Survival of the Weakest? General properties of many competing - PowerPoint PPT Presentation

MPI Dresden, July 2011 Survival of the Weakest? General properties of many competing species R.K.P. Zia Physics Department, Virginia Tech, Blacksburg, Virginia, USA S.O. Case, C.H. Durney, M. Pleimling EPL 92, 58003 (2010), PRE 83 , 051108 (2011) R.K.P. Zia, arXiv.org: 1101:0018 (2010-11) Supported by Materials Theory, Division of Materials Research

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): 𝜖 𝜐 𝑦 = 𝜇𝑦 – Verhulst (1838): 𝜖 𝜐 𝑦 = 𝜇𝑦(1 − 𝑦) …logistic map ( Feigenbaum , May, 1970’s) – Lotka- Volterra (1920’s) 𝜖 𝜐 𝑦 = −𝜀𝑦 + 𝛿𝑦𝑧 𝜖 𝜐 𝑧 = +𝛾𝑧 − 𝛿𝑦𝑧

Motivations • Cyclic competition of three species – Frey, et.al. : “Survival of the Weakest” – Easier, intuitive picture? and … – Does this apply in other situations?

Cyclic competition of 3 species p a A+B A+A A+A p b B+B B+C B+B p c C+C C+A C+C

Cyclic competition of 3 species Simple stochastic model : p a • No spatial structure A+B A+A A+A • Bag of N balls, of 3 colors p b (e.g., A zure, B lack, C innamon) B+B B+C B+B • Rule is easy: randomly pick a pair; change color of one ball according to given p ’s p c C+C N is conserved C+A C+C (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, p s , become rates: k s • Neglect correlations: e.g.,  AB    A  B  • Get ODEs for  A  , etc. (denoted by A , etc. ) • Result is … All (generic) initial populations … a couple of lines to see … evolve periodically ! not into absorbing states!

Cyclic competition of 3 species Mean Field (rate) Equations k b k c k a with rescaled time to normalize k a + k b + k c =1 Invariant: Fixed point: A = k b , B = k c , C = k a

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) p a < p b , p c 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 % !!

What about four Species ? Case, Durney, Pleimling and Zia, EPL 92 , 58003 (2010 )+… …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 4 Species ? Case, Durney, Pleimling and Zia, EPL 92 , 58003 (2010) D C A B

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: k a k c vs. k b k d • 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) D D B C A B A C 2( N +1) absorbing states: A-C vs. B-D

What about 4 Species ? Case, Durney, Pleimling and Zia, EPL 92 , 58003 (2010)   ≡ k a k c − k b k d  > 0: system ends up on A-C line .  < 0: system ends up on B-D line .

What’s special about  ? • Looks like a determinant… … is really a determinant ! (later) … Rate Equations (for averages  N m  t ) . • From Master Equation (for P { N m ; t } ) to …

What’s special about  ? • Looks like a determinant… … is really a determinant ! (later) … Rate Equations (for averages  N m  t ) . • From Master Equation (for P { N m ; t } ) to … 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 (0.45, 0.33, 0.14, 0.08) λ = 0.0366 (0.35, 0.42, 0.09, 0.14) λ = −0.0273

A better view of say,  > 0 a  In this region d  both a and c increase is a stable c  fixed line b  In this region both b and d decrease

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): 𝜖 𝜐 ℎ = 𝜕ℎ(1 − ℎ) • Other typical orbits spiral around this arrow. 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): 𝜖 𝜐 ℎ = 𝜕ℎ(1 − ℎ) • Other typical orbits spiral around this arrow. If you start anywhere on this line, you just move along it!

An example of  > 0 D D Backward orbit A C C B A Forward orbit

More special are  =0 cases ! Neutral !! Line of fixed points and Invariant manifolds

More special are  =0 cases ! A+C= γ B+D=1- γ If you start anywhere on this line, you just stay there!

More special are  =0 cases ! … are CONSTANTS under the evolution! • 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.

Two views of a  = 0 case D Fixed line A C B D rates: (0.4, 0.4, 0.1, 0.1) and initial values: (0.02, 0.1, 0.48, 0.4) A B C

Do invariants & Q s 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 & Q s 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 M • Get rate equations • Write in vector/matrix form

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 N m ’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