size. The theoretical study of population fluctuations and the effects of the environment upon them has been handled mainly by differential and difference equations in the past. With the development of digital computers a new approach to this problem is possible. Since interspecific relationships in nature are usually governed by a number of factors interacting in a complex way, any mathematical model that closely simulates nature is likely to involve a great deal of computation. Use of an Alwac III-E digital computer, allowed one to perform rapidly many more arithmetic calculations than would have been otherwise possible, enabling one to write rather elaborate programs. Three different situations have been considered: the case of one species by itself; the case of two species and their interactions; and the case of three species which are allowed to interact with each other in several ways. 12 T.B. 175
176
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1. The Single-species Model
First let us consider the changes in a one-species population over a period of time. The population is first assigned an age structure using the following notation : Ni is the number of individuals in the population which are i years old and were born in the year j. The unit of time, i and j, could of course be in weeks, days or hours if this better fitted the species in question. N’, is the number of young born in a particular year (j). is the average number of offspring an i-year-old organism will 4 produce in the ith year of its life. Since it is a mean value, K may be either greater or less than one. n is the maximum age attained by a member of the species. ” N’,= CKiN~-i=K1Nj,-‘+KzNj2-Z+K3Nj3-3+...+K,N~-”. i=l
The number of individuals one year old or older present at a particular time can be expressed in terms of the number of individuals of the same year class present the year before. Thus
Nj = N/-, where Ni- 1
A-1
a C
fiT1-cf:
NJ6+i-o-1-p-g
a=0
1
is the number of i-l-year-old born in the jth year (i.e. the number of members of thejth year class alive last year). is the probability that an organism that has lived i- 1 years will survive to the next year (the ith year) under condition of no predation or food deficiency and with a normal climate for the area. is age. is a factor representing the effects of overpopulation (compensatory mortality). The model assumes that at moderate levels of population density, mortality will not be significantly affected by density-dependent sources of mortality. Therefore when the population density falls within a designated density range, c is equal to zero. If the population is very sparse, so that the effects of underpopulation (depensatory mortality) appear, this term may be changed to
a=0
SOME 2
n;i+i-a-l
a=0 P
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is the total number of individuals alive in the population in the yearj+i-I. is a factor representing the reduction of the population by predators. It is a function of the size of the predator population. is a factor representing the effects of climate over the last year. It allows us to include the effects of extremes in weather; droughts, floods, hard winters, or by changing the sign, exceptionally favorable weather. In hypothetical situations, g may be set equal to a randomly chosen variable.
If there is evidence that the effects of population density, climate, or predation act more drastically on some age-classes than on others, the program may be easily modified so that different values of c, p or g are used for different age groups. In this case ci- r, pi-r, and gi- i would be used in finding the value of N!. Here a table of c values would be placed in the computer and for the number of one-year-olds the co value (i.e. the effect of crowding on this group of organisms during the previous time period) would be used. For the two-year-olds the c1 value would be used, etc. The same procedure would be used for values of p and g. By use of a jump switch it is possible to put in a different factor for g (climate) each year. This factor can come from a table of random numbers. By moving another switch the machine will stop at the end of any year and changes may be typed into the program. Thus the rate of predation, the effects of over- or under-population, and the basic probability of an organism living to the next year can be changed. The numbers in the year classes can also be changed in this manner, thus introducing immigration, emigration, or density-independent mortality. The program was written so as to keep
between zero and one. Since it is impossible for a negative number of organisms to be present, when this factor became negative, zero was substituted for it. Again when this factor became greater than one, one was substituted for it, since, barring immigrations, it is impossible to have more members in the Ni+r class next year than there were in the Ni class this year. It may be argued that even under ideal conditions all the organisms in one-year class would not live until the next year. By changing the program slightly,
178
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it could be made so that all values of fi-I
-ci
N;+‘-“-Lpm9 a=0
would be replaced by O-95.
1 >
0.95
2. A Model for the Study of Two Species
The two-species model is similar to the one-species model. This model is in somewhat the same general form that Volterra (1931) used, but differs from the Volterra equations in that the populations are separated into age classes, and in that there is a factor for climate and a factor for a predator on the predator. Also, in contrast to Volterra’s model, the changes are assumed to be discrete rather than continuous. The equations which were used are as follows:
N’, = i&N;-‘.
(1)
i=l
N:‘= Ni-, fiel-ci &
1
N~+i-a-1-d*~oM~ti-a-1-9 . a=0
= f KiMi-‘. i=l
(2)
(3)
M:‘=M{-,~i_l-~~M!+~-~-l+a~N~+i-~-l-~-~ . a=0 a=0 I (4) is the number of prey. is the number of predators. is the age of the organism. is the year of birth of the organism. are the number of young born in year j to the prey and predators, respectively. is the average number of offspring an i-year-old prey will produce in any given year. is the average number of offspring an i-year-old predator will produce in any given year. are the probabilities that an organism which has lived i-l years will survive to the ith year, when preypredator effects and climatic effects are not considered.
N
M i
i N’, and M’, Ki 4
J-1 anal.-1
c i
#ti-a-l
a=0
and
account for the effects of overpopulation.
&,f{ti-"-' a=0
n
1
is the maximum possible age attained by the prey.
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m d f
N!ti-a-l
a=0 g and jj P
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179
is the maximum possible age attained by the predator. j@ti-4-l
a=0 a i
MATHEMATICAL
is the effect of predation, which is directly related to the size of the predator population. is the effect of the food supply on the predator. are the effects of climate, and can be either positive or negative. is the effect of a predator on the predator if there is one. This factor may or may not exist, and can also be dealt with by putting in a third animal.
