A tight race between deterministic and stochastic dynamics of - PowerPoint PPT Presentation

A tight race between deterministic and stochastic dynamics of RPS-model Qian Yang Supervisors: Prof. Jonathan Dawes & Dr. Tim Rogers University of Bath Email: Q.Yang2@bath.ac.uk July 5, 2016 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Qian Yang (Uni. of Bath) Different dynamics in cyclic dominance July 5, 2016 1 / 16

Overview Motivation 1 RPS-Model 2 2.1 Rock-Paper-Scissors - It’s just a GAME. 2.2 ODEs and simulations for RPS-model - We found a RACE! 2.3 Three regions of average period of these cycles - WHO WINS? Summary 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Qian Yang (Uni. of Bath) Different dynamics in cyclic dominance July 5, 2016 2 / 16

1. Motivation Cyclic Dominance (Rock-Paper-Scissors)– 1 widely exists in nature, eg. Biology, Chemistry 2 describes the interactions between species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Qian Yang (Uni. of Bath) Different dynamics in cyclic dominance July 5, 2016 3 / 16

1. Motivation Analysis Method – 1 Deterministic: Continuous and infinite. 2 Stochastic: Discrete and finite Main work – 1 Agreement and disagreement of RPS-model with the two methods. 2 Noise slows down the evolution of cyclic dominance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Qian Yang (Uni. of Bath) Different dynamics in cyclic dominance July 5, 2016 4 / 16

2.1 Rock-Paper-Scissors Game RPS simplest model y : x beats y x B(Paper) C(Scissors) A(Rock) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Qian Yang (Uni. of Bath) Different dynamics in cyclic dominance July 5, 2016 5 / 16

2.1 Rock-Paper-Scissors Game RPS simplest model y : x beats y x B(Paper) C(Scissors) A(Rock) Payoff matrix A B C  0 − 1 1  A P = B 1 0 − 1   − 1 1 0 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Qian Yang (Uni. of Bath) Different dynamics in cyclic dominance July 5, 2016 5 / 16

2.1 Rock-Paper-Scissors Game RPS simplest model Differential equations for y : x beats y x deterministic study B(Paper) a = a ( c − b ) , ˙ ˙ b = b ( a − c ) , c = c ( b − a ) , ˙ and a ( t ) + b ( t ) + c ( t ) = 1 . C(Scissors) A(Rock) Payoff matrix A B C  0 − 1 1  A P = B 1 0 − 1   − 1 1 0 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Qian Yang (Uni. of Bath) Different dynamics in cyclic dominance July 5, 2016 5 / 16

2.1 Rock-Paper-Scissors Game RPS simplest model Differential equations for y : x beats y x deterministic study B(Paper) a = a ( c − b ) , ˙ ˙ b = b ( a − c ) , c = c ( b − a ) , ˙ and a ( t ) + b ( t ) + c ( t ) = 1 . C(Scissors) A(Rock) Chemical reactions for stochastic simulation Payoff matrix 1 A + B → B + B , − A B C 1 B + C − → C + C ,  0 − 1 1  A P = 1 B 1 0 − 1 C + A → A + A , −   − 1 1 0 C and A + B + C = N , N is fixed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Qian Yang (Uni. of Bath) Different dynamics in cyclic dominance July 5, 2016 5 / 16

2.1 Rock-Paper-Scissors Game Update the simplest RPS-model RPS-model with mutation and unbalanced payoff x y : x beats y x y : x mutates into y B(Paper) C(Scissors) A(Rock) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Qian Yang (Uni. of Bath) Different dynamics in cyclic dominance July 5, 2016 6 / 16

2.1 Rock-Paper-Scissors Game Update the simplest RPS-model RPS-model with mutation and unbalanced payoff x y : x beats y x y : x mutates into y B(Paper) C(Scissors) A(Rock) Payoff matrix A B C   0 − β − 1 1 A P = + Mutations B 1 0 − β − 1   − β − 1 1 0 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Qian Yang (Uni. of Bath) Different dynamics in cyclic dominance July 5, 2016 6 / 16

