the evolution of cooperation in 1-dimensional ......evolution of cooperation in 1-dimensional mobile...

Far East Journal of Applied Mathematics © 2016 Pushpa Publishing House, Allahabad, India Published Online: August 2016 http://dx.doi.org/10.17654/AM095010063 Volume 95, Number 1, 2016, Pages 63-88 ISSN: 0972-0960

Received: April 14, 2016; Accepted: June 1, 2016 2010 Mathematics Subject Classification: Primary 91A22. Keywords and phrases: evolution of cooperation, spatial structure, mobility, stochastic simulation.

Communicated by K. K. Azad

THE EVOLUTION OF COOPERATION IN 1-DIMENSIONAL MOBILE POPULATIONS

Igor V. Erovenko and Jan Rychtář

Department of Mathematics and Statistics University of North Carolina at Greensboro Greensboro NC 27402 U. S. A. e-mail: [email protected]

[email protected]

Abstract

We investigate the effects of spatial structure and mobility on the evolution of cooperation. We consider a finite fixed size population of mobile cooperators and free-riders. Cooperators provide benefits to all individuals in their vicinity at a cost to themselves, whereas free-riders do not provide any benefits and incur no cost. Individuals occupy positions in a 1-dimensional lattice and are allowed to move to try to maximize their payoff. Our model allows simultaneous interactions among multiple individuals as all individuals in a neighborhood compete for the same limited resource. We create an exact stochastic simulation of this Markov process and find that smaller neighborhood size, lower mobility and shorter migration range promote the fixation of cooperators in the population. On the other hand, sufficiently large migration range coupled with high mobility rate may result in breakdown of cooperation.

Igor V. Erovenko and Jan Rychtář 64

1. Introduction

Understanding the emergence and persistence of cooperation is a fundamental problem in evolutionary biology. Cooperation is needed for the evolution of higher levels of organization. Genomes, cells, multicellular organisms and human society are all based on cooperation. The essence of cooperation lies in an individual paying a cost for someone to receive a benefit. Since evolution is based on competition between individuals, it should reward only selfish behavior as cooperators forgo some of their reproductive potential to help others. In an infinite well-mixed population, replicator dynamics shows that cooperators are doomed since non-cooperators (also called defectors or free-riders) enjoy the benefits of cooperation but incur no cost. Yet cooperation abounds in nature and human society and one of the most remarkable aspects of evolution is its ability to generate and maintain cooperation in a competitive world. The evolution of cooperation is often studied in the context of evolutionary game theory [27, 13, 19] and one of the classical models for pairwise interactions between cooperators and defectors is the prisoner’s dilemma.

Potential benefits of cooperation alone are not sufficient for the evolutionary maintenance of cooperation since a population of cooperators is vulnerable to invasion by free-riders. Immense amount of research has been devoted to discovering and understanding additional mechanisms which help promote cooperation [5, 20]. One of the important observations is that real populations are not well-mixed since due to spatial structure or social networks, certain individuals interact more often than others. Since replicator dynamics neglects these circumstances, it has limited applicability to the study of the evolution of cooperation. Nowak and May [22] demonstrated that spatial structure, which allows cooperators to form clusters and to resist invasion of defectors, is one of the key mechanisms for the emergence and persistence of cooperation. The effect of spatial structure on the evolution of cooperation has been extensively studied [21, 18, 6, 23, 29, 25, 24] and it is usually found to be beneficial. On the other hand, there are also known instances when spatial structure inhibits cooperation [11].

Evolution of Cooperation in 1-dimensional Mobile Populations 65

Another key factor in studying the evolution of cooperation is mobility. This is motivated by the observation that individuals prefer better neighborhoods and are often willing to migrate to improve their situation. Once mobility is introduced into the mathematical model, the interaction structure constantly changes, and standard analytic models no longer apply. For the most accurate results, one usually employs agent-based stochastic simulations of the process. There are many different approaches to incorporating mobility into the mathematical model. Random diffusion [31, 26, 10, 15, 4, 30] may either promote or inhibit cooperation depending on parameters and rules of the model. Individuals may move if they are unhappy with their current interactions [1, 32] or if their payoff is lower than a certain threshold [17]. In reputation-based migration models, individuals move from low reputation neighborhoods [8]. Individuals may consider different alternative locations within their migration range and move to locations with better payoffs based on test interactions with individuals in possible new neighborhoods; such mobility model is called success-driven migration [12] (see also [7] for further elaboration on this model). It is also possible to incorporate mobility into the model implicitly by considering dynamic networks [3, 33].

