An introductory world food model

AN INTRODUCTORY W. Lehigh

University,

WORLD

MODEL

E. SCHIESSER*

Bethlehem,

(Rrcriord

FOOD

Pennsylvania

15 Novernhrr

18015. U.S.A.

1974)

1. INTRODUCTION

the present perspective of recent history, the year 1970 appears to have been pivotal, even to the degree that the world’s society has entered a new era. No doubt the tendency to view current and recent events as unique is a natural human trait; yet many observers of recent events have an uneasy feeling that a fundamental change has taken place with possibly ominous implications for the future. Not the least of the many manifestations of a significant change are the recurring reports of actual or impending scarcities in many of the basic physical elements essential for human life. No doubt any statement about “energy shortages”, “pollution crises”, “natural resource shortages” and “food shortages” would be anticlimatic, yet we reluctantly point out once again that these problem areas appear to be increasingly urgent and also appear to be long term. Thus these problems are appropriate topics for discussion at all levels of education since they will almost certainly be an important part of the foreseeable future, probably for the lifetime of most young people. In particular, inadequate worldwide production and distribution of food appears to be an increasingly critical problem; even with present-day highly efficient methods of food production which are based on the latest developments in technology, food experts are gravely concerned about the possibility of worldwide famine, or at least large-scale famine in some of the world’s most populous nations. To support this contention, experts point to the current food reserves which are at the lowest level in many years. A combination of low reserves, restricted supplies of energy and vital natural resources required for scientific farming, a world population growing at 2”/; a year (i.e., a doubling time of only 35 years) and the possibility of adverse weather conditions could come together simultaneously to provide a critical food shortage with devastating consequences for a major part of the world’s population. Clearly the world food problem is serious and warrants a better understanding by the younger generation which will face this problem in the years ahead and which will have the responsibility of developing orderly solutions to the problem. To this end an introductory world food model and computer simulation have been developed for use in education. The principal features of the model are outlined in the remainder of this paper. The food model is based on a simplified analysis of the dynamics of food and population changes with time. Consideration of the food/population interaction is limited to a single question: Is there enough food available at any given time to support the world’s population or is the food supply inadequate so that higher death rates (i.e., famine) and reduced birth rates bring population into balance with available food? Technically, the model is based on a differential equation for population, with the population derivative, dP/dt, determined by the difference in birth and death rates which in turn are determined by available per capita food. The model does not differentiate between various regions or countries of the world, i.e., worldwide average birth and death rates and per capita food are used within the model. Likewise, economic conditions, availability of energy and natural resources, agricultural chemicals and equipment, transportation systems and other factors which have a major influence on food production are not consideredt, i.e., for whatever reasons, either adequate food is available or compensating changes in birth and death rates must occur; in other words, the consequences of the existing food situation in terms of population, birth and death rates are defined by the model equations over time. The model can therefore be considered as a computerized Malthusian analysis which deals primarily with the Earth’s carrying capacity for population in terms of food.

From

*This work was supported in part by a National Science Foundation Faculty Fellowship. Grant No. 62-3767. t Several excellent surveys of the world food situation have appeared recently [l-S] which analyze the many complex interacting factors contributing to the current situation and the prospects for the future. 99

100

W. E. SCHIESSER

Within

the model, three basic relationships must be defined by the user: per capita birth rate as a function of available per capita food. (2) Worldwide per capita death rate as a function of available per capita food (3) Available per capita food as a function of level of population. These functional relationships are specified as tables of numbers. Values are provided with the program which are discussed in two subsequent sections of this paper. Basically these values reflect: (1) current worldwide birth and death rates corresponding to current available per capita food. and reasonable extrapolations of the birth and death rates to conditions of food abundance (i.e., higher and lower birth and death rates respectively) and food scarcity (i.e.. lower and higher birth and death rates respectively), and (2) available per capita food which declines with increasing population for three cases corresponding to food production constant at the present rate. increasing (optimistic) and decreasing (pessimistic) projections of food production with increasing population. These relationships are based on the best estimates and data available to the author, but should be refined through consultation with experts in nutrition and agricultural production. In fact, validation of the tabular functions would be well suited for student research. The program is completely llexible with respect to the specification of these functions so that they can even be defined as a function of time to simulate conditions of adverse weather, short-term energy shortages. significant breakthroughs in agricultural technology, improved health conditions, etc., as these situations might affect per capita food production and world birth and death rates. Next we consider the overall structure of the model and some of the details of implementation.

