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 20 . Moran and FW are equivalent in the limit of large populations, where both are well approximated by the same diffusion equation 21 . These are the simplest models that capture the key elements of population genetics (genetic drift, fixation probability, fixation time,…) with the fewest possible ingredients.

In the Moran model, a population of size N is composed of two types of individual, say A and S. Empty spots are created randomly with fixed rate α, increasing the carrying capacity by unity. Once an empty spot has been created, it will be colonized by the progeny of either an A or an S individual according to their proportion in the community. In order to keep the population constant, Moran added the constraint that the colonization of a new spot be followed immediately by the death of an individual in the community, restoring the population size to N. Moran is therefore a simultaneous model of duplication and annihilation; the transition probability densities for the A to increase or decrease their number n by one individual are

W + n → n + 1 = α n m ; W − n → n − 1 = c α n m

where m is the number of S individuals and c is the ‘cost’: 1/ c is the relative fitness of the A and c > 1 indicates a selective disadvantage. W ⁺ stands for the probability density that the new spot is colonized by an A and death occurs among the S. In principle, a similar set of equations must be written for the S individuals; however, as the population size is fixed, n + m = N , the quantity m in eq.(3) can be replaced by N-n and the whole stochastic process treated as a one dimensional random walk for the A.

We generalize this model by including two ingredients. First, the fixed size constraint can be relaxed and we let N vary between two bounds N _i and N _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 αn; in contrast, the death rate is proportional to the number of S individuals and is equal to α m. This is the simplest hypothesis that implies that the increase in the carrying capacity of the habitat is proportional to the number of altruists (see also Methods, mean field approximation).

The stochastic model that captures all these features is a two dimensional random walk with the following transition probability densities (Figure 1b):

W n , m → n , m + 1 = N f − n + m α n m

W n , m → n + 1 , m = N f − n + m α n n

W n , m → n , m − 1 = m + n − N i α m m

W n , m → n − 1 , m = c m + n − N i α m n

Consider for example the first two lines of the above equations, which are about birth events: the factor N f − n − m is the relaxation of Moran constraints and insures that population size remains below N _f ; the factor αn accounts for the fact that empty spot creations are proportional to the number of A; finally, once a birth event has occurred, the probability for it to be an A or an S is proportional to the number of the corresponding sub-populations present at this time. The last two lines, which govern population decrease, are similar: the factor m + n − N i ensures that population size remains above N _i ; the factor αm is the death rate (population decrease) for everybody due to the presence of selfish individuals. The cost of altruism is included in these equations: the proportion of A is n / m + n , but once a death event has occurred, the probability for it to be an A is:

c n m + c n > n m + n

if c > 1. The results below don’t change significantly if the cost of altruism is included in other rates. For example, a higher probability for an S to reproduce, or any combination that favors S over A. Note that if the increase/decrease rates were independent of m and n, we recover the Moran model by setting N f = N i + 1 , in which case each birth/death is succeeded by a death/birth event (see Methods, relation to Moran model).

The above rates ensure that if A are lost ( n = 0), the population size tends toward N _i and if S are lost ( m = 0), it tends toward N _f . Note that in the mean field approximation of the above process where fluctuations are neglected and the deterministic limit is taken, the A are always eliminated if c > 1 (see Methods, mean field approximation).

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 probability that altruists or selfish mutants take over the community. The fixation probability π( k) of a general stochastic process beginning with the initial state k and fixing either to k ₀ or k 1 is the solution of Kolmogorov backward equation which is a linear set of equations 22

∑ q π k − π q W k → q = 0

π k 0 = 0 ; π k 1 = 1

where the sum is over all the states q attainable from the state k with transition probabilities W k → q . For one dimensional, one step processes such as Moran, k =  n and the solution of the linear system is easily obtained 22 :

π n = 1 − c n 1 − c N ≈ e − N μ s − 1 e − N s − 1

where μ = n / N is the proportion of the A. The approximation corresponds to the Kimura solution obtained through a backward diffusion equation 19 and c = 1 / 1 + s . Expression (1) is the first order expansion of the above expression in s.

For the two dimensional process (4–7) where k = m , n is the initial number of the S and A, no closed form solution can be obtained. We can however solve equation (8) numerically by standard linear solvers or else resort to a Gillespie algorithm 23 to solve the stochastic equations (4–7) directly. Both these methods are used in this paper and the analytical approximations obtained below are compared to them.

