Chemical evolutionary games

Abstract

Inspired by the use of hybrid cellular automata in modeling cancer, we introduce a generalization of evolutionary games in which cells produce and absorb chemicals, and the chemical concentrations dictate the death rates of cells and their fitnesses. Our long term aim is to understand how the details of the interactions in a system with n species and m chemicals translate into the qualitative behavior of the system. Here, we study two simple 2 × 2 games with two chemicals and revisit the two and three species versions of the one chemical colicin system studied earlier by Durrett and Levin [28]. We find that in the 2 × 2 examples, the behavior of our new spatial model can be predicted from that of the mean field differential equation using ideas of [12]. However, in the three species colicin model, the system with diffusion does not have the coexistence which occurs in the lattices model in which sites interact with only their nearest neighbors.

1 Introduction

Game theory was invented by John von Neumann and Oscar Morgenstern [1] to study strategic and economic decisions of humans. Maynard Smith and Price [2], see also [3], introduced the concept into ecology in order to explain why conflicts over territory between male animals of the same species are usually of the “limited war” type and do not cause serious damage. Axelrod and Hamilton [4] studied the evolution of cooperation by investigating the Prisoner’s dilemma game. Since that time, evolutionary game theory has been used to study many biological problems including host-parasite interactions, ecosystems, animal behavior, social evolution, and human language. For surveys see [5]–[8].

All of the references in the last paragraph study evolutionary games in homogeneously mixing populations, in which case the frequencies (in continuous time) follow the replicator equation. One can argue that long distance connections in human social network imply that spatial effects can be ignored, but this is not true for systems in ecology and cancer. In his classic 1964 work, which introduced the notion of kin selection, Hamilton [9] mentions that the viscosity of spatial populations helps promote altruistic behavior. However, much of the more recent interest in the influence of space on ecological interaction can be traced to the important 1992 work of Nowak and May [10, 11] who showed that spatial structure enhanced the persistence of cooperators in Prisoner’s dilemma. Their competition was deterministic and took place on the square lattice, but others have considered stochastic systems and competitions taking place on graphs or in finite populations. In the references we list a representative sample of work of this type, [12]–[16]. Dozens of references can be found in [17] and [18].

There have been a number of studies of evolutionary games in spatially distributed populations. However, up to now spatial models have been constructed by declaring that game interactions occur only between a site x chosen at random and a set of neighbors y. Here, we will take a different approach: cells produce and absorb chemicals, and the local chemical concentrations dictate the death rates of cells and their fitness. This formulation is inspired by hybrid cellular automata models of cancer. Some examples of this type of modeling can be found in [19]–[25]. Hundreds more can be found in the book by Cristini and Lowengrub [26]. These papers, which study highly detailed models numerically, provide another motivation for this study: we will study simplified systems in order to understand how the details of the interaction translate into the qualitative behavior of the model.

1.1 Motivating example

To explain what we have in mind, we will consider an example first studied by Tomlinson [27]. In this system, there are cells of three types.

1. Ones that produce a toxic substance.

2. Others that are resistant to the toxin, but not produce it.

3. Wild type cells that are neither producers nor resistant.

Based on the verbal description, Tomlinson write down the following game matrix.

$$\begin{array}{cccc}i& j=1& 2& 3\\ 1& z-e-f+g& z-e& z-e+g\\ 2& z-h& z-h& z-h\\ 3& z-f& z& z\end{array}$$ (1)

Here G_ij is the payoff to a player who plays strategy i against an opponent playing strategy j. Taking the rows in reverse order, z is the baseline fitness while f is the cost to a sensitive cell due to the presence of the toxin. The cost of resistance to the toxin is h. In top row e is the cost of producing the toxin, and g is advantage to a producer when it subjects a sensitive cell.

It is interesting to note that in the same year [27] was published, Durrett and Levin [28] used a spatial model to model the competition two strains of E. coli, whose behaviors correspond to strategies 1 and 3 above: one produces colicin to which the other strain is sensitive. In their model, there are also empty cells (denoted by 0). Thinking of a petri dish, their system takes place on the two dimensional lattice. Individuals of type i > 0 give birth at rate β_i with their offspring sent to a site chosen at random from the four nearest neighbors of x Each species dies at rate δ_i due to natural causes, while type 3’s die at an additional rate γ times the fraction of neighbors of type 1 due to the effect of colicin.

