# Ecological theatre and the evolutionary game: how environmental and demographic factors determine payoffs in evolutionary games

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DOI: 10.1007/s00285-012-0573-2

## Abstract

In the standard approach to evolutionary games and replicator dynamics, differences in fitness can be interpreted as an excess from the mean Malthusian growth rate in the population. In the underlying reasoning, related to an analysis of “costs” and “benefits”, there is a silent assumption that fitness can be described in some type of units. However, in most cases these units of measure are not explicitly specified. Then the question arises: are these theories testable? How can we measure “benefit” or “cost”? A natural language, useful for describing and justifying comparisons of strategic “cost” versus “benefits”, is the terminology of demography, because the basic events that shape the outcome of natural selection are births and deaths. In this paper, we present the consequences of an explicit analysis of births and deaths in an evolutionary game theoretic framework. We will investigate different types of mortality pressures, their combinations and the possibility of trade-offs between mortality and fertility. We will show that within this new approach it is possible to model how strictly ecological factors such as density dependence and additive background fitness, which seem neutral in classical theory, can affect the outcomes of the game. We consider the example of the Hawk–Dove game, and show that when reformulated in terms of our new approach new details and new biological predictions are produced.

### Keywords

Replicator dynamics Mortality Fertility Eco-evolutionary feedback Trade-off Density dependence### Mathematics Subject Classification

92D40## 1 Introduction

## 2 The basic assumptions of classical theory

*replicator dynamics*can be defined. It describes changes of population state in time and can be derived in the following way. Assume that we have a finite number \(I\) arbitrary chosen strategies. For each strategy some payoff function \(r_{i}\) is assigned (for example matrix form, as in the classical Hawk–Dove game, however the form of payoff function depends on the modelled problem and can be more complicated). Then the growth of the population of \(i\)-strategists can be described by the Malthusian equation

## 3 Models of fertility and mortality

### 3.1 Framework I: fertility and post-reproduction mortality

Here we will extend the replicator dynamics to the case where mortality and fertility can be explicitly considered, not only the Malthusian parameter.

*sex and violence equations*,

### 3.2 Framework II: fertility and pre-reproduction mortality with mortality-fertility frequency dependent trade-offs

It is clear that in continuous models the probability of two independent events occurring within a given time interval tends to zero as the length of the time interval tends to zero. However, we should remember that biological reality can be very complicated and the outcomes of a single event can be very complex. For example, consider a male involved in a mating conflict where victory will lead to an immediate mating opportunity, i.e. there is a chain of conditional stages caused by a single interaction event. He may be killed instantly during a fight, or survive then mate successfully and die afterwards due to infection of his wounds. Death has occurred in both cases, but in the second case mating has also occurred, and it is important to distinguish the two different types of mortality. A similar idea has been known in population genetics for a long time (Prout 1965).

*sex or violence equations*, and here we approach unknown ground. Let us start from equations on the sizes of the subpopulations of the different types, where (5) is replaced by

### 3.3 Framework III: combining different mortality pressures

### 3.4 Framework IV: adding neutral density dependence

Populations cannot grow to infinity, therefore there should always be some density dependence. We can extend the model to include density dependence by adding offspring mortality described by the logistic suppression coefficient \((1-n/K)\), where we multiply all fertility parameters \(W_{i}\) by this coefficient. Such logistic suppression is used in population genetics (Hofbauer and Sigmund 1988, 1998), and in game theoretic models of the ideal free distribution (Cressman et al. 2004; Cressman and Krivan 2006, 2010). However, in the classical phenomenological form it may produce paradoxical and unrealistic predictions (Geritz and Kisdi 2011). Our approach to logistic growth was originated in Kozłowski (1982). Logistic suppression can be interpreted as the per capita mortality of juvenile individuals, which is selectively neutral, i.e. the same for each strategy. It was shown that without an additional mortality pressure on adult individuals that induces turnover of individuals, selection stops when the population reaches carrying capacity. However by the addition of some background neutral mortality of adult individuals this suppression of selection is avoided (Argasinski and Kozłowski 2008). We note that carrying capacity can be interpreted as a number of habitats rather that a population equilibrium. Therefore, under the assumption that individuals cannot live without a habitat, it makes no sense to consider an initial population size greater than the carrying capacity, and we shall assume that population size cannot exceed the carrying capacity.

