## Scientific Inspiration

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
•     Spatial distribution modeing using plot samples

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.
•     Partial differential equations – continuous time, continuous state space, spatial derivatives.
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.
•     Continuous Markov process – stochastic differential equations or a Fokker-Planck equation – continuous time, continuous state space, events occur continuously according to a random Wiener process.    