Theoretical Population Biology 73 (2008) 250–256www.elsevier.com/locate/tpb

How can we model selectively neutral density dependence inevolutionary games

Krzysztof Argasinski∗, Jan Kozłowski

Jagiellonian University, Institute of Environmental Sciences, Gronostajowa 7, 30-387 Krakow, Poland

Received 1 July 2007Available online 7 January 2008

Abstract

The problem of density dependence appears in all approaches to the modelling of population dynamics. It is pertinent to classic models(i.e., Lotka–Volterra’s), and also population genetics and game theoretical models related to the replicator dynamics. There is no densitydependence in the classic formulation of replicator dynamics, which means that population size may grow to infinity. Therefore the questionarises: How is unlimited population growth suppressed in frequency-dependent models? Two categories of solutions can be found in the literature.In the first, replicator dynamics is independent of background fitness. In the second type of solution, a multiplicative suppression coefficient isused, as in a logistic equation. Both approaches have disadvantages. The first one is incompatible with the methods of life history theory andbasic probabilistic intuitions. The logistic type of suppression of per capita growth rate stops trajectories of selection when population size reachesthe maximal value (carrying capacity); hence this method does not satisfy selective neutrality. To overcome these difficulties, we must explicitlyconsider turn-over of individuals dependent on mortality rate. This new approach leads to two interesting predictions. First, the equilibrium valueof population size is lower than carrying capacity and depends on the mortality rate. Second, although the phase portrait of selection trajectoriesis the same as in density-independent replicator dynamics, pace of selection slows down when population size approaches equilibrium, and thenremains constant and dependent on the rate of turn-over of individuals.c© 2007 Elsevier Inc. All rights reserved.

Keywords: Evolutionary game theory; Density dependence; Replicator dynamics; Mortality rate; Life history theory

In frequency-dependent models there is the problem ofavoiding explosion of the population size to infinity. Thisis called the neutral density dependence effect. In most ofthe current papers, solutions of this problem rely on makingreplicator dynamics independent of background fitness. Thispaper shows that this approach is at variance with elementaryprobabilistic intuition and the life history approach. A betterway is to use the logistic suppression coefficient, which ismultiplicative, and to include the additive mortality rate of theparent individual.

1. Basic definitions of an evolutionary game

In terms of evolutionary game theory, the players arespecies (populations) under natural selection, and strategies are

∗ Corresponding author.E-mail addresses: argas1@wp.pl, argass@poczta.onet.pl (K. Argasinski),

kozlo@eko.uj.edu.pl (J. Kozłowski).

0040-5809/$ - see front matter c© 2007 Elsevier Inc. All rights reserved.doi:10.1016/j.tpb.2007.11.006

interpreted as the behaviors of individuals during their lifetime(i.e., phenotype). The payoff is interpreted as Darwinian fitness(i.e., expected lifetime number of offspring).

Let us assume that all individuals can exhibit h possibletypes of behavior. We interpret them as pure strategies of agame. For simplicity assume that each individual can use onlya single pure strategy. The state of the whole population ora subpopulation can be described by a mixed strategy, whosecoefficients are frequencies of pure strategies. The set of mixedstrategies ∆h is an (h − 1)-dimensional standard simplex. Thepayoff (fitness) function is the function W : ∆h

× ∆h→ R.

Formal definitions of this structure are presented below.

Definition 1.1. A mixed strategy described by the vector σ =

[σ1, . . . , σh] ∈ ∆h is called the mean population strategy(population state). Symbol σk denotes the frequency of thekth pure strategy in the population. The simplex ∆h is calledthe set of population states. The set {i : σi > 0} is called thesupport of state σ and denoted supp σ . In general, for any vector

K. Argasinski, J. Kozłowski / Theoretical Population Biology 73 (2008) 250–256 251

v = (v1, . . . , vn) the set {i : vi 6= 0} is called the support of v

and denoted supp v.

Definition 1.2. The function W (P, σ ), which is linear in thefirst variable, is called the payoff (fitness) function of anevolutionary game. The value of this function describes thefitness of an individual with strategy P (where P is the kthpure strategy described by unit versor ek = [0, . . . , 1, . . . , 0])or the mean fitness of a subpopulation with mean strategy P inthe population in state σ .

