TURING INSTABILITY IN A PUBLIC GOODS GAME
Derick O. Poku
A Thesis Submitted to the University of North Carolina Wilmington in Partial Fulfillment of the Requirements for the Degree of Master of Science
Department of Mathematics and Statistics
University of North Carolina Wilmington
2010
Approved by
Advisory Committee
Nolan Mcmurray Wei Feng
Xiaojie Hou
Chair
Accepted by
Dean, Graduate School TABLE OF CONTENTS
ABSTRACT ...... iii DEDICATION ...... iv ACKNOWLEDGMENTS ...... v LIST OF FIGURES ...... vi 1 INTRODUCTION ...... 1 2 THE STABILITY OF THE EQUILIBRIA OF THE O.D.E SYSTEM 6 2.1 Classification of the Equilibrium Solutions ...... 6 2.2 Equilibria and their Stability ...... 10 3 THE TURING INSTABILITY ...... 15 3.1 Turing Mechanisms ...... 15 3.2 Linearization and Turing Instability of the Reaction-Diffusion model ...... 16 4 NUMERICAL EXAMPLE ...... 25 5 CONCLUSION ...... 30 6 APPENDIX ...... 31 6.1 Derivation of Bump Function B(x) ...... 31 REFERENCES ...... 33
ii ABSTRACT
In economic theory a good that is non-rivaled and non-excludable is referred to as a public good. For the sustainability of these public goods, cooperation among beneficiaries is paramount. Unfortunately, there are bound to be non-cooperators or free-riders and an associated cost for the altruistic behaviour of cooperators. We study a model that represents the interaction of cooperators and free-riders in the consumption of a public good. We determine the conditions under which spatially homogeneous equilibrium solutions to the model without the diffusion terms of cooperators and free-riders are stable to small perturbations. We extend our results to determine the threshold of the diffusion rate of the cooperators above which solutions to the full reaction-diffusion model is unstable (Turing Instability) to non-homogeneous perturbation.The analysis shows that under certain conditions with spatial diffusion, the stable coexistence of the cooperators and the free-riders is impossible in the long run.
iii DEDICATION
Dedicated to my Mum.
iv ACKNOWLEDGMENTS
I would like to thank my advisor Dr. Xiaojie Hou who guided me through this whole process. To him I owe my gratitude.
I would also like to thank Dr. Wei Feng for her help in streamlining the conditions (8) and (9) on page 11 and also thank her the many joyous discussions on population dynamics. Thanks also should be given to Dr. Nolan McMurray for agreeing to be on my thesis committee.
My gratitude also goes to the members and staff of the Department of Mathematics and Statistics, University of North Carolina, Wilmington, for making my period of studies a pleasant one.
I would also like to thank William Reid Peters for translating the sketches I had for some of the figures in this paper into postscript document format.
v LIST OF FIGURES
1 Unstable equilibrium point, trajectories going away from the origin. . 7 2 Stable equilibrium point, trajectories going towards the origin. . . . . 8 3 Saddle equilibrium point,trajectories going towards or away from the origin...... 8 4 a) Real part of both eigenvalues are negative. b) Real part of eigen- values have opposite signs...... 9
5 Intersection of null clines L1 and L2 formed by solving system ( 5 ). . 11 6 Vector fields of (5) under assumptions (6) , (7) , (8) and (9). . . . . 14
2 7 Plot of h(θ ) defined by (27). When the diffusion coefficient d2 of
2 the free-riders increases beyond the critical value dc, h(θ ) becomes negative...... 23 8 The coexistent state is stable to non-homogenous pertubation when the diffusion rate of the free-riders is below the diffusion threshold
dc: (a) initial state of the coexistent solution and (b) the asymptotic behaviour of the coexistent state...... 28 9 The coexistent state is unstable to non-homogenous pertubation when
the diffusion rate of the free-riders is above the diffusion threshold dc: (a) initial state of the coexistent solution and (b) the asymptotic behaviour of the coexistent state...... 29 10 The graph of c(t)...... 31 11 The graph of f(t)...... 32 12 The graph of g(x)...... 32 13 The graph of h(x)...... 33 14 The graph of m(x) ...... 33 15 The graph of B(x) ...... 33
vi 1 INTRODUCTION
In biology Turing instability is the presage of pattern formulation. Alan Turing (1952) postulated that chemicals called morphogens through reaction and diffusion could create spatial patterns in chemical concentrations through an instability pro- cess. Turing’s theory states that diffusion which is generally seen as a stabilizing process could destabilize an otherwise stable reaction-diffusion system and trigger a pattern formation. Spontaneous pattern formation has been observed in chemical and physical systems ranging from hydrodynamical phenomena like ripples in the sand and streaks of colours in animal furs [17]. Research on Turing instability in the context of competition for a public good is very new. This paper is devoted to the study of pattern formation in a social dilemma modelled as a game which represents the competition for resources.
