# Interaction rates, vital rates, background fitness and replicator dynamics: how to embed evolutionary game structure into realistic population dynamics

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## Abstract

In this paper we are concerned with how aggregated outcomes of individual behaviours, during interactions with other individuals (games) or with environmental factors, determine the vital rates constituting the growth rate of the population. This approach needs additional elements, namely the rates of event occurrence (interaction rates). Interaction rates describe the distribution of the interaction events in time, which seriously affects the population dynamics, as is shown in this paper. This leads to the model of a population of individuals playing different games, where focal game affected by the considered trait can be extracted from the general model, and the impact on the dynamics of other events (which is not neutral) can be described by an average background fertility and mortality. This leads to a distinction between two types of background fitness, strategically neutral elements of the focal games (correlated with the focal game events) and the aggregated outcomes of other interactions (independent of the focal game). The new approach is useful for clarification of the biological meaning of concepts such as weak selection. Results are illustrated by a Hawk–Dove example.

## Keywords

Replicator dynamics Evolutionary game Density dependence Interaction rate Eco evolutionary feedback Background fitness## Mathematics Subject Classification

91A22 92D15 37C10## Introduction

The cornerstone of building scientific theories is the proper choice of underlying terminology describing the objects and processes of interest; the mathematical structures used in the formalization of the theory can influence the underlying language. A good example is the impact of game theory on evolutionary theory, which has meant that strategic reasoning is common in works related to evolution, even if they are not supported by mathematical notions (Dawkins 1976; Williams 1996). However, the basic evolutionary game theoretic framework is described by abstract mathematical terms whose relations with observable biological processes is often unclear. The most influential concept, which is foundational for game theoretic methods in biology, is the game as a metaphor for the individual interactions. In this paper we will investigate how this aspect should be expressed in the context of the ecological population dynamics.

The modern approaches to evolutionary game modelling can essentially be divided into two classes. The first contains static models (see e.g. Maynard Smith 1982; Broom and Rychtar 2013), based on potentially complicated payoff functions describing some abstract parameter called “fitness”, while the second contains dynamic models based on replicator dynamics and simplified (mostly matrix) payoff functions (Maynard Smith 1982; Cressman et al. 1986; Hofbauer and Sigmund 1988, 1998). The first type is focused on the details of the interaction, while population dynamics aspects are lacking. There are no evolutionary processes in time, only causal outcomes of the particular interaction. Thus, while payoffs quantified by obtained resources or energetic gain have a clear biological interpretation, the impact of the game outcomes on the population state is not fully explained. In the second case, the situation is the opposite: the interactions are not explicitly depicted in the model but their outcomes are phenomenologically described by the excess from the average growth rate, and the dynamics of the selection process is explicitly analysed. In addition, from game theoretic methods have grown the field of adaptive dynamics (Dieckmann and Law 1996; Metz et al. 1996; Geritz et al. 1998; Dercole and Rinaldi 2008), focused on the long-term evolution of continuous traits, driven by mutations. This approach emphasises the importance of ecological context.

To fill the gap between the two approaches to games and investigate the ecological meaning of individual interactions we should answer the question: How do the outcomes of particular interactions affect the growth rates of the respective strategies? The methods related to game theory are also used in life history theory (Caswell 2001) to describe the competition between different life history strategies, but this framework does not assume interactions between individuals. In this approach fitness components are described as the vital rates (birth and death rates of the respective age or stage classes). We can use this approach to solve the posed problem and establish the link between interaction rates, describing the occurrence of the interaction events in time, and resulting vital rates of respective types of interactions, describing the changes of the population state. This question is important not only for game theoretic models. It is related to the problem of the general mathematical representation of fitness (Metz 2008; Roff 2008; Orr 2009) and the methodological interpretation of this term, discussed by biologists and philosophers of science (Mills and Beatty 1979; Rosenberg and Williams 1986; Horan 1994; Matthen and Ariew 2002; Brandon and Ramsey 2007; Matthen and Ariew 2009; Walsh 2010; Ramsey 2013).