Possibly 2 i Nj;+‘-‘-’ should be combined with the f’s, or the d’s varied a=0 from positive to negative as the food varies from a mean value. This will of course depend on how large a factor this particular prey is in the survival of the predator. The same general comments that apply to the one-species model still hold. Climate may be changed each year by use of random numbers. The climate factor for the prey does not need to have the same value as the climate factor for the predator. These two factors may be the same, completely independent of each other, or have some relationship between these two extremes. All the other factors in equations one to four may be changed at the end of each year (cycle). The factors
fi-l-c~~oN:ti-~-l--~~oM~ti-~-l-g] and ji-l-$oMyU-Lfaf: N~+i-~-‘-g-p] a=0 [ are held between zero and one in the manner already discussed. 3. A Model for the Study of Three Species
The three-species model is a general model which can be used for many types of interactions between different species of organisms. It is written in such a way that more than three species could easily be added without changing the form. The basic equations are as follows: (5) fi-l+C~L~+i-a-l+d~ a=0
a=0
N~ti-“-l+e~M!+i-“-l+p+g a=0
1 (6)
180
Ni = Ni-,
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1 (8)
In equations (5) to (lo), the notation resembles that for the previous situations; specifically the three species are divided into age classes. It is assumed that the period from i to i- 1 is the same for all three species, but this period is not necessarily a year. i is the “age” of the organism L{, N{, M; j is the “year” of birth of the organism. indicate the number of young born in a particular “year” L’,, N’,, M’, (time period), in each species. are the average number of offspring an i-year-old member Ki, Ri, Ri of each species will produce during the given “year” (time period). that an organism which has lived 1 are the probabilities will survive to the ith “year”, when the i- 1 “years” other factors such as c C L, d C N, e 2 M, p and g are not considered. These factors allow for the effects of infant mortality and of higher death rates in older animals. c is the effect of overpopulation or underpopulation of c,d, e species L; d and e are the predation effect or beneficial effect of species N and M on species L; c, d and e can be either positive or negative. a is the effect of overpopulation or underpopulation of f, a, G species N; E and Z are the predation effect or beneficial effect of species L and M on species N; E, a and E can be either positive or negative. E,a,i; k is the effect of overpopulation or underpopulation on species M; E and a are the predation effect or beneficial effect of species L and N on species M; F, 2 and Z can be either positive or negative. represent the effect of predation other than that consiP, 8, p’ dered above, or exploitation by man.
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- z 9, 93 9
are factors for climatic effect. They can be positive or negative or changed yearly. g, S and J are not necessarily equal. are the maximum possible ages of individuals in species I, n, m L, N and M, respectively. The same general comments which were given for the two-species model still hold. Climatic variations may be simulated by using random numbers. Any of the other factors can be easily changed at the end of each year (cycle) if so desired. The program may be modified so that different values of c, d, e, p or g are used for different age groups. This may be necessary where climatic variation, population density, and predation have selective effects on the very young and/or the very old. If p represents exploitation of a managed population by man, there may be no predation on individuals below a given minimum age or size class. The restriction that fiel+$
L.+i-o-l+dk
n=o 1 ~i_l+~C~i+i-“-l+;I~~d+i-“-l a=0
i
Ni+i-a-l+ef
Mj+i-a-l+p+g
n=O
a=0 +~j
n=O
f
I Mj;+i-o-1
+jj+g
a=0
and
1
have values between zero and one inclusive, must also hold. 4. Single-species Populations: Results
Using the program for one species, I chose the following values for the different factors and initial values : iv:
-
N;' N;= N;' N;' N;' Nib N;' Iv,' Np9 NC,," XN c
60 55 --- 50 45 40 30 20 10 -2 0 -- 412 -0.00 I
100
.f, J‘, f2 j; f4 f5 ,fb .1; fs f9 .f;o-
-
0.65 0.96 0.96 0.96 0.90 0.80 0.70 0.50 0.30 0.10 0
K, I\'2 K, K, K, K, K, K, K, K,o-
--
0.3 0.3 0,4 04 04 0.3 0.2 0.1 0 0
I'
-
0.002
n
-
0.0003
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Without making changes in any factor, the program was allowed to run for 198 cycles (years). It reached a steady population of 498.5, with very little variation from this value (Fig. 1). The age classes in this stable population are given in Table 1.
4100
, , , , I, D20xl405060708090
,
,
,
1
ime
FIG.
1. Development
of a stable population.
TABLE
1
Age classes in stable population
Using this stable population, the values of they’s were changed to simulate an epidemic. The new values for the f’s were: fb fi f2
-
f, f4 fs
-
0.33 0.48 0.48 0.48 0.45 0.40
fh 1; Js f9 .f*o
-~ --
0.35 0.25 0.15 0.05 0
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MATHEMATICAL
100 0
I IO
MODELS
I 20
I 40
I 30
IN
I 50
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I 60
183
I 70
Time
FIG. 2. Disease attacks stable population
for two years then recovery.
After two cycles (years) with the f’s at these values, they were returned to their original values. At the end of the second cycle the population reached a minimum of 117. Then it gradually returned to its original value of 498. After 86 cycles the population size was 493 (Fig. 2). Next, 500 individuals were placed in the zero year class to simulate the development of a population placed in a virgin territory. It should be noted parenthetically that colonization of newly available territory by juveniles is common in both natural and man-made situations. It is universal among benthic forms having pelagic larvae, and occurs in higher forms such as the beaver and muskrat where non-migratory adults set up restricted permanent home territories. Farm fish ponds and newly reclaimed lakes are usually stocked with young-of-the-year fish. The factors for climate, predation, probability of survival of each year class, and overpopulation were
560 E .@ g. 520 b 3 4
480
2 440 ‘.
400 I IO
FIG. 3. Development
I M
of a population
I x)
I 40 Time
1 SC
I 60
1 70
from 500 organisms in the zero year class.
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the same as in the first situation. The population reached a low of 387 the first cycle. Then it rose to a maximum of 609 in the fifth cycle. From there the population gradually returned to the stable size of 498.5. It was 498.97 after 82 cycles (Fig. 3). The population decreased from the fifth to the ninth cycle and then maintained approximately the same value of 562 and 561 in cycles nine, ten and eleven. This plateau is due to the larger number of “births” in cycles four and five which have now reached the older age groups and which act to retard the gradual decrease of the total population (Fig. 4). The values of g were then varied by use of random numbers while everything else was held in the stable condition of the first situation. This, of course, caused the population size to oscillate irregularly.
0
I
I IO
I 5
I 15
I 20
I 24
Tink
FIG. 4. Age composition
of a population
from 500 organisms in the zero year class.
5. Interactions between Two Species In the two-species model, there was a damped oscillation in the number of prey organisms and the number of predator organisms (Fig. 5). In this
829
92 0
5
I IO
I 15
I 20
FIG. 5. Fluctuations
1 25
I 30
I 36 Time
I 40
1 45
in a prey-predator
1430
1 50
population.
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185
case the constants in Table 2 were used. When the values for d and ~3were smaller the fluctuations were smaller, and if d and a were small enough TABLE
2
Constants for interactions between two species ~~ Prey
Predator
C
-
0~0001
(1
--
0.0003 0.0003
---
1.2 1.2 1.6
.(/ K, K, K, K i, K, K, K, K,
-
1.6
-
1.6 1.2 0% 0.4
K,
--
0 0 0.75
-..
0.98 0.98 0.98 0.97 0.92 0.9 0.8
--0
0.1 0.05
lo2 i:
f3 2
f6 2
7 ii $7 F KL h,7 R, R, R5 R6 x7
----~-
0~0001 0.0003 0.0003 om2 0.3 0.3 0.4 0.4 0.4 0.3 0.2
R!3 -R, -. ,; Lo- -
0.1 0 0 0.3
;;
-
0.5
f3 A
-
0.5 0.5
; h .!l?
--.. ~.