The first step in understanding the behavior of the system is to consider the mean-field version, which takes place on finite square with the neighborhood of each site being the entire grid. In this case if u_i is the frequency of sites in state i then in the limit as the size of the system goes to ∞ one arrives at differential equation

$$\begin{array}{l}\frac{{du}_1}{dt}={\beta }_1{u}_1{u}_0-{\delta }_1{u}_1\\ \frac{{du}_3}{dt}={\beta }_3{u}_3{u}_0-({\delta }_3+\gamma u_1)u_3\end{array}$$

In any equilibrium with u₁, u₃ > 0 we must have

$$u_0=\frac{{\delta }_1}{{\beta }_1}=\frac{{\delta }_3+\gamma u_1}{{\beta }_3}$$ (2)

If we assume δ_i < β_i so that the individual species survive, and

$$\frac{{\delta }_3}{{\beta }_3}<\frac{{\delta }_1}{{\beta }_1}<\frac{{\delta }_3+\gamma }{{\beta }_3}$$

then we can choose u₁ ∈ (0, 1) so that the second inequality holds and use the first to define u₀. Thus the dynamical system for the evolution of (u₁, u₃) has an interior fixed point in addition to the two boundary equilibria at (1 − δ₁/β,0) and (0, 1 − δ₃/β₃). Linearizing around the fixed points shows that the interior one is a saddle point while those on the boundary are stable.

Given the properties of the ordinary differential equation (ODE), it is hard to imagine how a colicin producer could arise by mutation in a sensitive population, since in that case their initial density will be small. However, as Durrett and Levin [28] showed, this can occur in a spatial model. This is consistent with the philosophy of their earlier work [12]: when the ODE du_i/dt = ϕ_i(u) is bistable, the winner in the spatial model is more accurately predicted by the behavior of the partial differential equation (PDE) du_i/dt = Δu_i + ϕ_i(u).

The situation becomes more interesting when we introduce a third competitor that is like Tomlinson’s type 2.

$$\begin{array}{l}\frac{{du}_1}{dt}={\beta }_1{u}_1{u}_0-{\delta }_1{u}_1\\ \frac{{du}_2}{dt}={\beta }_2{u}_2{u}_0-{\delta }_2{u}_2\\ \frac{{du}_3}{dt}={\beta }_3{u}_3{u}_0-({\delta }_3+\gamma u_1)u_3\end{array}$$

Based on the biology, it is natural to assume

$$\frac{{\delta }_1}{{\beta }_1}>\frac{{\delta }_2}{{\beta }_2}>\frac{{\delta }_3}{{\beta }_3}$$

That is, a cell pays a cost for colicin production or resistance which lowers its ability to compete. These inequalities imply that 3’s outcompete 2’s and 2’s outcompete 1’s. If γ is large enough, 1’s outcompete 3’s and the three species will have a rock-paper-scissors relationship.

If there is an equilibrium in which all u_i > 0, we must have

$$u_0=\frac{{\delta }_1}{{\beta }_1}=\frac{{\delta }_2}{{\beta }_2}=\frac{{\delta }_3+\gamma u_1}{{\beta }_3}$$

which is impossible. In contrast, as shown in [28], the three species coexist in the spatial model. It is interesting to note that after the three species coexistence was predicted it was found in laboratory experiments, see [29], [30].

In Tomlinson’ model, if we assume that the system is homogeneously mixing then the densities follow the replicator equation (see [5], Section 7.1.)

$$\frac{{du}_i}{dt}=u_i(F_i-\overline{F})$$

where F_i = Σ_j G_i,ju_j is the average fitness of the ith strategy and F̄ = Σ_j u_jF_j is the average fitness. As computed in [27], there is an interior equilibrium if

$$e<gh<f\frac{e}{g}>\frac{h}{f}$$ (3)

and its location is

$$\overline{u}=(h/f,1-e/g,e/g-h/f).$$ (4)

Strategy 3 always dominates 2 (in their two strategy subgame). If we make the natural assumption g > e (the benefit from toxin production outweighs the cost) then strategy 1 dominates 3. Finally, if we assume

$$h<e\text{and}h<e-g+f$$ (5)

then 2 dominates 1 and we again have a cyclic relationship between competitors. From the form of the replicator equation it is easy to see that the dynamics are not changed by adding a constant to each column in the game matrix G, so we can make the diagonal entries 0, When we do this, G has the form of generalized rock paper scissors game (here a_i, b_i > 0)

$$\left(\begin{array}{ccc}0& h-e& g-e\\ e-g+f-h& 0& -h\\ e-g& h& 0\end{array}\right)=\left(\begin{array}{ccc}0& -a_2& b_3\\ b_1& 0& -a_3\\ -a_1& b_2& 0\end{array}\right)$$

The initial assumptions h > 0 and g > e, with the conditions in (5) imply that our matrix has the desired sign pattern. In this case, Theorem 7.7.2 in [5], describes the asymptotic behavior

Theorem 1

Let Γ = b₁b₂b₃ − a₁a₂a₃.

