Analysis_and_Comparison_of_Mathematical_Population_Models[1] [Autosaved]

A Study on the Accuracy A Study on the Accuracy of Mathematical of Mathematical

Population Models and Population Models and Population PredictionPopulation Prediction

Student: James CainStudent: James CainAdvisor: Dr. Jianmin ZhuAdvisor: Dr. Jianmin Zhu

Department of Mathematics & Department of Mathematics & Computer ScienceComputer Science

Fort Valley State UniversityFort Valley State University

PurposePurpose To determine which population To determine which population

model is the most accurate in model is the most accurate in describing the population of a region describing the population of a region over a certain period of time. over a certain period of time.

To use this model to predict the To use this model to predict the population of the United States.population of the United States.

ObjectivesObjectives To consider mathematical models To consider mathematical models

describing the growth of a describing the growth of a populationpopulation

To analyze the behaviors of the To analyze the behaviors of the populationpopulation

To identify unknown parameters of a To identify unknown parameters of a mathematical model represented by mathematical model represented by a nonlinear differential equationa nonlinear differential equation

The ProcessThe Process Used three different population models Used three different population models

to approximate the population of the to approximate the population of the United StatesUnited States

Solved non-linear differential equations Solved non-linear differential equations using specific numerical methodsusing specific numerical methods

Minimized the error between estimated Minimized the error between estimated and observed valuesand observed values

Identify parameters of different Identify parameters of different mathematical population models mathematical population models

What is a Mathematical What is a Mathematical Model?Model?

The mathematical description of a The mathematical description of a system or phenomenon is called a system or phenomenon is called a mathematical model. mathematical model.

An Example of a An Example of a Mathematical ModelMathematical Model

If If P(t)P(t) is total is total population at time population at time t, then t, then

dP/dt = kPdP/dt = kP, where , where k is a constant. k is a constant.

This model is This model is important because important because it can be used to it can be used to model growth of model growth of small populations small populations over short over short intervals of time. intervals of time.

Population ModelsPopulation Models The Malthusian ModelThe Malthusian Model The Logistic ModelThe Logistic Model The Allee Effect ModelThe Allee Effect Model

The Malthusian ModelThe Malthusian Model Essentially Essentially

exponential growth exponential growth based on a based on a constant rate of constant rate of compound growth compound growth rate. rate. Po : Initial Po : Initial

PopulationPopulation r : growth rater : growth rate t : time.t : time.

The Logistic Population The Logistic Population ModelModel

A logistic function A logistic function or logistic curve or logistic curve models the S-curve models the S-curve of growth of some of growth of some set set PP. .

This model is used This model is used in a range of fields, in a range of fields, such as biology such as biology and economics. and economics.

The Allee Effect Population The Allee Effect Population ModelModel

The Allee effect is a phenomenon in The Allee effect is a phenomenon in biology characterized by a positive biology characterized by a positive correlation between population density correlation between population density and the per capita growth rate.and the per capita growth rate.

The general idea is that the The general idea is that the reproduction and survival of individuals reproduction and survival of individuals decrease for smaller populations.decrease for smaller populations.

What is a Differential What is a Differential Equation?Equation?

A differential equation is an equation A differential equation is an equation containing the derivatives of one or containing the derivatives of one or more dependent variables with more dependent variables with respect to one or more independent respect to one or more independent variables.variables.

Numerical Methods UsedNumerical Methods Used Runge Kutta MethodsRunge Kutta Methods

Runge Kutta MethodsRunge Kutta Methods An important family of methods for An important family of methods for

the approximation of solutions of a the approximation of solutions of a non-linear differential equationnon-linear differential equation

Least Squares MethodLeast Squares Method A method that determines the A method that determines the

difference between the estimated difference between the estimated and observed values squaredand observed values squared

Its purpose is to minimize error Its purpose is to minimize error between the estimated and observed between the estimated and observed values values

What is Fortran?What is Fortran? FortranFortran is a is a

general-purpose, general-purpose, procedural, procedural, imperative imperative programming programming language that is language that is especially suited to especially suited to numeric numeric computation and computation and scientific scientific computing. computing.