In the static game structure, a system of differentialequations called replicator dynamics can be defined, whichdescribes changes of the population state in time. It can bederived in the following way. Note that

σk(t) =Nk(t)

N (t), (1)

where Nk(t) is the number of individuals with the kth strategy,and

N = N (t) =

h∑k=1

Nk(t). (2)

Assume that the population growth is given by the equation

Nk(t) = Nk(t)W (ek, σ (t)) . (3)

From this assumption we can derive a system of ordinarydifferential equations (see, e.g. Cressman (1992)) defined onstrategy simplex ∆h :

σk(t) = σk(t) [W (ek, σ (t)) − W (σ (t), σ (t))]

for k = 1, . . . , h − 1. (4)

More details can be found in Maynard Smith (1982),Cressman (1992), Hofbauer and Sigmund (1988, 1998),Weibull (1995), and Taylor and Jonker (1978). A more generalmultipopulation and density-dependent approach is discussedby Argasinski (2006).

2. Interpretation of fitness function values

The value of fitness function W can be interpreted as the percapita growth rate, but then what is the per capita growth rate?We can write the Taylor expansion of the solution of Eq. (3):

Nk(t + ∆t) = Nk(t) + Nk(t)∆t + o(∆t)

= Nk(t) + Nk(t)W (ek, σ (t))∆t + o(∆t). (5)

Then:

W (ek, σ (t)) =

(Nk(t + ∆t)

Nk(t)− 1 −

o(∆t)

Nk(t)

)/∆t. (6)

For a suitably small ∆t = l factor o(∆t) is negligible. Whenwe change the time variable t =

tl , then for ∆t = 1 we obtain

W (ek, σ (t)) ≈Nk(t + 1)

Nk(t)− 1 (7)

or

Nk(t + 1) ≈ Nk(t)(1 + W (ek, σ (t))

)= Nk(t) + Nk(t)W (ek, σ (t)), (8)

which means that the payoff describes the per capita normalizedvalue, measuring how many individuals were added or removedfrom the population during one time unit. Note that in this casethe turnover of individuals is not considered explicitly, so we donot know how many births and deaths there really were in thispopulation. Because condition Nk (t+∆t)

Nk (t)≥ 0 must be satisfied,

W (ek, σ ) ∈ 〈−1, ∞) for any strategy. When W (ek, σ ) =

0, then population size is stable and every individual eithersurvives without reproduction or is replaced by a newbornindividual. When W (ek, σ ) = a > 0 then this value can beinterpreted as the number of additional individuals producedby an individual representing ek . The parental individual candie and be replaced by some newborn individual ek . WhenW (ek, σ ) = −a, then a can be interpreted as the fraction ofindividuals disappearing from the population (according to (7)and (8) we have Nk(t + 1) ≈ Nk(t)(1 − a)).

3. Alternative approaches to modelling of density depen-dence

Note that in density-independent models the overallpopulation size would grow to infinity, and therefore a selectiveneutral density effect must be considered. For simplicityassume that all models presented in this paper are forpopulations large enough that we can ignore random effectsof small or medium population size. There are two mutuallyincompatible approaches to the neutral density dependenceproblem: a multiplicative (logistic suppression coefficient) andan additive one. The multiplicative approach is expressed by

W (P, σ ) = W (P, σ )

(1 −

N

K

), (9)

where K is the carrying capacity of the environment. Sinceneutral density dependence is the subject of this paper, it isnatural for us to assume the same K for all strategies. Insome population genetics solutions however, different carryingcapacities for different strategies can be considered (seeHofbauer and Sigmund (1998), Section 19.4). The additiveapproach is related to background fitness insensitivity ofreplicator dynamics. In this case the fitness function has theform:

W (P, σ, N ) = Wb(σ, N ) + Wg (P, σ ) , (10)

where Wb(σ, N ) is the background fitness and Wg(P, σ ) isthe game-theoretical part of the fitness function. ComponentWb(σ, N ), which describes the density effect, vanishes underreplicator dynamics. This means that replicator dynamics isindependent of background fitness. A special case of thisapproach can be found, for example, in Cressman (1992),where:

W (P, σ ) = Wb(N ) + Wg (P, σ ) (11)

252 K. Argasinski, J. Kozłowski / Theoretical Population Biology 73 (2008) 250–256

Fig. 1. Intuitive illustration of the additive and multiplicative approach for density-dependence modeling. The figure shows that under the additive model the samenumber of offspring die and the death rate varies for different strategies. Under the multiplicative approach, the same fraction dies, so the death rates of the differentstrategies are equal.

and Wb(N ) is some negative-valued and decreasing function.An example in which it is phenomenologically assumed thatpopulation is currently at equilibrium density can be found inEshel and Sansone (2003):

“Assume now that all individuals in the population are purestrategists, each reproducing offspring equal to itself with a rateequal to its current payoff, and that a per capita death rate,average to per capita birth rate, keeps the total population sizenormalized.”

When we formalize this assumption, then Wb depends on thepopulation state (more exactly, Wb(σ ) = −Wg(σ, σ )). Hence,we have a fitness function of the form

W (P, σ ) = Wg (P, σ ) − Wg(σ, σ ). (12)

Note that in this case, when the average birth rate changesthen the average death rate must also change accordingly. Itis problematic how natural selection may put this mechanisminto effect. The advantage of the additive approach is that thecomponent Wb vanishes under replicator dynamics as a resultof background fitness insensitivity, and it is compatible withthe Nash axioms. Thus, the question is, which method is morerealistic for modelling density dependence?

4. Comparison with the life history approach

Let us take a look from the point of view of lifehistory theory (see Stearns (1992) and Roff (1992)). Wecan construct life tables for cohorts consisting of individuals

representing each strategy. Then we can count death rate d andsurvival rate s:

d =Nd

N, s =

N − Nd

N= 1 − d, (13)

where Nd is the number of individuals dying in the currenttime period and N is the number of all living individuals at thebeginning of that period. Therefore the death rate represents theproportion of dead individuals during the current time interval,and can be interpreted as a component of the selection force.The selective neutral density effect must be independent of theindividual’s strategy. This means that the probability of deathestimated as the death rate in the life table for all strategiesmust be the same. Now consider a population consisting ofindividuals with strategies e1 and e2 in population state σ .Assume that Wg (e1, σ ) = 4 and Wg (e2, σ ) = 8, and for theadditive method Wb = −2. Then the death rate of an offspringof an individual with strategy e1 is 2/4, and for e2 it is 2/8.Hence it depends on its strategy. Therefore the additive methoddoes not affect the game theoretical structure, but it is notselectively neutral in the sense of life history theory (differentdeath rates for different strategies). In the multiplicative case,death rates are the same for all strategies (for an intuitivepresentation, please see Fig. 1), so this approach is compatiblewith life history theory.

There are problems with the biological interpretation of theadditive method. In this approach, for any parent individualwith any strategy (which means with different numbers ofoffspring), the same number of its offspring individuals must

K. Argasinski, J. Kozłowski / Theoretical Population Biology 73 (2008) 250–256 253

Fig. 2. Numerical solution of Example 1 from Section 5; (a) and (b) represent the additive model of neutral density dependence, and (c) and (d) the multiplicativeapproach. Under the additive approach the rate of evolutionary process is constant. Under the multiplicative approach, evolution stops when the population numberreaches equilibrium, well before attaining an evolutionarily stable state.

be killed. This can be realized by some intentional factor,but not by random forces of nature. On the other hand,the multiplicative form can be interpreted as the mean value of abinomial distribution, where

(1 −

NK

)describes the probability

of survival to maturity of a single individual, and the number ofoffsprings produced (value of game theoretical fitness functionW (P, σ )) represents the number of trials. In this case, survival(or death) of any newly produced individual is a statisticallyindependent event. As we can see, only the multiplicativemethod is compatible with the life history approach andprobabilistic intuition, but it poses another problem, describedin the next section.

5. Is the multiplicative method really selectively neutral?

The multiplicative approach seems more realistic, but it alsocauses some problems, because we obtain the following formof replicator dynamics:

σk(t) = σk(t)

(1 −

N (t)

K

)[W (ek, σ (t)) − W (σ (t), σ (t))] .