(1) Worldwide

2. BASIC

The principal

components

STRUCTURE

of the model

THE

are summarized

in Fig. 2.1.

World avera e birth rote (I

L -

MODEL

and their interconnection

BRFM(t) Inltlal populotlon PN

OF

3

World population

P(t)

Y(‘) c Per capita production

food (3)

Wor Id average death rate(2)

Fig. 2.1 Simplified diagrammatic representation of the food model: (1) BRFM vs FR relationshiu: (2) DRFM vs FR relationshio: (3) FR vs P relationshio. BRFM birth-rate-from-food multipiier‘ (births/person/year); DRFM ‘death-rate-from-food rkultiplier (deaths/person/year); FR food ratio (per capita food); P: world’s population (people): PN: initial population = 4 x 10”. t = 1975 (people); t: time (calendar year).

Computer execution of the model begins with the specification of an initial population, e.g., P = 4,000,000,000 (4 x 109) people, t = 1975. This initial condition in turn defines the initial per capita food, taken as a unit value, FR = 1, and initial average per capita birth and death rates. BRFM = 0.030 and DRFM = 0.010 persons/person/year which correspond to present worldwide values.* Simultaneous solution of the model equations (or more precisely, numerical integration of the population differential equation) moves the model through time. A series of variables can be printed and plotted vs time, for example, P vs t for 1975 7 t 7 2075. Changes in the model relationships are easily made either initially at the beginning of a run, or at a specified point in time within a run on the computer. Multiple runs of the model within a single submission to the computer to investigate, for example, the effect of parameter changes is accomplished with multiple sets of data cards. The program is therefore structured to facilitate interactive computer simulation.

3. FOOD

PRODUCTlON

vs

POPULATION

RELATIONSHIP

The central mathematical relationship in the model is the per capita food available as a function of the level of population. This relationship is depicted qualitatively in Fig. 3.1 as FR, food ratio, vs P/PN, normalized population. The current (initial) condition is FR = 1. P/PN = 1. i.e.. unit per

*The worldwide birth and death rates have been reported by the Population Reference Bureau [9] as 0.033 and 0.013 persons/person/year respectively for 1973. However, the values 0.030 and 0.010 used in the present study, which were selected to simplify the tabular mathematical functions BRFM and DRFM, will give the same numerical results since the difference between the birth and death rates (0,020 initially in both cases corresponding to the present 2”,, growth rate) determines the rate of change of population.

101

An introductory world food model

capita food at unit normalized population where, for example, PN = 4 x 10” people. As population increases so that P/pN > 1, the world’s agricultural system is strained and eventually population may increase to such a high level that available per capita food begins to decline, i.e., FR < 1. This condition will decrease births and increase deaths so as to slow the rate of growth of population.

FR

I

P/PN

Fig. 3.1 The food-population

relationship with the initial condition FR = 1, P/PN = 1 indicated

The program is set up to run three cases of 3.1 World food production is constant at the proportional to population, i.e., FR = l/(P/PN). 3.1 which is part of the actual printout of the

the relationship in Fig. 3.1. current level so available per capita food is inversely This relationship is shown in tabular form in Table computer program.

