Survival of the Weakest? General properties of many competing species
S.O. Case, C.H. Durney, M. Pleimling EPL 92, 58003 (2010), PRE 83, 051108 (2011)
MPI Dresden, July 2011
R.K.P. Zia Physics Department, Virginia Tech,
Blacksburg, Virginia, USA
Supported by Materials Theory,
Division of Materials Research
R.K.P. Zia, arXiv.org: 1101:0018 (2010-11)
Outline
• Motivations – Michel’s fault + 2 students looking for summer projects.
– Population dynamics: Venerable, Interesting!
– Cyclic competition of 3 species: Survival of the Weakest!?!
• Competition of M species (NO spatial structure)
– M=4 cyclic competition: Other maxims and novel features
– Deterministic MFT vs. stochastic evolution
– General properties for any M with arbitrary pairwise
interactions
• Summary and Outlook
Motivations
• Population Dynamics… quick reminder
– Malthus (~1800): 𝜕𝜏𝑥 = 𝜆𝑥
– Verhulst (1838): 𝜕𝜏𝑥 = 𝜆𝑥(1 − 𝑥) …logistic map (Feigenbaum, May, 1970’s)
– Lotka-Volterra (1920’s)
𝜕𝜏𝑥 = −𝛿𝑥 + 𝛾𝑥𝑦
𝜕𝜏𝑦 = +𝛽𝑦 − 𝛾𝑥𝑦
Motivations
• Cyclic competition of three species
– Frey, et.al.: “Survival of the Weakest”
– Easier, intuitive picture? and …
– Does this apply in other situations?
A+A B+B C+C
A+B
B+C
C+A
A+A
B+B
C+C
pa
pb
pc
Cyclic competition of 3 species
A+A B+B C+C
A+B
B+C
C+A
A+A
B+B
C+C
pa
pb
pc
Cyclic competition of 3 species
Simple stochastic model: • No spatial structure
• Bag of N balls, of 3 colors
(e.g., Azure, Black, Cinnamon)
• Rule is easy: randomly pick a pair; change
color of one ball according to given p’s
N is conserved
(fractions) A + B + C = 1
• Three absorbing states.
Cyclic competition of 3 species
Simple stochastic model: • What we really want is:
Given the p’s and initial numbers,
…after t picks, what is the probability:
• Master equation it satisfies:
Cyclic competition of 3 species
Simple stochastic model: In particular, the eventual survival probabilities:
P(N,0,0; | …)
i.e., probability of (fraction) A = 1
P(0,N,0; | …) i.e., B = 1
…etc.
Cyclic competition of 3 species
Mean Field version:
• Take exact master equation for P(A,B,C ; t)
• …and consider averages: e.g.,
• Take N limit, get continuous time
• Probabilities, ps , become rates: ks
• Neglect correlations: e.g., AB AB
• Get ODEs for A, etc. (denoted by A, etc.)
• Result is …
… a couple of lines to see … All (generic) initial populations
evolve periodically !
not into absorbing states!
Cyclic competition of 3 species
Mean Field (rate) Equations
with rescaled time to normalize ka+kb+kc=1
Fixed point:
A = kb , B = kc , C = ka
Invariant:
kb
kc
ka
MFT predicts all will survive!
Invariant manifold:
R = const. Orbits are closed loops
Survival of the Weakest ??? Berr, Reichenbach, Schottenloher, and Frey, PRL 102 048102 (2009)
pa < pb , pc
A is the “weakest”
Survival of the Weakest!! Berr, Reichenbach, Schottenloher, and Frey, PRL 102 048102 (2009)
Stochastics
enlivens
the scene!!
Survival of the Weakest!! Berr, Reichenbach, Schottenloher, and Frey, PRL 102 048102 (2009)
100 % !!
…bottom line: Weakest do NOT always win!
Prey of the prey of the weakest lose.
…leads to weakest doing well in M=3 case!
What about four Species ? Case, Durney, Pleimling and Zia, EPL 92, 58003 (2010)+…
What about 4 Species ? Case, Durney, Pleimling and Zia, EPL 92, 58003 (2010)
A C
B
D
What about 4 Species ? Case, Durney, Pleimling and Zia, EPL 92, 58003 (2010)
Total number of balls, N, is constant.
• 2(N+1) absorbing states: A-C vs. B-D
• …forming opposing teams (like Bridge)
• Winner has larger rate product: kakc vs. kbkd
• Losers die out exponentially fast
• If competition is neutral, then there are
− two invariants
− one fixed line
− saddle shaped closed looped orbits
What about 4 Species ? Case, Durney, Pleimling and Zia, EPL 92, 58003 (2010)
A B
C
D
A
B
C
D
2(N+1) absorbing states: A-C vs. B-D
What about 4 Species ? Case, Durney, Pleimling and Zia, EPL 92, 58003 (2010)
> 0: system ends up on A-C line.
< 0: system ends up on B-D line.
≡ kakc − kbkd
What’s special about ?
• Looks like a determinant…
• From Master Equation (for P{Nm;t}) to …
… is really a determinant ! (later)
… Rate Equations (for averages Nmt).
What’s special about ?
• Looks like a determinant…
• From Master Equation (for P{Nm;t}) to …
… is really a determinant ! (later)
… Rate Equations (for averages Nmt).
linear combinations to…
What’s special about ?
What’s special about ?
> 0: system ends up on A-C line.
< 0: system ends up on B-D line.