(14)

Clearly, this system is defined not only on the strategysimplex but also on variable N . According to Argasinski (2006)N is a scaling parameter. Hence, an additional equation for theparameter N must be included:

N = N W (σ (t), σ (t)) , (15)

to obtain a properly defined system of differential equations,which depends not only on σk but also on N . Note that forN = K we have a stationary point of the dynamics whichmay be evolutionarily unstable from the static game point ofview. When population size (described by N ) increases, densitydependence slows down the speed of convergence, and maystop it before the trajectory of the system reaches an ESS restpoint. Let us consider two examples.

Example 1. We have a payoff matrix3 1 02 2 10 3 3

(16)

with initial conditions of σ1 = σ2 = 0.4, σ3 = 0.2, initialpopulation size N (t0) = 2, and the carrying capacity ofthe environment K = 150. When density dependence is notdirectly considered (or for the additive approach) we obtain thetrajectories from Fig. 2(a) (time trajectories) and 2(b) (phaseportrait). When the logistic suppression coefficient is used, wehave the trajectories represented in Fig. 2(c) (time trajectories)and 2(d) (phase portrait).

Example 2. For the rock–scissors–paper game we have apayoff matrix1 2 0

0 1 22 0 1

(17)

and initial conditions σ1 = 0.1, σ2 = 0.7, σ3 = 0.2, N (t0) = 2,and K = 10 000. For the density-independent dynamics or theadditive method, we obtain the trajectories shown in Fig. 3(a)(time trajectories) and 3(b) (phase portrait). In the case of thelogistic suppression coefficient, the trajectories are as presentedin Fig. 3(c) (time trajectories) and 3(d) (phase portrait).

To conclude, the additive method is incompatible withthe life history approach. The multiplicative method is morerealistic in this respect, but population growth toward thecarrying capacity suppresses the differences in fitness resultingfrom the game. In other words, density dependence mayslow down or stop the trajectories of evolution before theyreach an ESS (or may stop limit cycle). Does this mean thatovercrowding suppresses evolution and selection works only inthe growth phase? By no means.

254 K. Argasinski, J. Kozłowski / Theoretical Population Biology 73 (2008) 250–256

Fig. 3. Numerical solution of Example 2 from Section 5. Meaning of plots (a)–(d) as in Fig. 2.

6. Turn-over of individuals

In the classic approach to evolutionary game theory, payoff(fitness) means per capita growth rate. This is an effect of thedistinction between newborn and eliminated individuals. How-ever, in the classic approach there is no information about thethe exact number of newborns and exact number of dead indi-viduals. So, from the point of view of turnover of individuals,the classic model can be interpreted in such a way that the adultindividuals are immortal, and that there are different numbersof newborns produced by individuals with different strategies.Then the multiplicative logistic suppression coefficient resetsproduction of newborns for all individuals, but all adult individ-uals are still alive. That is why the trajectories of evolution stopin a state which is suboptimal from the game-theoretical pointof view, when the population is at its carrying capacity. So letus try to explicitly incorporate the turn-over of individuals intodensity-dependent replicator dynamics based on the multiplica-tive logistic suppression coefficient. To do this, we can applynotions from life history theory and demography (life tables).Assume that in time interval ∆t an individual with strategy eiproduces W (ei , σ ) offspring, which inherit his strategy. Everynewborn individual must find an empty spot, or dies homeless.Every adult individual may die after reproduction with proba-bility m. Therefore, the survival rate (Stearns, 1992) for new-borns is proportional to the fraction of free habitats. When pop-ulation size is N , then there are K − N empty spots. After divi-sion by K we obtain a fraction of free habitats which is equal tothe logistic coefficient, so

(1 −

NK

)describes the survival rate

of newborns, and for adult individuals the death rate is equal tom. Then the changes of the number of individuals with strategyei can be described by the equation

Ni (t) = Ni (t)W (ei , σ (t))

(1 −

N (t)

K

)− Ni (t)m (18)

or

Ni (t) = Ni (t)

(W (ei , σ (t))

(1 −

N (t)

K

)− m

). (19)

Thus we have a population growth equation in whichthe per capita instantaneous growth rate is described byW (ei , σ )