There is no clear-cut effect of mobility on the evolution of cooperation. The outcome depends on the model used, and if spatial structure already promotes cooperation by allowing cooperators to thrive in clusters, then introducing mobility into the model often allows free-riders to exploit and breakdown these clusters. Also, it is possible to create conditions where roaming free-riders keep exploiting unsuspecting cooperators [9]. On the other hand, if spatial structure alone is not sufficient for emergence and persistence of cooperation, then introducing mobility may help cooperators aggregate in clusters, which is essential for maintenance of cooperation.

In this paper, we investigate the effects of spatial structure and mobility on the evolution of cooperation. We consider a finite fixed size population of mobile individuals which are either cooperators or free-riders. Individuals never change their strategy throughout their lifetime. Individuals occupy


positions in a 1-dimensional lattice with periodic boundaries. In our model, any site can be occupied by multiple individuals, so it represents a fusion of lattice-structured populations, where any location can be occupied by at most one individual, and continuum-structured populations, where individuals may occupy any position in a continuum. The population is sparse in a sense that most locations remain vacant. Having empty sites may allow clusters of cooperators to isolate themselves from defectors and are known to help promote cooperation [2]. Individuals “sample” all possible locations within their migration range and are likely to move towards places with higher payoffs, i.e., towards cooperators and away from free-riders. In other words, we employ the success-driven migration model. We construct an exact stochastic simulation of this Markov process and run it over a wide range of parameters. This paper is an extension of preliminary work on this model reported in [28].

2. Methods

We consider a population of N individuals in a 1-dimensional lattice of length L with periodic boundaries. At the beginning of the simulation, each individual is assigned a random position in the lattice, and we allow multiple individuals to occupy the same position. Every individual is randomly assigned (with equal probability) the role of either cooperator or free-rider; individuals never change their strategy throughout their lifetime.

An individual mI is in a neighborhood of an individual nI if the

distance between them is at most D, which we call the neighborhood radius. So, a neighborhood of radius D of any individual consists of 12 +D points in the lattice. All individuals in the same neighborhood interact with each other by competing for local resources, and every individual in the neighborhood gets an equal share of the resource.

Let B be the benefit and C be the cost of cooperation. A cooperator provides the benefit B for each individual in its neighborhood (including itself) and pays a fixed cost C. The payoff of an individual nI is given by


,1n

nnnn CfccBp −

++⋅= (1)

where nc and nf are the numbers of cooperators and free-riders,

respectively, in the neighborhood of the individual nI (including the

individual itself), and

⎩⎨⎧=

rider-freeaisif0cooperatoraisif

n

nn I

ICC (2)

is the cost of providing (cooperator) or not providing (free-rider) benefits to the neighbors. The term ( )nn fc +1 represents the competition for local

resources among all individuals in the same neighborhood. For example, the payoff of an isolated free-rider is 1, while the payoff of an isolated cooperator is .1 CB −+ In our model ,CB < so that isolated cooperators have lower payoff than isolated free-riders and cooperators must form clusters in order to survive.

Our Markov chain process consists of two types of events: reproduction and movement. We implement reproduction as a birth-death process (Moran process), see for example [16], in order to keep the population size constant. We apply a “smoothing” function to payoffs of the individuals and define the propensity of the individual nI to reproduce as

( ) .2tan 1 π+= −nn pr (3)

This is done to ensure that individuals with negative payoffs can reproduce and that individuals outside a cluster of potentially many cooperators can reproduce. In particular, this allows potential formation of several independent clusters of cooperators.