The front page of a newspaper today may contain a report of violent attack by a group of people on a foreign soil, a domestic political scandal or a geopolitical issue of trade tariffs between an entrenched Superpower and a fledgling Economy. The inside pages may contain agitations by various social groups to sway the course of government decisions on specific policies. The business section may be full of deals relating to merger and acquisitions companies. The aforementioned scenarios have a common denominator which is a conflict of interest between groups of people such as governments, businesses and social networks . The theoretical models that represents these conflicts are termed games [6].
Mathematical theory of games termed Game theory was invented by John von Neumann and Oskar Morgenstern (1944). Game theory is the study of the ways in which strategic interactions among rational players produce outcomes with respect to the preferences of those players, none of which might have been intended by any
1 of them [16]. The interaction could be viewed from the angle of non-cooperation and cooperation amongst players. The development of “Prisoner’s Dilemma” [14] and John Nash’s papers on the definition and existence of equilibrium laid the foun- dations for modern non-cooperative game theory. Simultaneously, cooperative game theory reached important results in papers by Nash and Shapley on bargaining games [13].
An important question that arises in a game of conflict is the effect of cooperation among participants. Suppose a game represents the production/conservation and consumption of a public good. Countries have the opportunity to invest a certain amount of their Gross Domestic Product (GDP) to sustain the good. A return on investment would be an additional increase in GDP for the next fiscal year by the average of twice the sum of all contributions by investors. John Stuart Mill a 19th century political economist stipulated that the rational behaviour would be that individual countries in their selfish act to maximize utility would not make a contribution [3]. Evidently a non-contributor (“free rider”) tends to gain if at least one country makes a contribution. But the most prudent strategy is for each to make a contribution. The net return on investment is maximized for all.
Whereas some participants in a game similar to the scenario described earlier may realise the benefits of cooperation and act irrationally, others may act otherwise. We note that the cooperators (investors) may incur a cost for their altruistic behaviour. As an example of such a cost, suppose only 1 out of N (where N > 2) countries make a contribution of P of her GDP then the return on investment will be
2P N
2 and the cost of altruism incurred is
P (N − 2) N
Another twist to this game is the ability of players to migrate to other areas of a spatial domain to exploit a public good. The dynamics of this game raises some questions that needs to be addressed:
• Could the cooperators and “free riders” coexist in this game?
• Under what conditions will such coexistence occur?
• What is the effect of the mobility of the participants on the coexistence state?
• What is the effect of mobility on the spatial domain in which the game is played?
• What role does the cost of altruistic behaviour play in this game?
To answer these questions we examine the following public good game modelled as a reaction diffusion system:
∂u ∂2u u + av = d + u(1 − − α), 1 2 ∂t ∂x k(u) 2 ∂v ∂ v bu + v + = d2 + v(1 − ), (x, t) ∈ R × R , (1) ∂t ∂x2 k(u) u(x, 0) = u0(x), v(x, 0) = v0(x) where u = u(x, t), v = u(x, t) are two participating groups in the game and u0(x),v0(x) are nonnegative bounded smooth functions in R and a, b, α, d1, d2 are positive constants. The system is a continuous spatial-temporal version of a public
3 goods game [17] derived from the Lotka-Volterra competition model and describes the interaction between the group of players u(x, t) and v(x, t). The cooperators u employ an altruistic strategy in order to ensure sustenance of the public good; whiles the strategy of v (“free riders”) is to exploit the public good without investing. The function k(u) = 1 + ku (where k > 0) represents the investment contributed by the cooperators u and shared with the “free riders” v. The cost of altruism for the coop- erators is measured by α where 0 < α < 1 [2]. The effects of v on u is represented by the positive constant a and vice versa the constant b. The constants d1 > 0, d2 > 0 are respectively the diffusion rates (or mobility) of cooperators u and “free riders” v.
We assume zero flux boundary conditions for our system and restrict our domain to a one-dimensional finite interval where x ∈ R and 0 < x < L. The zero flux condition guarantees that there are no external input to our model. The model becomes self-organizing and all inferences made could be attributed to the domain under consideration [10]. An example of zero flux condition is when no new player can join nor leave the domain in which the game is played.
In Chapter 2, we determine the equilibrium states of the reaction diffusion model and conditions for asymptotic stability of these states without the diffusion coefficients. It will be shown that the homogeneous equilibrium state of coexistence is stable to small perturbations when some restrictions are imposed on the relationships between the positive constants a, b, k and α of our model.
In Chapter 3, we derive the necessary and sufficient conditions under which diffusion driven instability or Turing Instability may occur.
We present an example of our system that undergoes a Turing Instability in Chapter
4 4. The numerical example will show how our model goes from a homogeneous steady state to a Turing bifurcation.
Chapter 5 is a summary of this work and future direction and Chapter 6 is devoted to supporting computations in this paper.
