chemistry and the environment
Applying the Principles of Chemical Kinetics to Population Growth Problems Gerhard F. Swiegers Vista University, Private Bag X613, Port Elizabeth, 6000, South Africa It is interesting, particularly o' m an educational point of view, to note the many analogies between human and chemical behavior. Numerous technical terms or phrases are commonlyborrowed from chemistry to describe human behavior. Thus, people-like chemicals--choose the "path ofleast resistance" or "bond" when the "chemistry"is right. The mathematical modelling of population dynamics in communities of living organisms has been widely treated by both ecologists (1-5)and population geographers (68) (for human communities). The models used are generally statistical or matrix in nature. In the case of chemical kinetics, some authors have cited the connection between the principles governing rates of chemical reactions and those affecting other rate processes, such as oscillating humanlanimal population problems (9.10). Net Population Growth The Simplest Process
Comparison of Developed and Developing Countries Making the arbitrary assumption that these are firstorder processes (m, n = I), a comparison of the population growth between a developed and a developing country can be made. Uganda serves as an example of a developing country. The average life span is about 50 years The average woman will have six children during her lifetime (or 3 childrenlparent).
kb = 0.060 yF1 A plot of an initial sample of 100 people under these eonditions shows the population to grow to 111 (developed) and 5,460 (developing)within 100 years (Fig. 1). Success of the Process
In this study an application of the principles of simple chemical kinetics to the prediction of animate population growth is made with some interesting and educational results. To use these principles, the processes affecting an animate population must be schematically drawn as analogous to the writing of a chemical reaction. The simplest process would consider only two factors: the overall birth rate and the overall death rate.
kd P-tOP (death)
These very rough predictions are actually quite accurate, although one should not be too surprised. Population geographers, in essence, use the same method for predicting populations: Subtract the standardized death rate from the standardized birth rate. (These rates are not age- or sex-specific (12).) The model suggests a population doubling every so many years. 696 years (developed country) 17.5 years (developingcountry) Population geographers, by comparison, expect Britain's population to double in 700 years (11,121and Uganda's in 20 years (13).
where ka is the rate eonstant for death; kb is the rate eonstant for birth; and P represents a person. The net population growth will be
Initially: 100 People
6000
I
where m and n are the rate orders for the birth and death processes. To accurately solve this equation, empirically determined values for the variables must be used. The k b and kd values can be calculated most easily using widely available statistics. Given the British life expectancy of about 75 years, we get Developed Country
The average couple in Britain produces 2.14 children in their lifetimes or, statistically, one child every 70 years per parent. Thus, we get
0 0
20
40
60
80
100
120
Time (Years)
Figure 1. Predicted population growth: developed and developing country. 364
Journal of Chemical Education
Limitations of the Process But the processes described above are too generalized to allow for the prediction of the k values and rate orders from first principles. The birth rate can, for instance, be considered to be first-orderwith respect to the number of reproductively praductive married couples (in a strictly monogamous society) or second-order in total, that is, first-order with respect to the re~roductivelv male ~ooulation and first-order . oroductive . .. wl;h respect to the rrpmdunwrly p r o d u e r w frmalr populatam (mdlcatmga totally pmmtscuuus soriety) Sudden Death Model Moving to a more fundamental schematic, consider a model that differentiates between reproductively productive (P*) and unproductive (P) members of a society The young unproductive members mature a t a fixed rate into productive adults. During reproduction each couple may produce one, two, or three young. The reproduction step is accompanied by the death of the parents. kl P+P'
kz P*+ 0.5 P
k3