• If Γ = 0 then there is a one parameter family of periodic orbits.

• If Γ < 0, trajectories spiral out toward the boundary.

• If Γ > 0 then there is a globally attracting fixed point for H.

Tomlinson [27] gives simulation result for a number of examples in his paper. In one set g = 0.2, e = 0.15, and h = 0.1 while f varies. Here g > e > h. We will have

$$h<e-g+f\text{if}0.1<0.15-0.2+f\text{or}f>0.15$$

When this holds we have f > h and e/g = 3/4 > 2/3 > h/f. His table reports the following results about the limiting behavior:

f	u1	u2	u3
0.25	0.400	0.250	0.350
0.24	0.417	0.250	0.333
0.23	0.435	0.250	0.315
0.22	0.457	0.250	0.293
0.21	1	0	0

It is not hard to check that the values given for u₁, u₂ and u₃ in the first four rows correspond to the fixed point in (4), e.g., ū₂ = 1 − e/g = 0.25. To check the conditions of Theorem 1, we compute

$$\begin{array}{l}\mathrm{\Gamma }=b_1{b}_2{b}_3-a_1{a}_2{a}_3=(e+f-g-h)h(g-e)-(g-e)(e-h)h\\ =(g-e)h(f-g)\end{array}$$

Since g > e and h > 0, the interior fixed point is attracting if f > g, i.e., f > 0.2. The discrepancy in the last line is not surprising. When f = 0.2 the interior fixed point is surrounded by a one parameter family of periodic orbits, so when f ≈ 0.2 numerical solution of the differential equation becomes delicate, so it is not surprising that Tomlinson reached the wrong conclusion about the case f = 0.21.

2 Our model

In the two models considered above, only the nearest neighbors of a colicin producing cell are affected, while in reality the chemical will diffuse. With this in mind we reformulate the model. As before, each site of the lattice is occupied by a cell that is in state ξ(x) ∈ {1, 2, … m}. In addition, there are ℓ chemical species with concentration given by R_i, which are modeled by reaction diffusion partial differential equations:

$$\frac{{dR}_i}{dt}=D_i\mathrm{\Delta }{R}_i+{\lambda }_i-{\nu }_i{R}_i+{\mu }_{i,\xi (x)}-d_{i,\xi (x)}{R}_i.$$ (6)

Here D_i is the diffusion constant for chemical i, λ_i is the rate at which it enters the system, ν_i is the rate at which it breaks down, μ_i,k is the rate at which it is produced by cells of type k, and d_i,k is the rate it is consumed by cells of type k. To avoid boundary effects, we employ periodic boundary conditions on the PDE and the particle system model we will soon describe.

In the colicin example above, there is only one chemical which is produced by type 1’s and affects type 3’s. This is just one of many possibilities. In Gerlee and Anderson’s model [23] for the emergence of the glycolytic phenotype, there are three chemicals: glucose, oxygen, and hydrogen ion concentration. In their work, the PDEs are more biologically accurate and the behavior of the cells is dictated by a complicated network. However, as mentioned earlier, one of our motivations is to study simplified hybrid cellular automata in order to find general results to predict their qualitative behavior.

The sites in our model follow evolutionary game dynamics. In the terminology used in [16] (and many other works) these are generalized death-birth dynamics. The word generalized refers to the fact that death occurs at a rate that depends on the chemical concentrations:

$${\delta }_k={\eta }_k+\sum _i{t}_{k,i}{R}_i,$$ (7)

where η_k is a constant and t_k,i ≥ 0 is a measure of the toxicity of chemical i to species k. When a cell of type k dies it is replaced by a neighbor cell that is chosen with probability proportional to its fitness, with type j’s having fitness

$$f_j={\gamma }_j+\sum _i{b}_{j,i}{R}_i,$$ (8)

Note that in contrast to Tomlinson’s game dynamics, the fitness of a cell depends on the chemical concentrations rather than on the types of their neighbors.