5 2 THE STABILITY OF THE EQUILIBRIA OF THE O.D.E SYSTEM
In this chapter we study the stability of the coexistence equilibrium. First, the conditions for the coexistence equilibrium state is derived. We assume that the density of u and v are only time dependent and the basic global in time solutions to system (1) exist. Secondly, we linearise the vector field of (1) and further derive the stability of the co-existence equilibrium state. The existence and stability of the coexistent state shows that cooperators and “free riders” will sustain their winning strategy with the evolution of time.
2.1 Classification of the Equilibrium Solutions
We recall from Ordinary Differential Equations the classification of equilibrium solu- tions as stable, asymptotically stable or unstable by considering the following system
∂x = a x + a y + P (x, y), ∂t 11 12 (2) ∂y = a21x + a22y + Q(x, y) ∂t where |P (x, y)|, |Q(x, y)| ≤ β(|x|2 + |y|2) and β is some positive constant. The term (|x|2 + |y|2) is assumed to be very small and the origin, (0, 0), is an equilibrium solution1 of system (2).
To perform the classifications we state some definitions by considering the differen- tial equation
1 ∂u ∂v Let the system of ordinary differential equations, ∂t = f(u, v), ∂t = g(u, v) be defined in some n region U in R .Then a point (u0, v0) is called an equilibrium point of the system if f(u0, v0) = g(u0, v0) = 0 .
6 ∂x = f(x, t), x ∈ n. ∂t R
Definition 1 An equilibrium point x0 is stable iff for all > 0 there exist δ > 0 such that for a homogeneous pertubation p0 where |x − p0| < δ then |x0 − p0| < .
Definition 2 An equilibrium point x0 is asymptotically stable iff it is both stable and for a homogeneous pertubation p0 , |x0 − p0| → 0 as t → ∞.
We use the coefficients of system (2) to define matrix A
a a 11 12 A = (3) a21 a22
with the following eigenvalues λ1, λ2. Depending on the signs of the λ1 and λ2, we have the following classifications:
Case 1: 0 < λ1 < λ2
If 0 < λ1 < λ2 , then the equilibrium point (0, 0) is unstable. The local topological structure in the neighbourhood of (0, 0) is illustrated in figure (Fig. 1).
<1, λ > 1 eigenvectors y <1, > } λ2 <1, λ2>
<1, λ1>
0 x
Figure 1: Unstable equilibrium point, trajectories going away from the origin.
7 Case 2: λ1 < λ2 < 0
If λ1 < λ2 < 0 , then the equilibrium point (0, 0) is asymptotically stable. See figure (Fig. 2).
<1, λ > 1 eigenvectors y <1, > } λ2 <1, λ2>
<1, λ1>
0 x
Figure 2: Stable equilibrium point, trajectories going towards the origin.
Case 3: λ1 < 0 < λ2
If λ1 < 0 < λ2 , then the equilibrium (0, 0) is a saddle point. A saddle point is always unstable. See figure (Fig. 3).
y y
0 0 x x
a) b)
Figure 3: Saddle equilibrium point,trajectories going towards or away from the origin.
Case 4: λ1 and λ2 are two conjugate eigenvalues
If the real part of both eigenvalues denoted Re(λ1), Re(λ2) are negative then the equilibrium (0, 0) is an asymptotically stable point. Unstable otherwise. See figure
8 (Fig. 4) for the spiral trajectories.
y y
0 x 0 x
a) b)
Figure 4: a) Real part of both eigenvalues are negative. b) Real part of eigenvalues have opposite signs.
Define matrix J as ∂f ∂f (u0, v0) (u0, v0) ∂u ∂v J = ∂g ∂g (u , v ) (u , v ) ∂u 0 0 ∂v 0 0 where the ijth entry is the partial derivative of function f(u, v) or g(u, v) evaluated at an equilibrium point (u0, v0). The real parts Re(λ1), Re(λ2) of the eigenvalues of matrix J is negative (ie. (u0, v0) is stable) if J satisfies the following theorem.
Theorem 1 An equilibrium point (u0, v0) is stable if both conditions a) and b) si- multaneously hold
a) trace(J) = ∂f + ∂g < 0 ∂u u=u0,v=v0 ∂u u=u0,v=v0
b) det(J) = ∂f ∂g − ∂f ∂g > 0 ∂u ∂v u=u0,v=v0 ∂v ∂u u=u0,v=v0
Proof: see ([7]).
9 2.2 Equilibria and their Stability
Definition 3 Let the system of ordinary differential equations,
∂u = f(u, v), ∂t ∂v = g(u, v) ∂t
n be defined in some region U in R .Then a point (u0, v0) is called an equilibrium point of the system if f(u0, v0) = g(u0, v0) = 0 [11].
