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Session Length
~17 min
Adaptive Checks
15 questions
Transfer Probes
8
Mathematical biology, also known as biomathematics or mathematical and theoretical biology, is an interdisciplinary field that applies mathematical techniques, models, and theoretical frameworks to understand and predict biological phenomena. It draws on tools from differential equations, dynamical systems, probability theory, statistics, and computational methods to formalize biological processes ranging from molecular interactions within cells to the dynamics of entire ecosystems. By translating biological hypotheses into precise mathematical language, researchers can rigorously test assumptions, generate quantitative predictions, and uncover principles that are difficult to discern through experimentation alone.
The origins of mathematical biology can be traced to foundational works such as the Lotka-Volterra predator-prey equations in the 1920s and the Hodgkin-Huxley model of neuronal action potentials in the 1950s. The field expanded rapidly with the advent of computational power, allowing researchers to simulate complex systems such as protein folding, cardiac electrophysiology, and epidemiological outbreaks. Landmark contributions include Alan Turing's reaction-diffusion theory of morphogenesis, Ronald Fisher's mathematical framework for natural selection, and the SIR compartmental models of infectious disease that remain central to public health policy today.
In the modern era, mathematical biology has become indispensable across the life sciences. It drives advances in systems biology, where networks of genes and proteins are modeled as dynamical systems; in epidemiology, where differential equation models inform vaccination strategies and pandemic preparedness; in ecology, where population dynamics and evolutionary game theory guide conservation efforts; and in biomedical engineering, where tumor growth models support personalized cancer therapy. The field continues to evolve with the integration of machine learning, stochastic processes, and multi-scale modeling, making it one of the most vibrant and consequential areas of applied mathematics.
One step at a time.
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.
Example: Modeling the cyclical population fluctuations of Canadian lynx and snowshoe hares observed in Hudson's Bay Company fur-trapping records over two centuries.
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.
Example: Public health agencies used SIR-type models during the COVID-19 pandemic to estimate peak infection dates, hospital bed demand, and the impact of social distancing measures.
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.
Example: Alan Turing proposed that reaction-diffusion mechanisms could explain how a homogeneous embryo develops spatial patterns, such as the stripe and spot patterns on animal coats.
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.
Example: The model accurately reproduces the shape, threshold, and refractory period of the squid giant axon action potential, and it earned Hodgkin and Huxley the 1963 Nobel Prize in Physiology or Medicine.
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.
Example: Pharmacologists use Michaelis-Menten kinetics to model how quickly a drug is metabolized by liver enzymes, helping determine appropriate dosing intervals.
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.
Example: A population model may exhibit a Hopf bifurcation: as the carrying capacity increases past a critical value, a stable steady state gives way to sustained periodic oscillations in population size.
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.
Example: The Hawk-Dove game models animal conflict over resources: a mixed population of aggressive (hawk) and passive (dove) individuals can reach a stable equilibrium frequency predicted by payoff values.
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.
Example: Gene expression in individual cells is inherently noisy; stochastic models explain why genetically identical cells in the same environment can display different protein levels and phenotypes.
More terms are available in the glossary.
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