### State of the art. An event-based approach

This paper extends a novel approach to evolutionary games from Argasinski and Broom (2012). This approach is focused on ecological realism, falsifiability and a mechanistic interpretation of the results obtained. The main goal was to express individual fitness in terms of demographic parameters. This allows us to describe the terms, such as “costs” and “benefits”, by measurable parameters (mortality interpreted as the probability of death and fecundity interpreted as the number of newborns obtained in effect through an interaction) instead of an abstract, undefined “fitness” described by an infinitesimal rate of increase of the population (or single component of fitness such as fecundity as in Chakra et al. (2014), where the number of eggs laid constitutes fitness). This is realized by the explicit application of two payoff functions describing mortality and fecundity counted in the currencies of births and deaths, instead of one fitness function describing excess from the mean Malthusian growth rate. This new approach can be described as event-based because it describes cause and effect chains of underlying interaction events. For example mortality can act on adult individuals before or after reproduction, or the description of the structure of the interaction event can be more complex.

In addition, this approach emphasises the role of density dependence. The fertility payoff functions are not constant in time but can be affected by selectively neutral juvenile mortality leading to a more complex selection mechanism induced by eco-evolutionary feedback (Hauert et al. 2006, 2008; Argasinski and Kozłowski 2008; Zhang and Hui 2011; Argasinski and Broom 2012; Huang et al. 2015; Gokhale and Hauert 2016). Thus the fertility reward can decrease, due to the increase of the juvenile mortality, below the adult mortality costs. Population size does not converge to an arbitrary phenomenological carrying capacity (constant, as in for example Cressman and Křivan 2010; Křivan 2013, or affected by payoffs, as in Novak et al. 2013) as in many models, exploiting the classical logistic growth, but to a dynamic equilibrium between all mortality and fertility factors. A similar approach that can be found in epidemiological models is called the emergent carrying capacity (Bowers et al. 2003; Sieber et al. 2014). This is more realistic and provides a mechanistic interpretation in terms of demographic factors. The properties of the selection mechanism, induced by strategically neutral growth limitation, were analysed in Argasinski and Broom (2013). Here at the population size equilibrium, newborns form a pool of candidates from which survivors which will replace dead adults at their nest sites will be drawn; this was termed the nest site lottery.

### Two research goals of the paper

**(a) Role of event occurrence rates describing the distribution of interaction events in time:** We will analyse how the rates of event (or interaction) occurrence, associated with respective event-related mortality and fertility payoffs, constitute the vital rates (rates of change of the population state, Caswell 2001) driving the population dynamics. However, the main difference between ecological population dynamics and evolutionary game theory is that population dynamics is focused on how the population is shaped by different types of events (which can be described by different types of games), while evolutionary game theory generally analyses the selection of strategies in a single particular type of event. Thus we should be able to extract those focal interaction from our more complex general model, with the remaining events generating the corresponding background fitness. The application of event occurrence rates (or interaction rates) for evolutionary games was originated in Taylor and Nowak (2006). In their paper different strategy carriers can interact at different rates and thus can play different numbers of game rounds (the differences with our approach, related to the definition of fitness, are discussed in section “The event-based approach and rates of event occurrence” in the Discussion.

**(b) Classification of the types of background fitness:** Traditionally, background fitness has been modelled by a phenomenological additive element of the payoff function that vanishes under the replicator dynamics, with the associated dynamics being very simple. The growth rate is described by a single payoff function and it is not clear whether background fitness is a neutral element of the payoff or a separate factor acting at a different occurrence rate. Within classical evolutionary game theory, which is density independent, both approaches are equivalent and distinction between them is not necessary. In addition there is no clear biological interpretation of this factor and it has rather been interpreted as a technical element of mathematical notation. As was shown in Argasinski and Broom (2012), background fitness components can seriously affect the dynamics. However, the natural interpretation of those factors can be provided by the approach from point a). Thus, can we derive phenomenological neutral elements as the aggregated outcomes of background events?