Table 3.1 FR vs P/PN for the case of constant current world food production P/PN 0

0.25 1.25 2.25 3.25 4.25

L.50 1.50 2.50 3.50 C.50

G.75 1.75 2.75 3.75 4.75

2.0000 . 6667 .4OJ3 .2957 .2222

1.3330 .5714 .3636 .2667 .2105

1.00

2.00 3.00 it.iJo 5.00

FP L.rJOOO .8000 .44*4 .3077 .2353

10.0000

1.0000

.5000 .3333 .2500 .EOOO

Note that in general the elements of FR array are the reciprocal of the corresponding elements of the PipN array. The one exception is the first pair of elements, i.e., FR = 10, P/PN = 0, which is a limiting condition corresponding to ten times current per capita food at zero population. The tabular function of Table 3.1 can therefore be visualized as one special case of the general population/ per capita food (i.e., FR vs P/PN) relationship of Fig. 3.1. Case 3.1 represents a relatively pessimistic situation since world food production will increase, at least to some extent, through improved farming practices and technology. 3.2 Per capita world food production remains constant at FR = 1 until P/PN = 2 (i.e., twice the current world population), then declines at a rate somewhat less than inversely proportional to P/PN. This relationship is shown in Table 3.2. Table 3.2 FR vs P/PN for the case of increasing world food production for PjPN > 2 P/PN 0

0.25 1.25 2.2s 3.25 4.25

u.so 1.50 2.50 3.50 s.50

0.75 1.75 2.75 3.75 4.75

I.@0 2. oc 3.00 4.co 5.00

1.3330 1.0000 .9250 .75co .55ao

l.OOLO 1.0000 .9000 .7000 .5oco

F4 1l.00"0

~.OOOO 1.0000 .9?50 .8500 .65ilO

2.ILiDO 1.00115 .9550 .a000 .bOOO

Note that for low levels of population, P/PN < 1, per capita food, FR, is inversely proportional to P/PN which is probably conservative, e.g., if the population is half its present level, P/PN = 0.5, per capita food would probably be at least twice its current value, FR = 2. As population increases

beyond its current level, PjPN > 1, food production keeps pace so FR = 1. At P,!PN = 2. the limit of this productive capacity is reached and for PiPN > 2. FR < 1. However, the rate of decline of FR is less than inversely proportional to P/‘PN, e.g., with P/PN = 4 (four times the current world population), FR = 0.700 rather than 214 = 0.5. Thus this case represents a relatively optimistic situation in the sense that total food production keeps pace with increased population to relatively high levels of population (i.e.. P/PN = 2) and even continues to increase beyond P/PN = 2. but at a rate which cannot sustain per capita food production at FR = 1. 3.3 Per capita world food production remains constant at FR = I until P:PN = I.75 (i.e.. I.75 times the current world population). then dcclincs at a rate somewhat higher than inversely proportional to P/PN. This relationship is shown in Table 3.3. Table