Fortran FlowchartFortran Flowchart

Driver

Model

lmdif1 Error

lmdif1

modelFlag.LE.2

The Logistic Model The Allee Model The Malthusian Model

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)

Allee Model ChartAllee Model ChartYearsYears Actual Actual

Pop. (in Pop. (in millions)millions)

Estimated Estimated Pop. (in Pop. (in millions)millions)

Error Error (Square(Squared)d)

19201920 106.021106.021 108.410108.410 5.707325.7073211

19301930 123.202123.202 118.595118.595 21.224421.22444949

19401940 132.164132.164 133.054133.054 0.79210.792119501950 151.325151.325 152.941152.941 2.611452.61145

6619601960 179.323179.323 178.052178.052 1.615441.61544

1119701970 205.302205.302 204.653204.653 0.421200.42120

1119801980 226.542226.542 226.290226.290 0.063500.06350

44TotalTotal 32.435432.4354

7272

Estimating the Parameters Estimating the Parameters of the Allee Modelof the Allee Model

0

50

100

150

200

250

1920 1930 1940 1950 1960 1970 1980

ActualPopulationPopulationEstimate

Logistic Model ChartLogistic Model ChartYearsYears Actual Actual

Pop. (in Pop. (in millions)millions)

Estimated Estimated Pop. (in Pop. (in millions)millions)

Error Error (Squared)(Squared)

19201920 106.021106.021 105.173105.173 0.7191040.71910419301930 123.202123.202 119.716119.716 12.1521912.15219

6619401940 132.164132.164 136.262136.262 16.7936016.79360

4419501950 151.325151.325 155.081155.081 14.1075314.10753

6619601960 179.323179.323 176.486176.486 8.0485698.04856919701970 205.302205.302 200.807200.807 20.2050220.20502

5519801980 226.542226.542 228.432228.432 3.57213.5721TotalTotal 75.5981375.59813

44

Estimating the Parameters Estimating the Parameters of the Logistic Modelof the Logistic Model

0

50

100

150

200

250

1920 1930 1940 1950 1960 1970 1980

Actual PopulationPopulation Estimate

Malthusian Model ChartMalthusian Model ChartYearsYears Actual Actual

Pop. (in Pop. (in millions)millions)

Estimated Estimated Pop. (in Pop. (in millions)millions)

Error Error (Square(Squared)d)

19201920 106.021106.021 105.320105.320 0.491400.4914011

19301930 123.202123.202 119.877119.877 11.055611.05562525

19401940 132.164132.164 136.454136.454 18.404118.404119501950 151.325151.325 155.312155.312 15.896115.8961

696919601960 179.323179.323 176.775176.775 6.492306.49230

4419701970 205.302205.302 201.185201.185 16.949616.9496

898919801980 226.542226.542 229.010229.010 6.091026.09102

44TotalTotal 75.380375.3803

1212

Estimating the Parameters Estimating the Parameters of the Malthusian Modelof the Malthusian Model

0

50

100

150

200

250

1920 1930 1940 1950 1960 1970 1980

Actual PopulationPopulation Estimate

Population PredictionPopulation Prediction(with Allee Effect Model)(with Allee Effect Model)

050

100150200250300350400

1920

1940

1960

1980

2000

2010

2020

2030

Population Estimate Actual Population

ConclusionConclusion In this study, we compared three In this study, we compared three

population models in estimating the population models in estimating the population of the United States over a population of the United States over a certain period of time. We identified certain period of time. We identified parameters of mathematical population parameters of mathematical population models using numerical methods that models using numerical methods that solve nonlinear differential equations, solve nonlinear differential equations, and the least squares method with and the least squares method with Fortran programming language.Fortran programming language.The prediction of the population of the The prediction of the population of the United States is obtained with the Allee United States is obtained with the Allee Effect mathematical model, the best Effect mathematical model, the best model among three models.model among three models.

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 Dr. Dwayne Daniels, Department of

ChemistryChemistry

Questions?Questions?