Tangled nature model
The tangled nature model {{cite journal
| last1 = Christensen
| first1 = K.
| last2 = Di Collobiano
| first2 = S.A.
| last3 = Hall
| first3 = M.
| last4 = Jensen
| first4 = H.J.
| year = 2002
| title = Tangled nature: a model of evolutionary ecology
| journal = Journal of Theoretical Biology
| volume = 216
| issue = 1
| pages = 73–84
| doi = 10.1006/jtbi.2002.2530
| pmid = 12076129
| arxiv = cond-mat/0104116
| bibcode = 2002JThBi.216...73C
{{cite book
| editor1-last = Deutsch
| editor1-first = Andreas
| editor2-last = Bravo de la Parra
| editor2-first = Rafael
| editor3-last = de Boer
| editor3-first = Rob J.
| editor4-last = Diekmann
| editor4-first = Odo
| editor5-last = Jagers
| editor5-first = Peter
| editor6-last = Kisdi
| editor6-first = Eva
| editor7-last = Kretzschmar
| editor7-first = Mirjam
| editor8-last = Lansky
| editor8-first = Petr
| editor9-last = Metz
| editor9-first = Hans
| title = Mathematical Modeling of Biological Systems
| volume = II
| series = Modeling and Simulation in Science, Engineering and Technology
| publisher = Birkhäuser Boston
| year = 2008
| isbn = 9780817645557
}}
{{cite book
| last1 = Jensen
| first1 = Henrik Jeldtoft
| last2 = Sibani
| first2 = Paolo
| title = Stochastic Dynamics of Complex Systems: From Glasses to Evolution
| publisher = Imperial College Press
| year = 2013
| isbn = 9781848169951
}}
| last = Jensen
| first = Henrik Jeldtoft
| title = Complexity Science: The Study of Emergence
| publisher = Cambridge University Press
| year = 2022
| isbn = 9781108883160
| language = English
}} 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,{{cite journal
| last1 = Vázquez
| first1 = P.
| last2 = Del Río
| first2 = J.A.
| last3 = Cedano
| first3 = K.G.
| last4 = Martínez
| first4 = M.
| last5 = Jensen
| first5 = H.J.
| year = 2015
| title = An entangled model for sustainability indicators
| journal = PLOS ONE
| volume = 10
| issue = 8
| page = e0135250
| doi = 10.1371/journal.pone.0135250
| doi-access = free
| pmid = 26295948
| pmc = 4546502
| bibcode = 2015PLoSO..1035250V
}} organizational ecology,{{cite journal
| last1 = Arthur
| first1 = R.
| last2 = Nicholson
| first2 = A.
| last3 = Sibani
| first3 = P.
| last4 = Christensen
| first4 = M.
| year = 2017
| title = The tangled nature model for organizational ecology
| journal = Computational and Mathematical Organization Theory
| volume = 23
| pages = 1–31
| doi = 10.1007/s10588-016-9226-3
| doi-broken-date = 20 May 2025
}} the Gaia hypothesis{{cite journal
| last1 = Arthur
| first1 = R.
| last2 = Nicholson
| first2 = A.
| year = 2022
| title = Selection principles for Gaia
| journal = Journal of Theoretical Biology
| volume = 533
| pages = 110940
| doi = 10.1016/j.jtbi.2021.110940
| pmid = 34710434
| bibcode = 2022JThBi.53310940A
}} opinion dynamics{{cite journal
| last1 = Rajpal
| first1 = H.
| last2 = Rosas
| first2 = F.E.
| last3 = Jensen
| first3 = H.J.
| year = 2019
| title = Tangled worldview model of opinion dynamics
| journal = Frontiers in Physics
| volume = 7
| page = 163
| doi = 10.3389/fphy.2019.00163
| doi-access = free
| arxiv = 1901.06372
| bibcode = 2019FrP.....7..163R
}} and cultural evolution{{cite journal
| last1 = Nicholson
| first1 = A.E.
| last2 = Sibani
| first2 = P.
| year = 2016
| title = Cultural evolution as a nonstationary stochastic process
| journal = Complexity
| volume = 21
| issue = 6
| pages = 214–223
| doi = 10.1002/cplx.21743
| pmid = 28190951
| pmc = 5300684
}} among other topics.
Model Description
Individuals in the model are represented by binary 'genomes' of some fixed length . All individuals with the same genome are equivalent and combine into 'species' with populations where is the total population and the number of distinct species.
The individuals interact through a coupling matrix . Typically some fraction of the potential entries are set to zero, as well as the diagonals and for the non-zero elements .
In a single update step an individual is selected and reproduces with probability and dies with probability which is usually constant.
:
which is a sigmoid function of the fitness
:
This compares the interaction of every individual with every other individual as specified by the coupling matrix . 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 for each 'bit' in the genome to flip and thereby generate a new species.
Typically chances for reproduction and death are taken to constitute one generation and the model is run for many thousands of generations.
Model Dynamics
File:Tangled Nature Model population.png
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' {{cite journal
| last1 = Becker
| first1 = N.
| last2 = Sibani
| first2 = P.
| year = 2014
| title = Evolution and non-equilibrium physics: A study of the tangled nature model
| journal = Europhysics Letters
| volume = 105
| issue = 1
| page = 18005
}}
In a stable period a core species has . For a new species to arise and gain significant population requires . Solving for gives
:
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 . 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 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.