Univ. Grenoble 1/CNRS, LIPhy UMR 5588, Grenoble, F-38401, France

Abstract

Background

Altruistic behavior is defined as helping others at a cost to oneself and a lowered fitness. The lower fitness implies that altruists should be selected against, which is in contradiction with their widespread presence is nature. Present models of selection for altruism (kin or multilevel) show that altruistic behaviors can have ‘hidden’ advantages if the ‘common good’ produced by altruists is restricted to some related or unrelated groups. These models are mostly deterministic, or assume a frequency dependent fitness.

Results

Evolutionary dynamics is a competition between deterministic selection pressure and stochastic events due to random sampling from one generation to the next. We show here that an altruistic allele extending the carrying capacity of the habitat can win by increasing the random drift of “selfish” alleles. In other terms, the

Conclusions

The theory we present does not involve kin or multilevel selection, but is based on the existence of random drift in variable size populations. The model is a generalization of the original Fisher-Wright and Moran models where the carrying capacity depends on the number of altruists.

Background

Light production in

From the inception of evolution theory, the problem of the existence of altruists has been puzzling: how can a mutant with lower fitness prevail? And how does a community of altruists resist the spread of selfish allele (see

The above models are either deterministic,

It was established by the founding fathers of Population Genetics that a mutation that confers a relative fitness 1 +

The fixation probability is composed of two terms: even in the absence of selection, the population will become homogenic; in this neutral case, all individuals at generation zero have an equal probability 1/

For populations of fixed size, as can be seen from expression (1) or the more precise expression (10) obtained by Kimura

Consider an altruistic gene that by some means (production of a ‘common good’, limited grazing of natural resources, …) allows the carrying capacity to increase: if the community were composed only of altruists its population size would be _{
f
}; if it were composed only of selfish individuals the population size would be _{
i
} (_{
i
} < _{
f
}) (Figure

Variable carrying capacity

**Variable carrying capacity.**

(**a**): A community where the carrying capacity is an increasing function of altruist number, varying from _{i} when the population is composed only of selfish individuals to _{f} (_{f} > _{i}) when only altruists are present. (**b**) Two examples of random walks describing the stochastic behavior a such a system (transition probabilities 4–7), where

Consider now the fixation probability _{
A
} of one altruist mutant appearing in a community of _{
i
} selfish individuals. A crude use of expression (1) shows that _{
s
} of one selfish individual appearing in a community of _{
f
} altruists is

The above argument will be refined in the following. In the next section, we formulate precisely the stochastic process of altruism outlined above by generalizing the Moran model for non-structured, well mixed populations and we show that altruists can indeed be favored in their competition with selfish individuals. We outline the amplification of this advantage in geographically structured,

Results and discussion

Stochastic model for altruism

The fundamental aspects of population genetics were clarified in the framework of the classical Fisher-Wright (FW) stochastic model of non-overlapping generations or its continuous time alternative introduced by Moran

In the Moran model, a population of size

where ^{+} stands for the probability density that the new spot is colonized by an

We generalize this model by including two ingredients. First, the fixed size constraint can be relaxed and we let _{
i
} and _{
f
}: empty spots are created-colonized and individuals die, without these two events necessarily succeeding each other. More importantly, in order to include the effects of altruists, we suppose that the rate of creation of empty spots is proportional to the number of altruists and is equal to α

The stochastic model that captures all these features is a two dimensional random walk with the following transition probability densities (Figure

Consider for example the first two lines of the above equations, which are about birth events: the factor _{
f
}; the factor α_{
i
}; the factor α

if

The above rates ensure that if _{
i
} and if _{
f
}. Note that in the mean field approximation of the above process where fluctuations are neglected and the deterministic limit is taken, the

In finite size populations however, fluctuations play an important role. The focus of this paper is the computation of the fixation probability of the above process and the **k**) of a general stochastic process beginning with the initial state **k** and fixing either to **k**
_{
0
} or

where the sum is over all the states **q** attainable from the state **k** with transition probabilities **k =**

where

For the two dimensional process (4–7) where

For large populations, we use the usual diffusion equation approximation of eq.(8)

where

and

where

and

Fixation probabilities

**Fixation probabilities.**

Comparison of analytical solution (12) (solid lines) to numerical solution of eq.(8) for increasing selection pressure indicated by the arrows: _{f} = 100, _{i} = 90.