A common first step in understanding the workings of a spatial model is to see what happens when the spatial aspect is eliminated. Taking the diffusion rates D_i → ∞ in equation (6), and letting x_j be the fraction of cells of type j, the concentration of chemical i evolves according to the ODE:

$$\frac{{dR}_i}{dt}={\lambda }_i-{\nu }_i{R}_i+\sum _j({\mu }_{i,j}-R_i{d}_{i,j})x_j,$$ (9)

In equilibrium dR_i/dt = 0 so we have

$$R_i=\frac{{\lambda }_i+{\sum }_j{\mu }_{i,j}{x}_j}{{\nu }_i+{\sum }_j{d}_{i,j}{x}_j}.$$ (10)

If we now ignore correlations between neighbors, then we have the following ODE for the frequencies x_j,

$$\begin{array}{c}\frac{{dx}_j}{dt}=-{\delta }_j{x}_j\sum _{k\ne j}\frac{x_k{f}_k}{{\sum }_n{x}_n{f}_n}+\sum _{k\ne j}{\delta }_k{x}_k\frac{x_j{f}_j}{{\sum }_n{x}_n{f}_n},\\ =\sum _{k\ne j}\frac{x_k{x}_j({\delta }_k{f}_j-{\delta }_j{f}_k)}{{\sum }_n{x}_n{f}_n}.\end{array}$$ (11)

When δ_j ≡ 1 this looks like the replicator equation but the fitnesses are not linear in the frequencies

$$f_k={\gamma }_k+\sum _i{c}_{k,i}{R}_i={\gamma }_k+\sum _i{c}_{k,i}\frac{{\lambda }_i+{\sum }_j{\mu }_{i,j}{x}_j}{{\nu }_i+{\sum }_j{d}_{i,j}{x}_j}$$

Generalizing from the replicator equation, we have equilibrium if the ratio f_j/δ_j is constant. In general this is a system of cubic equations compared to the linear equations that come from the replicator equation. However, as the reader will see in the Section 4, the equations can be solved in some cases and accurately predict the behavior of the spatial model.

3 Implementation

Before turning to examples, we will describe our simulation methods. To solve the PDE in (6) that describing the chemical dynamics we use a two dimensional square lattice with spacing, h ≪ 1, between neighboring lattice nodes. For the temporal discretization of (6) we employ the modified Euler [31] method, which is of second order in time and can be written as an explicit Runge-Kutta. A standard second order finite difference scheme is also utilized for the spatial discretization of the Laplacian operator. The resulting numerical scheme does not require the inversion of a matrix at each time iteration hence it is fast. We have used h = 0.01 and τ = 0.001 as the stable time step for the PDE updates.

In order to simulate the continuous time Markov chain model describing the particle system dynamics, the corresponding discrete time approximation is used. Sites die with probability τ_s times their death rate and when they die are replaced by a neighbor proportional to its fitness. In doing our updates, we use a computation scheme that is equivalent to having two copies of the system. The previous state is used to compute the rates, while updates are made only in the second copy. For this study we have used τ_s = 0.01, that is we update the particle system once for every ten PDE updates. This is done to reduce the amount of work, while keeping the number of collisions (i.e., situations in which a site and its neighbor are both updated) relatively small, thus approximating better the behavior of the continuous time process.

4 Simple two species examples

We begin with two examples that illustrate the possibilities of coexistence and competitive exclusion in the case of two species and two chemicals. In these examples, neither species produces chemicals so μ_i,k = 0 and the toxicities t_k,i ≡ 0, so the death rates δ_k = η_k are frequency independent. In addition, we assume that the death rates η_k and the constants γ_k in the fitness function do not depend on k, and the decay rates ν_i = 0. With these conditions, the mean-field condition for equilibrium is f₁ = f₂ and we can drop the γ_k’s to write

$$\frac{c_{1,1}{\lambda }_1}{d_{1,1}x+d_{1,2}(1-x)}+\frac{c_{1,2}{\lambda }_2}{d_{2,1}x+d_{2,2}(1-x)}=\frac{c_{2,1}{\lambda }_1}{d_{1,1}x+d_{1,2}(1-x)}+\frac{c_{2,2}{\lambda }_2}{d_{2,1}x+d_{2,2}(1-x)}$$

In this case multiplying each side by the product of the denominators gives a linear equation:

$${\alpha }_{1,1}x+{\alpha }_{1,2}(1-x)={\alpha }_{2,1}x+{\alpha }_{2,2}(1-x)$$ (12)

where the constants

$${\alpha }_{i,j}=c_{i,1}{\lambda }_1{d}_{2,j}+c_{i,2}{\lambda }_2{d}_{1,j}$$

If when we set x = 0 in the denominator, we have

$$\frac{c_{1,1}{\lambda }_1}{d_{1,2}}+\frac{c_{1,2}{\lambda }_2}{d_{2,2}}>\frac{c_{2,1}{\lambda }_1}{d_{1,2}}+\frac{c_{2,2}{\lambda }_2}{d_{2,2}}$$ (13)

then 1 will be more fit than 2 when the frequency of 1’s is small, so the frequency of 1’s will increase. If in addition

$$\frac{c_{1,1}{\lambda }_1}{d_{1,1}}+\frac{c_{1,2}{\lambda }_2}{d_{2,1}}<\frac{c_{2,1}{\lambda }_1}{d_{1,1}}+\frac{c_{2,2}{\lambda }_2}{d_{2,1}}$$ (14)

then 2’s will be more fit than 1’s when the frequency of 1’s is close to 1, and the fitness of 1’s will decrease. When both (13) and (14) hold, (12) will have a solution in (0, 1) that will be an attracting fixed point. If we reverse the inequalities in (13) and (14) then there is an unstable fixed point in (0,1). We will now consider an example of each situation.