Two examples of ≠ 0
> 0, A-C wins
< 0, B-D wins
(0.35, 0.42, 0.09, 0.14) λ = −0.0273
(0.45, 0.33, 0.14, 0.08)
λ = 0.0366
A better view of say, > 0
a
c
d
b
In this region
both a and c
increase
In this region
both b and d
decrease
is a stable
fixed line
A better view of say, > 0
• Intersection is an irregular tetrahedron,
… in which orbits are monotonic.
• In particular, there is a straight-line
(dubbed “the arrow”) on which
the system evolves like the case
with just one species (Verhulst):
• Other typical orbits spiral around this arrow.
𝜕𝜏ℎ = 𝜔ℎ(1 − ℎ)
If you start anywhere on this line, you just move along it!
A better view of say, > 0
• Intersection is an irregular tetrahedron,
… in which orbits are monotonic.
• In particular, there is a straight-line
(dubbed “the arrow”) on which
the system evolves like the case
with just one species (Verhulst):
• Other typical orbits spiral around this arrow.
𝜕𝜏ℎ = 𝜔ℎ(1 − ℎ)
= /( )
If you start anywhere on this line, you just move along it!
An example of > 0
A B C
D D
C A
Forward orbit
Backward orbit
More special are =0 cases !
Line of fixed points and
Invariant manifolds
Neutral !!
More special are =0 cases !
If you start anywhere on this line, you just stay there!
A+C=γ
B+D=1- γ
More special are =0 cases !
• Each defines a (generalized) hyperbolic sheet.
• Intersection is a closed loop (~ edge of a saddle).
• Average (over an orbit) is a point on fixed line.
• Extremal points can be found analytically.
… are CONSTANTS under the evolution!
Two views of a = 0 case
rates: (0.4, 0.4, 0.1, 0.1)
and initial values:
(0.02, 0.1, 0.48, 0.4)
A B C
D
A
B C
D Fixed line
Do invariants & Qs always exist ? R.K.P.Zia, arXiv 1101.0018 (2010)
…insights from studying …
M species
with arbitrary pair-wise interactions
• Odd/Even M belong to different classes.
• Odd M
– Fixed point and R necessarily exist (“duality”)
– No other possibilities for cyclic competition
Do invariants & Qs always exist ?
• Even M
– Q necessarily exist!
– Λ, a determinant, generalizes (and plays same role)
– If Λ=0, there are subspaces of fixed points and
invariant manifolds (“duality”)
– No other possibilities besides fixed line and two invariants
for cyclic competition
– More interesting results if M are two ‘teams’
with M/2 players (ask me later!)
Brief glimpse of analysis
• Start with
• Get rate equations
• Write in vector/matrix form
M
Brief glimpse of analysis
• anti-symmetric, so odd M det = 0,
with at least one zero.
• Right e-vector gives fixed point
• Left e-vector provides invariant
…“duality”
Brief glimpse of analysis
• anti-symmetric, so even M det = Λ
can be anything.
• If Λ≠0, can invert to get
• So, and
• …evolves as
☺ Q in 4 species case is ! ☺
Brief glimpse of analysis
• anti-symmetric, so even M Λ=0 must
come with even number (2m) of zeros.
• Each zero corresponds to a fixed point and
an invariant.
• 2m-1 dimensional subspace of fp’s
…“dual” to…
• M-1-2m dimensional invariant manifold
☺ 4 species case has line of fp’s and invariant loop! ☺
Stochastics enlivens the scene !
• Not surprising:
– MF pretty good if all Nm’s are large.
– Unpredictable extinction probabilities
– Finding systematic behavior challenging
– Either pair may win in neutral (=0) case.
• Surprises:
– Evolution of Q distributions
– Distributions of surviving pairs
Stochastics enlivens the scene !
rates: (0.4, 0.4, 0.1, 0.1)…..
initial values:
(0.02, 0.1, 0.48, 0.4) 1000
Stochastics enlivens the scene !
Most interesting case we found:
– “Extreme” rates: 0.1, 0.0001, 0.1, 0.7999
– Initial values: 100, 700, 100, 100
– 10,000 runs, 90% ends on AC line (>0)
– Mostly, D dies first (B weakest!).
– MFT shows “3 spirals,” each coming close to
the ABC face (D=0) ...
– …corresponding to 3 distinct clusters
Stochastics enlivens the scene !
Most interesting case we found:
– “Extreme” rates: 0.1, 0.0001, 0.1, 0.7999
– Initial values: 100, 700, 100, 100
– 10,000 runs, 90% ends on AC line (>0)
– Mostly, D dies first (B weakest!).
– MFT shows “3 spirals,” each coming close to
the ABC face (D=0) ...
– …corresponding to 3 distinct clusters
Prey of the prey of the weakest lose.
Prey of the prey of the strongest win.
Stochastics enlivens the scene !
D nearly dies
in MF
Stochastics enlivens the scene !
D nearly dies
in MF
Stochastics enlivens the scene !
D dies in MF
Summary and Outlook
• Pairwise competition, ODE or stochastic,
provides many interesting issues to study
• Some aspects understood; but puzzles remain
• Many immediate extensions, e.g.,
spatial structures, networks, inhomogeneous environments,…
• Further generalizations and applications
• Many exciting things to do … many ways to
get involved …
S.O. Case, C.H. Durney, M. Pleimling and R.K.P. Zia,
EPL 92, 58003 (2010), PRE 83, 051108 (2011)
arXiv: 1101:0018 (2010-11)