0.4 0.3 0.2
.f .;I fl0
-----
0.03 0.07
-
0
no fluctuations occurred. Whenever the values for the f’s were larger, this species did not show oscillation. Such a situation occurred when the ,f’s were taken as J" .fl J2 j:, f4 fs
-----
0.3 0.6 0.6 0.6 0.6 0.5
f;& f8 :$y,
-
0.4 0.2 0.1 0.05 0
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and the values of d and a were both 04001. In this case neither species showed oscillation, but levelled off at a steady value for both species. The effect of climate as a random factor was then studied. Starting with the same values as were used for Fig. 5, the program was allowed to go through four cycles before changing the climate. From then on, for each cycle a different number was used for g and S. Numbers were obtained from a table of random numbers and were coded in the following way : Number
Value for g or 4 0
0~001 oaO2 0404 OGO8 -0GOl -0.002 -0404 -0X08
The number nine was skipped. The first number was used for the prey species, g; the second number was for the predator species, Lj. The third number was used in the next cycle for the prey, and so on. As can easily be seen in Fig. 6, the fluctuations are more erratic and are not damped as
Time
FIG. 6. Prey-predator
oscillations with climate as a random variable.
TABLE
3
Factors and initial values, where two species compete for the same food First species
-c tl rl
-0
N:: ,I.; Ivy" N;" Ni4 N,' N," N;' Ni9 Iv;:0 K, lc2 h’., K, K, h',
-.---~ -.~_
K, I\', K‘, K,, ,fo .f, ,f* .f3 Ik j-5 f6
--.-----
‘j&8
.f,
-
f8 J; s 10
---
04001 OGOOl
165 258 146 83 46 26 14 7 2 1 0 1.2 I.2 1.6 I.6 I.6 1.2 0.s 0.4 0 0 0.65 0.x 0,s 0.85 0.8 0.s 0% 0,4 0.2 0.1 0
Second species 7 II ii ;, %I:; M; ' My z iv;" M;j M;C M;' MT' Mi8 Mq9 M;,'O R,
-~- 0~00005 -~ -0.0002 ~. 0 ~~ 0 -- 750 -~~ 300 - I50 -- 90 -- 60 40 20 -- 10 -3 -~ I 0 -0.X
A2 7 h.: .x K, R,
-~---
.KC> A, 7 A, -7 A, -i h7 1" 10 .f, I2 .I; j4 j5 16 .1; I* .t?, .flO
-.... -.----1 -1 -1 -
0.8 0.9 0,s 0.7 0.6 0.5 0.4 0 0 0.15 0.9 0.9 0.9 0.9 0.s 0.6 0.4 0.3 0.1 0
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they are in Fig. 5. (Cycle 2 in Fig. 5 corresponds to cycle 1 in Fig. 6, i.e. the time numbers in Fig. 6 need to be moved one to the right to correspond exactly with Fig. 5.) Next the case where two species compete for the same food was examined. The factors and initial values in Table 3 were used. Since a is negative, an increase in the size of the population of species one has a negative effect on species two, as also an increase in the size of species two has a negative effect on species one. In this situation the population of species two went from 1424 individuals to zero individuals in 16 cycles (Table 4). TABLE
4
Population size, two species competing for the same food Cycle 0 1 2 3 4 5 6 I 8 9 10 11 12 13 14 15 16
First
species 1325 1362 1435 1602 1735 1878 2039 2197 2355 2491 2593 2675 2746 2769 2780 2786 2790
Second
species 1424 1190 1053 829 681 536 400 280 184 112 63 34 11 5 3 1 0
The value of c was changed to OGIO15 and of d to OWO15, and everything else kept as it was above. This time the size of the population of the first species went from 1325 individuals to 4 individuals in 13 cycles (Table 5). Next, the value of d was changed to OWOl and the other factors kept as they were in the second case. Now the population of species two went from 1424 individuals to 92 individuals in 27 cycles (Table 6). The number of individuals in species two formed a monotonic decreasing sequence of terms with each one smaller than the one before. These three mathematical experiments point to the precarious nature of co-existence when two species having very similar ecological requirements are placed in the same environment. In each case, very small changes in the
TABLE
5
Population size, two species competing for the same food Cycle
First
species 1325 924 174 713 651 583 506 422 277 187 110 52 18 4
0 1 2 3 4 5 6 7 8 9 10 11 12 13
Second
species 1424 1191 1316 1408 1523 1667 1848 2070 2528 2885 3288 3722 4146
TABLE 6 Population size, two species competing for the same food Cycle 0 1 2 3 4 5 6 I 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
First
species 1325 1141 1111 1105 1109 1119 1132 1166 1199 1219 1260 1302 1334 1366 1398 1432 1467 1502 1537 1568 1605 1636 1665 1692 1717 1736 1758 1716
Second
species 1424 1191 1141 1113 1088 1062 1031 967 910 871 805 733 686 637 587 537 486 436 389 343 300 259 223 189 160 134 111 92
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parameters governing the relationships between the two competing species resulted in one species or the other going to extinction. It would probably be quite difficult to find a set of values that would allow both species to survive under the conditions of these experiments. In a more complex situation where differential effects of climate and predation were introduced co-existence might be prolonged, perhaps indefinitely where changes in these effects tended to favor first one species, then the other. In nature it is most unusual to find two species attempting to occupy closely similar niches in the same territory. In the long run such situations tend to be stabilized either by the elimination of one species or by the survival of mutant forms having sufficiently different ecological requirements. 6. Interactions among Three Species In the first three-species situation species L was eaten by species N and species M, while species M also ate species N. The factors and initial values of Table 7 were used. In this case the sign is important, since all factors are assumed positive unless there is a negative sign. Damped oscillations were obtained in all three species, as shown in Fig. 7. Next, the reproductive rates of the predator species A4 were changed to R, = 0.4, K, = 0*5, R, = 0.4, z, = 0.2, K, = 0, and all the other factors left the same as in the former situation. In this case species M went to extinction. After 112 cycles there were five organisms left, whereas in the former situation the number of organisms in species M oscillated about 1100 organisms. Species N averaged about 830 organisms, while in the previous 834
2346 5 2. 39.
560 :
s i 3 2. 1408 3 720
0
5
1 lo
I 1 15 20
I 25
I 30
I 35
I 40
I 45
I 50
Time
FIG.
7. Three
interrelated
species.
469
5 2 z. $u)
I
f
1
1;
i
N
169
20 -
1200
M 240
TABLE 7 Initial factors for three species Species L (’
-
d L' ,I
--
Y LiI L; ’ LLZ
-0~0001
-0.0003 -0~0001 -~~ 0 0.00’
-.... 847 -- 290 -~-
164
~-_
57
L];% LY4 I* ( i L,” I.-’
92
-~
29 15 7
L,H
3
LI,"
I
1.2 t.2
I.6 1% I.6 I.2 0.S 0.4 0 0 0.75 0.95 0.95 0.95 0.95 0.92 0.9 0.8 0.7 0.3 0 T.B.