Consider system (1) with the diffusion coefficients set to zero,
∂u u + av = u 1 − − α , ∂t k(u) ∂v bu + v = v 1 − , (4) ∂t k(u) u(x, 0) = u0(x), v(x, 0) = v0(x)
The equilibrium states are determined by
. u+av f(u, v) = u 1 − k(u) − α = 0, (5) . bu+v g(u, v) = v 1 − k(u) = 0
We solve (5) to get the following equilibria: 1 − α (0, 0) , (0, 1) , , 0 and 1 − k + αk α + a − 1 b − αb − 1 , ab + k − ak − αk − 1 ab + k − ak − αk − 1 which are illustrated in figure (5).
The point (0, 0) which corresponds to the origin of (5) is when no participants are in the game, (0, 1) is the state of the game that is purely made up of “free riders” v
10 1−α and have reached their saturation and ( 1−k+αk , 0) is when the participants are solely α+a−1 b−αb−1 cooperators u. The last scenario, ( ab+k−ak−αk−1 , ab+k−ak−αk−1 ), is the coexistence between u and v.
α + a – 1 u0 = ab + k – ak – αk – 1 v b – αb – 1 v0 = ab + k – ak – αk – 1 1 – α a
1
(u0 , v0 )
0 1 – α 1 u 1 – k + αk b – k L1 L2
Figure 5: Intersection of null clines L1 and L2 formed by solving system ( 5 ).
Throughout the rest of this paper, we make the following assumptions:
b > k (6)
1 − α 1 1 < , or b < (7) 1 − k + αk b − k 1 − α 1 − α > 1, or a < 1 − α (8) a
ab < 1 (9)
Next, we analyse the stability of the equilibria corresponding to the notions of section (2.1). Consider matrix J where the ijth entry is the partial derivative of function f(u, v) or g(u, v) of system (5),
2u+av+ku2 au 1 − 2 − α − J = (1+ku) 1+ku . v(b−kv) 1+ku−bu−2v − (1+ku)2 1+ku
11 To determine the stabilities of the four equilibrium points found in section (2.1), we apply theorem (2) to derive the following propositions.
Proposition 1 Under the conditions (6) , (7) , (8) and (9), the equilibrium point (0, 0) is unstable.
Proof 1 − α 0 J(0, 0) = 0 1 trace(J(0, 0)) = 2 − α > 0 since by the earlier assumption 0 < α < 1 . Theorem (2) implies that (u0, v0) = (0, 0) is an unstable equilibrium point.
Proposition 2 Assuming the conditions of proposition (1), the equilibrium point (0, 1) is unstable
Proof 1 − a − α 0 J(0, 1) = . k − b −1
The determinant det(J(0, 1)) = a + α − 1 < 0, since by equation (8), a < 1 − α .
Evidently, theorem (2) makes (0, 1) an unstable equilibrium point.
Proposition 3 Assuming the conditions of proposition (1), the equilibrium point
1−α (u0, v0) = 1−k+α , 0 is unstable.
Proof (α − 1)(1 − k + αk) a(α − 1) J( 1−α , 0) = 1−k+α 0 1 + b(α − 1)
1−α The determninant det(J 1−k+α , 0 ) = (α − 1)(1 − k + αk)(1 + b(α − 1)) < 0, the earlier assumptions have the term 1−k+αk > 0 and by equation 7, (1+b(α−1)) > 0
1−α . We conclude that the equilibrium solution (u0, v0) = 1−k+α , 0 is unstable.
12 We now state the stability of the coexistent state.
Proposition 4 Assuming the conditions of proposition (1), the equilibrium point
α+a−1 b−αb−1 ab+k−ak−αk−1 , ab+k−ak−αk−1 is asymptotically stable.
Proof − (1−k+αk)(a+α−1) − a(α+a−1) α+a−1 b−αb−1 ab−1 ab−1 J( ab+k−ak−αk−1 , ab+k−ak−αk−1 ) = (b−k)(1−b+αb) 1−b+αb ab−1 ab−1
α+a−1 b−αb−1 (1−k+αk)(a+α−1)−(1−b+αb) The trace trace(J( ab+k−ak−αk−1 , ab+k−ak−αk−1 )) = − ab−1 < 0, this is obvious from the fact that (1 − k + αk) > 0 and by equation (2.1) and (2.2) (a + α − 1) < 0, ab − 1 < 0, (1 − b + αb) > 0. The determinant
α+a−1 b−αb−1 det(J( ab+k−ak−αk−1 , ab+k−ak−αk−1 ))
(1 − b + αb)(−a + 2ak − 2αak + 1 − k + 2αk − α − α2k + abα + a2k − ab) = (ab − 1)2
(1 − b + αb)(α + a − 1)(ab + k − ak − αk − 1) = (ab − 1)2
> 0
The above inequality holds since from the earlier results (1−b+αb) > 0 , (α+a−1) < 0 and (ab + k − ak − αk − 1) < 0. Theorem (2) implies that the equilibrium point
α+a−1 b−αb−1 (u0, v0) = ab+k−ak−αk−1 , ab+k−ak−αk−1 , is stable. Figure (Fig. 6) represents the vector fields of the trajectories of system (5). The figure clearly indicates that the coexistence equilibrium is stable while other equilibrium states are unstable. We sum up this chapter by noting that for the O.D.E form of system (1), the cooperators and “free riders” can coexist provided the following conditions holds
