A Finite Population Destroys a Traveling Wave in Spatial Replicator Dynamics
FFinite and Infinite Population Spatial Rock-Paper-Scissors in OneDimension
Christopher Griffin ∗ Riley Mummah † Russ deForest ‡ June 2, 2020
Abstract
We derive both the finite and infinite population spatial replicator dynamics as the fluidlimit of a stochastic cellular automaton. The infinite population spatial replicator is identicalto the model used by Vickers and our derivation justifies the addition of a diffusion to thereplicator. The finite population form generalizes the results by Durett and Levin on finitespatial replicator games. We study the differences in the two equations as they pertain to theone-dimensional rock-paper-scissors game. In particular, we show that a constant amplitudetraveling wave solution exists for the infinite population case and show how population collapseprevents its formation in the finite population case. Additional solution classes in variations onrock-paper-scissors are also studied.
Evolutionary games using the replicator dynamic have been studied extensively and are now welldocumented [1–4]. Variations on the classical replicator dynamic include discrete time dynamics [5]and mutations [6, 7]. Additional evolutionary dynamics, such as imitation [1, 4, 8, 9] and exchangemodels [10] have been studied. Alternatively evolutionary games have been extended to includespatial models by a number of authors [11–18]. Most of these papers append a spatial componentto the classical replicator dynamics (see e.g., [16]) or discuss finite population replicator dynamicsin which total population counts are used (see e.g., [11]). In the latter case, a spatial term is againappended to the classical replicator structure.In [19], Durrett and Levin study discrete and spatial evolutionary game models and comparethem to their continuous, aspatial analogs. For their study the authors focus on a specific classof two-player two-strategy games using a hawk-dove payoff matrix. Because their payoff matrixis 2 ×
2, a single (spatial) variable p ( x, t ) can be used to denote the proportion of the populationplaying hawk (Strategy 1) while a second variable s ( x, t ) denotes the total population. Using thepayoff matrix: A = (cid:20) a bc d (cid:21) , ∗ Applied Research Laboratory, Penn State University, University Park, PA 16802 † Department of Ecology and Evolutionary Biology, UCLA, Los Angeles, CA 90095 ‡ Department of Mathematics, Penn State University, University Park, PA 16802 a r X i v : . [ q - b i o . P E ] M a y he remarkable reaction-diffusion equation is analyzed: ∂p∂t = ∆ p + 2 s ∇ p · ∇ s + pq (( a − c ) p + ( b − d )(1 − p )) (1) ∂s∂t = ∆ s + s (cid:0) ap + ( b + c ) p (1 − p ) + (1 − p ) d (cid:1) − κs . (2)Here κ is a death rate due to overcrowding and q = (1 − p ). Let e i ∈ R n be the i th unit vector. Let u = (cid:104) p, q (cid:105) . When κ = 0, we can rewrite these equation as: ∂p∂t = ∆ p + 2 s ∇ p · ∇ s + p (cid:0) e T − u T (cid:1) Au (3) ∂s∂t = ∆ s + s · u T Au . (4)That is, Durrett and Levin have encoded the replicator dynamic into a finite population spatialpartial differential equation, which differs from the one used by Vickers [16] because it assumes afinite population in its derivation.In this paper we show that Durrett and Levin’s finite population spatial replicator is gen-eralizable to an arbitrary payoff matrix. We then focus our attention on the one-dimensionalrock-paper-scissors game, which has interesting properties in both the finite and infinite populationcases. In particular, we show: (i) we show that the model used by Vicker’s [16] arises naturallyas the infinite population limit of the generalization of Durrett and Levin’s model, which in turncan be derived from a stochastic cellular automaton (particle) model as a fluid limit. (ii) Theone dimensional infinite population spatial rock-paper-scissors dynamic has a constant amplitudetraveling wave solution for all time. However, the finite population version does not exhibit suchsolutions, but does seem to exhibit an attracting stationary solution. We