Fisher-KPP wave; Levin metapopulation; Wright-Fisher drift; Kimura neutral theory; Kingman coalescent; cell-lineage branching; Allee effect; May stability-complexity; Bak-Sneppen evolution; MacArthur resource-competition; replicator…
Mathematical Biology
Fisher-KPP traveling wave
Fisher 1937 / KPP 1937 reaction-diffusion u_t = D u_xx + r u (1-u); admits stable traveling-wavefront with speed c* = 2 sqrt(rD); models…
Levin metapopulation
Levin 1969 dp/dt = c p (1-p) - e p where p = patch-occupancy; persistence requires c > e; Hanski 1999 spatially-realistic…
Wright-Fisher genetic drift
Discrete N-individual reproduction with binomial sampling; allele-frequency variance Var(p) = p(1-p)/(2N) per generation; diffusion limit…
Kimura neutral theory of evolution
Kimura 1968: most molecular variation is selectively-neutral, fixed by drift not selection; substitution rate = mutation rate, independent…
Kingman coalescent
Kingman 1982 backward-in-time: in sample of n, two lineages coalesce at rate (n choose 2) / 2N; waiting-time exponential(C(n,2)/2N);…
Cell-lineage branching process
Stem-cell + transit-amplifying + differentiated states modeled by multi-type Galton-Watson; cell-fate decisions modeled stochastically;…
Allee effect
Allee 1931 / Stephens 1999: per-capita growth rate decreases at low density (mate-limitation, group-defense loss); creates extinction…
May stability vs complexity
May 1972 random-matrix model: large-S diverse community (n>10) generally unstable since lambda_max scales as sqrt(S C sigma^2); paradox of…
Bak-Sneppen evolution
Bak-Sneppen 1993 self-organized-criticality model of macroevolution; lowest-fitness species + neighbors mutated at each step; produces…
MacArthur resource competition
MacArthur 1970 mechanistic competitive-exclusion: n consumer species competing for k resources; competitive exclusion principle (n cannot…
Replicator equation (evolutionary game)
x_i' = x_i (f_i(x) - <f>): population frequency follows fitness-deviation from mean; basis of evolutionary-game-theory and ESS (Maynard…
Demographic noise (population)
Population-size fluctuations from finite-N effects scale as 1/sqrt(N); modeled via master-equation or Langevin-equation with multiplicative…
Lotka-Volterra (1925-1926)
A Lotka 1925 + V Volterra 1926 predator-prey ODEs; modern Hofbauer-Sigmund 1998 evolutionary-game extensions.
Turing morphogenesis (1952)
A Turing 1952 reaction-diffusion morphogen-patterns; modern animal-coat-pattern Murray + Othmer extensions; experimental BZ-reaction…
Hodgkin-Huxley (1952)
Hodgkin-Huxley 1952 (Nobel 1963) action-potential ODEs; foundation of computational-neuroscience + cardiac-electrophysiology.
SIR (Kermack-McKendrick 1927)
Kermack-McKendrick 1927 SIR-model; basic-reproduction-number R0; modern compartmental + agent-based COVID-19 models.
ESS (Maynard Smith 1973)
Maynard Smith-Price 1973 ESS; modern replicator-dynamics + adaptive-dynamics; basis of evolutionary-game-theory mathematical foundations.
ABM (Grimm-Railsback 2005)
Grimm-Railsback 2005 'Individual-Based Modeling and Ecology'; ODD-protocol; modern epidemiology + ecology + economics applications.