4.1 Two species example with coexistence

In this example, we take the diffusion constant D_i = 0.003, which is inspired by the diffusion constant for oxygen in the model of [23]. Both chemicals enter the system at rate λ_i = 0.1, neither is produced μ_i,j ≡ 0, and the decay rates are ν_i = 0. The toxicity coefficients t_k,i = 0 and the death rates η_k = 1. The consumption rates d_i,k and the coefficients c_k,i that determine the fitnesses are given by

$$d=\left(\begin{array}{cc}0.525& 0.225\\ 0.125& 0.425\end{array}\right)c=\left(\begin{array}{cc}0.8& 0.7\\ 0.6& 0.9\end{array}\right)$$

with the constants in the definition of f_k in (8) given by γ_k = 0.1. Since 0.8 + 0.7 = 1.5 = 0.6 + 0.9 it is easy to see that in this example the inequalities in (13) and (14) hold

$$\frac{0.8}{0.125}+\frac{0.7}{0.425}<\frac{0.6}{0.125}+\frac{0.9}{0.425}\text{and}\frac{0.8}{0.525}+\frac{0.7}{0.225}<\frac{0.6}{0.525}+\frac{0.9}{0.225}$$

so we expect coexistence. For an intuitive explanation note that type i is more fit when chemical i is present but type i also consumes more of chemical i so it will prefer to be around the opposite type. Simulations in Figure 1 show coexistence and confirm that that chemical 2 has a greater concentration near sites occupied by species 1 and vice versa.

Figure 1.

Simulation of the system described in Section 4.1. The mean-field differential equation has a fixed point in (0,1) that is stable. As the graphs of densities in panel (a) show the system converges to equilibrium. At first glance it may seem that there are large fluctuations in the densities but the numbers on the axis go from 7.9 to 8.25 × 10⁴. Panel (b) shows the system at the final time t = 200. Panels (c) and (d) show the concentration of the two chemicals at the final time.

4.2 Two species, competitive exclusion

The only change from the first example is that in the matrix of consumption rates d_ik has been changed to have larger entries off the diagonal

$$d=\left(\begin{array}{cc}0.225& 0.525\\ 0.425& 0.225\end{array}\right)c=\left(\begin{array}{cc}0.8& 0.7\\ 0.6& 0.9\end{array}\right).$$

In this example that the inequalities in (13) and (14) are reversed

$$\frac{0.8}{0.525}+\frac{0.7}{0.225}<\frac{0.6}{0.425}+\frac{0.9}{0.225}\text{and}\frac{0.8}{0.225}+\frac{0.7}{0.425}>\frac{0.6}{0.225}+\frac{0.9}{0.425}.$$

The interior equilibrium is unstable, so this is case 2 of Durrett and Levin [12]. Based on this analogy we predict that the two species will separate and the winner will be dictated by the direction of interface separating the two species.

Figure 2 confirms the segregation. Intuitively this occurs since species i consumes more of chemical i but is more fit when the other chemical is high, so species i will prefer to be around its type. Panels (b) and (c) of Figure 3 show the concentration of the two chemical species at the final time confirming that chemical i has a greater concentration near sites occupied by species i.

Figure 2.

Simulation of the system described in Section 4.2. The mean-field differential equation has a fixed point in (0,1) that is unstable.

Figure 3.

Graph of densities versus time and the final chemical concentrates for the system from Figure 2. From the snapshots in that figure the direction of movement of the interface is not clear, but panel (a) here shows that the number of type 2’s is increasing. Panels (b) and (c) give the concentrations of the two chemicals at the final time.

5 Colicin

We now return to the example that motivated our investigation.