illustrate the latter resultnumerically. Let A ∈ R n × n be a payoff matrix for a symmetric game [4]. All vectors are column vectors unlessotherwise noted. Below we construct a stochastic cellular automaton and show that the fluidlimit of this system yields a generalization of Durrett and Levin’s specific finite population spatialreplicator.The state of cell i at time index k of the cellular automaton is a tuple (cid:104) U ( i, k ) , . . . , U n ( i, k ) (cid:105) where U j ( i, k ) provides the size of the population of species j at position i at time k . For simplic-ity, we assume that species interaction may only happen between cells and not within cells; i.e.,the U ( i, k ) members of species 1 will not play against the U n ( i, k ) members of species n . Thisassumption will become irrelevant in the limit.During state update, an agent A at cell i chooses a random direction (cell i (cid:48) ) and a randommember of the population (agent A (cid:48) ) within that cell. Assume Agent A uses strategy r while Agent A (cid:48) uses strategy s . After play, there are α · A rs additional agents at cell i using strategy r and β · A rs additional agents playing strategy r at cell i (cid:48) , where α + β = 1 is the probability of motion from cell i to cell i (cid:48) . If A rs <
0, then agents are removed from their respective cells. To avoid computationalissues when more members of a species die than are present, A can be modified so that A rs ≥ − r, s ∈ { , . . . , n } , without altering the evolutionary dynamics [2]. The update rule for a singleagent is illustrated in Fig. 1. The process described above and illustrated in Fig. 1 is assumed to behappening simultaneously for each agent and we assume that the replication/death as a result of2igure 1: Illustration of a single interaction in a spatial game on a one-dimensional lattice; α + β = 1game play along with the migration are happening on (roughly) the same time scale. Using theseassumptions, we can construct mean-field equations for the population U r at position i and time k + 1 in a 1-D cellular automaton, assuming an equal likelihood that agents diffuse left or right.The mean number of agents playing strategy r present at position i at time k + 1 is determined by:1. The expected number of agents who remain at cell i : αU r ( i, k )2. The expected number of new agents created at cell i who remain at cell i : α U r ( i, k ) (cid:32)(cid:88) s A rs ( u s ( i + 1 , k ) + u s ( i − , k )) (cid:33) .
3. The expected number of agents who migrate to position i from neighboring cells: β U r ( i + 1 , k ) + U r ( i − , k )) .
4. The expected number of agents created in a neighboring cell who migrate to cell i : β U r ( i + 1 , k ) + U r ( i − , k )) (cid:88) s A rs u s ( i, k ) . Here u s ( i, k ) is the proportion of the population at cell i playing strategy s at time step k .Let u ( i, k ) = (cid:104) u ( i, k ) , . . . , u n ( i, k ) (cid:105) . Re-writing sums as matrix products, the expected numberof agents playing strategy r at cell i at time k + 1 is: U r ( i, k + 1) = αU r ( i, k ) + β U r ( i + 1 , k ) + U r ( i − , k )) + α U r ( i, k ) (cid:0) e Tr A ( u ( i + 1 , k ) + u ( i − , k )) (cid:1) + β U r ( i + 1 , k ) + U r ( i − , k )) e Tr Au ( i, k ) . (5)3ssume the cellular grid has lattice spacing ∆ x . Following [20] and using a Taylor approxima-tion, we can write: U r ( x, t + ∆ t ) ≈ U r ( x, t ) + ∆ t ∂U r ( x, t ) ∂t + O (∆ t ) U r ( x + ∆ x, t ) ≈ (cid:88) j =0 ∆ x j j ! ∂ j U r ( x, t ) ∂x j + O (∆ x ) . We proceed to derive the mean-field approximation. Passing to the continuous case and assumingthat interaction rates decrease linearly with ∆ t , we can write a second order approximation ofEq. (5) as:∆ t ∂U r ( x, t ) ∂t = αU r ( x, t ) + β (cid:18) U r ( x, t ) + 12 ∆ x ∂ U r ∂x (cid:19) + α ∆ tU r ( x, t ) · e Tr A (cid:18) u ( x, t ) + 12 ∆ x ∂ u ∂x (cid:19) + β ∆ t (cid:18) U r ( x, t ) + 12 ∆ x ∂ U r ∂x (cid:19) e Tr Au ( x, t ) − U r ( x, t ) . (6)Where the − U r ( x, t ) on the right-hand-side arises from the formation of the Newton quotient onthe left-hand-side. Expanding