----
---
Species N -~--
Species RI
0.000 I
0.0003 -09001 - 0~0002 0 -0Wl 203
-0.0003 -0~0001 - 0.003 -0,002 200 115 85 50 20 i
134 II5 9s s5
73 56 37
~:!I
-
K, K2 R, R, K,
-
0.8 1 0.8 0.4 0
i
-
0.35 0.5
;2 2
1 -
0.5 0.3 0.2
15
-
0
20 9
4 834 0.5 0.3 0.4 0.4 04 0.3 0.2 0.1 0 0 0.3 0.5 0.5 0.5 0.5 0.4 0.3 0.2 0.1 0.05 0
415
13
TABLE
8
Initial factors for one predator and two prey species Species
L oaOO2 0
Species
N
Species
M
0~0001
0
c
--
tl
---
e 1’
--~-
- omO3 0
!/
L" L;l
-~--
0 847
424
212
-
290
I 50
IS
L;_’ L;’
--
164 92
82
41 2s
LiJ L;5 Lp
---
52 29 15
L;’
-~_
7
3
3
Li8-
3
2
L;'L;;O -
I 0
I 0
2 I
K, -K, KS -K, KS -_ KS CL
K, K8 K, ;I0 ; k ii fl f8
-
-
-;o-
1500
- 0.000 1 - 04002 0 0.00 I
50 2s 15 7
759 1.4
0~0002 - oaIo3 0 -0~001
13 8 4
0 384
0.7 0.9
0.4 0.5
I I
0.5
1.x 1.4
0.9
0.5 0.4
0.9
0.5 0.3
1% 2 2
0.5 0.2 0 0.6 0.95 0.95 O-95 0.95 0.92 0.9 O-8 0.7 0.3 0
0.7
0.1 0
0.5
0.3 0.2 0.1
0.75
0 0.3
0.9 0.9
0.5 0.5
0.9 0.9
0.5 0.5
0.9 0.9 0.S
0.4 0.3 0.2
0.7 0.3 0
0.1 0.05 0
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example it had varied around 280 organisms. Species L fluctuated around 1480 individuals, while in the first case its eventual value was around 2000. Thus the presence of species A4 kept the population of species iV low and allowed species L to reach a higher value. This outcome makes good sense, since the program was arranged to make the predation of species N greater on species L than the predation of species M (Figs 7 and 8).
919
1802
N
,,,,,,,,,I0 0
5
IO
15
L
0 20
25
30
35
40 Time
45
203
M
0
50
FIG. 8. Three interrelated species, one going to extinction.
IO 0
1 5
I IO
I 15
I 20
I1 25 30
11 35 40 Time
11 45 50
0
FIG. 9. One predator, A4, attacks two prey.
0
0
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The next situation considered was that in which one predator, species M, eats two prey, species L and N. There is no relationship between the number of individuals in species L and N. The factors and initial values which were used are given in Table 8. Lotka (1925) reported that it was possible for one prey species to go to extinction in this type of situation. Neither of the prey species in this program became extinct, but they do have rather violent fluctuations. If Fig. 9 is studied closely, it will be seen that the size of species M increases after species L and/or species N become large. When the size of species L and N is small, the population of species M immediately decreases. The many large fluctuations seem to indicate that this population structure is not very stable and that one species could become extinct rather easily. 7. Further Developments and Comments The three-species model can easily be extended to include any number of species. The only limit is the memory size of the computer. On the Alwac III-E there is room to store at least 20 species. The general form of the three-species model would be used.
constitutes a general form of the equations. Here N, stands for the first species, not an age group, NZ stands for the second species, and so forth. The types of relationships between any two species would be determined by the size and sign of the a’s. If two or more organisms reproduce at different intervals this situation can also be handled. Assume one organism has three litters per year while another has only one, The program could be so coded that the computer would go through three cycles for the first species each time it went through one cycle for the second. To consider a more complex case, let us assume one species has three litters per year while another has two. In this case the numbers one and two would be placed so that on each cycle they would be reduced by one. When one of these numbers became zero the computer would calculate the population of the species corresponding to that number and replace the zero with a one or two. Here six cycles would correspond to one year. In some fields, especially fisheries, instantaneous mortality rates are important. As an example, let us assume that a fish population has a natural mortality rate of 25 % and a fishing mortality rate of 30 %. At the end of a year the total mortality rate would not be 55 % of the population at the beginning of the year. The population size would be gradually lowered by natural deaths and fishing during the year. Therefore the 30% fishing
SOME
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mortality would be acting on a different size population at one time than at another time. To handle this type of situation the time intervals would have to be shortened. Since this is a discrete model, only close approximations could be made. The time interval could be taken as a week or a month instead of a year. Here, if the population reproduced annually, the N,, class would be added only every 52 or 12 cycles. Taking the time interval as a day would present difficulties, since the program must go through 365 cycles to cover one year. This could be prohibitive because of the time involved. If a population has a high birth rate with a large infant mortality, the first year could be better handled if it were divided into smaller time intervals, say monthly intervals. Thus the N,, class would go through 12 cycles to become the N, class. After these cycles the rest of the population would go through one cycle and a new N,, class would be formed. Without the above consideration, it would be very difficult to find a satisfactory value for fO. This is true because first-year mortalities are likely to occur at a shifting instantaneous rate. If the sizes of the different age classes in a population are known, or can be estimated over a period of years, this program can be modified to give estimates of other factors. In other words, if everything but d is known or estimated in the equation Ni = Ni- ,[fi- I +cc N+dx M], it can be solved for d. By repeating this procedure over several years and age classes, estimates may be made of compensatory mortality, predation pressure, and other factors. REFERENCES LOTKA, A. J. (1956). “Elements of Mathematical Biology”, 465 pp. New York: Dover. (First published under the title “Elements of Physical Biology”, Baltimore: Williams & Wilkins, 1925.) VOLTERRA, V. (1931). “Lqons sur la theorie mathematique de la lutte pour la vie”, 214 pp. Paris: Gauthier-Villars.