## Results

A list of important symbols

Symbol | Description |
---|---|

| Population size |

\(W^{i}\) | Fertility payoff function of the |

\(d^{i}\) | Mortality payoff function of the |

| Carrying capacity (maximal environmental load) |

\(q_{i}\) | Frequency of the |

\(W_{i}(q)\) | Fertility payoff of the |

\(s_{i}(q)\) | Pre-reproductive survival payoff function of the |

\(V_{i}(q)\) | Mortality-fertility trade-off function for the |

\(\tau _{i}\) | Rate of occurrence (intensity) of the |

\(\tau _{F}\) | Rate of occurrence (intensity) of the focal game event |

\(\tau _{B}\) | Rate of occurrence of the background event |

\(\tau \) | Interaction rate from Argasinski and Broom (2012)—see “Appendix 1” |

\(\theta =\tau _{B}/\tau _{F}\) | Average number of background events between two focal events |

\(W_{b}\) | Focal game background fertility (payoff based approach) |

\(d_{b}=1-s_{b}\) | focal game background post-reproductive mortality (payoff-based approach) |

\(W_{B}\) | Average background event fertility (dynamics-based approach) |

\(d_{B}=1-s_{B}\) | Average background event mortality (dynamics-based approach) |

\(\Phi =\theta W_{B}\) | Rate of the average background fertility |

\(\Psi =\theta (1-s_{B})\) | Rate of the average background mortality |

| Hawk–Dove example survival payoff matrix |

\(F=WP\) | Hawk–Dove example fertility payoff matrix |

\(d=1-s\) | Probability of death during a Hawk–Dove contest |

\(\tilde{q}_{h}(n)\) | Frequency nullcline describing the Nash equilibria |

\(\tilde{n}(q_{h})\) | Density nullcline describing the ecological equilibria |

### The general model

#### Introduction of the rates of event occurrence and derivation of the vital rates

*i*-th type occur at rate \(\tau ^{i}\) (the superscript describes the event type since later the subscript will describe the strategy), where its outcomes are described by respective fertility and mortality payoff functions \(W^{i}\) and \(d^{i}\), where \(W^{i}\) is the average number of newborns produced and \(d^{i}\) is the probability of death during this type of interaction event. If the

*i*-th event type is a safe mating opportunity then the respective death probability \(d^{i}\) equals zero. On the other hand, if the event is not related to mating or reproduction but is dangerous, then \(W^{i}=0\). The general growth equation thus has the following form:

*i*-th type of event. The sum of the respective vital rates over all types of events will constitute the crude mortality and fertility rates (Caswell 2001).

We will next apply the approach presented in this section and summarized by Eq. (1) to obtain the evolutionary dynamics framework centred on a particular focal game. We will extract one particular type of event from our general model to analyse the selection of individual strategies related to that game. The aggregated impact of all other types of events will constitute the background fitness.

#### Background fitness as the aggregated outcomes of background events

Individuals enter an arbitrarily chosen focal game (with payoffs \(W_{F}\) and \(d_{F}\) where auxiliary lower index *F* means “focal event”) at rate \(\tau _{F}\) as in Eq. (1), and engage in other activities at rates described by \(\tau _{B}^{i}\); we can consider a single class of all such activities, as we show below.

Each of the background events can be characterised by outcomes which include a fertility \(W_{B}^{i}\) and mortality \(d_{B}^{i}\) component (lower index *B* means “background event”). We can calculate the outcomes of the average background event. \(W_{B}=\sum _{i}\tau _{B}^{i}W_{B}^{i}/\tau _{B}\) is the average fertility per event (where \(\tau _{B}=\sum _{i}\tau _{B}^{i}\)) and \( d_{B}=\sum _{i}\tau _{B}^{i}d_{B}^{i}/\tau _{B}\) is the average death probability per event.