and simplifying yields:∆ t ∂U r ( x, t ) ∂t = β x ∂ U r ∂x + ∆ tU r ( x, t ) e Tr Au ( x, t )+ α ∆ t x U r ( x, t ) e Tr A ∂ u ∂x + β ∆ t x ∂ U r ∂x e Tr Au ( x, t ) . (7)Assume β ∈ (0 , t and assuming that lim ∆ t → ∆ x / ∆ t = 2 D/β yields: ∂U r ( x, t ) ∂t = U r ( x, t ) e Tr Au ( x, t ) + D ∂ U r ∂x . (8)The constant D is the diffusion constant and the assumption thatlim ∆ t → ∆ x / ∆ t = 2 D/β is a variant of the assumption used to derive Fick’s Law [21] and identical when β = 1.These are the spatial dynamics used by Durrett and Levin, (in the first part of their paper),but are derived only by adding a diffusion term to the standard finite population growth equations.In [19], Durrett and Levin note that they derive a set of equations they feel are more appropriatefor modeling finite spatial systems. Their derivation at the end of [19] (for a specific hawk-dovesystem) rests on the assumption that migration happens “on a much faster timescale” than gameinteractions. Our model assumes that migration and game interactions occur on approximatelythe same time scale. Under this assumption, Eq. (8) is the correct spatial adaptation for finitepopulations; i.e., one simply adds a diffusion term. In the case where migration happens morequickly, then the derivation in [19] may be used instead.4 Spatial Replicator with Finite Population
The derivation of Eqs. (1) and (2) are not given in [19]. They can be generalized for an arbitraryevolutionary game using Eq. (8) as the starting point. Let: M ( x, t ) = (cid:88) s U s ( x, t ) . Differentiating we have: ∂∂t U r ( x, t ) M ( x, t ) = 1 M ( x, t ) ∂U r ( x, t ) ∂t − u r ( x, t ) (cid:88) s M ( x, t ) ∂U s ( x, t ) ∂t . Substituting from Eq. (8) we obtain: ∂u r ∂t = u r · (cid:0) e Tr Au − u T Au (cid:1) + DM (cid:32) ∂ U r ∂x − u r · (cid:88) s ∂ U s ∂x (cid:33) . (9)Unlike in the derivation of the standard replicator dynamic, the rate of change of the populationproportion is not solely a function of the proportions themselves.We can remove dependence on the individual populations to derive an independent (coupled)system of differential equations that includes only the total population. For arbitrary strategy r ,we can apply the quotient rule to obtain: M ∂u r ∂x = ∂U r ∂x − u r ∂M∂x Differentiating again, multiplying by 1 /M and re-arranging yields the expression:1 M ∂ U r ∂x = u r M ∂ M∂x + 2 M ∂u r ∂x ∂M∂x + ∂ u r ∂x . Using this we can write DM (cid:32) ∂ U r ∂x − u r (cid:88) s ∂ U s ∂x (cid:33) = D (cid:18) M ∂M∂x ∂u r ∂x + ∂ u r ∂x (cid:19) Using this we can re-write Eq. (9) as: ∂u r ∂t = u r · (cid:0) e Tr Au − u T Au (cid:1) + D (cid:18) M ∂M∂x ∂u r ∂x + ∂ u r ∂x (cid:19) . (10)The dynamics of M can be derived (by addition) from Eq. (8): ∂M∂t = M u T Au + D ∂ M∂x . Thus, we have a coupled set of differential equations written entirely in terms of u and M , ratherthan U r , M and u : ∀ r ∂u r ∂t = u r · (cid:0) e Tr Au − u T Au (cid:1) + D (cid:18) M ∂M∂x ∂u r ∂x + ∂ u r ∂x (cid:19) ∂M∂t = M u T Au + D ∂ M∂x . (11)5his is the spatial replicator equation for finite populations. Letting M = s and D = 1, we recoverthe dynamics of Durrett and Levin. In contrast to the aspatial replicator, the inclusion of dynamicsfor M yields a linearly independent system of differential equations.Allowing M to approach infinity uniformly in x , we arrive at the fluid limit in terms of u alone;this is the 1D nonlinear reaction-diffusion equation used by Vicker’s [16, 17]: ∀ r (cid:26) ∂u r ∂t = u r · (cid:0) e Tr Au − u T Au (cid:1) + D ∂ u r ∂x . (12)Generalization to N -dimensions is straightforward by replacing ∂ x with the Laplacian ∆. The N -dimensional spatial replicator with finite population is given by: ∀ r (cid:26) ∂u r ∂t = u r · (cid:0) e Tr Au − u T Au (cid:1) + D (cid:18) M ∇ M · ∇ u r + ∆ u r (cid:19) ∂M∂t = M u T Au + D ∆ M. Thus, the finite population case adds a nonlinear convection term that forces u r to follow thepopulation gradient. A similar system is studied by deForest and Belmonte in [11], where thepayoff gradient is followed instead of the population gradient.In