27, 175-195
Predation, Competition and EnvironmentalVariables: SomeMathematical Models VERNA LOUISE ENGSTROM-HEG Department of Mathematics, State University College, Oneonta, New York, U.S.A. (Received 20 February 1969, and in revisedform 9 September 1969) Mathematical models adapted to computer programming were prepared to simulate various ecological interactions. The models were tested on an Alwac III-E computer, using various hypothetical population parameters. Single-species populations in a constant environment became stabilized in number of individuals and age-composition. Prey-predator interactions resulted in damped oscillations in both populations. However, when the effect of climate was introduced as a random variable, the oscillations were no longer damped. Simulations of two species competing on the same trophic level resulted in the extinction of one species or the other. Interactions among three species had various outcomes, depending on the trophic relationships among the species. The two- and three-species models depart from the classical approach in that the age composition of each species is taken into account. The three-species model can be expanded to handle larger numbers of species, and can be modified to handle reproductive cycles of various lengths and to approximate instantaneous mortality rates. If desired, the model can be programmed to solve for unknowns other than the population size. The theoretical study of population fluctuations and the effects of the environment upon them has been handled mainly by differential and difference equations in the past. With the development of digital computers a new approach to this problem is possible. Since interspecific relationships in nature are usually governed by a number of factors interacting in a complex way, any mathematical model that closely simulates nature is likely to involve a great deal of computation. Use of an Alwac III-E digital computer, allowed one to perform rapidly many more arithmetic calculations than would have been otherwise possible, enabling one to write rather elaborate programs. Three different situations have been considered: the case of one species by itself; the case of two species and their interactions; and the case of three species which are allowed to interact with each other in several ways. 12 T.B. 175
176
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1. The Single-species Model
First let us consider the changes in a one-species population over a period of time. The population is first assigned an age structure using the following notation : Ni is the number of individuals in the population which are i years old and were born in the year j. The unit of time, i and j, could of course be in weeks, days or hours if this better fitted the species in question. N’, is the number of young born in a particular year (j). is the average number of offspring an i-year-old organism will 4 produce in the ith year of its life. Since it is a mean value, K may be either greater or less than one. n is the maximum age attained by a member of the species. ” N’,= CKiN~-i=K1Nj,-‘+KzNj2-Z+K3Nj3-3+...+K,N~-”. i=l
The number of individuals one year old or older present at a particular time can be expressed in terms of the number of individuals of the same year class present the year before. Thus
Nj = N/-, where Ni- 1
A-1
a C
fiT1-cf:
NJ6+i-o-1-p-g
a=0
1
is the number of i-l-year-old born in the jth year (i.e. the number of members of thejth year class alive last year). is the probability that an organism that has lived i- 1 years will survive to the next year (the ith year) under condition of no predation or food deficiency and with a normal climate for the area. is age. is a factor representing the effects of overpopulation (compensatory mortality). The model assumes that at moderate levels of population density, mortality will not be significantly affected by density-dependent sources of mortality. Therefore when the population density falls within a designated density range, c is equal to zero. If the population is very sparse, so that the effects of underpopulation (depensatory mortality) appear, this term may be changed to
a=0
SOME 2
n;i+i-a-l
a=0 P
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is the total number of individuals alive in the population in the yearj+i-I. is a factor representing the reduction of the population by predators. It is a function of the size of the predator population. is a factor representing the effects of climate over the last year. It allows us to include the effects of extremes in weather; droughts, floods, hard winters, or by changing the sign, exceptionally favorable weather. In hypothetical situations, g may be set equal to a randomly chosen variable.
If there is evidence that the effects of population density, climate, or predation act more drastically on some age-classes than on others, the program may be easily modified so that different values of c, p or g are used for different age groups. In this case ci- r, pi-r, and gi- i would be used in finding the value of N!. Here a table of c values would be placed in the computer and for the number of one-year-olds the co value (i.e. the effect of crowding on this group of organisms during the previous time period) would be used. For the two-year-olds the c1 value would be used, etc. The same procedure would be used for values of p and g. By use of a jump switch it is possible to put in a different factor for g (climate) each year. This factor can come from a table of random numbers. By moving another switch the machine will stop at the end of any year and changes may be typed into the program. Thus the rate of predation, the effects of over- or under-population, and the basic probability of an organism living to the next year can be changed. The numbers in the year classes can also be changed in this manner, thus introducing immigration, emigration, or density-independent mortality. The program was written so as to keep
between zero and one. Since it is impossible for a negative number of organisms to be present, when this factor became negative, zero was substituted for it. Again when this factor became greater than one, one was substituted for it, since, barring immigrations, it is impossible to have more members in the Ni+r class next year than there were in the Ni class this year. It may be argued that even under ideal conditions all the organisms in one-year class would not live until the next year. By changing the program slightly,
178
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it could be made so that all values of fi-I
-ci
N;+‘-“-Lpm9 a=0
would be replaced by O-95.
1 >
0.95
2. A Model for the Study of Two Species
The two-species model is similar to the one-species model. This model is in somewhat the same general form that Volterra (1931) used, but differs from the Volterra equations in that the populations are separated into age classes, and in that there is a factor for climate and a factor for a predator on the predator. Also, in contrast to Volterra’s model, the changes are assumed to be discrete rather than continuous. The equations which were used are as follows:
N’, = i&N;-‘.
(1)
i=l
N:‘= Ni-, fiel-ci &
1
N~+i-a-1-d*~oM~ti-a-1-9 . a=0
= f KiMi-‘. i=l
(2)
(3)
M:‘=M{-,~i_l-~~M!+~-~-l+a~N~+i-~-l-~-~ . a=0 a=0 I (4) is the number of prey. is the number of predators. is the age of the organism. is the year of birth of the organism. are the number of young born in year j to the prey and predators, respectively. is the average number of offspring an i-year-old prey will produce in any given year. is the average number of offspring an i-year-old predator will produce in any given year. are the probabilities that an organism which has lived i-l years will survive to the ith year, when preypredator effects and climatic effects are not considered.
N
M i
i N’, and M’, Ki 4
J-1 anal.-1
c i
#ti-a-l
a=0
and
account for the effects of overpopulation.
&,f{ti-"-' a=0
n
1
is the maximum possible age attained by the prey.
SOME
m d f
N!ti-a-l
a=0 g and jj P
MODELS
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179
is the maximum possible age attained by the predator. j@ti-4-l
a=0 a i
MATHEMATICAL
is the effect of predation, which is directly related to the size of the predator population. is the effect of the food supply on the predator. are the effects of climate, and can be either positive or negative. is the effect of a predator on the predator if there is one. This factor may or may not exist, and can also be dealt with by putting in a third animal.
Possibly 2 i Nj;+‘-‘-’ should be combined with the f’s, or the d’s varied a=0 from positive to negative as the food varies from a mean value. This will of course depend on how large a factor this particular prey is in the survival of the predator. The same general comments that apply to the one-species model still hold. Climate may be changed each year by use of random numbers. The climate factor for the prey does not need to have the same value as the climate factor for the predator. These two factors may be the same, completely independent of each other, or have some relationship between these two extremes. All the other factors in equations one to four may be changed at the end of each year (cycle). The factors
fi-l-c~~oN:ti-~-l--~~oM~ti-~-l-g] and ji-l-$oMyU-Lfaf: N~+i-~-‘-g-p] a=0 [ are held between zero and one in the manner already discussed. 3. A Model for the Study of Three Species
The three-species model is a general model which can be used for many types of interactions between different species of organisms. It is written in such a way that more than three species could easily be added without changing the form. The basic equations are as follows: (5) fi-l+C~L~+i-a-l+d~ a=0
a=0
N~ti-“-l+e~M!+i-“-l+p+g a=0
1 (6)
180
Ni = Ni-,
V.