5.1 Two species

In this case, we have one chemical (colicin) with λ₁ = 0, ν₁ > 0, and μ_1,1 > 0. The constants c_1,j = d_1,j = 0 t_1,1 = 0 and t_3,1 > 0. In words the first species produces colicin which increases the death rate of the second one, which for consistency with earlier discussion and in preparation for the next example, we call species 3. Our choices imply that

$$f_i={\gamma }_i{\delta }_1={\eta }_1{\delta }_3={\eta }_3+t_{3,1}{R}_1$$

There is a cost of producing colicin so we assume

$$\frac{{\gamma }_1}{{\eta }_1}<\frac{{\gamma }_3}{{\eta }_3}$$ (15)

Using (10), we see that in equilibrium R₁ = μ_1,1x/ν₁. So by (11) the mean-field differential equation for fraction of type 1 is

$$\frac{dx}{dt}=\frac{x(1-x)({\delta }_3{f}_1-{\delta }_1{f}_3)}{x{f}_1+(1-x)f_3}$$ (16)

In equilibrium

$$({\eta }_3+t_{3,1}{\mu }_{1,1}x/{\nu }_1){\gamma }_1={\eta }_1{\gamma }_3$$

Let α = t_3,1μ_1,1/ν₁. In order to have an interior fixed point we need

$$\frac{{\gamma }_1}{{\eta }_1}>\frac{{\gamma }_3}{{\eta }_3+\alpha }$$ (17)

When x ≈ 0, dx/dt < 0 in (16) by (15), while for x ≈ 1, dx/dt > 0 by (17), so the interior fixed point is unstable. Thus by the reasoning in the previous example, in the spatial model the victor will be determined by the direction of movement of interfaces.

To check this prediction by simulation, we investigate a concrete example. As usual, we take the diffusion constant D₁ = 0.003. The other positive parameters are μ_1,1 = 0.525, ν₁ = 0.4, γ₁ = 0.25, γ₃ = 0.6, t_3,1 = 0.65, η₁ = 0.25, and η₃ = 0.2. With the current parameters

$$\begin{array}{ccc}{\gamma }_3/{\eta }_3& {\gamma }_1/{\eta }_1& {\gamma }_3/({\eta }_3+\alpha )\\ \frac{0.6}{0.2}=3& \frac{0.25}{0.25}=1& \frac{0.6}{0.2+0.853125}=0.5697\end{array}$$

so there is an interior fixed point. Figure 4 shows a simulation on a 200 × 200 grid. The colicin producers rapidly eliminate the sensitive strain.

Figure 4.

Simulation of the two species colicin chemical game described in Section 5.1. Note that the colicin producer rapidly eliminates the sensitive strain.

5.2 Three species

We turn now to the three species system. Generalizing the previous example λ₁ = 0, ν₁ > 0, and μ_1,1 > 0, c_1,j = d_1,j = 0, t_1,1 = t_2,1 = 0 and t_3,1 > 0. Our choices imply that

$$f_i={\gamma }_i{\delta }_1={\eta }_1,{\delta }_2={\eta }_2,{\delta }_3={\eta }_3+t_{3,1}{R}_1$$

There is a cost of producing colicin, and the sensitive strain would be doomed if it was not stronger than the resistant strain in the absence of colicin, so we assume

$$r_1<r_2<r_3$$ (18)

where r_i = γ_i/η_i

Again in equilibrium R₁ = μ_1,1x₁/ν₁. Ignoring the denominator in the mean-field equation, which is just a time change, and writing α = t_3,1μ_1,1/ν₁ the three equations for equilibrium are

$$\begin{array}{l}0=\frac{{dx}_1}{dt}=x_1{x}_2({\eta }_2{\gamma }_1-{\eta }_1{\gamma }_2)+x_1{x}_3[({\eta }_3+\alpha x_1){\gamma }_1-{\eta }_1{\gamma }_3]\\ 0=\frac{{dx}_2}{dt}=x_2{x}_1({\eta }_1{\gamma }_2-{\eta }_2{\gamma }_1)+x_2{x}_3[({\eta }_3+\alpha x_1){\gamma }_2-{\eta }_2{\gamma }_3]\\ 0=\frac{{dx}_3}{dt}=x_3{x}_1[{\eta }_1{\gamma }_3-({\eta }_3+\alpha x_1){\gamma }_2]+x_3{x}_2[{\eta }_2{\gamma }_3-({\eta }_3+\alpha x_1){\gamma }_2]\end{array}$$

Moving the first term in the first equation and the second term in the second equation to the left, we see that for this to hold we want:

$$x_1{x}_2({\eta }_1{\gamma }_2-{\eta }_2{\gamma }_1)=x_1{x}_3[({\eta }_3+\alpha x_1){\gamma }_1-{\eta }_1{\gamma }_3]=x_2{x}_3[{\eta }_2{\gamma }_3-({\eta }_3+\alpha x_1){\gamma }_2]$$

The first term is positive since r₂ > r₁. If we let r̄₃ = γ₃/(η₃ + αx₁) then for the second and third to be positive we need r₁ > r̄₃ and r̄₃ > r₂, which is impossible, so there is no interior fixed point.