(1 −

NK

)− m. A similar method can be found in

Kozlowski (1980). Condition (8) in this case has the form

Nk(t + 1)

Nk(t)≈ W (ek, σ (t))

(1 −

N (t)

K

)+ (1 − m). (20)

In this case, W (ek, σ (t)) can be clearly interpreted as themean per capita number of offspring of individual ek producedduring a single time unit. Also, m can be interpreted as theprobability that this individual will not survive this time period,so according to (13) formula (1 − m) can be interpreted assurvival rate of an adult individual. Note that in principle both(1 −

NK

)and m are multiplicative coefficients. The difference

between them is that(1 −

NK

)works on the newborn offspring

of individual ei , and m works on the parent individual ei . Thatis why in a formula that describes the total per capita growthrate m is in additive form. When we derive replicator dynamics,m vanishes (because it is additive) and

(1 −

NK

)appears as a

multiplicative coefficient at the right side of all equations. As aresult we obtain a system of Eq. (14). The difference betweenthe new and the previous approach appears only in an equationon the total number of individuals N :

N (t) =

h∑n=1

Nn(t) =

h∑n=1

Nn(t)

(W (en, σ )

(1−

N (t)

K

)− m

)

= N (t)h∑

n=1

σn

(W (en, σ )

(1 −

N (t)

K

)− m

). (21)

By the linearity of function W on the first variable and thefact that

∑hn=1 σn = 1 we have

N (t) = N (t)

(W (σ (t), σ (t))

(1 −

N (t)

K

)− m

). (22)

The stationary points of this equation (described by N )satisfy the following conditions:

K. Argasinski, J. Kozłowski / Theoretical Population Biology 73 (2008) 250–256 255

Fig. 4. Numerical solution of Example 1 from Section 5 when the new model of density dependence is applied. Evolution is very rapid before the populationdensity reaches equilibrium, then slows down to a level dependent on the turn-over of adult individuals (a). Because the population attains an evolutionarily stablestate, the phase portrait is identical to that under the additive approach (b). At equilibrium population number, described by N on figure (c), is lower than carryingcapacity, and depends on both adult turn-over and the average payoff in the population at evolutionary equilibrium.

Fig. 5. Numerical solution of Example 2 in Section 5 when the new model for density dependence is applied. Meaning of (a)–(c) as in Fig. 4.

N (t) = 0 or W (σ (t), σ (t))

(1 −

N (t)

K

)− m = 0, (23)

so

N (t) = K

(1 −

m

W (σ (t), σ (t))

). (24)

For m = W (σ (t), σ (t)) we have N (t) = 0, and form = 0 we have N (t) = K , and for any other morerealistic case the stationary point N is lower than carryingcapacity K . The value of N depends on the mean productionof individuals W (σ (t), σ (t)) and mortality m. In this case theevolutionary trajectories will not stop, but only slow down whenthe population size reaches equilibrium. This effect can be seenin the examples from Section 5. The mortality rate of adultindividuals is set to 0.2. When we apply the new method to

Example 1, we obtain the trajectories illustrated in Fig. 4(a) and(b). Figure Fig. 4(c) shows the dynamics of the total numberof individuals. For Example 2 (rock–scissors–paper game) thetrajectories are as represented in Fig. 5(a) and (b), and thedynamics of population size as in Fig. 5(c).

As we can see in the figures, in this case evolution is notfully suppressed, as it was in the classic logistic approach (seeSection 5), but only slows down rapidly when the numberof individuals reaches equilibrium. It is also important thatthe stable size of the population depends on the valuesof mean fitness and the death rate of adult individuals m;

this equilibrium is lower than K , unless adult mortality isnegligible. Note that the phase portraits are the same asthat for the density-independent case or additive method(see Section 5). However, the dynamics of the processesare completely different. As we can see, the new approach

256 K. Argasinski, J. Kozłowski / Theoretical Population Biology 73 (2008) 250–256

brings game-theoretical models closer to the classic Malthusianapproach to population dynamics.

Acknowledgments

The authors were supported by the Polish ScienceFoundation. K. Argasinski was a participant in the YoungScientist Programme organized by the British Council and thePolish Ministry of Science and Higher Education.

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