For large populations, we use the usual diffusion equation approximation of eq.(8) 19 22 . For weak selection pressure, the diffusion approximation error for the simple Moran process is O ( 1 / N ) 24 ; for more general cases, the validity of the approximation has been discussed by Zhou and Qian 25 . Setting x = m / N f , y = n / N f , k = N i / N f , and denoting π( x, y) the fixation probability for the initial composition ( x, y), the diffusion equation reads:

F x ∂ x π + y ∂ y π + 1 / 2 N f G x ∂ xx 2 π + y ∂ yy 2 π + c − 1 H − ∂ y π + 1 / 2 N f ∂ yy 2 π = 0

where

F = y + k x − x + y 2 G = y − k x + x 2 − y 2 H = x y x + y − k

and π x , 0 = 0 π 0 , y = 1 . This is a complicated elliptic partial differential equation. In the absence of selection (c = 1) however, the trivial neutral solution is π x , y = y / x + y which as expected, is just the proportion of altruists. Building upon this solution, and denoting μ = y / x + y for the proportion of altruists and η = x + y , we can check that to the first order of perturbation s ¯ = c − 1 , the solution reads

π μ , η = e N f s ¯ μ g η − 1 e N f s ¯ g η − 1

where

g η = γ 1 − 1 / η N f

and γ is a numerical coefficient: γ = 1 / N f + 1 + k / 2 . The first order perturbation solution (12), which was derived for small selection pressures N f s ¯ ≪ 1 , proves in fact to be an excellent approximation for selection pressure as high as N f s ¯ = 2 , (Figure 2).

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 s ¯ , the fixation probabilities π _A of one A introduced in a community of S reads:

π A = π m = N i − 1 , n = 1 = 1 N i − γ 2 N i − 1 2 N f N i 3 s ¯

and the fixation probabilities π _s of one S introduced in a community of A is

π S = 1 − π m = 1 , n = N f − 1 = 1 N f + γ 2 N f − 1 2 N f 2 s ¯

Figure (3a) shows the evolution of these probabilities as a function of selection pressure for various N _i and N _f . Equations (1314) show that the condition for the altruist to be favored, π A > π S , is simply

N ¯ s ¯ < N ¯ s ¯ ∗ = Δ N N ¯

where Δ N = N f − N i and N ¯ = N f + N i / 2 and we have kept only the leading terms. s ¯ ∗ is the equilibrium relative excess cost of altruism at which A and S individuals become equivalent. Figure 3b shows the excellent agreement between the above results and exact numerical results. Altruists have a selective advantage if the selection pressure against them, i.e. the combined effect of fitness and population size, is smaller than the relative increase in population size. Unlike a Hamilton rule, criterion (15) is a finite size effect and is of purely stochastic nature : because of the demographic effect, selfish mutants are submitted to a higher stochastic noise than altruist; this can be sufficient to prevent them from prevailing. Note that the above computations were performed for the limiting case of weak selection (N_fs < < 1), which is considered by most, but not all, scientists, to be the relevant limit of evolutionary dynamics 26 27 . Direct numerical resolution of eq. (8) shows however that an equilibrium excess fitness exists even at high selection pressure, given a high enough relative increase in population size.

The altruists’ advantage can be enhanced for large structured populations 28 29 30 31 . Geographically structured populations can be modeled as divided into colonies that exchange migrants 32 . The Moran model on graph is a non trivial problem 33 ; we restrict our treatment here to the simplest case where the migration time scale is small compared to fixation time of one mutant (viscous populations): a migrant is either lost or fixed before a new migration event happens. The argument we develop below is similar to the two level model of Traulsen and Nowak 34 . Consider a one dimensional community subdivided into M colonies (Figure 4), exchanging migrants with neighboring patches at rate m. As the migration event is rare, these colonies are fixed either into an A or S state. The probability density per unit time p _SA for an S colony on the border to become an A colony is to receive one migrant from the neighboring A colony multiplied by the probability that this mutant gets fixed:

p SA = m N f × π A

Similarly the probability density for an A colony on the border to become S is

p AS = m N i × π S

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:

Π A = π A 1 − r 1 − r M − 1

where r = p AS / p SA . If the criterion (15) is satisfied, then obviously r < 1 and for large number of communities M > >  1,

Π A ≈ π A − N i N f π S > 0

On the other hand, the probability Π _s for a selfish mutant to be fixed is

Π S = π S 1 − 1 / r 1 − 1 / r M − 1

and Π S → 0 for M > >  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 ≈ M × N f . Criterion (15) shows that in this regime, altruists cannot emerge; this is indeed equivalent to the deterministic case where emergence of altruists calls for other mechanisms. Between these two regimes of high and low migration rate, there is a rich interval where migration rate is a key ingredient in the competition between altruists and cheaters.