This was the case in the model of Durrett and Levin, but their model had coexistence. If we assume that

$${\overline{r}}_3<r_1<r_2<r_3$$

then the competitors have the same rock-paper scissors relationship. 2 beats 1, 3 beats 2, and 1 beats 3. For this reason we expected to find coexistence in the new model with diffusion but after many tries we have failed to find even one example for which this holds. Figure 5 shows the results of a typical simulation. As usual, we take the diffusion constant D₁ = 0.003. Most of the other positive parameters are the same as in the previous example μ_1,1 = 0.525, ν₁ = 0.4, γ₁ = 0.25, γ₃ = 0.6, η₁ = 0.25, and η₃ = 0.2. Here we have increased the death rate colicin coefficient, t_3,1, from 0.65 to 0.8 in order to compensate for the fewer initial number of colicin producers. The new species has γ₂ = 0.37 and η₂ = 0.29, i.e., r₂ = 0.37/0.29 = 1.276 so (18) holds. The parameters here were chosen by simulating a system with strips of each of the three types on a torus in order to confirm that when the types were put down in the order 1, 2, 3, all of the interfaces move to the left (Fig. 6). However in the simulation starting from a random initial configuration the 1’s cannot advance against the 3’s because the 2’s are mixed in with them and reduce the colicin concentration.

Figure 5.

Simulation of the three species colicin chemical game described in Section 5.2. Note that the 3’s separate from the 1’s and 2’s and then win the competition.

Figure 6.

Simulation of the three species colicin chemical game with parameters as in Figure 5.2, starting from three strips on a torus. Note that all three interfaces move to the left with positive speeds.

5.3 Diffusion kills coexistence

For the three species colicin system, there is no coexistence in our chemical game model, but there is in Durrett and Levin’s interacting particle system. To understand the reason for this difference, we will allow the colicin in latter system to be produced at rate μ_1,1 = 0.525, decays at rate ν₁ = 0.4, and have a diffusion constant of D₁ = 0.003. It is. We start with an initial chemical concentration of R₁(x, 0) ≡ 0.1. For a site of type i births occur with rate γ_i, and death with rate δ_i, i = 1, 2, 3 where δ₁ = η₁, δ₂ = η₂ and δ₃ = η₃ + t_3,1R₁, i.e., we use the concentration of colicin rather than the fraction of colicin producing neighbors. Note that in contrast to [28] we have set t_2,1 = 0. In the next section we will show that this is not a significant change.

Simulations are done with the scheme described in Section 3, except that now fitnesses are not calculated but birth and deaths occur according to the indicated rates. In the next two simulations the grid is 200 × 200. Figure 7 shows that when t_3,1 = 2.4, the frequency of type 3’s decreases rapidly until it reaches a low level. When their density of type 3’s is small, the type 2’s drive the type 1’s to extinction and then the type 3’s take over the system. Figure 8 shows that when t_3,1 = 2.8, the type 3’s are driven to extinction by type 1, who then lose out to the type 2’s.

Figure 7.

Simulation of Durrett and Levin’s three species interacting particle system with diffusion of colicin as described in Section 5.3 with t_3,1 = 2.4. Note that the 1’s separate from the 2’s and 3’s and the 3’s are present at a low density. The 2’s then drive the 1’s to extinction, and in the end lose to the 3’s.

Figure 8.

Simulation of the system considered in Figure 7, with t_3,1 increased from 2.4 to 2.8. Note that now the 3’s die out and then the 2’s outcompete the 1’s.

Figure 9 shows a simulation with the rest of the parameters as before and t_3,1 = 2.8. Type 1’s initially drive type 3’s close to extinction, but due to the larger grid size the type 3’s do not die out. With the 3’s a low level, type 2’s drive the numbers of type 1’s to extinction and then lose to the type 3’s. To investigate the effect of diffusion in the previous competition we reduced D₁ from 0.003 to 0.001, keeping the rest of the parameters fixed. Figure 10 shows that for this setup the system initially behaves in the same way as before, but instead of the type 1’s survive in a patch that has 2’s on the inside and 3’s on the outside. The result is that the 1’s drive the 3’s to extinction and then the 2’s outcompete the 1’s.

Figure 9.

Simulation of the system considered in Figure 8 on a 400 × 400 grid. This time, due to the larger grid, the type 3’s escape extinction. While the 3’s are at a low level, the 2’s drive the 1’s to extinction and then lose in competition with the 3’s.