both Eqs. (11) and (12), we see that the aspatial replicator dynamics appear on the righthand side perturbed by a spatial term. It is well known that the dynamics of the aspatial replicatorare confined to the n -dimensional simplex ∆ n . This remains true for the spatial replicator dynam-ics with finite populations. Moreover, the solution u r = 1 (i.e., there is only one population) is afixed point for the spatial replicator dynamic since the spatial derivative of the probability distri-bution of the population proportions is zero and the time derivative is identically zero as expected.Thus, pure populations are constant stationary solutions for these dynamics. Lastly, if ˜ u ( x, t ) isa constant solution at a Nash equilibrium for the game defined by A , then the right-hand-side isagain identically zero by the Folk Theorem [4] of evolutionary game theory together with the factthat there is no spatial variation. Thus every Nash equilibrium of the matrix game correspondsto a spatially constant stationary solution of the spatial replicator dynamic in both the finite andinfinite population cases. Durrett and Levin’s analysis of Hawk-Dove was aided by the fact that one strategy can be elim-inated, leaving a coupled system of two partial differential equations. In the remainder of thispaper, we analyze variations of rock-paper-scissors, which yield more interesting results because ofits cyclic three-strategy nature and because it can be easily parameterized as discussed in [2].The generalized rock-paper-scissors (RPS) payoff matrix is given by: A = − a a − − a . When a = 0, this is the standard RPS game which has Nash equilibrium (cid:104) , , (cid:105) . This is theunique interior fixed point and the aspatial replicator exhibits an elliptic fixed point at this Nashequilibrium. This Nash equilibrium is preserved for a (cid:54) = 0; and corresponds to an asymptoticallystable interior fixed point when a > a < u = (cid:104) u r , u p , u s (cid:105) and note: ζ ( u r , u p , u s ) ∆ = u T Au = a ( u r u s + u r u p + u s u p ) . There are at least two classes of global solutions to Eq. (11):
Stationary Solution
Here, u r = u p = u s = and M solves:0 = a M + M (cid:48)(cid:48) or there is a single population (e.g., u r = 1) and M is constant in x . Oscillating Solution
Here, u ∗ ( x, t ) ≡ υ ∗ ( t ) with ∗ ∈ { r, p, s } where υ r , υ p , υ s are solutions to thestandard RPS replicator and M ( x, t ) = µ ( t ) satisfies linear equation:˙ µ = ζ ( υ r , υ p , υ s ) µ. Solutions of this kind are also present for Eq. (12). The stationary solution class is relevant to ouranalysis of the a > a = 0 caseunder finite population dynamics. We consider the a < a = 0 When a = 0, we have ζ ( u r , u s , u p ) = 0 so that M is governed by the heat equation with diffusionconstant D . Assuming appropriate initial and boundary conditions, there is a global solution, M ( x, t ). We express the convection coefficient of ∂ x u ∗ in Eq. (11) in terms of this solution: (cid:15) ( x, t ) = 2 D∂ x M ( x, t ) M ( x, t ) . Under appropriate boundary conditions, diffusion of the (locally) finite population implies that:lim t →∞ M ( x, t ) = M ∗ > t →∞ (cid:15) ( x, t ) = 0 . Therefore, asymptotically (in time), any solution of Eq. (11) must also satisfy Eq. (12). Interest-ingly, the convection term in the finite population case drives the long-term behavior of the systemto the oscillating solution while the infinite population spatial replicator equation converges to thestationary solution. To see this, let a = 0, D = 1 and assume a periodic boundary condition(evolution on an annulus or S ) with M ( − π, t ) = M ( π, t ), u ∗ ( − π, t ) = u ∗ ( π, t ) ( ∗ ∈ { r, p, s } ).Assume: M ( x,
0) = 2 + cos( x )so that: M ( x, t ) = 2 + e − t cos( x ) (cid:15) ( x, t ) = − e − t sin( x )2 + e − t cos( x ) . u r ( x,
0) = (1 + sin( x − π/ u p ( x,
0) = (1 + sin( x − π/ u s ( x,