13 Figure 6: Vector fields of (5) under assumptions (6) , (7) , (8) and (9).
1 • b < 1 − α
• a < 1 − α
14 3 THE TURING INSTABILITY
In this chapter we study the Turing instability and derive the conditions under which the reaction-diffusion model (1) exhibits such dynamical behaviour. To accomplish this, we nondimensionalize the diffusion coefficient parameters to translate d1 to 1. An advantage of nondimensionalization is to reduce the number of parameters in our system under consideration. We determine the threshold of the diffusion coefficient
2 d2 of the “free riders” beyond which the Turing instability of our system occurs .
3.1 Turing Mechanisms
Alan Turing (1952) postulated that chemicals can react and diffuse to produce a steady state of heterogeneous spatial patterns based on their diffusion rates. His idea was that if without diffusion some chemicals tend to a linearly uniform stable state, then under certain conditions, spatially uniform pattern formation can form due to diffusion instability. This may occur when the diffusion coefficients of the chemicals are non-zero and unequal [15].A recent application of Turing instability has been in Forestry in the study of pioneer/climax species interaction [4].
Definition 4 A reaction diffusion system exhibits diffusion driven instability or Turing instability if the homogeneous steady state is stable to small perturbations in the absence of diffusion but unstable to small spatial perturbations when diffusion is present.
We note that without spatial variation the equilibrium state of coexistence in our model is linearly stable as shown in the previous chapter.
For the convenience of later use, we recall the following lemma:
2Our analysis follows closely the standard discussion in (Murray,1993, Chapter 14)
15 Lemma 1 Let Ω be a bounded region with boundary S and let u be harmonic3 in Ω¯. If M and m are, respectively, the maximum and minimum values of u(x) for x on S, (Weak Maximum-Minimum Principle) m ≤ u(x) ≤ M for all x in Ω¯ or more precisely (Strong Maximum-Minimum Principle) either m < u(x) < M for all x in Ω or else m = u(x) = M for all x in Ω¯ [8]
3.2 Linearization and Turing Instability of the Reaction-Diffusion model
We introduce the following transformation
π π 2 x¯ = x, t¯= d t (10) L 1 L to nondimensionalize system (1) [4] into the form
∂u ∂2u u+av = 2 + u(1 − − α), ∂t ∂x k(u) ∂v ∂2v bu+v , (11) ∂t = d2 ∂x2 + v(1 − k(u) ), u(x, 0) = u0(x), v(x, 0) = v0(x) where the bar notation has been dropped for simplicity. We still assume zero flux boundary conditions for this transformation and we restrict our domain to a finite interval 0 ≤ x ≤ π. We now consider the equilibrium point of coexistence
3A function u = u(x) is harmonic in an open region Ω if u is twice continuously differentiable in Ω and satisfies Laplace’s equation in Ω. u is harmonic in Ω,¯ the closure of Ω, if u is harmonic in Ω and continuous in Ω [8]
16 α+a−1 b−αb−1 (u0, v0) = ab+k−ak−αk−1 , ab+k−ak−αk−1 and introduce the following perturbations
u = u0 + w1(x, t) (12) v = v0 + w2(x, t)
where the perturbation |w(x, t)i| is assumed to be very small. Without loss of gen- erality we let w(x, t)i ≡ wi. Substituting equation (12) into the nondimensionalized system (11) yields the following
∂w ∂2w u + w + a(v + w ) 1 = 1 + (u + w )(1 − 0 1 0 2 − α), 2 0 1 ∂t ∂x k(u0 + w1) 2 ∂w2 ∂ w2 b(u0 + w1) + v0 + w2 (13) = d2 2 + (v0 + w2)(1 − ), ∂t ∂x k(u0 + w1) w1(x, 0) = w1,0(x), w2(x, 0) = w2,0(x).
For brevity we rewrite system (13) as
2 ∂w1 ∂ w1 = + f(u0 + w1, v0 + w2), ∂t ∂x2 2 ∂w2 ∂ w2 = d2 + g(u0 + w1, v0 + w2), (14) ∂t ∂x2 w1(x, 0) = w1,0(x), w2(x, 0) = w2,0(x).