Figure 10.

Simulation of the system considered in Figure 9, but with the diffusion constant reduced from 0.003 to 0.001. As in the previous simulation, the 3’s are initially driven to a low level, but this time the 2’s do not drive the 1’s to extinction. The 1’s that survive end in a small patch with 2’s on the inside and 3’s on the outside, so the 1’s will drive the 3’s to extinction and then lose to the 2’s. Panel (f) shows the concentration of colicin at the final time.

5.4 Removing diffusion restores coexistence

To seek further confirmation that diffusion of colicin is eliminating coexistence, we now remove diffusion from both processes. In the interacting particle system studied by Durrett and Levin [28], we redefine the death rate of type 3 to be δ₃ := η₃ + t_3,1ϕ₁, where ϕ₁ is the fraction of the four nearest neighbors in state 1. Figure 11 shows a simulation on a 200 × 200 grid with γ₁ = 0.3, γ₂ = 0.32, γ₃ = 0.4, η₁ = η₂ = η₃ = 0.1 and t_3,1 = 0.3 so we have the rock-paper scissors relationship. As the figure shows coexistence has been restored.

Figure 11.

Simulation of the Durrett-Levin three species colicin interacting particle system with no diffusion. The three species coexist as in the continuous process simulation in [28].

Figure 12 shows a simulation of the colicin chemical game system with γ₁ = 1, γ₂ = 3, γ₃ = 7, η₁ = 0.8, η₂ = 1.6, η₃ = 1, and t_3,1 = 16.5.

Figure 12.

Simulation of the three species colicin chemical game with no diffusion. This time the three species coexist.

$$\frac{{\gamma }_3}{{\eta }_3+t_{3,1}}=\frac{7}{17.5}<\frac{{\gamma }_1}{{\eta }_1}=1.25<\frac{{\gamma }_2}{{\eta }_2}=1.6<\frac{{\gamma }_3}{{\eta }_3}=7,$$

so we have the rock-paper scissors relationship. As the simulation shows the three species segregate and coexist. Here as in [28] the densities oscillate but with higher frequency and at a much smaller amplitude.

6 Discussion

Comparing results in Section 5, we see that in the Durrett-Levin three species colicin interacting particle system and in our chemical game system, there is coexistence when sites interact only with their nearest neighbors, but not when the colicin diffuses. If we think of spatial separation being the same as genetic relatedness, then this is consistent with the kin selection view point of Hamilton [9]. Diffusion means that colicin production benefits more distant relatives so the system will be less effective as maintaining coexistence.

It is an interesting question whether any positive amount of diffusion destroys coexistence or is there a threshold level? We believe that there is no threshold. If we decrease the diffusion rate, then we can speed up time to return it to its previous values. When diffusion is held constant, the ratio of the production and decay rates controls how far the chemical will diffuse before it is destroyed. The concentrations will decay exponentially with distance, so if the chemical is present in significant levels at nearest neighbors, then it will also occur at similar levels at the next nearest neighbors, etc. When we decrease the diffusion, we also increase the birth and death rates and it is far from obvious how these two rescalings interact. However, we have simulated a large number of parameter combinations and have not found coexistence.

Based on the results of simulations, we have some ideas about why diffusion destroys coexistence. It seems that diffusion and relatively high t_3,1 values prevent the formation of sharp boundaries between type 1’s and 3’s regions. We believe that this is due to diffusion allowing colicin to penetrate into areas that do not have producers (type 1’s). Thus, diffusion helps the type 1’s when they are in relatively large numbers, since colicin diffuses from many locations and results in the fast extinction of type 3’s (see figure 8). In contrast, when the population of type 1’s falls to a low number (due to the action of type 2’s and lower values of t_3,1), diffusion reduces the ability of type 1’s to poison type 3’s and allows the latter to rebound. See figures 7 and 9. Reducing the diffusion coefficient can reduce this effect and concentrate more nutrient to the immediate neighborhood of type 1s. In this case even though we do not have coexistence, we observe that a small number of type 1’s manage to reemerge and form a clear front with the 3’s (see figure 10).

While we cannot say for certain that any positive amount of diffusion is enough to eliminate coexistence, we have shown that replacing simple interactions of a site with its nearest neighbors, by the more realistic situation of sites being influenced by diffusing chemicals can lead to dramatic changes in the behavior of a spatial model. There are many ecological systems involving competition or cooperation between small organisms such as bacteria to which this research will be relevant. However, our main interest is to use these systems to understand the ecology of cancer, i.e., to analyze models of interactions between tumors and their microenvironment, in order to be able to predict the properties of the model from their structure without having to study each example by simulation.