Mathematical Biology Cheat Sheet

The core ideas of Mathematical Biology distilled into a single, scannable reference — perfect for review or quick lookup.

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Quick Reference

Lotka-Volterra Equations

A pair of first-order nonlinear ordinary differential equations that model the dynamics of predator-prey interactions. The prey population grows exponentially in the absence of predators, while predators decline without prey, and their interaction produces characteristic oscillatory cycles.

SIR Model

A compartmental model in epidemiology that divides a population into Susceptible, Infected, and Recovered compartments. Ordinary differential equations govern the flow rates between compartments based on transmission and recovery rates.

Reaction-Diffusion Systems

Partial differential equations that describe how the concentration of one or more substances changes in space and time under the influence of local chemical reactions and diffusion. They are fundamental to pattern formation theory in biology.

Hodgkin-Huxley Model

A mathematical model that describes how action potentials in neurons are initiated and propagated, using a set of nonlinear ordinary differential equations representing ionic currents through voltage-gated channels in the cell membrane.

Michaelis-Menten Kinetics

A mathematical model of enzyme kinetics relating the reaction rate to substrate concentration via two parameters: the maximum rate ($V_{\text{max}}$) and the Michaelis constant ($K_m$), which is the substrate concentration at which the rate is half-maximal.

Bifurcation Theory

The study of qualitative changes in the behavior of a dynamical system as a parameter is varied. Bifurcations can cause a system to transition from a stable equilibrium to oscillations, chaos, or other complex behaviors.

Evolutionary Game Theory

A framework that applies game theory to evolving populations, where strategies are heritable traits and payoffs affect reproductive fitness. The central concept is the Evolutionarily Stable Strategy (ESS), a strategy that cannot be invaded by a rare mutant.

Stochastic Processes in Biology

Mathematical models that incorporate randomness to capture the inherent variability in biological systems. These include birth-death processes, Markov chains, and stochastic differential equations applied to gene expression, population dynamics, and molecular interactions.

Fisher's Equation

A partial differential equation combining logistic growth with spatial diffusion, originally proposed by R.A. Fisher in 1937 to model the spread of an advantageous gene through a population. It produces traveling wave solutions with a characteristic minimum wave speed.

Systems Biology and Network Modeling

An approach that uses mathematical and computational models to analyze complex biological networks, including gene regulatory networks, metabolic pathways, and signal transduction cascades, treating them as integrated systems rather than isolated components.

Key Terms at a Glance

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

Attractor: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.

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

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

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

Compartmental Model: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).

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

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

Dynamical System: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.

Eigenvalue: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.

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

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

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

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

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

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