0) = (1 + sin( x )) . (15)Unless otherwise noted, these are the initial conditions for all numerical examples. The (numericallycomputed) solution to Eq. (11) is illustrated in Fig. 2. Notice the long-run behavior is consistentwith the oscillating solution. Since ζ ( υ r , υ p , υ s ) ≡ M must approach a trivial stationary solution; i.e., M ( x, ∞ ) = 2. By way of comparison a (a) Space-time Evolution u r , u p , u s Solution to Finite Population Spatial Replicator at x = RockPaperScissors (b) Time Evolution ( x = 0) Figure 2: Evolution of the finite population spatial replicator converges to an oscillating solutionwhen a = 0.(numerical) solution to the infinite population spatial replicator equation is shown in Fig. 3. In (a) Space-time Evolution u r , u p , u s Solution to Infinite Population Spatial Replicator at x = RockPaperScissors (b) Time Evolution ( x = 0) Figure 3: Evolution of the infinite population spatial replicator converges to a steady state solutionwhen a = 0. This is surprising since the Nash equilibrium fixed point is neutrally stable in theaspatial replicator.this case, the long-run behavior is consistent with the stationary solution. In the aspatial replicatorthe fixed point corresponding to the Nash equilibrium is neutrally stable. Thus, the fixed point isstabilized by the diffusion terms in Eq. (12). 8 .2 The case when a > When a >
0, the stationary solution for M must satisfy: M ( x ) = C sin (cid:16)(cid:113) a x (cid:17) + C cos (cid:16)(cid:113) a x (cid:17) . However, this solution is non-physical because it admits negative population counts and it is clearfrom Eq. (11) that for nonnegative initial conditions, solutions will remain nonnegative. Therefore,when a >
0, the finite population spatial replicator will not admit a stationary solution. Insteadwe find that M ( x, t ) → ∞ and t → ∞ so that solutions asymptotically satisfy Eq. (12)To see this we combine the fact that (cid:104) , , (cid:105) is an asymptotically stable fixed point of theaspatial replicator with a maximum principle argument for ∂u r ∂t in Eq. (11) and note that for anyinitial condition in the interior of ∆ there is some ζ > ζ ( u r , u p , u s ) ≥ ζ . It follows thatthe solution M ( x, t ) is bounded below by the solution of˙ m ( t ) = ζ m, m (0) = min x M ( x,
0) (16)In particular, M is increasing everywhere and M ( x, t ) → ∞ as n → ∞ Consequently, (cid:15) ( x, t ) → t and any solution to Eq. (11) must asymptotically satisfy Eq. (12). This is illustratedin Fig. 4 (here D = , a = ). Furthermore, linear stability analysis of Eq. (12) suggests that u r , u p , u s Solution to Finite Population Spatial Replicator at x = ( a > ) RockPaperScissors (a) Time Evolution ( x = 0) (b) Space-time Evolution(c) Population Evolution Figure 4: The evolution of the finite population spatial replicator is shown for a >
0. The populationproportions u r , u p , u s quickly approach the oscillating solution. As M increases, these oscillationsdecay and the population proportions approach the stationary solution of the infinite populationcase. 9he steady state solution u r = u p = u s = should be asymptotically stable for Eq. (12) (seeTheorem 1 of [23]), since the aspatial replicator dynamics with a > u r , u p , u s then slowly decay toward the fixed point of the aspatial replicator as M increases. When a <
0, the dynamics are more interesting, as we demonstrate below. a < Consider the infinite population model first. Letting z = x − ct , we can re-write Eq. (12) in compactform as: ∀ i (cid:40) Dv (cid:48) i = − u i (cid:0) e Ti Au − u T Au (cid:1) − cv i u (cid:48) i = v i . (17)For RPS we have the six dimensional linear system: Dv (cid:48) r = au r ζ ( u r , u s , u p ) − u r ( u s − u p ) − au r u s − cv r u (cid:48) r = v r Dv (cid:48) p = au p ζ ( u r , u s , u p ) − u p ( u r − u s ) − au p u r − cv p u (cid:48) p = v p Dv (cid:48) s = au s ζ ( u r , u s , u p ) − u s ( u p − u r ) − au s u p − cv s u (cid:48) s = v s . If u ∗ is a Nash equilibrium of A , then the pair u = u ∗ and v = is a fixed point of Eq. (17).Linearizing about u r = u p = u s = , v r = v p = v s = 0, we obtain the eigenvalues: λ , = − c ± √ c + 12 aD Dλ , = − c ± (cid:113) c + 6 aD + 6 D √ (cid:112) − ( a + 2) Dλ , = − c ± (cid:113) c + 6 aD − D √ (cid:112) − ( a + 2) D .
