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The scalar equation $n^{\prime }(t)=rn(t-\tau )$ makes the present growth rate depend on the population one delay time earlier. A history function on $[-\tau ,0]$ is required in place of a single initial value.

This is one linear delay differential equation, rather than the entire delayed-evolution class.

The population iterationcombines annual survival with density-dependent recruitment. Its positive equilibrium iswhich exists when $k>1-r$.

If a population $N$ has per-capita birth rate $b{e}^{-aN}$ and per-capita death rate $d$, thenWhen $b>d$, its positive equilibrium isand the derivative of the scalar vector field there is $-ad{N}^{\ast }<0$, so the equilibrium is locally asymptotically stable.

The equationhas an unstable zero equilibrium and a stable positive equilibrium $K$. Its per-capita density correction is quadratic rather than the logistic model's linear correction.

Autocatalysis is a reaction mechanism in which a product promotes its own production.

Enzyme kinetics models how free enzyme, substrate, enzyme-substrate complexes, and product concentrations evolve through a reaction mechanism.

The law of mass action assigns an elementary reaction a rate constant times the product of its reactant concentrations, with each concentration raised to its stoichiometric multiplicity.

An age-structured population model tracks a density $n(a,t)$ transported toward greater age, reduced by age-dependent mortality, and replenished at age zero by births.

The continuous-age transport version is the Von Foerster equation. Discrete age-class models are other members of this broader modeling class.

A per-capita quantity is a total quantity divided by the number of individuals in the population.

As a heuristic approximation, inserting $r=6GM/c^2$ into the circular orbit form of Kepler's third law, with $M$ here denoting the total mass, givesThe dominant quadrupole gravitational-wave frequency is twice the orbital frequency, $f_{\mathrm{GW}}=2/T$.

The Tzitzeica equation in light-cone coordinates is $u_{xy}=e^u-e^{-2u}$ and admits a matrix Lax pair.

A Lax pair encodes a nonlinear equation as compatibility of two linear equations depending on an auxiliary spectral parameter.

Commuting mixed derivatives impose an equation on the coefficient matrices of an overdetermined auxiliary system.

For $F=(p^2+W^2{q}^2+aq)/2$, shifting $Q=q+a/(2W^2)$ gives oscillator energy $E=F+a^2/(8W^2)$ and action $I=E/|W|$.

A Hamiltonian that is a sum of one-coordinate quadratic potentials plus linear terms becomes a collection of independent harmonic oscillators after translating each equilibrium position.

Hamiltonians that split into terms $F_i(p_i,q_i)$ have $\{F_i,F_j\}=0$ for $i\ne j$, and their differentials are generically independent because they use disjoint coordinate pairs.

The inverse scattering transform evolves the scattering data of an auxiliary linear operator and reconstructs the potential from the evolved data.

The Korteweg-De Vries equationis an integrable nonlinear dispersive equation. Its sign convention here admits negative solitary waves.

Writing a defocusing nonlinear Schrödinger field as $\psi =\sqrt{\rho }{e}^{iS}$ turns it into a continuity equation for $\rho$ and a velocity equation for $v=S_x$, including a quantum-pressure term.

In a common normalization, the defocusing case is $i{\psi }_t+{\psi }_{xx}-2|\psi {|}^2\psi =0$. The sign distinguishes it from the focusing nonlinear Schrödinger equation.

The focusing and defocusing equationshave Hamiltoniansunder the convention $i{\psi }_t=\delta H_{\pm }/\delta \overline{\psi }$.