If the reproduction event is going to occur, then one individual in the population is going to reproduce and the probability that an individual nI is

chosen for reproduction is given by

.

1∑ =

Nk k

n

r

r (4)


The offspring inherits the strategy (cooperator or free-rider) of its parent and is placed randomly in the neighborhood of the parent. (The size of the neighborhood is determined by the neighborhood radius parameter D.) Finally, a random individual of the original population (potentially including the parent) is removed in order to maintain the constant size of the population.

The other event in our stochastic process is movement. One individual is going to move and the individual for the movement event is chosen randomly. Once an individual nI is chosen to move, it “samples” all possible

locations it can move to from its current position based on the migration range R. We assume that the cost of such test interactions with potentially new neighbors is negligible and they do not affect an individual’s payoff. An individual then picks its new location among 12 +R possible ones with a probability positively correlated to the difference between the payoff at the potentially new location and its current payoff. More specifically, if the payoff at a potentially new location is ,np′ then the propensity of the

individual nI to move from its current position with payoff np to this new

location is given by ( ).exp nn pp −′ (Note that this is similar to the Fermi

rule, which is often used in spatially structured models [25].)

The probabilistic nature of movement means that an individual may move to a position with lower payoff. However, such a move is less likely than the one that improves the individual’s payoff, and the non-deterministic migration pattern gives individuals a chance to find global maxima by not getting trapped in the local ones.

The mobility of the population is controlled by the mobility rate parameter M. It represents an average number of individual movement events for every reproduction event. The probability that the next event is reproduction is thus ( )11 +M and the probability that the next event

is movement is ( ).1+MM Table 1 summarizes all parameters of the

simulation.


Table 1. Summary of the parameters of the model Parameter Meaning Range of values

B Benefits of cooperation⎭⎬⎫

⎩⎨⎧ +++ 4

32,2

12,4

12,2

CCCC

C Cost of cooperation { }4,3,2

D Neighborhood radius { }40,30,25,20,15,10,5,1

M Mobility rate { }10,1,0

N Size of the population { }40,30,20,10

R Migration range { }5,3,1

We run each simulation until the Markov chain process reaches one of its two absorbing states: all cooperators or all free-riders. For each set of parameters, we run the simulation 10,000 times. The fraction of times the population ends up consisting of cooperators only is called the fixation probability of cooperators.

The length L of the lattice is fixed at 100. The size of the population N ranges over { }.40,30,20,10 In test simulations, we have found that

increasing population size past 40 does not produce much difference in the results, so in actual full simulations, we capped the population size at 40. The neighborhood radius D ranges from 1 to 40; the actual values it assumes are { }.40,30,25,20,15,10,5,1 The mobility rate M of the population ranges

over { }10,1,0 with 0=M corresponding to non-mobile population. We

allow the possibility of zero mobility in order to isolate the effect of the spatial structure of the model on the fixation probability of cooperators. The migration range R ranges over { }5,3,1 and can be thought of as the speed of

movement. The faster individuals can move, the farther they may migrate.

We will now explain our choice of values for the benefit and cost parameters. First, CB < so that isolated cooperators have lower payoff than isolated free-riders. The difference between B and C determines the payoff of an isolated cooperator. Once an isolated cooperator is chosen for reproduction, a cluster of two cooperators is formed (provided that same cooperator is also not chosen to be removed from the population). Viability


of this cluster is in part determined by comparing payoffs of two cooperators with an isolated free-rider. If a cluster of as few as two cooperators provides higher reproduction propensity than that of isolated free-riders, then even a single reproduction event of an isolated cooperator may give rise to a thriving cooperating cluster. On the other hand, if more than two cooperators are needed to become favorites over free-riders, then fewer viable cooperating clusters will be formed.

