This study compared three population models - Malthusian, logistic, and Allee effect - to determine the most accurate in describing US population growth over time. The models were used to estimate population data from 1920-1980 and solve nonlinear differential equations numerically. The Allee effect model minimized error between estimated and observed populations, making it the best model for predicting future US population growth.

1. A Study on the Accuracy ofA Study on the Accuracy of
Mathematical Population ModelsMathematical Population Models
and Population Predictionand Population Prediction
Student: James CainStudent: James Cain
Advisor: Dr. Jianmin ZhuAdvisor: Dr. Jianmin Zhu
Department of Mathematics & ComputerDepartment of Mathematics & Computer
ScienceScience
Fort Valley State UniversityFort Valley State University

2. PurposePurpose
 To determine which population model is theTo determine which population model is the
most accurate in describing the population of amost accurate in describing the population of a
region over a certain period of time.region over a certain period of time.
 To use this model to predict the population ofTo use this model to predict the population of
the United States.the United States.

3. ObjectivesObjectives
 To consider mathematical models describing theTo consider mathematical models describing the
growth of a populationgrowth of a population
 To analyze the behaviors of the populationTo analyze the behaviors of the population
 To identify unknown parameters of aTo identify unknown parameters of a
mathematical model represented by a nonlinearmathematical model represented by a nonlinear
differential equationdifferential equation

4. The ProcessThe Process
 Used three different population models toUsed three different population models to
approximate the population of the Unitedapproximate the population of the United
StatesStates
 Solved non-linear differential equations usingSolved non-linear differential equations using
specific numerical methodsspecific numerical methods
 Minimized the error between estimated andMinimized the error between estimated and
observed valuesobserved values
 Identify parameters of different mathematicalIdentify parameters of different mathematical
population modelspopulation models

5. What is a Mathematical Model?What is a Mathematical Model?
 The mathematical description of a system orThe mathematical description of a system or
phenomenon is called a mathematical model.phenomenon is called a mathematical model.

6. An Example of a MathematicalAn Example of a Mathematical
ModelModel
 IfIf P(t)P(t) is total populationis total population
at time t, thenat time t, then
 dP/dt = kPdP/dt = kP, where k is a, where k is a
constant.constant.
 This model is importantThis model is important
because it can be used tobecause it can be used to
model growth of smallmodel growth of small
populations over shortpopulations over short
intervals of time.intervals of time.

7. Population ModelsPopulation Models
 The Malthusian ModelThe Malthusian Model
 The Logistic ModelThe Logistic Model
 The Allee Effect ModelThe Allee Effect Model

8. The Malthusian ModelThe Malthusian Model
 Essentially exponentialEssentially exponential
growth based on agrowth based on a
constant rate ofconstant rate of
compound growth rate.compound growth rate.
 Po : Initial PopulationPo : Initial Population
 r : growth rater : growth rate
 t : time.t : time.

9. The Logistic Population ModelThe Logistic Population Model
 A logistic function orA logistic function or
logistic curve models thelogistic curve models the
S-curve of growth ofS-curve of growth of
some setsome set PP..
 This model is used in aThis model is used in a
range of fields, such asrange of fields, such as
biology and economics.biology and economics.

10. The Allee Effect Population ModelThe Allee Effect Population Model
 The Allee effect is a phenomenon in biologyThe Allee effect is a phenomenon in biology
characterized by a positive correlation betweencharacterized by a positive correlation between
population density and the per capita growth rate.population density and the per capita growth rate.
 The general idea is that the reproduction and survivalThe general idea is that the reproduction and survival
of individuals decrease for smaller populations.of individuals decrease for smaller populations.

11. What is a Differential Equation?What is a Differential Equation?
 A differential equation is an equation containingA differential equation is an equation containing
the derivatives of one or more dependentthe derivatives of one or more dependent
variables with respect to one or morevariables with respect to one or more
independent variables.independent variables.

12. Numerical Methods UsedNumerical Methods Used
 Runge Kutta MethodsRunge Kutta Methods

13. Runge Kutta MethodsRunge Kutta Methods
 An important family of methods for theAn important family of methods for the
approximation of solutions of a non-linearapproximation of solutions of a non-linear
differential equationdifferential equation

14. Least Squares MethodLeast Squares Method
 A method that determines the differenceA method that determines the difference
between the estimated and observed valuesbetween the estimated and observed values
squaredsquared
 Its purpose is to minimize error between theIts purpose is to minimize error between the
estimated and observed valuesestimated and observed values

15. What is Fortran?What is Fortran?
 FortranFortran is a general-is a general-
purpose, procedural,purpose, procedural,
imperative programmingimperative programming
language that is especiallylanguage that is especially
suited to numericsuited to numeric
computation andcomputation and
scientific computing.scientific computing.

