Introduction

Any theory aims towards recovering the landscape of possibilities of what can happen in nature, it is thus always the basis of any thinking about the causes of observed patterns.

When conceptualizing these possibilities, we build models, i.e. the formal representation of particular set of possibilities. Models can be geometric, revealing formal (mathematical) links between patterns, or dynamical.

Dynamical models

Dynamical models conceptualize the processes in nature as a movement between different states, driven by particular rules that represent the key assumed ecological processes. The transition between states can be deterministic or stochastic (probabilistic).

Attractors

For given sets of states and rules, there are always attractors of the dynamic, represented by states towards which the dynamic converges. We are often interested in these attractors of a dynamic.

The attractor can be just one state (point attractor), or can consist in cycling among several states (cyclic attractor), or in a more complex movement among states which is not predictable without an exact knowledge of all parameters and the whole dynamic (chaotic behaviour), due to a strange attractor.

The very popular images of the butterfly effect are an example of a strange attractor called the Lorenz attractor.

Attractors then represent stable points of the dynamics, so that any movement away from it leads ultimately back to the attractor.

The set of states which lead to a particular attractor is called the basin of attraction.

States of a system are often characterized by a combination of particular continuous variables of interest (e.g. the number of individuals of each species population). Then the dynamics are modelled as movement within the state space. Besides the combination of the values of the variables in focus, each state is characterized by a vector of its future change, i.e. by its tendency to transition to another state.

Extra credits

For a fun, interactive explanation of attractor landscapes, take a look at this webpage https://ncase.me/attractors/