We have three possible values for the cost of cooperation, and for each value of the cost, we have four possible values of the benefits. The values of the benefit B are chosen in such a manner that the payoff of the cluster of two cooperators does not depend on the cost C. Indeed, if a neighborhood consists of k cooperators, then the payoff of each cooperator in this neighborhood is ,1 kCkB +− and for ,2=k the C parameter cancels out

since B is of the form ,2 α+C where { }.43,21,41,0∈α It follows

that the B parameter controls viability of a cluster of two cooperators (and hence how likely reproduction of an isolated cooperator is to give rise to a competitive cluster of cooperators), and the C parameter controls payoffs of isolated cooperators (and hence how likely an isolated cooperator is to reproduce and to give rise to a cluster of two cooperators) and payoffs in clusters with three or more cooperators. Keep in mind that the effect of changing B or C parameters is not linear due to the smoothing function (3) used to compute reproduction propensities.

3. Results

We ran the exact stochastic simulation of our Markov process for (almost) all possible combinations of parameters of the model. In this section, we illustrate the results of the simulation by choosing certain combinations of parameters only. In deciding which graphs are to be included in the paper, we simply tried to represent a wide variety of combinations of different parameters. The general observations we make and conclusions we draw remain valid for other combinations of parameters as well.


3.1. Effect of spatial structure on the fixation probability of cooperators

We begin by studying the effect of spatial structure and other parameters in our model on the fixation probability of cooperators. For this, we take the mobility out of the picture by letting .0=M The only events in our stochastic process are now reproduction events.

3.1.1. Smaller neighborhoods promote cooperation

A simple straightforward pattern of the effect of spatial structure emerges: the smaller the neighborhood size is, the higher the fixation probability of cooperators is. Also, once the neighborhood size becomes sufficiently large - about half the size of the entire lattice - cooperators become underdogs regardless of the other parameters. This is due to the fact that for such neighborhood size, there can be at most two clusters of individuals in the entire population and it essentially becomes a well-mixed population. Since free-riders pay no cost, they have slightly higher payoff than cooperators and this causes free-riders to dominate more often than cooperators. Cooperators cannot win more than 50% of the time in a well-mixed population case. This effect of the neighborhood size on the fixation probability of cooperators is consistent with observations in [14].

3.1.2. Higher population density helps cooperators

Next, we investigate the effect of population size on the fixation probability of cooperators. Since the lattice length in our simulations is fixed at 100, one can also think of the population size as population density. The general pattern is that higher density of the population helps cooperators; see Figure 1. Increasing the population size from 10 to 20 significantly improves cooperators’ chances, while increasing the population size past 30 results in marginal gains. As we already mentioned in Section 2, increasing the population size from 40 to 50 does not produce any meaningful difference in the fixation probability of cooperators. Also, we do not consider populations with more than 50 individuals since our model assumes sparse populations with the majority of lattice locations remaining vacant.


Figure 1. Higher population density increases the fixation probability of cooperators. The left diagram represents the best possible combination of benefit and cost parameters for cooperators: lowest cost with highest benefits. The right diagram shows the worst possible combination of benefit and cost parameters for cooperators: highest cost with lowest benefits.

In a small population, cooperators often do not have sufficient time to aggregate in stable clusters before they become extinct. Figure 2 shows an average number of generations the simulation takes before the stochastic process reaches one of its two absorbing states. (We define a generation in population of size N as N consecutive reproduction events.) Even if a cluster of cooperators is formed, then it might be exploited by a defector who is more likely to reproduce and replace cooperators with its offspring. Cooperators often do not have either time or the numbers to form another cluster and are forced to extinction. For example, in the case of the worst possible combination of benefit and cost parameters for cooperators: highest cost ( )4=C and lowest corresponding benefit ( ),2=B cooperators are

never favorites over defectors in a population of size 10 (see the right diagram in Figure 1). If we increase the benefits to ,25.2=B then

cooperators are slight favorites only when the neighborhood radius is minimal possible with .1=D


Figure 2. The average number of generations the simulation takes for the stochastic process to reach one of its two absorbing states. A generation in a population of size N is defined as N consecutive reproduction events. The data is robust regardless of mobility rate.