*q*) will not be included in the numbered equations. Then the interaction rates multiplied by demographic payoffs will constitute the “vital rates” (per capita rates of change of the population state, see Caswell 2001). Now we can extend our model to the detailed description of the evolutionary game including the different strategies. Each strategy should be represented by its respective equation of type (2) and assigned demographic payoff functions \(W_{F}(q)\) and \(d_{F}(q)\). We can use the structure of the demographic payoff functions from Argasinski and Broom (2012) (see “Appendix 1” for the necessary details) allowing us to describe the mortality and fertility outcomes of the elements of the causal chain underlying the single interaction event. We limit ourselves to the simple trade-off between a single pre-reproductive mortality stage and a single fertility stage (this happens for example in mating conflicts when males fight and the surviving winners can mate; thus mortality acts before reproduction). For each strategy \(W_{F}(q)\mathbf {\ }\)will be the mortality–fertility trade-off function \(V_{i}(q)= \sum _{j}q_{j}s_{i}(e_{j})W_{i}(e_{j})\) describing the reproductive success of the survivors of the mortality stage described by survival payoff \( s_{i}(q)=1-d_{i}(q)\) (where index

*i*describes the strategy number) acting as \(1-d_{F}(q)\). In addition, we will include the density dependent juvenile survival function \((1-n/K)\) to introduce the nest site lottery mechanism (Argasinski and Broom 2013). Thus the general growth equation for the

*i*-th strategy will be as follows:

*dynamics based approach*since it is not related to the game theoretic structure. Note that this approach is related to the methodology used for the separation of ecological equations from selection dynamics (Cressman and Garay 2003a, b; Cressman et al. 2001). However, here we do not want to separate the ecological dynamics from the selection dynamics, since we believe that the relationship between ecology and selection is extremely important.

#### Two distinct approaches to background fitness

*m*in Argasinski and Broom 2012) will appear together in the multiplicative factor \(\left( W_{b}\left( 1-\frac{n}{K}\right) +s_{b}\right) \) of the survival payoffs \(s_{i}\) (this approach was used in Argasinski and Broom 2012). Note that here we use the lower case subscript

*b*for the neutral elements of the payoff functions (that can be termed the

*payoff-based approach*), to distinguish them from the payoffs from the alternative approach, where we use

*B*. The replicator dynamics will be

*n*-nullcline, which is the attractor in the

*n*-subspace) is

The difference between the two approaches relies on the different distributions of events in time. In the dynamics-based background fitness \( \Psi \) all background deaths gradually aggregate according to the intensities of all other games. In the payoff-based approach, all background deaths occur at the same time with the focal interaction as the last element of the causal chain (some survivors of the game are killed). Theorem 1 (see “Appendix 2”) shows that this mortality can be interpreted as the aggregated mortality between two focal game events.

If we limit analysis to the static case, then the interpretation of the background fitness as the mortality between two focal games is more natural and allows us to get rid of the instantaneous rates of occurrence from our reasoning. In addition this allows us to remove the abstract terminology of differential equations. In effect, the static reasoning can be expressed in clear, intuitive and empirically measurable terms, describing the respective causal stages of the interaction.

However, if we are interested in the dynamics, the differences related to the different distribution of deaths in time can seriously affect the predictions. This will be illustrated in the next section.

### Formulation of a Hawk–Dove game as an example

*S*(survival probability) and

*P*, where the fertility matrix is \(F=WP\), below: Open image in new window Here \(s=1-d<1\) is the survival probability of a fight between Hawks, and the fertility matrix contains the expected number of newborns

*W*produced from the interaction. This leads to the following set of replicator equations (see “Appendix 3” for a detailed derivation):

We see that when \(\Phi \) and \(\Psi \) are small as in Fig. 2, and to a lesser extent in Fig. 3, the trajectory is clearly distinct from the density nullcline \(\tilde{n}(q)\), whereas for large \(\Phi \) and \(\Psi \), as in Fig. 4, the trajectory converges quickly to this nullcline, and then follows it to the equilibrium point. Thus, only for large \(\Phi \) and \(\Psi \) , i.e. in the weak selection limit, can we obtain a separation of timescales for slow frequency and fast size dynamics, which can be described by its equilibrium value.

