Tangled nature model

The tangled nature model ^[1][2][3][4] is a model of evolutionary ecology developed by Christensen, Di Collobiano, Hall and Jensen. It is an agent-based model where individual 'organisms' interact, reproduce, mutate and die across many generations. A notable feature of the model is punctuated equilibrium, abrupt and spontaneous transitions between long lived stable states. In addition to evolutionary ecology the model has been used to study sustainability,^[5] organizational ecology,^[6] the Gaia hypothesis^[7] opinion dynamics^[8] and cultural evolution^[9] among other topics.

Individuals in the model are represented by binary 'genomes' ${\displaystyle a}$ of some fixed length ${\displaystyle L}$. All individuals with the same genome are equivalent and combine into 'species' with populations ${\displaystyle N_{a}}$ where ${\displaystyle N=\sum _{a}^{D}N_{a}}$ is the total population and ${\displaystyle D}$ the number of distinct species.

The individuals interact through a coupling matrix ${\displaystyle J}$. Typically some fraction of the potential entries are set to zero, as well as the diagonals ${\displaystyle J_{aa}=0}$ and for the non-zero elements ${\displaystyle J_{ab}\neq J_{ba}}$.

In a single update step an individual is selected and reproduces with probability ${\displaystyle p_{off}(H_{a})}$ and dies with probability ${\displaystyle p_{d}}$ which is usually constant.

${\displaystyle p_{off}(H_{a})={\frac {e^{H_{a}}}{1+e^{H_{a}}}}}$

which is a sigmoid function of the fitness

${\displaystyle H_{a}=-\mu N+{\frac {1}{N}}\sum _{b}J_{ab}N_{b}}$

This compares the interaction of every individual with every other individual as specified by the coupling matrix ${\displaystyle J}$. ${\displaystyle \mu }$ is the inverse of the carrying capacity and controls the total number of individuals which can exist in the model. When an individual reproduces asexually there is some small, fixed probability ${\displaystyle p_{mut}}$ for each 'bit' in the genome to flip and thereby generate a new species.

Typically ${\displaystyle {\frac {N}{p_{d}}}}$ chances for reproduction and death are taken to constitute one generation and the model is run for many thousands of generations.

Plotting the model population over time demonstrates punctuated equilibrium, long lived quasi stable states which abruptly terminate and are replaced with new ones. During a stable period the model generates a network of mutualistic interactions between a small number of populous species, often called the 'core' and 'cloud' ^[10]

In a stable period a core species has ${\displaystyle p_{off}(H_{a})\simeq p_{d}}$. For a new species ${\displaystyle c}$ to arise and gain significant population requires ${\displaystyle p_{off}(H_{c})>p_{d}}$. Solving for ${\displaystyle H_{c}}$ gives

${\displaystyle \sum _{b}J_{cb}{\frac {N_{b}}{N}}>\mu N-\log(1/p_{d}-1)}$

as the requirement for the new species to be viable. This means the new species has to have sufficiently strong net positive interactions, especially with the core species, which are the only ones with large values of ${\displaystyle {\frac {N_{b}}{N}}}$. The right hand side represents a 'barrier', controlled by the total population, which makes large population states harder to invade.

If a new species can overcome the barrier then it will grow rapidly, at the expense of the existing species either through parasitic couplings ${\displaystyle J_{cb}<0}$ or by using up the carrying capacity of the system. This can precipitate either a core rearrangement, with the incorporation of the new species into the core and a readjustment of populations, or a total collapse of the state.