Increasing the population size to 20 significantly improves cooperators’ chances. First, cooperators have more time to regroup and produce clusters. Second, with higher population size, at least several clusters of cooperators are likely to form. As long as at least one of them is not exploited by defectors, it has a great chance to take over the entire population. Suppose at some point in the simulation, there are two clusters of cooperators, one of them contains only cooperators while another one also contains one or more defectors. With this arrangement, the defectors exploiting the second cluster will reproduce more often than cooperators in that same cluster. So, defectors are favored to drive cooperators in the second cluster to extinction. However, as soon as there are no more cooperators in the cluster, or simply the number of cooperators in the cluster is reduced significantly, the payoff of those defectors drops drastically. As a result, cooperators in the first unexploited cluster now enjoy the highest reproduction propensity and quickly take over the entire population.

Increasing the population size to 30 results in slight improvement of the fixation probability of cooperators, while increasing the size further to 40 provides only marginal gains. With the most favorable combination of benefit and cost parameters for cooperators: lowest cost ( )2=C and highest

corresponding benefit ( ),75.1=B cooperators simply dominate defectors in


sufficiently dense populations for small neighborhood radius (see the left diagram in Figure 1). As explained in the previous paragraph, having several independent clusters of cooperators allows cooperators to better resist exploitation by defectors since there is a higher chance that at least one of the clusters will remain unexploited. Also, with higher population density, each cluster will have more individuals on average and due to the smoothing function (3), the difference in reproduction propensities of cooperators and exploiting defectors is very small. This makes it more difficult for defectors to exploit dense clusters of cooperators.

3.1.3. Effects of benefit and cost of cooperation

Increasing benefits of cooperation while keeping the cost fixed naturally increases the fixation probability of cooperators. Figure 3 shows how the benefit parameter affects the reproduction propensity of cooperators in a cluster consisting of two individuals both of which are cooperators. The reproduction propensity of an isolated defector is the same as the reproduction propensity of a cluster of two cooperators for the second value of B, which in case 3=C is equal to 1.75. The effect of increasing benefits past this value is diminishing due to the smoothing function (3) used to transform an individual’s payoff into its reproduction propensity.

Figure 3. Increasing benefits of cooperation has diminishing effect on the reproduction propensity of a cluster of two cooperators. The graph shows benefit values corresponding to cost of cooperation ,3=C but the graph does not depend on the cost parameter.


The effect of the benefit parameter is higher for sparser population and smaller neighborhood radii; see Figure 4. In low density populations, most cooperators are isolated after the initial random set up and first clusters of cooperators that might arise will usually consist of two cooperators. Since the benefit parameter controls the vitality of a cluster of two cooperators, the effect is stronger if many cooperating clusters start with only two individuals. Also, with smaller neighborhood radius, the average size of a cluster is small, and hence clusters of only two cooperators are more likely to arise than in the case of larger value of neighborhood radius.

Figure 4. Increasing benefits of cooperation while keeping the cost fixed helps promote cooperation. The effect is stronger for sparse populations and for small neighborhoods. The effect of increasing the benefits is diminishing.

Having lower cost while maintaining the same relative level of benefits increases the fixation probability of cooperators; see Figure 5. Recall that


the cost parameter is designed to control the reproduction propensity of an isolated cooperator and a cluster of three or more cooperators. Lower cost helps isolated cooperators, but hurts large clusters of cooperators, while higher cost hurts isolated cooperators and helps large clusters of cooperators.

Figure 5. Lowering the cost of cooperation while maintaining the same relative level of benefits helps cooperators. The effect of lower cost is stronger for sparse populations and small neighborhoods.

Due to the smoothing function (3), which converts an individual’s payoff to its reproduction propensity, increasing the payoff of an isolated cooperator has much stronger impact on promotion of cooperation than increasing payoff of a large cluster of cooperators. For example, lowering the cost from 3 to 2 while keeping the smallest benefit value for each cost increases an isolated cooperator’s payoff from –0.5 to 0, which translates into an increase of its reproduction propensity from 1.1071 to 1.5708, for a gain of almost


42%. On the other hand, it decreases the payoff of a cluster of three cooperators from 611 to ,34 which translates into a decrease in reproduction

propensity from 2.6422 to 2.4981, for a loss of only 5.5%.

