10.2 Mathematical models for population projection

Mathematical models for population projection are essential tools in demography. They help predict future population sizes and growth patterns, ranging from simple exponential models to complex polynomial equations.

These models vary in their assumptions and applications. Choosing the right model depends on factors like available resources, time horizon, and data patterns. Understanding these models is crucial for accurate population forecasting and informed decision-making.

Population Projection Models

Exponential and Logistic Growth Models

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• The exponential growth model assumes a constant growth rate and is used for populations with unlimited resources
  • Expressed as $P(t) = P_0 * e^{rt}$, where $P_0$ is the initial population size, $r$ is the growth rate, and $t$ is time
  • Provides accurate short-term projections for populations with high growth rates (bacteria in a petri dish)
• The logistic growth model accounts for the carrying capacity of the environment and is used for populations with limited resources
  • Expressed as $P(t) = K / (1 + (K - P_0) / P_0 * e^{-rt})$, where $K$ is the carrying capacity
  • Captures the slowing down of population growth as the population approaches the carrying capacity (deer population in a forest)

Polynomial and Other Growth Models

• Polynomial models, such as quadratic and cubic models, capture more complex population dynamics and account for changing growth rates over time
  • Expressed as $P(t) = a_0 + a_1t + a_2t^2 + ... + a_nt^n$, where $a_0, a_1, ..., a_n$ are coefficients
  • Flexible and can be fitted to historical data, but may not have a clear biological interpretation (human population growth)
• The Gompertz model assumes that the growth rate decreases with time and is used for modeling the growth of individual organisms
• The Von Bertalanffy model is also used for modeling the growth of individual organisms (fish growth)

Model Assumptions and Characteristics

Exponential and Logistic Model Assumptions

• The exponential growth model assumes a constant growth rate and unlimited resources, which is unrealistic for most populations
  • Simple to use and can provide accurate short-term projections for populations with high growth rates
• The logistic growth model assumes a carrying capacity and limited resources, which is more realistic for most populations
  • Captures the slowing down of population growth as the population approaches the carrying capacity
  • May not account for external factors that can affect the carrying capacity (environmental changes, predation)

Polynomial and Other Model Assumptions

• Polynomial models can capture more complex population dynamics and changing growth rates over time
  • Flexible and can be fitted to historical data
  • May not have a clear biological interpretation and can be sensitive to the choice of polynomial degree
• The Gompertz and Von Bertalanffy models have specific assumptions about the growth rate and are more suitable for modeling the growth of individual organisms rather than entire populations

Model Selection for Projections

Factors to Consider in Model Selection

• Consider the growth characteristics of the population (constant, increasing, or decreasing growth rate over time)
• Evaluate the resources available to the population (limited or unlimited) to determine if a logistic model is more appropriate than an exponential model
• Assess the time horizon of the projection (short-term or long-term) to choose between exponential, logistic, or polynomial models
• Consider the purpose of the projection (planning, policy-making, or research) to align the choice of model with the intended use of the results

Data-Driven Model Selection

• Analyze the historical population data to identify patterns or trends that can guide the selection of an appropriate mathematical model
• Use model selection criteria (AIC, BIC) to compare the fit of different models to the data and select the most appropriate one
• Validate the selected model using out-of-sample data or cross-validation techniques to ensure its predictive performance and robustness

Parameter Estimation for Models

Regression Techniques for Parameter Estimation

• Use least squares regression to estimate the parameters of the chosen mathematical model by minimizing the sum of squared differences between the observed and predicted population sizes
  • For the exponential growth model, linearize the equation by taking the natural logarithm of both sides and estimate the growth rate

    r

    using linear regression
  • For the logistic growth model, use non-linear least squares regression to estimate the carrying capacity

    K

    and the growth rate

    r

    simultaneously
  • For polynomial models, use multiple linear regression to estimate the coefficients

    a0, a1, ..., an

    by fitting the model to the historical population data

Model Evaluation and Validation

• Assess the goodness of fit of the estimated models using measures such as the coefficient of determination (R^2), root mean square error (RMSE), or Akaike information criterion (AIC) to compare different models and select the best-fitting one
• Validate the estimated models using out-of-sample data or cross-validation techniques to ensure their predictive performance and robustness
• Analyze the residuals of the fitted models to check for any systematic patterns or deviations from the model assumptions (homoscedasticity, normality)
• Perform sensitivity analysis to assess the impact of changes in the estimated parameters on the population projections and quantify the uncertainty associated with the projections