Since |wi| is assumed to be small, we can expand (14) in its Taylor series about
(u0, v0). This yields the following linearized system
17 2 ∂w1 ∂ w1 = + f(u0, v0) + w1fu(u0, v0) + w2fv(u0, v0)// ∂t ∂x2 + 1 [(w )2f (u , v )...], 2! 1 uu 0 0 2 ∂w2 ∂ w2 = d2 + g(u0, v0) + w1gu(u0, v0) + w2gv(u0, v0)// (15) ∂t ∂x2 + 1 [(w )2g (u , v )...], 2! 1 uu 0 0 w1(x, 0) = w1,0(x), w2(x, 0) = w2,0(x)
The smallness of |wi| allows us to keep only the degree 1 terms in (15). We also observe that f(u0, v0) = g(u0, v0) = 0 due to the earlier assertion that the parameter
(u0, v0) is an equilibrium point of the ODE of system (1). The final linearised system becomes
2 ∂w1 = ∂ w1 + w f (u , v ) + w f (u , v ), ∂t ∂x2 1 w1 0 0 2 w2 0 0 2 ∂w2 ∂ w2 , (16) ∂t = d2 ∂x2 + w1gw1 (u0, v0) + w2gw2 (u0, v0) w1(x, 0) = w1,0(x), w2(x, 0) = w2,0(x)
We rewrite (16) into a compact form
∂w ∂2w = D + Aw (17) ∂t ∂x2 where 1 0 w f f 1 w1 w2 D = , w = ,A = (u0, v0) 0 d2 w2 gw1 gw2
To solve (17), we find a time independent solution to the eigenvalue problem
18 ∂2W + θ2W = 0 ∂x2 (18) Wx(0) = Wx(π) = 0 where θ is the eigenvalue. We have the following cases to deal with,
Case 1: θ ≤ 0 If θ2 ≤ 0 then by lemma 1 we can easily show that the only solution to (18) is the trivial solution.
Case 2: θ > 0 If θ2 > 0 then µ2 + θ2 = 0 is the characteristic polynomial of (18) and µ = ±θi is the solution of the polynomial. The general solution to the eigenvalue problem (18) then becomes
W (x) = c1 cos θx + c2 sin θx
where c1, c2 are positive constants.
We solve for c1, c2 by applying the boundary condition in (18) as follows
Wx(x) = −c1θ sin θx + c2θ cos θx
Wx(0) = c2 = 0
W (x) = c1 cos θx (19)
Wx(x) = −c1θ sin θx
Wx(π) = −c1θ sin(θπ) = 0 we then derive that θπ = nπ, n = 1, 2, 3... (20) θ = n and let c1 = 1, then Wθ(x) = cos θx, is the eigenfunction of the eigenvalue problem
19 (18). For our linearized reaction diffusion system, we look for solutions of the form
X λt W (x, t) = cθe Wθ(x) (21) θ
The constant cθ can be found by a Fourier expansion of the initial conditions in terms of Wθ(x) and λ is the eigenvalue which determines temporal growth [10].
We substitute (21) into (17),
X X ∂2W X c λeλtW (x) = D c eλt + A c eλtW (x) (22) θ θ θ ∂x2 θ θ θ θ θ and cancel out like terms on both sides of the equation to yield
∂2W λW (x) = D + AW (x) (23) θ ∂x2 θ
∂2W 2 From (18) we subsitute ∂x2 = −θ Wθ into (23) resulting in
2 λWθ(x) = −Dθ Wθ(x) + AWθ(x)
2 λWθ(x) + Dθ Wθ(x) − AWθ(x) = 0 (24)
2 (λI + Dθ − A)Wθ(x) = 0
2 But from earlier results Wθ(x) is non-trivial therefore the term (λI + Dθ − A) is a degenerate matrix with determinant equal to zero. That is |λI + Dθ2 − A| = 0 or equivalently
2 λ 0 θ 0 fw fw 1 2 + − = 0 (25) 0 λ 0 d θ2 g g 2 w1 w2
20 We evaluate (25) to yield
2 2 2 4 2 λ + λd2θ − λgw2 + θ λ + θ d2 − θ gw2 − fw1 λ − //
2 fw1 d2θ + fw1 gw2 − fw2 gw1 = 0
2 2 2 4 2 2 λ + λ [d2θ − gw2 + θ − fw1 ] + θ d2 − θ gw2 − fw1 d2θ + // (26) fw1 gw2 − fw2 gw1 = 0
2 2 2 4 2 λ + λ [d2θ − gw2 + θ − fw1 ] + θ d2 − θ [gw2 − fw1 d2] + //
fw1 gw2 − fw2 gw1 = 0
2 2 2 4 2 λ + λ [d2θ − gw2 + θ − fw1 ] + θ d2 − θ [gw2 − fw1 d2] + |A| = 0
From (26) we have the following polynomial which is quadratic in λ
2 2 2 2 λ + λ [d2θ − gw + θ − fw ] + h(θ ) = 0 2 1 (27) 2 4 2 where h(θ ) = θ d2 − θ [gw2 − fw1 d2] + |A|
The roots of (27) are the eigenvalues λ(θ) which are functions of θ. The coexistence equilibrium point (u0, v0) is unstable if one of the roots of (27) has a positive real part. We apply the quadratic formula to get the roots of (27) as
q 2 2 2 2 − [θ (d2 + 1) − (gw2 + fw1 )] ± [θ (d2 + 1) − (gw2 + fw1 )] − 4h(θ ) λ(θ) = (28) 2
For the real part Re(λ(θ)) > 0 and θ 6= 0, it suffices that
2 a) θ (d2 + 1) − (gw2 + fw1 ) < 0 or b) h(θ2) < 0
2 For the case of a), the terms θ (d2 +1) and (gw2 +fw1 ) satifies the following inequal-
21 ities
2 θ (d2 + 1) > 0, (29)
(gw2 + fw1 ) < 0 (30) we recall that (30) is exactly the trace of the Jacobian of the ODE. We conclude that condition a) does not apply in the search for instability. The only possibility for an instability of our reaction-diffusion public good game model is when condition b) is met. We explore that scenario in our subsequent computations.