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1 (8)
In equations (5) to (lo), the notation resembles that for the previous situations; specifically the three species are divided into age classes. It is assumed that the period from i to i- 1 is the same for all three species, but this period is not necessarily a year. i is the “age” of the organism L{, N{, M; j is the “year” of birth of the organism. indicate the number of young born in a particular “year” L’,, N’,, M’, (time period), in each species. are the average number of offspring an i-year-old member Ki, Ri, Ri of each species will produce during the given “year” (time period). that an organism which has lived 1 are the probabilities will survive to the ith “year”, when the i- 1 “years” other factors such as c C L, d C N, e 2 M, p and g are not considered. These factors allow for the effects of infant mortality and of higher death rates in older animals. c is the effect of overpopulation or underpopulation of c,d, e species L; d and e are the predation effect or beneficial effect of species N and M on species L; c, d and e can be either positive or negative. a is the effect of overpopulation or underpopulation of f, a, G species N; E and Z are the predation effect or beneficial effect of species L and M on species N; E, a and E can be either positive or negative. E,a,i; k is the effect of overpopulation or underpopulation on species M; E and a are the predation effect or beneficial effect of species L and N on species M; F, 2 and Z can be either positive or negative. represent the effect of predation other than that consiP, 8, p’ dered above, or exploitation by man.
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- z 9, 93 9
are factors for climatic effect. They can be positive or negative or changed yearly. g, S and J are not necessarily equal. are the maximum possible ages of individuals in species I, n, m L, N and M, respectively. The same general comments which were given for the two-species model still hold. Climatic variations may be simulated by using random numbers. Any of the other factors can be easily changed at the end of each year (cycle) if so desired. The program may be modified so that different values of c, d, e, p or g are used for different age groups. This may be necessary where climatic variation, population density, and predation have selective effects on the very young and/or the very old. If p represents exploitation of a managed population by man, there may be no predation on individuals below a given minimum age or size class. The restriction that fiel+$
L.+i-o-l+dk
n=o 1 ~i_l+~C~i+i-“-l+;I~~d+i-“-l a=0
i
Ni+i-a-l+ef
Mj+i-a-l+p+g
n=O
a=0 +~j
n=O
f
I Mj;+i-o-1
+jj+g
a=0
and
1
have values between zero and one inclusive, must also hold. 4. Single-species Populations: Results
Using the program for one species, I chose the following values for the different factors and initial values : iv:
-
N;' N;= N;' N;' N;' Nib N;' Iv,' Np9 NC,," XN c
60 55 --- 50 45 40 30 20 10 -2 0 -- 412 -0.00 I
100
.f, J‘, f2 j; f4 f5 ,fb .1; fs f9 .f;o-
-
0.65 0.96 0.96 0.96 0.90 0.80 0.70 0.50 0.30 0.10 0
K, I\'2 K, K, K, K, K, K, K, K,o-
--
0.3 0.3 0,4 04 04 0.3 0.2 0.1 0 0
I'
-
0.002
n
-
0.0003
182
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Without making changes in any factor, the program was allowed to run for 198 cycles (years). It reached a steady population of 498.5, with very little variation from this value (Fig. 1). The age classes in this stable population are given in Table 1.
4100
, , , , I, D20xl405060708090
,
,
,
1
ime
FIG.
1. Development
of a stable population.
TABLE
1
Age classes in stable population
Using this stable population, the values of they’s were changed to simulate an epidemic. The new values for the f’s were: fb fi f2
-
f, f4 fs
-
0.33 0.48 0.48 0.48 0.45 0.40
fh 1; Js f9 .f*o
-~ --
0.35 0.25 0.15 0.05 0
SOME
MATHEMATICAL
100 0
I IO
MODELS
I 20
I 40
I 30
IN
I 50
ECOLOGY
I 60
183
I 70
Time
FIG. 2. Disease attacks stable population
for two years then recovery.
After two cycles (years) with the f’s at these values, they were returned to their original values. At the end of the second cycle the population reached a minimum of 117. Then it gradually returned to its original value of 498. After 86 cycles the population size was 493 (Fig. 2). Next, 500 individuals were placed in the zero year class to simulate the development of a population placed in a virgin territory. It should be noted parenthetically that colonization of newly available territory by juveniles is common in both natural and man-made situations. It is universal among benthic forms having pelagic larvae, and occurs in higher forms such as the beaver and muskrat where non-migratory adults set up restricted permanent home territories. Farm fish ponds and newly reclaimed lakes are usually stocked with young-of-the-year fish. The factors for climate, predation, probability of survival of each year class, and overpopulation were
560 E .@ g. 520 b 3 4
480
2 440 ‘.
400 I IO
FIG. 3. Development
I M
of a population
I x)
I 40 Time
1 SC
I 60
1 70
from 500 organisms in the zero year class.
184
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the same as in the first situation. The population reached a low of 387 the first cycle. Then it rose to a maximum of 609 in the fifth cycle. From there the population gradually returned to the stable size of 498.5. It was 498.97 after 82 cycles (Fig. 3). The population decreased from the fifth to the ninth cycle and then maintained approximately the same value of 562 and 561 in cycles nine, ten and eleven. This plateau is due to the larger number of “births” in cycles four and five which have now reached the older age groups and which act to retard the gradual decrease of the total population (Fig. 4). The values of g were then varied by use of random numbers while everything else was held in the stable condition of the first situation. This, of course, caused the population size to oscillate irregularly.
0
I
I IO
I 5
I 15
I 20
I 24
Tink
FIG. 4. Age composition
of a population
from 500 organisms in the zero year class.
5. Interactions between Two Species In the two-species model, there was a damped oscillation in the number of prey organisms and the number of predator organisms (Fig. 5). In this
829
92 0
5
I IO
I 15
I 20
FIG. 5. Fluctuations
1 25
I 30
I 36 Time
I 40
1 45
in a prey-predator
1430
1 50
population.
SOME
MATHEMATICAL
MODELS
IN
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185
case the constants in Table 2 were used. When the values for d and ~3were smaller the fluctuations were smaller, and if d and a were small enough TABLE
2
Constants for interactions between two species ~~ Prey
Predator
C
-
0~0001
(1
--
0.0003 0.0003
---
1.2 1.2 1.6
.(/ K, K, K, K i, K, K, K, K,
-
1.6
-
1.6 1.2 0% 0.4
K,
--
0 0 0.75
-..