The general solution (12) allows for the computation of the fixation probability of one individual introduced into a community of the other type. To the first order of perturbation in _{
A
} of one

and the fixation probabilities _{
s
} of one

Figure (3a) shows the evolution of these probabilities as a function of selection pressure for various _{
i
} and _{
f
}. Equations (1314) show that the condition for the altruist to be favored,

where _{f}s < < 1), which is considered by most, but not all, scientists, to be the relevant limit of evolutionary dynamics

Criterion for Altruists selection

**Criterion for Altruists selection.** ( **a**) fixation probabilities _{s} (red squares) and _{A} (blue circles) as a function of selection pressure _{f} = 100 and _{i} are indicated by the arrow. (**b**) equilibrium selection pressure _{A} = _{s} for multiple combinations of

Geographically structured populations

The altruists’ advantage can be enhanced for large structured populations _{
SA
} for an

Geographically structured populations

**Geographically structured populations.**

Geographically structured population where patches can exchange migrants. For low migration rates, the border between

Similarly the probability density for an

Therefore, the movement of the border itself can be considered a biased random walk. The probability Π_{
A
} for an altruist mutant to take over the whole community is thus the probability for a mutant to take over one colony and then for this colony to take over the whole community:

where **1**,

On the other hand, the probability Π_{
s
} for a selfish mutant to be fixed is

and **1**: once altruists dominate, the chances for a selfish mutant to invade the community is close to zero! Increased random noise due to production of common good and a small migration rate are an efficient way of keeping selfishness in check.

The above computation concerns the low migration limit. In the high migration limit, the community is non-structured and its effective size is

Conclusions

The main concepts of Population Genetics were clarified in the framework of the original model of Fisher-Wright and Moran (FWM). These models introduced the key ingredient of population size and its role in the randomness of selection. It became clear in the 1920-30’s that a beneficial mutation does not spread automatically to the whole population, but has to overcome the “random noise” of population sampling over generations. The idea that random noise plays also a role for the selection of altruism has been introduced in two kind of models, which have a marked difference with the model we present here. The first class of models, formulated mostly through evolutionary game theory formalism, concerns fixed size populations, where the transition rates are frequency dependent

The model we present here is not frequency dependent: an

In summary, we have shown, by a slight generalization of the Moran model, that in finite size populations, the fixation probability of altruists can be higher than that of selfish ones, even though their

The aim of the present article is not to contest the merits of kin/group selection models which have been investigated during the last forty years with a large number of case studies. We believe we are providing an alternative way of thinking about altruism which is complementary to the above models and which restores the key ingredients of population genetic to this topic.

Methods

Diffusion equation derivation

In the discrete backward Kolmogorov eq. (8) set **q** all the states reachable from **k**, **1**. The equations read

For large populations

where

Mean field approximation

In the deterministic approximation, fluctuations are neglected. Denoting by

It is more fruitful to write directly the evolution of the proportion of

where

The equation for total population reads

for

which shows that the increase in carrying capacity of the habitat **1**.

Relation to Moran model

In a simple model where population size is variable, but birth and death rates are independent of the number of altruists and selfish individuals, a constant α will replace (α_{
i
} and _{
f
}. The analog of the Moran process is obtained by computing the two steps transition probabilities

The same expression is obtained if

Numerical resolution of fixation probabilities

Two different kinds of numerical resolution were used to check the validity of our analytical results on the fixation probabilities: A Gillespie stochastic algorithm and direct resolution of eq. (8).

Gillespie algorithm

The stochastic equations given by the rates (4–7) can be seen as 2 chemical reactions for the species

which we solve by the classical Gillespie algorithm ^{6} stochastic trajectories are generated.

Direct resolution

Equation (8) constitutes a linear system and can therefore be solved by standard numerical packages. For the present case however, the unknowns,

where **1**, **1**.

Tensor reindexation

**Tensor reindexation.**

(**a**) To each 2 d index ( **b**) The re-indexation transforms the tensorial equation (8) into a normal linear system ^{k} are the unknowns.

The re-indexation transforms the equation (8) into a normal linear system

where

where ^{
k
} is a sparse vector provided by the limit conditions ^{
k
}. Note that because we index the interior of the trapezoid, the index

Once the linear system (20) has been constituted, it can be solved by any linear solver. We have used the commercial package matlab for these manipulations.

Authors’ contributions

BH designed the reasearch, performed the numerical work and wrote the article. BH and MV performed the analytical computations. Both authors read and approved the final manuscript.

Acknowledgements

We are grateful to O. Rivoire and E. Geissler for fruitful discussions.