## Background Info

## 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.**

**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.**

**• 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.**