0.98 0.98 0.98 0.97 0.92 0.9 0.8
--0
0.1 0.05
lo2 i:
f3 2
f6 2
7 ii $7 F KL h,7 R, R, R5 R6 x7
----~-
0~0001 0.0003 0.0003 om2 0.3 0.3 0.4 0.4 0.4 0.3 0.2
R!3 -R, -. ,; Lo- -
0.1 0 0 0.3
;;
-
0.5
f3 A
-
0.5 0.5
; h .!l?
--.. ~.
0.4 0.3 0.2
.f .;I fl0
-----
0.03 0.07
-
0
no fluctuations occurred. Whenever the values for the f’s were larger, this species did not show oscillation. Such a situation occurred when the ,f’s were taken as J" .fl J2 j:, f4 fs
-----
0.3 0.6 0.6 0.6 0.6 0.5
f;& f8 :$y,
-
0.4 0.2 0.1 0.05 0
186
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and the values of d and a were both 04001. In this case neither species showed oscillation, but levelled off at a steady value for both species. The effect of climate as a random factor was then studied. Starting with the same values as were used for Fig. 5, the program was allowed to go through four cycles before changing the climate. From then on, for each cycle a different number was used for g and S. Numbers were obtained from a table of random numbers and were coded in the following way : Number
Value for g or 4 0
0~001 oaO2 0404 OGO8 -0GOl -0.002 -0404 -0X08
The number nine was skipped. The first number was used for the prey species, g; the second number was for the predator species, Lj. The third number was used in the next cycle for the prey, and so on. As can easily be seen in Fig. 6, the fluctuations are more erratic and are not damped as
Time
FIG. 6. Prey-predator
oscillations with climate as a random variable.
TABLE
3
Factors and initial values, where two species compete for the same food First species
-c tl rl
-0
N:: ,I.; Ivy" N;" Ni4 N,' N," N;' Ni9 Iv;:0 K, lc2 h’., K, K, h',
-.---~ -.~_
K, I\', K‘, K,, ,fo .f, ,f* .f3 Ik j-5 f6
--.-----
‘j&8
.f,
-
f8 J; s 10
---
04001 OGOOl
165 258 146 83 46 26 14 7 2 1 0 1.2 I.2 1.6 I.6 I.6 1.2 0.s 0.4 0 0 0.65 0.x 0,s 0.85 0.8 0.s 0% 0,4 0.2 0.1 0
Second species 7 II ii ;, %I:; M; ' My z iv;" M;j M;C M;' MT' Mi8 Mq9 M;,'O R,
-~- 0~00005 -~ -0.0002 ~. 0 ~~ 0 -- 750 -~~ 300 - I50 -- 90 -- 60 40 20 -- 10 -3 -~ I 0 -0.X
A2 7 h.: .x K, R,
-~---
.KC> A, 7 A, -7 A, -i h7 1" 10 .f, I2 .I; j4 j5 16 .1; I* .t?, .flO
-.... -.----1 -1 -1 -
0.8 0.9 0,s 0.7 0.6 0.5 0.4 0 0 0.15 0.9 0.9 0.9 0.9 0.s 0.6 0.4 0.3 0.1 0
188
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they are in Fig. 5. (Cycle 2 in Fig. 5 corresponds to cycle 1 in Fig. 6, i.e. the time numbers in Fig. 6 need to be moved one to the right to correspond exactly with Fig. 5.) Next the case where two species compete for the same food was examined. The factors and initial values in Table 3 were used. Since a is negative, an increase in the size of the population of species one has a negative effect on species two, as also an increase in the size of species two has a negative effect on species one. In this situation the population of species two went from 1424 individuals to zero individuals in 16 cycles (Table 4). TABLE
4
Population size, two species competing for the same food Cycle 0 1 2 3 4 5 6 I 8 9 10 11 12 13 14 15 16
First
species 1325 1362 1435 1602 1735 1878 2039 2197 2355 2491 2593 2675 2746 2769 2780 2786 2790
Second
species 1424 1190 1053 829 681 536 400 280 184 112 63 34 11 5 3 1 0
The value of c was changed to OGIO15 and of d to OWO15, and everything else kept as it was above. This time the size of the population of the first species went from 1325 individuals to 4 individuals in 13 cycles (Table 5). Next, the value of d was changed to OWOl and the other factors kept as they were in the second case. Now the population of species two went from 1424 individuals to 92 individuals in 27 cycles (Table 6). The number of individuals in species two formed a monotonic decreasing sequence of terms with each one smaller than the one before. These three mathematical experiments point to the precarious nature of co-existence when two species having very similar ecological requirements are placed in the same environment. In each case, very small changes in the
TABLE
5
Population size, two species competing for the same food Cycle
First
species 1325 924 174 713 651 583 506 422 277 187 110 52 18 4
0 1 2 3 4 5 6 7 8 9 10 11 12 13
Second
species 1424 1191 1316 1408 1523 1667 1848 2070 2528 2885 3288 3722 4146
TABLE 6 Population size, two species competing for the same food Cycle 0 1 2 3 4 5 6 I 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
First
species 1325 1141 1111 1105 1109 1119 1132 1166 1199 1219 1260 1302 1334 1366 1398 1432 1467 1502 1537 1568 1605 1636 1665 1692 1717 1736 1758 1716
Second
species 1424 1191 1141 1113 1088 1062 1031 967 910 871 805 733 686 637 587 537 486 436 389 343 300 259 223 189 160 134 111 92
190
V.
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ENGSTROM-HEG
parameters governing the relationships between the two competing species resulted in one species or the other going to extinction. It would probably be quite difficult to find a set of values that would allow both species to survive under the conditions of these experiments. In a more complex situation where differential effects of climate and predation were introduced co-existence might be prolonged, perhaps indefinitely where changes in these effects tended to favor first one species, then the other. In nature it is most unusual to find two species attempting to occupy closely similar niches in the same territory. In the long run such situations tend to be stabilized either by the elimination of one species or by the survival of mutant forms having sufficiently different ecological requirements. 6. Interactions among Three Species In the first three-species situation species L was eaten by species N and species M, while species M also ate species N. The factors and initial values of Table 7 were used. In this case the sign is important, since all factors are assumed positive unless there is a negative sign. Damped oscillations were obtained in all three species, as shown in Fig. 7. Next, the reproductive rates of the predator species A4 were changed to R, = 0.4, K, = 0*5, R, = 0.4, z, = 0.2, K, = 0, and all the other factors left the same as in the former situation. In this case species M went to extinction. After 112 cycles there were five organisms left, whereas in the former situation the number of organisms in species M oscillated about 1100 organisms. Species N averaged about 830 organisms, while in the previous 834
2346 5 2. 39.
560 :
s i 3 2. 1408 3 720
0
5
1 lo
I 1 15 20
I 25
I 30
I 35
I 40
I 45
I 50
Time
FIG.
7. Three
interrelated
species.