3.3 FR vs P;PN for the case of decreasing

wjorld food production

for P!PN > 1.7.5

P/PN 0

6.25 1.25 2.25 3.25 4.25

U.50

1.50 7.50 3.50 4.50

0.75 i.7< 2.75 3.75 +.75

1.00 2.co 3.00 4.00 5. DO

FR 10.0000

Ir. 3055 l.CO70 .8011(1 . 45dC

.2500

7.0030

1.0030 .7GOO .+OO? .ZOJO

1.3330 1.0000 .6OciO .35co .1500

1.0000 .-Jo00 .5orlJ .3000 .lGOO

Note that again for low levels of population, P;PN < 1. per capita food. FR. is inversely proportional to P/PN for the same reason as in case 3.2. As population increases beyond its current level, P/PN > 1, food production keeps pace so FR = I. At P/PN = 1.75. the limit of this productive capacity is reached and for PI’PN > 1.75. FR < 1. The rate of decline in this case is somewhat greater than inversely proportional to P/PN. e.g., with P!PN = 4, FR = 0.300 rather than 1,75/4GI = 0.4375. This lower level of food represents a relatively pessimistic situation which could result, for example, from impaired food production at higher levels of population due to overcrowding, excessive pollution, depletion of soil and essential natural resources and insufficient supplies of energy. fertilizers and pesticides to sustain high agricultural production rates. In other words, for P/PN > 1.75, total food production declines because of the reduced capacity of the world’s agricultural system, and therefore per capita food production is less than inversely proportional to population. Obviously the FR vs P,/PN function is a fundamental relationship which characterizes the ultimate carrying capacity of the world’s agricultural system. The three cases which are part of the program were selected to illustrate important limiting conditions. They can, however. very easily be modified by changes in the program data cards and should. in fact, ideally represent a consensus of opinion of agricultural experts. Clearly a critical aspect of this relationship is how much food production can be increased in an attempt to keep pace with expanding population. e.g.. will the curve of Fig. 3.1 break at P,/PN = 1, 2 or 1.75 as in cases 3.1. 3.2 and 3.3 respectively, and when it does break how rapidly will it decline? These are factors which probably cannot be specified precisely even by agricultural experts, but which are of critical importance for the future of mankind as long as population continues to increase. Therefore, even semiquantitative estimates of these characteristics of ultimate world food productive capacity are well worth additional research for refinement and validation. The critical breakpoint in the FR vs P/PN relationship in the neighborhood of P,/PN = 2 is suggested by a number of agricultural experts. For example 171. “Many agricultural scientists feel that if these reqtnrements (generally assistan e to the farmers of underdeveloped countries) are met, it should he possible to more than double ~Coodproduction in underdeveloped countries using technology that is now ayailahle.” “A key problem over the long run 1s the rate of population growth. While agricultural progress over the last l&IS years has been spectacular. the growth in population has largely wiped it out.” “Whether fertility can be controlled in time to abert chronic widespread famine IS a matter of wide debate. There is general agreement that while a rising standard of living based on agricultural advances tends to reduce the birth rate, the effect probably is not rapid enough in the more populous countries to control numbers before famines exact devastating tolls.”

The FR vs P./PN relationship in Fig. 3.1 is a quantitative description of this race between growth in population and food production with the outcome characterized by the breakpoint after which per capita food, FR, declines with increasing population, and by the slope of the curve after the breakpoint signifying the rate at which food production falls behind population growth.

An introductory world food model 4. BIRTH

AND

DEATH

RATES

IO3

vs FOOD

Time variations in the available per capita food, FR, will in turn set the worldwide birth and death rates, at least in a Malthusian-type analysis. Of particular interest at the present times is the situation of food scarcity, FR < 1, when near-famine conditions will lead to decreased and increased birth and death rates respectively. Table 4.1 BRFM vs FR FR 0

0.125 0.625 1.125 1.625 2.125

0.750 0.750 1.250 1.750 2.250

0.375 0.875 1.375 l.R75 2.375

0.560 l.GOO 1.500 Z.OCO 2.500

.002u .0150 .0350 .0525 .0690

.0030 .0200 .n400 .955c .ObOO

.0050 .a250 .Li450 .c575 .ObCO

.OlOO .0300 .05co .ObOO .ObOO

BRFHl .OOlO

The per capita birth rate, BRFM, which has been assumed as a function of available food, FR, is indicated in Table 4.1. Of particular interest is the portion of the table for FR < 1 (food scarcity) for which the birth rate is assumed to drop sharply from the current condition of BRFM = 0.030, FR = 1. For example, for one-quarter of the current food supply, FR = 0.250, the birth rate is one-tenth of the current value, BRFM = OGI3. At the other end of the table corresponding to food abundance, FR > 1, the birth rate is assumed to approach a limiting value of 0.060; this would correspond to a “saturation” condition for which food is so readily available it presents no serious problem and birth rates are therefore determined by other factors such as economic conditions, pollution, crowding, etc. The maximum biological birth rate is approximately 0.088, and rarely have birth rates above 0.06 been observed [lo]. The death rate, DRFM, as a function of available per capita food, FR, is given in Table 4.2. Table 4.2 DRFM vs FR FR 0