The shape of the density nullcline shows the strength of the impact of the focal game, via eco-evolutionary feedback, on the ecology of the population. In Figs. 2 and 3, where the focal game is quite a frequent event (since background fitness is relatively low), the density nullcline (as a function of \(q_{h}\)) depends strongly on the strategic composition. On the other hand, in the case when the focal game is rare (as in Fig. 4), and so its impact is weak, the density nullcline is nearly flat.

#### Impact of the distribution of interaction events in time on the dynamics

Numerical simulations show that for small background mortalities the two approaches produce similar trajectories of strategy frequencies, but ecological predictions differ significantly (see Figs. 5 and 6). The dynamics-based model can predict extinction in the case when the payoff-based model shows a positive stable population size (Fig. 6). These numerical results support the analytical results from Theorem 1 in “Appendix 2”, which shows that the stable sizes predicted by the model of Argasinski and Broom (2012) are biased, while the frequency levels of the intersections are the same for both approaches. With an increase of the background mortality, the differences between the models also increase and can affect frequency trajectories and phase portraits (Fig. 7). Thus the above example supports the claims that background fitness and the background payoff are distinct, although related, concepts.

## Discussion

### Two types of background fitness

Background fitness is traditionally interpreted as some phenomenological constant (or function) added to the payoffs of all strategies which vanishes from the continuous replicator equations. This concept can be found in many papers (for example see Cressman et al. 1986; Houston and McNamara 1991; Claussen and Traulsen 2005), but it is treated as a technical element of the mathematical notation and these works are not primarily focused on the biological meaning of it. Background fitness can be interpreted in two ways: First, as an element of the game theoretic structure (a generalization of the background payoff from classical game theory). Second, as an element of the dynamics occurring independently from the focal game at a separate rate of occurrence. In the basic approach to evolutionary game theory, the two approaches are indistinguishable. Some researchers, for instance those working explicitly using discrete dynamics or weak selection models, interpret the classical background fitness in rather a similar way to the first approach (Broom and Rychtar 2013; Taylor and Nowak 2006; Wu et al. 2010), while for others, for instance in optimal foraging/ diet choices models, the underlying logic involves the second approach (Křivan 1998, 2003; Cressman and Křivan 2010). However there was no rigorous formalization of this aspect and it is rather an example of “folk” knowledge.

Our work shows that there are essentially two types of background fitness (or more precisely, of background mortality and fertility), the payoff-based and the dynamics-based approaches. The dynamics-based approach that we focus on in this paper acts as the classical background fitness and is derived from the general ecological model, not phenomenologically postulated. While the payoff-based approach is a good tool to describe the selectively neutral factors related to the game interaction, which is clearly shown by the example of density dependent juvenile mortality (Argasinski and Kozłowski 2008; Zhang and Hui 2011; Argasinski and Broom 2012, 2013), we have shown here that such an application to the factors not related to the focal game can be problematic. The payoff-based approach does not take into account the distribution of the background events in time. The outcomes of all background events which occurred between two focal events affect the population state simultaneously when a single focal event occurs, since they are the final element of the focal game’s causal chain. The dynamics-based background fitness is free from this disadvantage. Note that both types of background fitness are not selectively neutral and affect the dynamics of the system via strategically neutral juvenile mortality (as is shown by a Hawk-Dove example). This impact is nontrivial and will probably affect the general stability conditions. This is a question which is analysed in a subsequent paper (Argasinski and Broom 2017).