Since adjusting cost has significant effect on reproduction propensity of isolated cooperators and only marginal effect on the reproduction propensity of large clusters of cooperators, it is only natural that the effect of cost parameter is higher for sparse populations and small neighborhood radii. For dense populations and large neighborhoods, lowering cost of cooperation has marginal impact since there are fewer isolated cooperators and more large clusters of individuals.

3.2. Effect of mobility on the fixation probability of cooperators

We discovered how neighborhood size, population density, benefit and cost parameters affect the fixation probability of cooperators. Now we bring mobility into the model and investigate how different mobility rates and migration range affect cooperators. The general pattern turns out to be that mobility does not help cooperators and may, in fact, lead to an almost complete breakdown of cooperation.

3.2.1. Increasing mobility rate decreases the fixation probability of cooperators

If cooperators dominate free-riders, which is the case for small neighborhoods and dense populations, then increasing mobility rate decreases the fixation probability of cooperators. This is consistent with our expectation that if the spatial structure alone is sufficient to promote cooperation, then introducing additional factors, such as mobility, into the model would only hurt cooperators. On the other hand, for sparse populations and large neighborhoods, when cooperators do not rate so well, mobility has no visible effect on the outcome of the simulation. When free-riders are favorites over cooperators, then even allowing individuals to migrate in efforts to improve their payoff does not increase the fixation probability of cooperators.


Figure 6. Increasing mobility rate does not help cooperators. In cases when cooperators dominate free-riders - small neighborhoods and dense populations - higher mobility rates decrease the fixation probability of cooperators. If free-riders are favorites over cooperators, then allowing individuals to migrate trying to improve their payoff still does not help cooperators to do better.

Figure 6 shows the effect of different mobility rates on the evolution of cooperation in case the migration range is minimal possible at 1. Notice that the negative effect of mobility is stronger in cases when cooperators do best: small neighborhoods and dense populations. Once the neighborhoods become sufficiently large and our model essentially becomes identical to a well-mixed population case, then mobility naturally has no effect on cooperation.


Notice how in Figure 6 in the diagram for population size 40, cooperators do slightly better with increased mobility rate when the neighborhood radius is equal to 20. We have observed similar, though less pronounced, patterns for certain other combinations of parameters for

.20=D However, for many other combinations of parameters with ,20=D this does not happen, so we refrain from drawing any conclusions.

This can be the effect of stochastic noise. But there is a strong pattern that regardless of other parameters of the model, increasing mobility rate does not hurt cooperators if the neighborhood radius is at least 20.

3.2.2. Increasing migration range may result in a breakdown of cooperation

Since increasing mobility rate often allows free-riders to exploit cooperators, it is only natural to expect that increasing migration range would only make this exploitation mechanism more effective for free-riders. While Figure 6 shows that increasing mobility rate has little negative effect on the fixation probability of cooperators when the migration range is minimal possible at ,1=R Figure 7 demonstrates that with longer migration range,

the situation is notably worse for cooperators.

Increasing the migration range from 1=R to 5=R while keeping the mobility rate fixed at 1=M is almost equivalent to increasing the mobility rate from 1=M to 10=M while keeping the migration range fixed at

.1=R Also, notice that in the top left diagram in Figure 7, corresponding to the case of population size 10, the graph dips when the neighborhood radius is at .1=D This is the only combination of parameters (lowest benefit to cost ratio and smallest population) where this phenomenon occurs if the mobility rate is fixed at 1. We discuss it further in the case of mobility rate 10 where it is even more pronounced causing a breakdown of cooperation for some combinations of parameters.


Figure 7. Increasing migration range has visible negative impact on the fixation probability of cooperators even when the mobility rate is equal to 1.