4 The term d2θ is positive and from the ODE calculations in chapter 2 the deter-
2 minant |A| is also positive. For h(θ ) to be negative requires [gw2 − fw1 d2] > 0 but
2 2 [gw2 − fw1 ] < 0 and this implies that d2 6= 1. We have h(θ ) to be quadratic in θ
2 2 and the necessary condition for h(θ ) < 0 is for its global minimum hmin(θ ) to be negative. We determine the critical point of h(θ2) as
dh(θ2) = 2d θ2 − (g − f d ) dθ2 2 w2 w1 2
dh(θ2) 2 where the critical point is found by setting dθ2 = 0 and solving for θ
g − f d θ2 = w2 w1 2 2d2
We evaluate h(θ2) at this critical point to get
2 2 gw2 +fw1 d2 gw2 +fw1 d2 h(θ ) = d2 − (gw + fw d2) + |A| 2d2 2d2 2 1 2 2 (gw +fw d2) (gw +fw d2) = 2 1 − 2 1 + |A| 4d2 2d2 2 (gw +fw d2) = − 2 1 + |A| 4d2
22 Since |A| > 0, then for h(θ2) < 0 it suffices that
(g + f d )2 w2 w1 2 > |A| (31) 4d2
2 2 2 At bifurcation dc fw1 +2dcfw1 gw2 −4dc|A|+gw2 = 0 and the positive root dc becomes the threshold for which if d2 > dc then Turing instability occurs. We solve for the roots.
p 2 2 2 (4|A| − 2fw1 gw2 ) ± (4|A| − 2fw1 gw2 ) − 4fw1 gw2 dc = 2 (32) 2fw1 At the global minimum where h(θ2) is negative requires that θ2 is positive so we let
p 2 2 2 (4|A| − 2fw1 gw2 ) + (4|A| − 2fw1 gw2 ) − 4fw1 gw2 dc = 2 2fw1
p 2 (4|A| − 2fw1 gw2 ) + 4 (|A| − |A|fw1 gw2 ) = 2 (33) 2fw1 be the positive root.
h(θ2)
|A| d2 < dc
d2 = dc
d2 > dc
2 2 2 2 0 θ1 θc θ2 θ
2 Figure 7: Plot of h(θ ) defined by (27). When the diffusion coefficient d2 of the 2 free-riders increases beyond the critical value dc, h(θ ) becomes negative.
23 The following theorem is a summary of our main result in this chapter.
Theorem 2 System (1) undergoes Turing instability if the diffusion coefficient d2 of the “free-riders” exceeds the critical threshold dc. That is Turing instability occurs when the following inequality holds.
p 2 (2|A| − fw1 gw2 ) + 2 (|A| − |A|fw1 gw2 ) d2 > dc = 2 (34) fw1
24 4 NUMERICAL EXAMPLE
In this chapter we present an example of the public good game model that undergoes a Turing Instability. The numerical example will show how the model goes from a homogeneous steady state to a Turing bifurcation. To demonstrate Turing process we define a - the effects of v on u , b - the effects of u on v , α - the cost of altruism of the cooperators u , k - the carrying capacity of the ecosystem and d2 - the diffusion coefficient of the “free-riders” such that the assumptions referenced by (6), (7), (8) and (9) are satisfied.
Let α := 1/2; a := 2/5; b := 1; k := 1/4; (35) then the non-dimensionalized model becomes
2 u+ 1 v ∂u = ∂ u + u 1 − 11 − 1 , ∂t ∂x2 1+ 1 u 2 4 (36) 2 11 u+v ∂v ∂ v 10 ∂t = d2 ∂x2 + v 1 − 1 1+ 4 u
α+a−1 b−αb−1 Substituting (35) into the equilibrium point ab+k−ak−αk−1 , ab+k−ak−αk−1 we have
(u0, v0) = (20/107, 90/107). Next, we calculate the threshold dc above which the diffusion coefficient d2 of the “free-riders” would cause an instability.
The previous chapter had dc as
p 2 (2|A| − fw1 gw2 ) + 2 (|A| − |A|fw1 gw2 ) dc = 2 (37) fw1
where |A| is the determinant of the jacobian matrix A and fw1 , gw2 are entries in A.