two pesticides. One reduces the population of unproductive young, but does not affect the adults, whereas the other reduces the population of adults, but does not affect the young. Assume also that an application of each pesticide maintains the affected population at low and constant levels for an extended period of time. Choice of Pesticide Under these circumstances, the best choice of pesticide would depend on the rate a t which immature grubs mature into adults (i.e., kl) and the relative rates a t which reproduction (and death) occur. The exterminator must hope that the reduced population will be so low that diffision control sets in, and the population finally dies out. If this does not occur, the total population of bugs will increase despite the low population of adults or young. The effect of the time required for maturing from grubs to adults can then best be illustrated by considering a situation in which the total bug population grows despite an artificially low grub or adult population. For example, in a colony of bugs initially comprising 100 adults and 100young, the "&-killer" is used to maintain the population of young bugs a t 100. Then the total bug population will still increase from 200 to 1,450 in one year, provided that
P'+lP
0.5 k2 = days 70
k4
k S = l days
P'
1.5 P
For simplicity, the reproductive steps in each of the models has been abbreviated as shown below. For the step involving k z above P'+P*+IP
becomes 2pt+1p
which is then reduced by dividing throughout by 2 to give P*+ 0.5 P
From this scheme, we get
Thus, the total population growth reduces to
k,=-
70 1.5 days 70
If one chooses the "adult killer" and the rate of maturation (i.e., kl)is 1/40 days, the total population will increase to only 556 in one year. However, if the adult killer is chosen, given a kl of 1/20 days, the total population will grow to 2,190 in one year. Clearly the adult killer would have been better in the former case, but the grub killer in the latter. Figure 2 shows the effect on the total insect population of the adult killer and the grub killer over a number of years under the conditions discussed above. These graphs, and those of Figure 1 match the expected increase in the population of a group with high survival probabilities as described by Leslie (3) and Poole (4). Adult Population kept at 100 20 Dav Maturine 40 Day Maturing
Applying the steady state approximation, we get
Thus,
Similarly, if [PI is considered constant one arrives a t
Population Control of Insects
This exercise leads to some interesting insights into the world of bug extermination. Assume one has the choice of
Time (Years)
FfgLre 2. Pred cled popu ation growth in an insect colony in tial y conssttng 01 200 insects (100 adults. 100 grdbs). Volume 70 Number 5
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365
The Windshield Effect
B r t h Rate 0 0143/[pop] Death Rate 00133
Then what prevents a n overpopulation of insects on the planet? Obviously a crucial step in population control was omitted from the kinetic scheme: Both productive and unproductive members of the species may die without repmducing. When we include this "windshield effect", we get the following scheme. kl P + P'
kz P'
+ 0.5 P
k3 P'+ 1 P
kd P + death
[/yr/prsnl
I/Y~
\
loo
k4 P'-+ 1.5 P
kd P'
+ death
Thus,
The population growth will become zero when
1
kd[P]= 0 5k4 0 5k2 - kd [P*]
( . - .
(9) Encounter Model Arefinement of the above model can be made to differentiate between males (m, Mj and females (f, Fj. Reproduction in such a model would then involve a "collision", or more accurately, in the case of humanlanimal repmduction, an "encounter" of a productive male (MI with a productive female (Fj to produce an immature male (m) or female (0 without the death of the parents. kl
kl m-+M
kd
m + death
f+F
kd M +death
kd
f + death
kd
F + death
and
where m represents a n immature, unproductive male; f represent a n immature unproductive female; M represents a productive adult male; and F represents a productive adult female. The reproduction step is second-order, so the value of kz must have units of yr' person-'. The best solution seems to be to consider k z to be
.
F ~ g ~3r ePreotcteo populanon oechne in a colony ln~tiallyconslstlng of 99 ad~ltwomen an0 t adult man
and
Figure 3 shows the decline predicted in an initial population of 99 adult women and 1 adult man. Figure 4 illustrates the total population change in a group consisting initially of 50 males and 50 females. When all the males and females are adults, the decline in the total population is less severe than when all the males and females are children. Interestingly, the decline for 50 boys and 50 girls will occur more rapidly than under the conditions described in Figure 3. This is due to the 20 years required for the children to become productive. After that, the rate of decline is much the same as the adult group. The shapes of these graphs are consistent with the empirically determined "survivorship curve" developed by Rabinovich (14).
