Theoretical Population Biology 73 (2008) 250–256 www.elsevier.com/locate/tpb How can we model selectively neutral density dependence in evolutionary games Krzysztof Argasinski ∗ , Jan Kozlowski Jagiellonian University, Institute of Environmental Sciences, Gronostajowa 7, 30-387 Krak´ ow, Poland Received 1 July 2007 Available 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 density dependence in the classic formulation of replicator dynamics, which means that population size may grow to infinity. Therefore the question arises: 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 is used, as in a logistic equation. Both approaches have disadvantages. The first one is incompatible with the methods of life history theory and basic probabilistic intuitions. The logistic type of suppression of per capita growth rate stops trajectories of selection when population size reaches the maximal value (carrying capacity); hence this method does not satisfy selective neutrality. To overcome these difficulties, we must explicitly consider turn-over of individuals dependent on mortality rate. This new approach leads to two interesting predictions. First, the equilibrium value of population size is lower than carrying capacity and depends on the mortality rate. Second, although the phase portrait of selection trajectories is the same as in density-independent replicator dynamics, pace of selection slows down when population size approaches equilibrium, and then remains 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 of avoiding explosion of the population size to infinity. This is called the neutral density dependence effect. In most of the current papers, solutions of this problem rely on making replicator dynamics independent of background fitness. This paper shows that this approach is at variance with elementary probabilistic intuition and the life history approach. A better way is to use the logistic suppression coefficient, which is multiplicative, and to include the additive mortality rate of the parent individual. 1. Basic definitions of an evolutionary game In terms of evolutionary game theory, the players are species (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. Kozlowski). 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 possible types of behavior. We interpret them as pure strategies of a game. For simplicity assume that each individual can use only a single pure strategy. The state of the whole population or a subpopulation can be described by a mixed strategy, whose coefficients are frequencies of pure strategies. The set of mixed strategies Δ h is an (h − 1)-dimensional standard simplex. The payoff (fitness) function is the function W : Δ h × Δ h → R. Formal definitions of this structure are presented below. Definition 1.1. A mixed strategy described by the vector σ = [σ 1 ,...,σ h ] ∈ Δ h is called the mean population strategy (population state). Symbol σ k denotes the frequency of the k th pure strategy in the population. The simplex Δ h is called the set of population states. The set {i : σ i > 0} is called the support of state σ and denoted supp σ . In general, for any vector 0040-5809/$ - see front matter c  2007 Elsevier Inc. All rights reserved. doi:10.1016/j.tpb.2007.11.006