One classic work in this area is Alan Turing’s paper on morphogenesis entitled The Chemical Basis of Morphogenesis, published in 1952 in the Philosophical Transactions of the Royal Society.
• Travelling waves in a wound-healing assay
• Swarming behaviour
• A mechano-chemical theory of morphogenesis
• Biological pattern formation
The earlier stages of mathematical biology were dominated by mathematical biophysics, described as the application of mathematics in biophysics, often involving specific physical/mathematical models of bio-systems and their components or compartments.
The following is a list of mathematical descriptions and their assumptions.
Deterministic processes (dynamical systems)
A fixed mapping between an initial state and a final state. Starting from an initial condition and moving forward in time, a deterministic process will always generate the same trajectory and no two trajectories cross in state space.
• Difference equations/Maps – discrete time, continuous state space.
• Ordinary differential equations – continuous time, continuous state space, no spatial derivatives.
See also: Numerical ordinary differential equations.
• Partial differential equations – continuous time, continuous state space, spatial derivatives.
See also: Numerical partial differential equations.
Stochastic processes (random dynamical systems)
A random mapping between an initial state and a final state, making the state of the system a random variable with a corresponding probability distribution.
• Non-Markovian processes – generalised master equation – continuous time with memory of past events, discrete state space, waiting times of events (or transitions between states) discretely occur and have a generalised probability distribution.
• Jump Markov process – master equation – continuous time with no memory of past events, discrete state space, waiting times between events discretely occur and are exponentially distributed. See also: Monte Carlo method for numerical simulation methods, specifically dynamic Monte Carlo method and Gillespie algorithm.