The situation is much worse for cooperators if the migration range is increased when the mobility rate is higher with ;10=M see Figure 8. Notice

how increasing migration range flattens the fixation probability curves. If individuals are allowed to migrate further, then even small neighborhoods -cooperator’s paradise - are no longer sufficient for domination of cooperation. This is an expected result since increasing migration range gives free-riders ample opportunities to locate and exploit clusters of cooperators with small footprint. So it effectively negates the advantage of isolation of small neighborhood clusters.


Figure 8. Increasing migration range has significant negative impact on the fixation probability of cooperators when the mobility rate is equal to 10. Longer migration range has “flattening” effect on the fixation probability curves with higher mobility rate. The negative impact of longer migration range coupled with high mobility rate on evolution of cooperation is extremely strong for small neighborhoods and dense populations.

Also, the negative impact of longer migration range on the fixation probability of cooperators is extremely strong for dense populations, where cooperators dominate defectors if mobility rate and migration range are limited. The effect of “flattening” the fixation probability curves is most dramatic in the situations where they used to be fairly steep. Of course, when neighborhoods are sufficiently large (at least half the size of the lattice), then increasing migration range has no impact on the evolution of cooperation. In this case, the model is essentially equivalent to the well-mixed population case where mobility factors play no role.


A truly unexpected result of our simulations is that a combination of sparse population, small neighborhoods, low benefit to cost ratio, high mobility, and long migration range causes an almost complete breakdown of cooperation; see Figure 9.

Figure 9. A combination of sparse population and small neighborhood radius causes breakdown of cooperation for small benefit to cost ratios if both mobility rate and migration range are large. There is no such effect on denser populations or if the benefit to cost ratio is sufficiently high.

While we do not have an exhaustive explanation of this phenomenon, we will try to provide some insight. First of all, under the best of circumstances, sparse populations have lower fixation probabilities of cooperators than dense populations; this has been discussed earlier. Combining sparse population with small neighborhood radius results in clusters with relatively few individuals. With high mobility rate and long migration radius, free-


riders can easily locate and exploit such clusters. The fewer cooperators there are in a cluster the easier they are to exploit. This is because due to the smoothing function (3), free-riders have only marginally better payoffs than cooperators in a cluster with sufficiently many cooperators. Dense populations usually have larger clusters and these are more difficult to exploit even if free-riders can locate them.

If the neighborhood size is increased, then the average size of a cluster becomes larger even for sparse populations. Large clusters are harder to exploit, so certain stable levels of cooperation can be maintained.

The impact of low benefit to cost ratio on breakdown of cooperation is more difficult to explain. Recall that varying the benefit parameter affects viability of a cluster of two cooperators. In sparse populations, most cooperating clusters start off as clusters of two cooperators only. Therefore, it is extremely important for such clusters to have sufficiently high payoff compared to free-riders. Recall from Figure 3 that the effect of increasing benefits is stronger for sparse populations. Free-riders can easily locate and breakdown small cooperating clusters when mobility rate and migration range are sufficiently high. Cooperators need to be able to create and develop new clusters once old ones had been exploited by free-riders. Such newly formed clusters have a better chance of developing if the benefit parameter is as high as possible. By the time remaining, free-riders are able to locate them, they may grow in size and hence be able to resist invasion.

4. Discussion

We have created an exact stochastic simulation to investigate how spatial structure and mobility affect evolution of cooperation. In our model, individuals never change their strategy throughout their lifetime: individuals that are born as cooperators remain cooperators and individuals that are born as free-riders remain free-riders. Individuals reside on a 1-dimensional lattice with periodic boundaries and are allowed to migrate at a certain rate. Our model differs from standard lattice-structures models in that multiple


individuals may occupy the same location. This affects the outcome of the simulation, as we will explain below.

We have found that cooperators dominate free-riders when the population is sufficiently dense, neighborhoods are small, and mobility rate is set to zero. The effect of small neighborhoods is well understood and the outcome of the simulation illustrates that having small neighborhoods allows cooperative clusters to isolate themselves from free-riders; this is crucial for the maintenance of cooperation. On the other hand, it might not be intuitionally clear why increasing the density of the population helps cooperators. It seems that with higher population, density cooperative clusters will find it difficult to isolate themselves from free-riders. However, in our model, lattice points can be occupied by more than one individual. This allows formation of dense compact cooperative clusters which are hard to exploit. The effect of the density of the population on the evolution of cooperation in our model contrasts with previous standard lattice-structured models where lower density of the population was usually found to be beneficial for cooperators. However, even for standard lattice-structured models, higher density of the population may be beneficial for the evolution of cooperation under certain circumstances [30].