1.125

0.250 0.750 1.250

1.625

1.750

2.125

2.250

0.375 0.875 1.375 1.875 ?.375

.9000 .I500 .u L9(I -0055 .co35

.5000 .OBllO .ooso .0050 .0030

.3500 .0300 .GO70 .GO45 .0030

0.125

C.625

ORFWT 10.G000

0.5co

.2400 .OlOO

1.000 1.500 2.000 2.51*0

.OUbU .0040 .0030

Again the section of major interest corresponds to food scarcity, FR < 1. Death rates increase very rapidly as available food declines from the current condition DRFM = 0.010, FR = 1. With only one-half the current food available, FR = 0.5, the death rate is DRFM = 0.240 which is 24 times the current value. When FR = 0, DRFM = 10 indicating that if no food is available, 10,000 deaths/l,000 persons/year or roughly 1,000 deaths/l,000 persons/month occur so that essentially the entire population dies in one month. At the other extreme of food abundance, FR > 1, DRFM = OGO3 as a limiting condition. Again food has no significant effect on the death rate which is determined by other factors (e.g.. old age, disease, accidental deaths). Again the birth and death rate functions of Tables 4.1 and 4.2 are considered reasonable, semiquantitative estimates, but they should be reviewed by nutritionists who have some idea of the effect of available food on birth and death rates. Recent statements give some indication of what form these relationships may take. For example [ll]. “There is some evidence indicating that where death rates go up because of starvation, there is a compensating rise in birth rates.” Presumably as couples observe higher death rates (i.e., higher DRFM) for their children from starvation (i.e., lower FR), they have children at an increased rate (i.e., higher BRFM) with the hope that at least some will survive. Thus the BRFM-FR relationship may actually have a maximum in the interval 0 7 FR 7 1 since it must pass through the point BRFM = 0, FR = 0 (i.e., births without food, on the average, are impossible).

IO1

W. E. SCHIESSER

In any case, experimentation with the tabular functions of the model to investigate these important alternatives and to assess their implications ability of food and the growth in population. 5. REPRESENTATIVE

MODEL

provides the opportunity with respect to the avail-

OUTPUT

Once the three tabular functions FR vs P/PN, BRFM vs FR and DRFM vs FR are specified, the model is completely defined for a given initial condition, e.g., P = 4 x 109, t = 1975. The model equations can then be solved numerically on a digital computer and the output displayed as tables of numbers and as plots. Some representative model output is given in Figs. 5.1, 5.2, 5.3 and 5.4. Figure 5.1 is a plot of world population, P, vs time, t, for 1975 7 t 7 2075 for the three cases of the FR vs P/PN function considered in Section 3. Note that in all three cases, population reaches a limiting level, printed in the caption, as per capita food declines so that the birth rate is balanced by the death rate. This latter condition is indicated in Figs. 5.2 and 5.3 which show the decreasing and increasing birth and death rates respectively until the two eventually equal 0.0260 births (or deaths),iperson/year. This zero population growth condition (ZPG) results from declining per capita food plotted in Fig. 5.4 which reaches a limiting value of 0.900 (i.e., according to the model only a 10”~; decrease in per capita food will impose a ZPG condition which may not be unrealistic since so many people in the world now survive on a near-starvation diet). The 0.900 figure is in reasonably close agreement with two recent statements 16,121. “According to U.N. estimates, Chinese (on the mainland) get only 91”; of their caloric requirements.” “The developed countries as a whole had 23”” more food-energy supplies available than were required, but the developing countries had five percent fewer supplies than were required.” In addition to the population, birth rate, death rate and food variations with time shown in Figs. 5.1 to 5.4, the time scale for these plots is of interest. For example, Fig. 5.3 indicates sharply increased death rates from insufficient food starting in 2010 (only 35 years from now) even for the optimistic case 3.2 (i.e., curve 2). The point at which this worldwide famine condition would occur is, of course, dependent on the tabular functions which have been assumed. Note also that the additional area under the DRFM vs time curve between two tunes gives the total number of deaths from famine for that time interval.

