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An Introduction to Population Modeling

Anu Buddhikot

Quantitative Biology and Chemical Engineering ’26

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Mentor: Arnab Roy

Directed Reading Program Fall 2023

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Overview

  • Deriving the Malthus Model
  • Carrying Capacity and the Logistic Model
  • Modifications to the Malthus Model: Predation
    • Scaling
  • Additional Modifications to the Malthus Model

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The Malthus Model

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Assumptions of the Malthus Model

  • Homogeneous
  • Isolated
  • Invariant habitat
  • Very large population

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Components of the Malthus Model

  • Birth rate
    • β Δt

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  • Mortality
    • μ Δt

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  • Instantaneous growth rate
    • r = β- μ

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The Malthus Model

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Solutions to the Malthus Model

  • r > 0

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  • r = 0

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  • r > 0

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The Logistic Model

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Assumptions of The Logistic Effect

  • Similar to the assumptions of the Malthus Model
  • Intraspecific competition
  • Limited resources
  • Carrying capacity (K = N*)

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The Logistic Model

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Solutions of the Logistic Model

  • N0 < K

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  • N0 > K

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Predation

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Developing A Predation Model

  • Method of Encounters
  • Probability of encounter (P*)
  • Rate of successful attack (a)

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A New Technique: Scaling

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Solutions of the Predation Model

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Further Applications of the Malthus Model

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Additional Models

  • Immigration/Migration
  • The Allee Effect (low populations)
  • Harvesting

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Conclusions

  • Derived the Malthus and Logistic population models and obtained their general solutions
  • Explored several extensions of the basic population modeling equations
  • Learned how scaling can be used to simplify complex differential equation models

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References

  • M. Ianelli and A. Pugliese, An Introduction to Mathematical Population Dynamics: Along the trail of Volterra and Lotka, Series: UNITEXT, Vol. 79., Springer International, Switzerland, 2014.

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Thank You

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