We can simultaneously show that for appropriate choice of wave speed, a center manifold exists andtherefore a non-decaying traveling wave solution exists for the PDE. As a by-product, we computethe wave speed for a non-decaying traveling wave in terms of a and D . Assume a ∈ ( − , b (to be determined), let:(3 c ± bi ) = 9 c + 6 aD ± D √ (cid:112) − ( a + 2) = 9 c + i aD ± D ( a + 2) √ . Expanding the left hand side and relating real and imaginary parts we have:9 c − b = 9 c + 6 aD bc = 6 D ( a + 2) √ . Solving for b and c yields: b = √− aD ± ˜ c = ± ( a + 2) √ k √− a c . Using this information, we can obtain:˜ λ , = − c ± √ c + 12 aD D ˜ λ = − c − bi D ˜ λ = bi D ˜ λ = − c + bi D ˜ λ = − bi D .
Our assumption that a < c > λ , ) < D > a ∈ ( − , c (cid:54) = ˜ c , there may still be a traveling wave, but its amplitude will decay ( c > ˜ c )or grow in time ( c < ˜ c ). Starting from the PDE it is difficult to determine what wave speed willemerge (and therefore its relationship to a and D ) given the initial conditions. However, travelingwaves can be observed with both increasing and decreasing amplitude; fine tuning the parametersleads to a numerically stable traveling wave over the region of integration. This is illustrated in thePDE system in Fig. 5 using a = − .
79 and D = . For completeness, we note the negative waveFigure 5: An example of a stable amplitude traveling wave solution is illustrated for RPS when a < − ˜ c implies an unstable fixed point (consistent with a < a <
0, there is a value ζ > ζ ( u r , u p , u s ) ≤ − ζ onthe interior of ∆ . It follows from Eq. (16) (substituting − ζ for ζ ) that the population on the realline (or an annulus) will collapse. As M ( x, t ) → a = − .
79 and D = . The finite(collapsing) population seems to cause the variations in the wave front boundaries. Additionally,the wave seen in Figure 6b is not stable. The amplitudes of u r , u s , u t are increasing in time (seeFig. 7 This is most likely due to the fact that the finite population (convection) term is acting likeadditional diffusion as the population collapse. 11 a) Infinite Population (b) Finite Population Figure 6: A comparison of the dynamics of a stable traveling wave (left) and the resulting unstablewave that results when finite populations are considered.