16. Fortran FlowchartFortran Flowchart
Driver
Model
lmdif1 Error

17. lmdif1
modelFlag.LE.2
The Logistic Model The Allee Model The Malthusian Model

18. Error
modelFlag.LE.2
Rungkuta (x, t, h, alpha, nn, LogisticFunc)
Rungkuta (x, t, h, alpha, nn, AleeFunc)
Rungkuta (x, t, h, alpha, nn, MalthusianFunc)

19. Allee Model ChartAllee Model Chart
YearsYears Actual Pop.Actual Pop.
(in millions)(in millions)
Estimated Pop.Estimated Pop.
(in millions)(in millions)
ErrorError
(Squared)(Squared)
19201920 106.021106.021 108.410108.410 5.7073215.707321
19301930 123.202123.202 118.595118.595 21.22444921.224449
19401940 132.164132.164 133.054133.054 0.79210.7921
19501950 151.325151.325 152.941152.941 2.6114562.611456
19601960 179.323179.323 178.052178.052 1.6154411.615441
19701970 205.302205.302 204.653204.653 0.4212010.421201
19801980 226.542226.542 226.290226.290 0.0635040.063504
TotalTotal 32.43547232.435472

20. Estimating the Parameters of theEstimating the Parameters of the
Allee ModelAllee Model
0
50
100
150
200
250
1920 1930 1940 1950 1960 1970 1980
Actual
Population
Population
Estimate

21. Logistic Model ChartLogistic Model Chart
YearsYears Actual Pop.Actual Pop.
(in millions)(in millions)
Estimated Pop.Estimated Pop.
(in millions)(in millions)
ErrorError
(Squared)(Squared)
19201920 106.021106.021 105.173105.173 0.7191040.719104
19301930 123.202123.202 119.716119.716 12.15219612.152196
19401940 132.164132.164 136.262136.262 16.79360416.793604
19501950 151.325151.325 155.081155.081 14.10753614.107536
19601960 179.323179.323 176.486176.486 8.0485698.048569
19701970 205.302205.302 200.807200.807 20.20502520.205025
19801980 226.542226.542 228.432228.432 3.57213.5721
TotalTotal 75.59813475.598134

22. Estimating the Parameters of theEstimating the Parameters of the
Logistic ModelLogistic Model
0
50
100
150
200
250
1920 1930 1940 1950 1960 1970 1980
Actual Population
Population Estimate

23. Malthusian Model ChartMalthusian Model Chart
YearsYears Actual Pop.Actual Pop.
(in millions)(in millions)
Estimated Pop.Estimated Pop.
(in millions)(in millions)
ErrorError
(Squared)(Squared)
19201920 106.021106.021 105.320105.320 0.4914010.491401
19301930 123.202123.202 119.877119.877 11.05562511.055625
19401940 132.164132.164 136.454136.454 18.404118.4041
19501950 151.325151.325 155.312155.312 15.89616915.896169
19601960 179.323179.323 176.775176.775 6.4923046.492304
19701970 205.302205.302 201.185201.185 16.94968916.949689
19801980 226.542226.542 229.010229.010 6.0910246.091024
TotalTotal 75.38031275.380312

24. Estimating the Parameters of theEstimating the Parameters of the
Malthusian ModelMalthusian Model
0
50
100
150
200
250
1920 1930 1940 1950 1960 1970 1980
Actual Population
Population Estimate

25. Population PredictionPopulation Prediction
(with Allee Effect Model)(with Allee Effect Model)
0
50
100
150
200
250
300
350
400
1920
1940
1960
1980
2000
2010
2020
2030
Population Estimate Actual Population

26. ConclusionConclusion
 In this study, we compared three populationIn this study, we compared three population
models in estimating the population of themodels in estimating the population of the
United States over a certain period of time. WeUnited States over a certain period of time. We
identified parameters of mathematicalidentified parameters of mathematical
population models using numerical methodspopulation models using numerical methods
that solve nonlinear differential equations, andthat solve nonlinear differential equations, and
the least squares method with Fortranthe least squares method with Fortran
programming language.programming language.
The prediction of the population of the UnitedThe prediction of the population of the United
States is obtained with the Allee EffectStates is obtained with the Allee Effect
mathematical model, the best model amongmathematical model, the best model among
three models.three models.

27. I’d Like to Thank…I’d Like to Thank…
 National Science FoundationNational Science Foundation
 Peach State LSAMPPeach State LSAMP
 Dr. Dwayne Daniels, Department of ChemistryDr. Dwayne Daniels, Department of Chemistry

28. Questions?Questions?