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For maximizing $f$ subject to $g_i(x)\ge 0$ and $h_j(x)=0$, suppose a feasible $x^{\ast }$ and multipliers ${\lambda }_i\ge 0,{\mu }_j$ satisfy complementary slackness and $x^{\ast }$ globally maximizesThen $x^{\ast }$ globally maximizes $f$ on the feasible set. Convexity or concavity hypotheses are commonly used to prove the required global extremum of the Lagrangian from first-order conditions.

Biopolymer mechanics studies bending, stretching, thermal fluctuations, and force-induced deformation of filamentous biological macromolecules.

Chemotaxis is directed cell motion up or down a chemical concentration gradient.

Biological fluid dynamics studies flows generated by and acting on organisms, cells, tissues, and biological transport networks.

The Kramers-Moyal expansion Taylor-expands the gain terms of a jump-process master equation in the jump size. Keeping its first two terms yields a Fokker-Planck equation whose drift and diffusion coefficients are the first two infinitesimal jump moments.

A cross-infection model's disease-free equilibrium becomes a saddle when the product of infection gains exceeds the product of recovery rates.

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 $I=E/|W|$.