25 Referencing (35), A is evaluated as
f f −5/32 −1/14 w1 w2 A = = (38) gw1 gw2 −153/224 −45/56
Evidently, (u0, v0) is a stable equilibrium point since the determinant |A| = 963/12544 >
0 and the trace , trace(A) = −215/224 < 0. Substituting |A|, fw1 and gw2 into (37), yields dc = 1782/36125
The next step would be to apply a non-homogeneous perturbation to the solution point (u0, v0). To do that we first determine the initial conditions of the participants u - cooperators and v - “free-riders” playing the game. We carefully choose the following initial conditions for system (36) as
u0(x) = 20/107 + B(x) (39) v0(x) = 90/107 + B(x) where B(x) is a C∞ bump function [1] on the interval [p, q]. That is, B(x) satisfies
a. B(x) = 1 for p ≤ x ≤ q b. B(x) = 0 for x ≤ µ and x > ξ where µ < p and ξ > b. (40) c. D0 (x) 6= 0 on the intervals(µ, p) and (q, ξ).
A construction4 of B(x) results in
2 2 − p −q e x−2p−p2 B(x) = 1 + 2 2 2 2 (41) − p −q − p −q e x−2p−p2 + e q2−2p2−2p−x
We finally apply Matlab PDE solver - pdepe - to solve system (36). For the diffusion coefficient d2 = 1/50 < dc, Figure (Fig. 8) shows that the coexistent state is stable
4See Chapter 6 - For derivation
26 to non-homogenous perturbation. In figure (Fig. 9) the coexistent state is unstable to non-homogenous perturbation when d2 > dc.
27 (a)
(b)
Figure 8: The coexistent state is stable to non-homogenous pertubation when the diffusion rate of the free-riders is below the diffusion threshold dc: (a) initial state of the coexistent solution and (b) the asymptotic behaviour of the coexistent state.
28 (a)
(b)
Figure 9: The coexistent state is unstable to non-homogenous pertubation when the diffusion rate of the free-riders is above the diffusion threshold dc: (a) initial state of the coexistent solution and (b) the asymptotic behaviour of the coexistent state.
29 5 CONCLUSION
The dynamics of the public good game in the absence of space, reduced the model to an ordinary differential equation. In Chapter 2 of this paper we showed that the homogeneous coexistent state is stable without diffusion. The coexistence between the cooperators and the “free riders” is guaranteed provided the effects each has on the other is bounded within the game. This coexistence shows that each player in the game is able to sustain their winning strategy in the long run. Evidently the cost of altruism α charged against cooperators for such a strategy was mitigated by the utility drawn in partaking in the game with the evolution of time.
When spatial extension is introduced into the model the diffusion rate of “free rid- ers” was shown to be the cause of Turing instability or spatial pattern formation. As demonstrated in the numerical example, when the initial densities of the two groups represented in the game is close enough to the coexistent state, then a small perturbation drives the coexistence to an instability state. This occurs when the diffusion coefficient of the “free riders” exceeds the threshold defined in theorem 2
The public good game model as presented falls under classical game theory and does not deal with the dynamical aspects of the strategies of the players in the game. Groups have to decide at the onset of the game their strategy ( i.e cooperating or non-cooperating) without the possibility to learn from their opponents and hence make adjustments in the course of play. For future investigation of spatial dynamics the model could be modified to capture the learning behaviour of players and their ability to alter their strategy.
30 6 APPENDIX
6.1 Derivation of Bump Function B(x)
Consider the following C∞ function
e−1/t for t > 0, c(t) = (42) 0 for t ≤ 0 with graph shown in Figure 10
Figure 10: The graph of c(t).
Define
c(t) f(t) = (43) c(t) + c(1 − t)
It can be shown that the denominator c(t) + c(1 − t) cannot take the value zero. As g(t) is derived from the quotient of two C∞ functions, then g(t) is also a C∞. Moreover, for t ≤ 0, c(t) = 0 and c(1−t) > 0, c(t) ≡ 0 for t ≤ 0. A similar argument can be made to show that c(t) ≡ 1 when t ≥ 1 Consider two positive real numbers p < q, we perform a linear change of variables to map [p2, q2] to [0, 1] : x − p2 x 7→ . (44) q2 − p2
31 Figure 11: The graph of f(t).
Let x − p2 g(x) = f (45) q2 − p2
Then g : R → [0, 1] is C∞ and
0 for x ≤ p2, g(x) = (46) 1 for x ≥ q2.
Figure 12: The graph of g(x).
We now replace x by x2 to make g(x) symmetric in x: h(x) = g(x2) (Figure 13) We then apply the following transformation
x − p2 m(x) = 1 − h (47) q2 − p2
Finally B(x) = m(x − 2q)
32 Figure 13: The graph of h(x).
Figure 14: The graph of m(x) .
Figure 15: The graph of B(x) .
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