An introductory

world

cccccccccccccc:zzz

.IIOE-llt

..‘....,....‘.........‘.........’.........’.........‘.......;;~;........‘.......;;~;........’.......;;~;. 1915.

1995.

2015.

Fig. 5.3

food model

;:

Fig. 5.4

If the functions of Tables 3.2. 4.1 and 4.1 are reasonable. Fig. 5.3 indicates that not much time is left to curb population growth and/or sharply increase world food supplies. This result does indicate that the refinement of the tabular functions to rellect the best current expert opinion is urgent. Additionally, a sensitivity analysis to determine how much the model output changes with variations in the tables would be worthwhile. The author intends to undertake this research. and would welcome the opportunity to work with other educators who may be interested in using the model and computer program (v. Section 7). 6. SUMMARY

The preceding discussion indicates that the introductory world food model is based on a simplified. aggregated. first-order analysis. The simplicity of the model is requisite for use in education. yet the model is considered to be sufficiently detailed to provide at least semiquantitative estimates of the ultimate carrying capacity of the Earth with respect to food. For example. the model ia semiquantitatively in agreement with the results reported by Frejka[l.i] which are based on a more elaborate demographic model. Refinements of the relationships between (1) birth and death rates and food. and (7) per capita food and population should improve the efficacy of the model. Also, this work can be considered as a starting point for the development of more detailed models to be used for the purposes of analysis and planning of agricultural production. Certainly this approach to a quantitative understanding of the world food problem will become increasingly important in the future. 7. AVAILABILITY

Ok. THE

COMPUTER

PROGRAM

The introductory world food model is available as a FORTRAN IV program which is thoroughly documented with comments that define all variables and explain the details of the model. A package is available for a nominal preparation charge consisting of: (1) a source deck in standard FORTRAN IV of approximately 1500 cards including the line printer routine which gcneratcd Figures 5.1 to 5.4 and data cards, (2) a source listing and the output for cases 3.1. 3.2 and 3.3. and (3) a covering letter with suggestions for the computer installation of the program. Inquiries concerning the program should be directed to Dr. W. E. Schiesser. Department of Chemical Engineering, Whitaker Laboratory, Building No. 5. Lehigh University. Bethlehem. Pennsylvania 18015.

107

W. E. SCHIESSER

Ackno~~ledymte~ltsMs. for use in the model.

Roysdon

of the Lehigh Mart Library

provided

information

from the current

literature

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 0.

I I. 2. 3.

Brown L. R.. Tl~e Futurist 56-64, April (1974). Brown L. R.. In Thr Hurtm Interest. W. W. Norton & Company. New York (1974). Brown L. R. and Eckholm E. P., Bp Bread Alone. Praeger. New York, (1974). The World Food Situation and Prospects to 1985, Economic Research Service. U.S. Department of Agriculture. Foreign Agricultural Economic Report No. 98, December (1974). Scrimshaw N. S.. Teck~tol. Rrc. 77. I2 (1974). Tinte. pp. 66 X3, November 11 (1974). Rcnsberger B.. B’orltl Foot! Crisis: Brrsic Wa,v of Lti Fuce Upheucul. The New York Tinzes, November 5 (1974) (one of a series of major articles on the world food problem). Science (entire special issue), Vol. 188. No. 4188. 9 May (1975). 1973 World Population Data Sheet, available from Population Reference Bureau, Inc., 1755 Massachusetts Ave., N.W., Washington D.C. 20036. Meadows D. L.. Behrens W. W. III, Meadows D. H., Nail1 R. F., Randers J. and Zahn E. K. O., The D~wr~~~ics ctf Grwth ii1 a Finite World. Wright-Allen Press, Inc., Cambridge. Massachusetts (1974). Pragmatic Immorality. The New York Times, 5 January (1975). Brtwrrt~ thr Liw.s. Notes on Population Reference Bureau Publications, March (1975). Frejka T.. T/n Future of Population Growth: Altrrnative Paths to Equilihriurn, p, 54. Wiley, New York (1973).