Cooperators thrive if they aggregate in clusters and resist exploitation by free-riders. In order to resist exploitation, the cooperative clusters should be (a) numerous, and (b) hard to breakdown if located by free-riders. Combination of small neighborhoods and dense population assures that both (a) and (b) hold. With small neighborhood size, the entire population gets split into many independent clusters; the larger the population, the more clusters there are. The more cooperative clusters there are the higher the chance that some of them will not be exploited and continue to thrive. Higher density of the population also results in larger average size of clusters, and due to the smoothing function (3), the difference in reproduction propensity between cooperators and free-riders in such clusters is negligible. Hence, dense cooperating clusters can resist invasion by free-riders.


Increasing mobility rate of the population in general decreases the fixation probability of cooperators. Mobility has no effect on the evolution of cooperation for large neighborhoods since in this case, we are essentially dealing with a well-mixed population case. However, high mobility rate visibly decreases cooperators’ chances for small neighborhoods. This is because allowing individuals to migrate somewhat negates the advantage of isolation of compact cooperative clusters. Cooperators are still big favorites over free-riders in dense populations when neighborhoods are small, but the fixation probability of cooperators is decreasing as the mobility rate is increasing.

The surprising result of our simulation is that increasing the migration range may result in breakdown of cooperation. Long migration range coupled with high mobility rate allows free-riders to dominate cooperators if the population is sparse, neighborhoods are minimal in size, and benefits of cooperation are low. This does not occur if the population is relatively dense (30 or 40 individuals), if the neighborhood radius is 5 or more, or if the benefits of cooperation are high. In cases when cooperation does not breakdown, longer migration range flattens the fixation probability curves since it negates the advantage of possible isolation of cooperative clusters when neighborhoods are small.

We believe that one of the main reasons that high mobility rate has such a negative effect on the fixation probability of cooperators is the probabilistic nature of movement in our model. An individual may improve its payoff drastically by joining a cluster of cooperators. So, if there is a chance that a given move results in joining a cooperating cluster, then such a move is very likely to occur since the difference in payoff is going to be fairly large. This makes it relatively easy for free-riders to locate and try to exploit cooperators. On the other hand, an individual improves its payoff only marginally by moving away from free-riders. So, if such a move is possible, then it has only slightly higher likelihood of being chosen than less optimal moves. This makes “running away” from free-riders much more difficult.


The success-driven migration mechanism, we employ in this paper somewhat differs from the model in [28]. In [28], individuals who can improve their payoff the most were more likely to move, while in our current model, a random individual is chosen to move. Our motivation for this adjustment is two-fold. First, joining a cluster of cooperators improves individual’s payoff drastically, while running away from a free-rider improves individual’s payoff only marginally. So, the model in [28] made it much easier for free-riders to exploit cooperators than for cooperators to resist such exploitation. As a result, introducing even the minimal amount of mobility into the model significantly reduced fixation probability of cooperators. Our current model represents a uniformly mobile population, where every individual is given equal opportunity to migrate. Second, this model significantly reduces computational time.

There are several directions in which we plan to extend this work. First, we would like to investigate the effect of other migration models on the fixation probability of cooperators. For example, we may employ a deterministic movement model where every individual moves deterministically to a location with maximum possible payoff. This makes both “chasing cooperators” and “running away from free-riders” equally likely to occur in the migration patterns. We hope that this would reduce the negative effect of high mobility rates, and, perhaps, even help cooperators in certain circumstances. Second, we would like to create a similar simulation for 2-dimensional lattices. It would be interesting to see if there are any significant quantitative and qualitative differences between 1-dimensional and 2-dimensional models.

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