In the discrete backward Kolmogorov eq. (8) set k = m , n and q all the states reachable from k, i.e. all states of the form ( m ± 1, n) and m, n ±  1. The equations read

W m , n → m + 1 , n π m , n − π m + 1 , n + W m , n → m − 1 , n π m , n − π m − 1 , n + … = 0

For large populations N f ≫ 1 , we set x = m / N f , y = n / N f and develop the above expression to the second order in d x = d y = 1 / N f (Kramers-Moyal expansion). Combining all the resulting terms leads to the partial differential equation (11). It is fruitful to express this equation in terms of total relative population η = x + y and proportion of altruists μ = y / x + y ; the inside domain shown in Figure 1 then maps into the k , 1 × 0 , 1 rectangle, where k = N i / N f . In these coordinates, the diffusion equation reads:

F η ∂ η π + 1 2 N f G η 2 ∂ ηη π + μ 1 − μ ∂ μμ π − ( c − 1 ) H η ∂ η π + 1 − μ ∂ μ π + O c − 1 N f = 0

where

F = η k − η + 1 − k μ G = − k + η 1 − 2 μ + 1 + k μ H = η η − k μ 1 − μ

In the deterministic approximation, fluctuations are neglected. Denoting by m and n the ensemble average of the number of S and A individuals, their deterministic evolution equation reads:

d m d t = W m , n → m + 1 , n − W m , n → m − 1 , n d n d t = W m , n → m , n + 1 − W m , n → m , n − 1

It is more fruitful to write directly the evolution of the proportion of A-individuals μ = n / m + n . Using the expression for transition probabilities (4–7), we have

1 N f 2 α d μ d t = − ( c − 1 ) η ( η − k ) μ ( 1 − μ ) 2

where η = ( m + n ) / N f and k = N i / N f . It is then obvious that for c > 1 , d μ / d t < 0 . In the deterministic model, A-individuals always disappear.

The equation for total population reads

1 N f 2 α d η d t = η 2 1 − k μ + k − η − c − 1 η − k μ 1 − μ

for c − 1 ≪ 1 , the stationary solution of this equation, assuming that μ is held constant is

η = k + 1 − k μ − c − 1 1 − k μ 2 1 − μ k + 1 − k μ 2

which shows that the increase in carrying capacity of the habitat η - k, at small selection pressure, is mostly proportional to the number of A-individuals. A closer look at the above equations (16,18) shows that η = k , μ = 0 is the only stable fixed point when c >  1.

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 (αn) and (α m) in equations (4–7). In the case where N f = N i + 1 , the stochastic movement pictured in Figure 1b reduces to a movement on the anti-diagonal staircase: births and deaths occur only when the total population N = m + n is respectively equal to N _i and N _f . The analog of the Moran process is obtained by computing the two steps transition probabilities W M n , m → n ± 1 , m ∓ 1 . If m + n = N i , this implies first the birth of one individual of one type and then the death of an individual of the other type. Combining the rates given by eqs.(4–7) where birth and death rates are constant, we obtain

W M n , m → n + 1 , m − 1 = α 2 m n W M n , m → n − 1 , m + 1 = c α 2 m n

The same expression is obtained if m + n = N f .

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).

The stochastic equations given by the rates (4–7) can be seen as 2 chemical reactions for the species A i = A , S :

A i → k i + 2 A i ; A i → k i − Ø

which we solve by the classical Gillespie algorithm 23 written in C++. We are interested here only in the fixation probability and not in the fixation time; the program can therefore be accelerated by computing only the nature of the event that occurs at each turn (and not its time of occurrence). In general, to solve for the fixation probability, R = 10⁶ stochastic trajectories are generated.

Equation (8) constitutes a linear system and can therefore be solved by standard numerical packages. For the present case however, the unknowns, i.e. the fixation probabilities π( m,n) don’t constitute a vector but a second rank tensor; the tensor formed from the rates W is of rank 4. To adapt our linear system to standard linear solvers, we have to re-index the unknowns and decrease their rank by one: m , n ↦ k . We have chosen the following scheme, which corresponds to a sequential scanning of the anti-diagonal lines (Figure 5a):

k m , n = δ N i − 1 + δ − 1 / 2 + n

where δ = m + n − N i . The m , n points belong to the interior of the trapezoid N i ≤ m + n ≤ N f , n ≥  1, m ≥  1.

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

∑ k ′ ∈ I ( k ) π k ′ − π k W k → k ′ = 0

where I(k) designates the 1 d indexes of the four nearest neighbors of the point ( m,n), where k = k m , n . The above equations can be written in standard matrix notation

W k ′ k π k ′ = B k

where π k ′ are the unknowns. W k ′ k is a sparse matrix, which apart from the diagonal elements, has at most four non-zero elements per line: if k is the image of element ( m,n), then W k ′ k ≠ 0 only if k′ is the image of one of the four nearest neighbors of ( m,n), in which case its value is given according to rates (4–7). The right hand side vector B ^k is a sparse vector provided by the limit conditions π ( m = 0 , n ) = 1 : if k′, one of the 4 nearest neighbors of the element k belongs to the border m = 0, then π k ′ = 1 and the corresponding W k ′ k is transferred to the right hand side to constitute the vector B ^k . Note that because we index the interior of the trapezoid, the index k itself can never belong to the border.

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.