Mathematical Biology

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.