SLIDE 1 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)
SLIDE 2 Outline
– 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
SLIDE 3 Motivations
- Population Dynamics… quick reminder
– Malthus (~1800): 𝜖𝜐𝑦 = 𝜇𝑦 – Verhulst (1838): 𝜖𝜐𝑦 = 𝜇𝑦(1 − 𝑦) …logistic map (Feigenbaum, May, 1970’s) – Lotka-Volterra (1920’s) 𝜖𝜐𝑦 = −𝜀𝑦 + 𝛿𝑦𝑧 𝜖𝜐𝑧 = +𝛾𝑧 − 𝛿𝑦𝑧
SLIDE 4 Motivations
- Cyclic competition of three species
– Frey, et.al.: “Survival of the Weakest” – Easier, intuitive picture? and … – Does this apply in other situations?
SLIDE 5
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
SLIDE 6 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
SLIDE 7 Cyclic competition of 3 species
Simple stochastic model:
Given the p’s and initial numbers, …after t picks, what is the probability:
- Master equation it satisfies:
SLIDE 8 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.
SLIDE 9 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!
SLIDE 10
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
SLIDE 11
MFT predicts all will survive!
Invariant manifold:
R = const.
Orbits are closed loops
SLIDE 12 Survival of the Weakest ???
Berr, Reichenbach, Schottenloher, and Frey, PRL 102 048102 (2009)
pa < pb , pc
A is the “weakest”
SLIDE 13 Survival of the Weakest!!
Berr, Reichenbach, Schottenloher, and Frey, PRL 102 048102 (2009)
Stochastics enlivens the scene!!
SLIDE 14 Survival of the Weakest!!
Berr, Reichenbach, Schottenloher, and Frey, PRL 102 048102 (2009)
100 % !!
SLIDE 15 …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)+…
SLIDE 16 What about 4 Species ?
Case, Durney, Pleimling and Zia, EPL 92, 58003 (2010)
A C B D
SLIDE 17 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
SLIDE 18 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
SLIDE 19 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
SLIDE 20 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).
SLIDE 21 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…
SLIDE 22
What’s special about ?
SLIDE 23
What’s special about ?
> 0: system ends up on A-C line. < 0: system ends up on B-D line.
SLIDE 24 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
SLIDE 25 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
SLIDE 26 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.
𝜖𝜐ℎ = 𝜕ℎ(1 − ℎ)
If you start anywhere on this line, you just move along it!
SLIDE 27 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.
𝜖𝜐ℎ = 𝜕ℎ(1 − ℎ) = /( )
If you start anywhere on this line, you just move along it!
SLIDE 28 An example of > 0
A B C D D C A
Forward orbit Backward orbit
SLIDE 29 More special are =0 cases !
Line of fixed points
and
Invariant manifolds
Neutral !!
SLIDE 30 More special are =0 cases !
If you start anywhere on this line, you just stay there!
A+C=γ B+D=1- γ
SLIDE 31 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!
SLIDE 32 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
SLIDE 33 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
SLIDE 34 Do invariants & Qs always exist ?
– 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!)
SLIDE 35 Brief glimpse of analysis
- Start with
- Get rate equations
- Write in vector/matrix form
M
SLIDE 36 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”
SLIDE 37 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 ! ☺
SLIDE 38 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! ☺
SLIDE 39 Stochastics enlivens the scene !
– MF pretty good if all Nm’s are large. – Unpredictable extinction probabilities – Finding systematic behavior challenging – Either pair may win in neutral (=0) case.
– Evolution of Q distributions – Distributions of surviving pairs
SLIDE 40 Stochastics enlivens the scene !
rates: (0.4, 0.4, 0.1, 0.1)….. initial values: (0.02, 0.1, 0.48, 0.4) 1000
SLIDE 41
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
SLIDE 42
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.
SLIDE 43 Stochastics enlivens the scene !
D nearly dies in MF
SLIDE 44 Stochastics enlivens the scene !
D nearly dies in MF
SLIDE 45 Stochastics enlivens the scene !
D dies in MF
SLIDE 46 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 …
SLIDE 47 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)