### The event-based approach and rates of event occurrence

The most general and important claim resulting from our approach is that the payoff is not equivalent to the population growth rate. Game-theoretic notions describe the causal structure and the resulting outcomes of the specific single interaction. The interactions aggregate with some rate and the product of this rate with demographic outcomes constitute the vital rates. This is different to the approach from Taylor and Nowak (2006), where the fitness is expressed as the outcome of the average interaction (the sum of payoffs from interactions divided by the number of those interactions). This assumption does not take into account the impact of the different numbers of interactions on the fitness of the particular strategy. We can imagine an example when one strategy will obtain on average lower reproductive success per interaction than a competitor, but will participate in more interactions, so that its aggregated reproductive success is larger. The approach proposed here explicitly takes this into account. We thus believe that the number of games played should be explicitly considered. Then the demographic payoffs will describe the outcomes of the average interaction event (similarly to the stoichiometric coefficients in chemical kinetics, Upadhyay 2006) not the growth rates as in traditional evolutionary games.

In static models there is no time but causal consequences of the strategic “decisions” of individuals described by their reproductive success. Thus, the rigorous derivation of the population growth rate as an aggregated outcome of individual interactions needs rates of occurrence as the necessary element to make the framework consistent (the problem of the consistency and realism of modelling frameworks was discussed in Houston and McNamara 2005; McNamara 2013). Traditionally, interaction rates are not explicitly analysed in game-theoretic selection models (some exceptions will be discussed later). However, they can be a practical analytic tool. We can imagine a population of individuals where the type of games played depends upon their situation. Then the probabilities of finding particular situations associated with the respective type of game can be described by different rates of occurrence. The new approach corrects intuitions inspired by classical birth and death processes (see e.g. Haigh 2002) where birth and death events are described by different intensities, which implies statistical independence of births and deaths. In this case trade-offs between mortality and fertility are impossible. In evolutionary theory benefit is linked to reproductive success while expected cost is related to the associated mortality risk. The approach from this paper can be used to describe the correlations between mortality and fertility factors associated with particular activities.

In addition, rates of occurrence are not necessarily constants. For example they can be functions of the population size (more individuals implies potentially more interactions per unit time) or population state. A good example of this is the battle of the sexes with the problem of pair formation (Mylius 1999) or dynamic model of sex ratio evolution (see Argasinski 2012, 2013 and 2017). There an elementary event (a Bernoulli trial) is the production of a single offspring with a randomly chosen partner. Then females interact at a constant rate, while males interact proportionally to the number of available females described by the actual sex ratio (female to male). Thus interaction rates constitute a crucial element of the strategy selection mechanism. We can also imagine the situation when interaction rates can depend on an individual’s strategy. This can be illustrated by the example of non-uniform interaction rates in the models of social dilemmas, such as the models of upstream reciprocity (Nowak and Roch 2007; Pena et al. 2011). In this case there are different interaction rates for different individual strategies within the single game.

There is also an interesting relationship between the so-called weak selection concept and our eco-evolutionary feedback. Traditionally, in population genetics and models based on continuous traits, the weak selection limit assumes very small differences between strategic agents or alleles resulting in small selective advantage (Kimura 1968; Ohta 2002). This assumption was also used in matrix game models, where assumption of small differences is not necessarily applicable (for example in models with contrasting strategies such as Hawks and Doves which by definition will obtain different demographic payoffs). Then, the weak selection limit is introduced via a selection constant (see Wild and Traulsen 2007 for the comparison of both approaches). Note that the selection constant (Nowak et al. 2004; Antal et al. 2009; Taylor et al. 2004; Ohtsuki et al. 2006; Taylor et al. 2007; Fu et al. 2009; Tarnita et al. 2009a; Wild and Traulsen 2007) can be interpreted as the rate of the focal game’s occurrence. Under weak selection, a focal game described by relatively high demographic parameters will be a rare event, and its impact on the population dynamics will be small. Our interpretation embeds this concept in a clear biological context. The weak selection limit can be applied only in the case that the focal events are really rare. Thus it cannot be applied in common types of interactions such as mating conflicts or resource conflicts during foraging. Note that, in the weak selection limit, where our parameter \(\theta \) tends to infinity, the impact of the frequency dynamics on the population size vanishes, but the second element of the eco-evolutionary feedback is still present. The frequency dynamics is affected by population size via juvenile survival inducing the nest site lottery mechanism on the density nullcline (Argasinski and Broom 2013).

