Biology
Population ecology
Exponential and logistic growth, carrying capacity, life history, predator–prey sketch.
Basics
Exponential growth
With spare resources and low density, numbers can grow as N = N₀ e^{rt}. r is births minus deaths, the intrinsic rate. Doubling time is ln 2 / r. Real populations soon hit food, space, or disease. Bacterial flasks and some invasions look like this early on. Humans shift r and limits with technology, but the exponential is not endless.
Logistic growth and K
Carrying capacity K is a rough head-count the habitat can support for a while. The logistic is dN/dt = r N (1 − N/K); growth slows as N nears K. An S-curve is the ideal. K moves with season, competitors, and people. Overshoot can mean unsustainable harvest or a crash. It is a model, not a full forecast.
Life history
r-leaning species breed fast with little care. K-leaning species have few young and long care. Reality is a spectrum. Survivorship curves split into heavy early death (oysters), steadier loss (some birds), and late death (large mammals). Age pyramids hint at future births. Conservation counts recruitment, not only adults.
Predator–prey sketch
A Lotka–Volterra sketch shows prey rising, then predators, then prey falling — an oscillation. Real systems damp it with refuges, alternative food, and disease. Competitive exclusion says two species on the exact same resource rarely coexist for long. Field work measures how much niches overlap. The equations are for intuition.
Formulas
Exponential growth
Continuous time. t₂ is doubling time.
Symbols
-
rintrinsic growth rate -
t₂doubling time
Logistic
Fastest increase at N=K/2 in this model.
Symbols
-
Kcarrying capacity
Lotka–Volterra (prey)
N prey, P predators. a growth, b attack rate. A sketch.
Symbols
-
Ppredator density -
battack coefficient
Key table
| MSY sketch | near K/2 in the logistic — uncertain and easy to abuse |
|---|---|
| Density dependence | higher density cuts births or raises deaths |
| Metapopulation | local populations linked across habitat patches |
In this field
The cell
Prokaryotes vs eukaryotes, organelles, membranes.
Genetics and molecular biology
DNA, transcription/translation, Mendel, meiosis.
Metabolism
Photosynthesis, respiration, ATP, enzymes.
Evolution and ecology
Natural selection, phylogeny, trophic levels, cycles.
Human physiology basics
Homeostasis, circulation, nerves, immunity.
Plants and development
Plant tissues, hormones, developmental stages.
Immunity and microbes
Pathogens, innate and adaptive immunity, antibiotics and vaccines.
Nervous and endocrine systems
Neurons, synapses, hormones, and homeostasis — fast electrical and slower chemical signals.
Molecular lab sketch
Restriction enzymes, electrophoresis, PCR, and sequencing — a short principle sketch.
Life’s diversity
Domains, a phylogeny sketch, homology versus analogy, and binomial names.
Biological macromolecules
Carbohydrates, lipids, proteins, and nucleic acids — structure and role sketches.