History of pattern formation dates back to 1952 when A. M. Turing pointed
out that diffusion can destabilize an otherwise stable system to give rise to spatial
patterns. Since then the instability has been studied in ecological, chemical and biochemical
systems. An alternative to reaction–diffusion systems is meta–population
models which assume that species can be thought as distributed in different spatial
pockets connected by spatial processes such as migration and dispersal. Murdoch et al.
(1992) explored the model proposed by Godfray and Pacala who assumed that within
patch dynamics is described by Lotka–Volterra model. Spatial differences were created
by making the prey birth rate in patch 2 (α2) greater than that in patch 1(α1). Prey
moves symmetrically from one patch to the other; i.e., z1 = z2 . The meta-population is
neutrally stable when birth rates of prey are equal. When significant difference in birth
rates is created, oscillations in prey abundance in two patches become increasingly less
correlated. This is associated with per capita prey immigration into a patch becoming
increasingly temporally density–dependent. The density dependence arises as the
number of immigrants into a patch is weakly correlated with the number of residents in
the patch. Cellular automata simulation of a reaction–diffusion system obeying rules by
Ebenhoh shows that fractals are present in fish school motion. Lewis and Collaborators
developed a modeling approach which enables us to find out invasion speed of a
biological invasion. This approach involves setting up an integro–differential equation
which needs a dispersal kernel to be specified.
Keywords: Spatial patterns, Crystal lattices, Host–parasitoid models, Reaction–
diffusion systems, Turing instability, Hopf–bifurcation, Turing–hopf bifurcation,
Population stability, Meta–population models, Animal movements, Stochastic
differential equations, Fokker–plank equation, Ebenhon rules, Fish school motion,
Cellular automata, Fractals, Biological invasion, Dispersal kernel, Integro–
differential equations, Invasion speed.
