Mathematical Biology

1
Mathematical Biology

• Lecture 1

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What is Mathematical Biology?

• Mathematical Biology
• Deterministic Models (ODE, DE, PDE)
• Stochastic Models (Probability, Stochastic)
• Bio Informatics (Computation Statistics)
• What do we do in this course?
• Deterministic Models (ODE, DE, PDE)

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Outline of the Course I

• Population Biology (3 weeks)
• Single and Interacting Species Models,
• (First Order ODE and Systems of First order ODES,
  Difference Equations)
• Systems and Diseases (3 weeks)
• Modelling of HIV/AIDS, infectious diseases and
  cancer
• (Systems of ODE)
• Communication Nerves (2 weeks)
• Models of Neuronal and Synaptic communication
• (Systems of ODE)

4
Outline of the Course II

• Random Movements in Space and Time (2 weeks)
• Diffusion, and communication within the organism.
• (Random Walk, Diffusion Equation)
• Cellular Biochemistry (3 weeks)
• Chemical Kinetics
• (Perturbation Theory)
• Genomics (Time Permitting)
• (Probability Statistics)

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Population Models

• Readings S-H Chap 3,4 EK Chap 1,2,3
• How to model population growth?
• A Simple Assumption
• Rate of Change of Population Existing
  Population
• Here (Biotic
  Potential)
• What about approximating time as continuous?

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Examples of Population Growth

• - Human Population over the Past 600 yrs

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Discrete Time Model

• Assume time changes discretely
• Formulate a Difference Equation to model Growth
  (Exponential Growth Model)
• is the population at time step n
  
• Relation between ?

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Examples of Population Growth

• Bacterial Growth

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Environmental Limitations

• Realistically r the growth rate is not constant
• As population grows ..growth rates fall
• What kind of function should f be?
• (Exponential
  Growth)
• (No Growth
  over Time)
• Why did growth stop at ?
• Environment reaches Carrying
  Capacity K

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Logistic Growth Model

• The growth rate is modeled as
• (Satisfies both
  conditions)
• Population growth is then governed by
• In Continuous time

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Models for Interacting Species

• A Simple Host Parasite Model (Nicholson Bailey)
• 
  Host Population
• 
  
• 
  Parasite Population
• Assumes random encounters between host and
  parasite
• Encounters are Poisson Events (Details when we
  discuss the model later)

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Model for Prey Predator Interaction (Lotka
Voterra Model)

• Prey (Moose M) have exponential Growth
• Predators (Wolves W) depend on Prey to survive
  with growth rate proportional to predation
• Rate of predation depends on mutual encounters

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Predator Prey Model
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The Plan

• Study Single Species Discrete Growth Models
• Study Two Species Discrete Growth Models
• Move on to Continuous Population Growth Models
• Some discussion of Discrete vs. Continuous