Solution to Finite Population Spatial Replicator at x = ( a < ) RockPaperScissors
Figure 7: The amplitude of the traveling wave increases in the finite population case ( x = 0). Thisis consistent with the asymptotically unstable nature of the equilibrium when a < Stable Steady State with a < If the population size is held constant at the boundary (i.e., we replace the periodic boundaryconditions with Dirichlet boundary conditions) then steady state analysis becomes possible in thefinite population. Consider the steady state solution u r = u p = u s = . The population steadystate equation is: 0 = − a M + D ∂ M∂x . Imposing the non-zero symmetric boundary conditions M ( − L, t ) = M ( L, t ) = m > M ( x ) = α cosh (cid:18) √− a √ D x (cid:19) , where: α = m sech (cid:18) √− a √ D L (cid:19) . Unlike the steady state solution for M when a >
0, this is physically realizable. When L = π andwe operate on an annulus, we may imagine a point (at x = ± π ) where population is held constantand then moves around the annulus to die. Suppose that u ∗ ( L, t ) = u ∗ ( − L, t ) = ( ∗ ∈ { r, p, s } ). (a) Spacetime Plot u r , u p , u s Solution to Finite Population Spatial Replicator at x = ( a < ) RockPaperScissors (b) Time Plot - - - M Population Population Spatial Replicator at t = ( a < ) Numerical M Computed SteadyState (c) Population
Figure 8: A stable stationary solution is illustrated with a “valley of death” for the population.We conjecture that the steady state solution is stable for the finite population replicator. This is13llustrated in Fig. 8 using the modified initial condition: u r ( x,
0) = f r ( x ) f r ( x ) + f p ( x ) + f s ( x ) u p ( x,
0) = f p ( x ) f r ( x ) + f p ( x ) + f s ( x ) u s ( x,
0) = f s ( x ) f r ( x ) + f p ( x ) + f s ( x ) M ( x,
0) = f r ( x ) + f p ( x ) + f s ( x ) , with f r ( x ) = 25 + x ( x − π )( x + π ) f p ( x ) = 25 − x ( x − π )( x + π ) f s ( x ) = 25 − ( x − π )( x + π ) . These equations satisfies the Dirichlet boundary condition u ∗ ( − π ) = u ∗ ( π ) = and M ( − π ) = M ( π ) = 75. (The exact value of m is irrelevant.) In Fig. 8, we set a = − and D = . Thelong-run behavior of M ( x, t ) is shown in Fig. 8c. It is clear that the solution has converged to thestationary state.It is worth nothing that the stability of the steady state solution is entirely a function of theDirichlet boundary conditions. To see this, we can use use the initial population distributions fromEqs. (13) to (15) with consistent boundary conditions. In this case, a standing wave emerges in thefinite population spatial replicator as the long-run behavior and the long-run population behavioris no longer described by a hyperbolic cosine. This is shown in Fig. 9a. By way of comparison,the corresponding Dirichlet boundary conditions in infinite population spatial replicator equationyield a highly complex wave structure that is similar to the oscillating solutions identified earlier(see Fig. 9b). (a) Finite Population (b) Infinite Population Figure 9: A comparison of the finite and infinite population replicator with Dirichlet boundaryconditions and a <
Conclusion
In this paper we studied a finite and infinite population spatial replicator. We showed how thefinite population spatial replicator can be derived from first principles from a stochastic cellularautomaton model and from there how the infinite population replicator used by Vickers [16, 17]follows from this. This result generalizes the work of Durrett and Levin [19] who first derived andstudied the finite population spatial replicator for a specific game. We then compared the finiteand infinite population spatial replicator for rock-paper-scissors on a one dimensional annulus onedimension ( S ). Most interestingly, we showed that for a certain rock-paper-scissors variant stableamplitude traveling waves can emerge as solutions to the infinite population spatial replicator, butthese are destroyed by population collapse in the finite population spatial replicator. We studiedpopulation collapse using Dirichlet boundary conditions on S and illustrated the structure ofsteady state solutions, including standing waves.The finite population spatial replicator is intriguing because it is a highly non-linear reaction-convection-diffusion equation where convection is governed by the per capita bulk population mo-tion. Consequently, it may be particularly useful for studying human behavior, where large popula-tions tend to move as a result of regional crises and strategies (i.e., behaviors) may represent humaninteraction patterns. Studying a simpler game (as Durrett and Levin did with Hawk-Dove) may il-lustrate additional behaviors. Certainly proving our conjecture that when u ∗ ( L, t ) = u ∗ ( − L, t ) = ( ∗ ∈ { r, p, s } ) and M ( L, t ) = M ( − L, t ) = m the finite population spatial replicator convergesto the proposed steady state solution is a clear future direction. In addition to this, identifyingmore ways in which the finite and infinite spatial replicators differ in terms of solutions would beinteresting. The most interesting differences in the case of RPS seem to occur when the popula-tion is collapsing. It would be intriguing to find cases where population collapse is not the maindriver in behavioral differences, since tautologically this must cause differences in finite vs. infinitepopulation equations. Acknowledgement
Portions of CG’s work were supported by the National Science Foundation Grant CMMI-1932991.
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