2.1 Rock-Paper-Scissors Game Update the simplest RPS-model Ordinary Differential Equations a = a [ c − (1 + β ) b + β ( ab + bc + ac )] + µ ( b + c − 2 a ) , ˙ ˙ b = b [ a − (1 + β ) c + β ( ab + bc + ac )] + µ ( c + a − 2 b ) , c = c [ b − (1 + β ) a + β ( ab + bc + ac )] + µ ( a + b − 2 c ) . ˙ with β > 0 and µ > 0, µ is mutation rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Qian Yang (Uni. of Bath) Different dynamics in cyclic dominance July 5, 2016 7 / 16

2.1 Rock-Paper-Scissors Game Update the simplest RPS-model Ordinary Differential Equations a = a [ c − (1 + β ) b + β ( ab + bc + ac )] + µ ( b + c − 2 a ) , ˙ ˙ b = b [ a − (1 + β ) c + β ( ab + bc + ac )] + µ ( c + a − 2 b ) , c = c [ b − (1 + β ) a + β ( ab + bc + ac )] + µ ( a + b − 2 c ) . ˙ with β > 0 and µ > 0, µ is mutation rate. Chemical reactions for stochastic simulations 1 1 1 A + B → B + B , B + C → C + C , C + A → A + A , − − − β µ µ A + B + B → B + B + B , − A − → B , A − → C , β µ µ A + A + C → A + A + A , − B − → C , B − → A , β µ µ B + C + C → C + C + C , − C − → A , C − → B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Qian Yang (Uni. of Bath) Different dynamics in cyclic dominance July 5, 2016 7 / 16

2.2 Analysis of ODEs Jacobian Matrix of ODEs 1 The only interior equilibrium: x ∗ = ( a ∗ , b ∗ , c ∗ ) = (1 / 3 , 1 / 3 , 1 / 3) 2 Jacobian matrix around the equilibrium ( − 1 − 2 3 − 1 ) 3 − 3 µ 3 β J ∗ = 2 3 + 1 1 3 + 1 3 β 3 β − 3 µ 3 The critical value of µ : µ c = β 18 . 4 When µ < µ c , Hopf bifurcation happens. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Qian Yang (Uni. of Bath) Different dynamics in cyclic dominance July 5, 2016 8 / 16

2.2 Numerical solution of ODEs Formation of a robust cycle - limit cycle √ Figure: β = 1 216 < µ c = 1 1 36 , y 1 = a + 1 3 2 , µ = 2 b , y 2 = 2 b . 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 The coordinates transformation makes the flow spiraling outwards visible. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Qian Yang (Uni. of Bath) Different dynamics in cyclic dominance July 5, 2016 9 / 16

2.2 Stochasitic simulations Comparison with numerical solution to ODEs Figure: β = 1 1 2 , µ = 216 . N is total of individuals in simulation. Behave differently. SSA - Stochastic Simulation Algorithm. Size N controls the level of randomness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Qian Yang (Uni. of Bath) Different dynamics in cyclic dominance July 5, 2016 10 / 16

2.2 Quasi-periodic cycles Average period of these quasi-periodic cycles An interesting comparison: There is a RACE between stochastic dynamic and deterministic dynamic! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Qian Yang (Uni. of Bath) Different dynamics in cyclic dominance July 5, 2016 11 / 16

2.3 Region I - the right part Determinsitic dynamic governs the game Compose local and global map: (0,1,0) global map (1,0,0) (0,0,1) local map T ODE ∝ − ln µ is proved theoretically. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Qian Yang (Uni. of Bath) Different dynamics in cyclic dominance July 5, 2016 12 / 16

2.3 Region III - the left part Stochastic dynamic prevails the opponent The cycle in this region looks like: It is a 1-D death-birth process. 1 1 When µ ≪ N ln N , T SSA ∝ N µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Qian Yang (Uni. of Bath) Different dynamics in cyclic dominance July 5, 2016 13 / 16

2.3 Region II - the middle part A tight race between the two dynamics Explanation: the idea of asymptotic phase (the picture, cited from J. M. Newby) and SDE. N ln N , T SDE ≈ − 3 ln µ + C 3 1 When µ ∼ N µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Qian Yang (Uni. of Bath) Different dynamics in cyclic dominance July 5, 2016 14 / 16