469
5 2 z. $u)
I
f
1
1;
i
N
169
20 -
1200
M 240
TABLE 7 Initial factors for three species Species L (’
-
d L' ,I
--
Y LiI L; ’ LLZ
-0~0001
-0.0003 -0~0001 -~~ 0 0.00’
-.... 847 -- 290 -~-
164
~-_
57
L];% LY4 I* ( i L,” I.-’
92
-~
29 15 7
L,H
3
LI,"
I
1.2 t.2
I.6 1% I.6 I.2 0.S 0.4 0 0 0.75 0.95 0.95 0.95 0.95 0.92 0.9 0.8 0.7 0.3 0 T.B.
----
---
Species N -~--
Species RI
0.000 I
0.0003 -09001 - 0~0002 0 -0Wl 203
-0.0003 -0~0001 - 0.003 -0,002 200 115 85 50 20 i
134 II5 9s s5
73 56 37
~:!I
-
K, K2 R, R, K,
-
0.8 1 0.8 0.4 0
i
-
0.35 0.5
;2 2
1 -
0.5 0.3 0.2
15
-
0
20 9
4 834 0.5 0.3 0.4 0.4 04 0.3 0.2 0.1 0 0 0.3 0.5 0.5 0.5 0.5 0.4 0.3 0.2 0.1 0.05 0
415
13
TABLE
8
Initial factors for one predator and two prey species Species
L oaOO2 0
Species
N
Species
M
0~0001
0
c
--
tl
---
e 1’
--~-
- omO3 0
!/
L" L;l
-~--
0 847
424
212
-
290
I 50
IS
L;_’ L;’
--
164 92
82
41 2s
LiJ L;5 Lp
---
52 29 15
L;’
-~_
7
3
3
Li8-
3
2
L;'L;;O -
I 0
I 0
2 I
K, -K, KS -K, KS -_ KS CL
K, K8 K, ;I0 ; k ii fl f8
-
-
-;o-
1500
- 0.000 1 - 04002 0 0.00 I
50 2s 15 7
759 1.4
0~0002 - oaIo3 0 -0~001
13 8 4
0 384
0.7 0.9
0.4 0.5
I I
0.5
1.x 1.4
0.9
0.5 0.4
0.9
0.5 0.3
1% 2 2
0.5 0.2 0 0.6 0.95 0.95 O-95 0.95 0.92 0.9 O-8 0.7 0.3 0
0.7
0.1 0
0.5
0.3 0.2 0.1
0.75
0 0.3
0.9 0.9
0.5 0.5
0.9 0.9
0.5 0.5
0.9 0.9 0.S
0.4 0.3 0.2
0.7 0.3 0
0.1 0.05 0
SOME
MATHEMATICAL
MODELS
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193
ECOLOGY
example it had varied around 280 organisms. Species L fluctuated around 1480 individuals, while in the first case its eventual value was around 2000. Thus the presence of species A4 kept the population of species iV low and allowed species L to reach a higher value. This outcome makes good sense, since the program was arranged to make the predation of species N greater on species L than the predation of species M (Figs 7 and 8).
919
1802
N
,,,,,,,,,I0 0
5
IO
15
L
0 20
25
30
35
40 Time
45
203
M
0
50
FIG. 8. Three interrelated species, one going to extinction.
IO 0
1 5
I IO
I 15
I 20
I1 25 30
11 35 40 Time
11 45 50
0
FIG. 9. One predator, A4, attacks two prey.
0
0
194
V.
L.
ENGSTROM-HEG
The next situation considered was that in which one predator, species M, eats two prey, species L and N. There is no relationship between the number of individuals in species L and N. The factors and initial values which were used are given in Table 8. Lotka (1925) reported that it was possible for one prey species to go to extinction in this type of situation. Neither of the prey species in this program became extinct, but they do have rather violent fluctuations. If Fig. 9 is studied closely, it will be seen that the size of species M increases after species L and/or species N become large. When the size of species L and N is small, the population of species M immediately decreases. The many large fluctuations seem to indicate that this population structure is not very stable and that one species could become extinct rather easily. 7. Further Developments and Comments The three-species model can easily be extended to include any number of species. The only limit is the memory size of the computer. On the Alwac III-E there is room to store at least 20 species. The general form of the three-species model would be used.
constitutes a general form of the equations. Here N, stands for the first species, not an age group, NZ stands for the second species, and so forth. The types of relationships between any two species would be determined by the size and sign of the a’s. If two or more organisms reproduce at different intervals this situation can also be handled. Assume one organism has three litters per year while another has only one, The program could be so coded that the computer would go through three cycles for the first species each time it went through one cycle for the second. To consider a more complex case, let us assume one species has three litters per year while another has two. In this case the numbers one and two would be placed so that on each cycle they would be reduced by one. When one of these numbers became zero the computer would calculate the population of the species corresponding to that number and replace the zero with a one or two. Here six cycles would correspond to one year. In some fields, especially fisheries, instantaneous mortality rates are important. As an example, let us assume that a fish population has a natural mortality rate of 25 % and a fishing mortality rate of 30 %. At the end of a year the total mortality rate would not be 55 % of the population at the beginning of the year. The population size would be gradually lowered by natural deaths and fishing during the year. Therefore the 30% fishing
SOME
MATHEMATICAL
MODELS
IN
ECOLOGY
195
mortality would be acting on a different size population at one time than at another time. To handle this type of situation the time intervals would have to be shortened. Since this is a discrete model, only close approximations could be made. The time interval could be taken as a week or a month instead of a year. Here, if the population reproduced annually, the N,, class would be added only every 52 or 12 cycles. Taking the time interval as a day would present difficulties, since the program must go through 365 cycles to cover one year. This could be prohibitive because of the time involved. If a population has a high birth rate with a large infant mortality, the first year could be better handled if it were divided into smaller time intervals, say monthly intervals. Thus the N,, class would go through 12 cycles to become the N, class. After these cycles the rest of the population would go through one cycle and a new N,, class would be formed. Without the above consideration, it would be very difficult to find a satisfactory value for fO. This is true because first-year mortalities are likely to occur at a shifting instantaneous rate. If the sizes of the different age classes in a population are known, or can be estimated over a period of years, this program can be modified to give estimates of other factors. In other words, if everything but d is known or estimated in the equation Ni = Ni- ,[fi- I +cc N+dx M], it can be solved for d. By repeating this procedure over several years and age classes, estimates may be made of compensatory mortality, predation pressure, and other factors. REFERENCES LOTKA, A. J. (1956). “Elements of Mathematical Biology”, 465 pp. New York: Dover. (First published under the title “Elements of Physical Biology”, Baltimore: Williams & Wilkins, 1925.) VOLTERRA, V. (1931). “Lqons sur la theorie mathematique de la lutte pour la vie”, 214 pp. Paris: Gauthier-Villars.