Course Overview

  • Focus on maximizing fitness and tessellations
  • Assignment 2 due Tuesday night
  • Test on material covered in classes 1-4 next week

Key Concepts

  • Optimization of space and resources discussed previously.
  • Bees conserve energy producing honeycombs to allocate more for reproduction.

Fitness and Survival

  • Darwin's concept of survival of the fittest in natural selection.
  • Fitness consists of:
    • Survival: Probability of reaching reproductive age.
    • Reproduction: Number of offspring produced.
  • Loggerhead turtles:
    • Endangered due to low survival rates to maturity (1 in 10,000).
    • Long lifespan and high egg clutches (approximately 100 eggs every ~2 years).
    • Tradeoff between longevity, reproductive output, and resources.

Pathogen Fitness

  • Pathogens have varying virulence:
    • Avirulent forms cause no harm; virulent forms can be deadly.
  • For viruses, fitness defined by:
    • Survival: Duration of infection (keeping host alive).
    • Reproduction: Successful transmission between hosts.
  • Virulence-transmission tradeoff: increasing transmission can increase host harm.

Myxoma Virus Example

  • Introduced in Australia to control rabbit populations.
  • Initial high virulence leading to significant rabbit mortality, decreased over time.
  • Fitness defined as:
    • Fitness = 𝛽 × (1 / v + d + γ)
    • where v is virulence, γ is recovery rate, d is natural death rate.
  • Intermediate virulence optimizes fitness for the pathogen.

Marek's Disease in Poultry

  • Virulence increases with more infectious strains.
  • Trade-off between transmission and virulence.
  • Vaccination interferes with pathogen dynamics, leading to shifts in virulence.

Tessellations Overview

  • Definition: Covering a surface with flat shapes with no overlaps or gaps.
  • Regular polygons (triangles, squares, hexagons) tessellate effectively.
  • Irregular shapes can also tessellate.
  • Edge-to-edge tessellations: Tiles touch along entire edges.
  • Interior angles of polygons influence tessellation capabilities.

Transformations

  • Types of transformations in tessellations include:
    • Rotation
    • Reflection
    • Translation
  • Each transformation preserves size and shape in tessellating process.