Mathematical Biology Glossary

A rapid, transient electrical signal propagated along a neuron's axon, generated by the sequential opening and closing of voltage-gated ion channels.

A set of states toward which a dynamical system evolves over time from nearby initial conditions. Types include fixed points, limit cycles, and strange attractors.

A qualitative change in the dynamics of a system as a parameter passes through a critical threshold, potentially creating or destroying equilibria or oscillations.

The maximum population size that can be sustained indefinitely by the available resources in an environment, denoted $K$ in the logistic equation.

The directed movement of an organism or cell in response to a chemical concentration gradient.

A model that divides a population into distinct groups (compartments) and uses differential equations to describe flows between them. Widely used in epidemiology (SIR, SEIR).

A mathematical model in which the outcome is fully determined by the initial conditions and parameters, with no random elements.

The net movement of particles from regions of high concentration to low concentration, described mathematically by Fick's laws and the diffusion equation.

A mathematical framework describing how a point in a state space evolves over time according to a fixed rule, typically expressed as differential equations or difference equations.

A scalar associated with a linear transformation of a vector space. In stability analysis, eigenvalues of the Jacobian matrix determine whether a fixed point is stable or unstable.

The study of the distribution and determinants of disease in populations. Mathematical epidemiology uses models like SIR to predict and control outbreaks.

A state of a dynamical system where all variables remain constant over time; mathematically, where all derivatives equal zero.

A strategy in evolutionary game theory that, if adopted by an entire population, cannot be displaced by any alternative mutant strategy through natural selection.

A measure of an organism's reproductive success, often expressed as the expected number of offspring. In mathematical models, it determines selection dynamics.

The matrix of all first-order partial derivatives of a vector-valued function, used in linearization of nonlinear dynamical systems around equilibrium points.

A closed trajectory in phase space to which nearby trajectories converge (stable limit cycle) or diverge (unstable limit cycle). Represents sustained periodic oscillations.

A population growth model where the per capita growth rate decreases linearly as the population approaches carrying capacity: $\frac{dN}{dt} = rN(1 - \frac{N}{K})$.

The biological process that causes an organism to develop its shape. Mathematical models, particularly reaction-diffusion systems, explain how spatial patterns emerge during development.

A curve in the phase plane along which one variable's rate of change is zero. The intersections of nullclines for different variables identify equilibrium points.

An equation involving derivatives of an unknown function with respect to a single independent variable (usually time). The primary tool for modeling temporal dynamics in biology.

An equation involving partial derivatives of an unknown function with respect to multiple independent variables (e.g., time and space). Used for spatiotemporal biological models.

A two-dimensional graphical representation of a dynamical system where each axis corresponds to one state variable, and trajectories show the system's evolution over time.

The study of how and why the size and structure of populations change over time, modeled using differential equations, difference equations, or stochastic processes.

A mathematical model incorporating random variables or processes, used when inherent biological variability or small population sizes make deterministic approximations inadequate.

An interdisciplinary approach using mathematical and computational models to understand complex biological systems as integrated networks of interacting components.