Inadequacies There are obviously some inadequacies in this simplified scheme. According to the model, there is a n equal likelihood of producing a male or female when reproducing, and Birth Rate: 00143/[popl Death Rate 0 0133
j/yr/prsnl
I/Y~
100
k6 tatal population
50 Men. 50 Women 0 Children
where k i is a first-order wnstant For a couple having two children in 70 years, k 2 bewmes 2 k2=y~'. person-' 70 total population
--
0.0286 *I.,, total population
erson - ~
The value of kl is taken a s 1/20 years = 0.050 y r l . Thus,
a 2 40 50 Boys, 50 G w l s
o
C
0 Adults 20
0
Consider a situation in which 100 adults and no children are present. Then, using initial rates, we see that a population with 64 of one sex and 36 of the other will be static, for 366
Journal of Chemical Education
1
0
10
20 30 Time (Years)
40
Figure 4. Predicted population decline in a colony initially mnsisting of 50 males and 50 females (either allchildren or all adults).
males and females mature into repmductively productive adults at the same rate. A ~ooulation comorisine 99 adult women and 1 adult . . man is predicted to decline as fast (-1.27 persons. yr-'I as one comorisinr 99 adult men and 1 adult woman. (The scheme does l o t take into account a pertinent fact: Althoueh a successhl encounter between a male and female adult may produce a child, the female loses her repmductive capacity during the eestation period and for a time immediate?; after the birth itself.) Also the scheme does not allow for the possibility that the one male or female in the group of 100 will die. This is a statistically possible event: Adult males and females die with a rate constant of k d . Should this occur, the predictions made according to this scheme would be incorrect.
Refinements More complicated but possible refinements of the model would be rewired to take these factors into account. The need to apply these schemes and equations, and indeed to apply the principles of chemical kinetics, to much larger sample groups also becomes obvious. As more refinements are added, the models would become increasingly more complex, leading eventually to the results (and complexity) of the ecological models described earlier. Conclusion
Exercises such as these models potentially allow us to predict the ~ o ~ u l a t i o erowth n of a e r o u ~ without reauirine that we foiiod the popkation change i f that gmupbver oeriod of time (as used in statistical models). Thus. the growth in population of, for example, a new species of bacteria can potentially be predicted simply by drawing up a detailed and accurate model of their life cycle and then ap-
plying the principles of chemical kinetics as demonstrated above. Experimental All graphs were derived from the equations shown. Equation 1was used to produce Figure 1, and eqs 6 and 7 were used for Figure 2. Equations 10 and 11and other related equations were used for Figures 3 and 4. The data for Figures 3 and 4 was calculated using a BASIC program that sequentially calculated new populations using the initial-rates method. Copies of the programs are available on request. Acknowledgment The author wishes to thank Professor Andrew Garton of the Chemistry Department at the University of Connecticut for his comments and assistance. Literature Cited 1. JeBern, J. N. R. Roetiti0"wa Honibaok on thp M&lli* of @nornrc change in E w s t e m ; John Wiley and Sona: Chicbester, 1988. ; M. S.; 2. The Matkmllcol Theory ofthellparnics o f B i o l a g i e d P o p ~ h t i 6Badett, Hiom, R. W,Eds.;Arademic Re-: London, 1973. 3. Leslie, P H. Blomptdo 1946.33, 181;Biomolriko 1918,35,213;Biomtri!a I%%, d.-, i if
4. Pwle R. WAn Intmdvctiovctioto QwnIiMiwEmlogy:MeGraw-Hil1:NewYaL. 1974. 5. Ludrip, J.k: FWnold8. J . F. Slotistiml Em10gy:A P I i w ; John Wdeyand Sons: NeviYmk. 1988. R. W.: Huggeh, R J. Modelling in CPogmphy:A MotlumolImlAppmh; 6. Tho-, Harper snd Row:Londan, 1980. 7. Hugget&R.Sy#twmAMlysis in Cmgmphy;Claren&n: Oxfoxfrd.1980. Helm: 8. Goiledge, R. G.; Stimson. R. J. Anolytid Bahoviouml Gwgmphy; C-
a 1976 12 Jonc8.H Populalron Cmpmph,: Wul andChapman landon. 194) 13. MrlGow7.T The M,&m Rcr~ofPopWonm.Edvardh o l d London, 1977 I4 Ral~lanmrh.J E (\nn Enromol. b c Am 1970 :I. 749
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