### General discussion

The event-based approach constitutes a clearly defined area of application of game-theoretic notions within the evolutionary dynamics framework. The mathematical structure describing the focal interaction can be very complex (e.g. see Broom and Ruxton 1998; Gokhale and Traulsen 2010; Broom 2002; Broom and Cannings 2002, and in general the book Broom and Rychtar 2013) and a clear methodology how to incorporate the game into a population dynamic model can be important. The approach proposed in this paper shows that evolutionary dynamics under growth, limited by nest site availability, is a synergistic product of different games played by individuals, not only a simple aggregated sum of the outcomes of those games. Thus, the dynamical approach is more than an extension of the static game structure, as in classical theory. An important aspect of this approach is that it can be easily interpreted, which is a significant advantage over abstract simplified models , as argued Geritz and Kisdi (2012). Further, the proposed approach allows for more precise modelling of the outcomes of selection dynamics on ecological parameters such as population size. This is very important, because the relationship between ecological mechanisms (regulation of the population size) and the process of natural selection is one of the major problems of modern evolutionary biology (Birch 1960; Hutchinson 1965; Ginzburg 1983) and is still at the centre of debate (Pelletier et al. 2009; Morris 2011; Post and Palkovacs 2009; Schoener 2011). We note that aspects discussed above are important from the point of view of the general definition of fitness (see Metz 2008; Roff 2008; Orr 2009) and its interpretation within evolutionary theory (Mills and Beaty 1979; Rosenberg and Williams 1986; Horan 1994; Matthen and Ariew 2002; Brandon and Ramsey 2007; Matthen and Ariew 2009; Walsh 2010; Ramsey 2013).

The proposed approach shows how the game theoretic notions, causal structure underlying the interactions that shape the population dynamics, can be used within many theoretical frameworks. It can be easily extended to Adaptive Dynamics (Dercole and Rinaldi 2008) due to its clear description of the underlying ecology. On the other hand, decomposition of fitness into separate demographic payoffs creates the possibility of incorporating more detailed population genetic mechanisms (Crow and Kimura 1970; Hartl and Clark 1997; Bürger 2000), which will affect the fertility payoffs. Then the fertility payoff will describe the number of mating attempts which can be weighted by the probability of gene transfer determined by the underlying genetic system.

In addition the impact of the proposed methodology can be broader and more general. Note that in the case of abstract model parameters, that are “fitted” to data, it can be hard to falsify the obtained outcomes, if the model is “flexible” enough with respect to the “fitted” parameters, to cover different types of datasets. The clear demographic meaning of our model parameters can allow for easy falsification according to empirical data or the outcomes of individual-based simulations (which will have the status of in-silico experiments, see Uchmański and Grimm 1996; Grimm and Railsback 2005) parameterized by the same values. The predictions of the analytical model can be helpful in a mechanistic explanation of the patterns produced by the simulation (see, for example, Gerlee and Lundh 2010) or observed empirical data. This will constitute important progression in the direction of the development of theoretical notions related to the individual level, originated by Łomnicki (1988), and in particular related to research on animal personalities (Dall et al. 2004; Wolf et al. 2007; Wolf and Weissing 2010, 2012; Wolf and McNamara 2012). Thus, the event-based terminology not only extends the mathematical notions, but also influences the general theory and contributes to the understanding of the causal structure of the evolutionary process.

## Notes

### Acknowledgements

In memory of Krzysztof Chodasewicz. We want to thank Jan Kozłowski, John McNamara and Ryszard Rudnicki for their support of the project and helpful suggestions. K. Argasinski was supported by Grant 2013/08/S/NZ8/00821 FUGA 2 by the Polish National Science Centre.

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