Important symbols for our modelling framework

Symbols |
Meaning |
---|---|

\(B\) |
The benefit in the classical Hawk–Dove game |

\(C\) |
The cost in the classical Hawk–Dove game |

\(T\) |
The classical Hawk–Dove game payoff matrix |

\(S\) |
The survival (mortality) matrix |

\(F\) |
The fertility matrix |

\(n_{i}\) |
The number of individuals of the \(i\)-th type |

\(r_{i}\) |
Function describing the Malthusian parameter of the \(i\)-th type |

\(n\) |
The total population size |

\(K\) |
The population carrying capacity |

\(q_{i}=n_{i}/n\) |
The relative frequency of the \(i\)-th type |

\(W_{i}\) |
Reproductive success function of the \(i\)-th type |

\(s_{i}=1-d_{i}\) |
The pre-reproduction survival probability function of the \(i\)-th type |

\(b_{i}=1-m_{i}\) |
The post-reproduction survival probability function of the \(i\)-th type |

\(V_{i}(q,s_{i},W_{i})\) |
The frequency dependent mortality-fertility tradeoff of the \(i\)-th type |

\(W_{b}\) |
The background fertility |

### 3.5 The system in ecological equilibrium

Then from the mathematical point of view the system is simplified because, instead of a system of two differential equations, we obtain a single differential equation (24) and the function of the population state \( \check{n}\) (22), describing the population size. On the other hand the equation on frequency dynamics is more complicated and behaviour may be more complex. There is a unique restpoint for the density independent case which now become a function of \(n\), because then the fertility bracket is multiplied by formula (23). Thus, stationary states should be described by stable frequencies and densities.

### 3.6 Is background fitness really background?

## 4 An application: a Hawk–Dove example game

Now we will analyze the relationship between the classical approach and our new approach.

###
**Theorem 1**

The classical Hawk–Dove game is equivalent to the new model without density dependence if \(W_{b}=0\) and \(b=1.\) Then the benefit \(G=W\) and cost \(C=d(W+2)\).

(for a proof see Appendix 4).

Thus the benefit is the expected fertility \(W\). It is interesting that cost is a function of benefit, which is an effect of the application of pre reproduction mortality. Now we can express abstract costs and benefits from the classical Hawk–Dove game in empirically measurable demographic parameters. For example consider the simplified case without density dependence and with \(W_{b}=0, W=2\) and \(b=1\). Then inequality (35) collapses to \(d>0.5\), which means that a Hawk’s probability of death from a Hawk versus Hawk contest should be \(>\)0.5. However a lack of details such as density dependence or background mortality supported by our new model suggests that this classical approach cannot be realistic. From the point of view of the new approach, in classical model Doves are immortal, Hawks can die during fight only and population growth is unlimited. The absence of density dependence is especially problematic. In the next section the model in ecological equilibrium will be analyzed and differences between unlimited exponential growth and the more realistic new approach will be shown.

### 4.1 The system in ecological equilibrium

###
**Theorem 2**

### 4.2 Numerical examples

The model can be extended to the case when individuals are foraging for some resource and when two individuals find a resource then conflict begins. Then the probability of conflict will depend on the density and the availability of the resource. In this case the number of free resources can be described by the background fertility \(W_{b}\) and should decrease as the population grows, while the number of resources that can be obtained by conflict \((W)\) should increase. In this case Hawks will decrease in low densities and increase in high ones. This case is worthy of rigorous analysis in subsequent research.

In the preceding paragraph only the frequency dynamics was analyzed in detail. However, the model produces another interesting prediction on density dependence. At ecological equilibrium the frequency dynamics can affect the population size, for example the spread of Hawks can reduce the population size (even to extinction); this is shown in Fig. 2b.

The relationship described by (34) is plotted in Fig. 2c. It shows why the Hawk population becomes fixed in ecological equilibrium. Simply, with the increase of the Hawk frequency, the population size decreases which causes an increase in newborn survivability. Therefore a reduction of the population size makes the Hawk strategy profitable again. This mechanism is shown in Fig. 3. For the same frequency and initial conditions, at low densities Hawks outcompete Doves, whilst at ecological equilibrium the system converges to a mixed equilibrium. This example shows how sensitive the game theoretic structure is to ecological mechanisms such as density dependence.

## 5 Summary of results

### 5.1 Game theoretic results

In our new approach the game theoretic structure splits into two stages: the mortality and fertility stages. The equilibrium is an effect of the interplay between both stages.

Background fitness can be expressed as background mortality and fertility and it does not vanish from the replicator equations but appears as a multiplicative factor of the mortality stage which has the form \((1-\frac{n}{K})W_{b}+b\).

When the population reaches a stable size manifold consisting of equilibria of the size equation (a type of stationary density surface; Cressman and Garay 2003), conditional on actual strategy frequencies, then the game theoretic structure may change. “Benefit” is affected by density dependent juvenile mortality described by the modifier (23), and in effect the respoint becomes an attractor manifold. Thus, global stationary points are intersections of size and frequency manifolds, and this allows a population of Hawks to become stable. This mechanism is induced by the decrease of population size caused by increased average mortality, which causes a decrease of newborn mortality. The invasion barrier is the Hawk frequency that when surpassed makes the number of offspring surviving to maturity high enough to make the Hawk strategy profitable. Therefore the value of the “benefit” can be modified by neutral density dependence which can in effect seriously change the rules of the game.

This new approach allows us to explicitly describe the cause and effect structure of the modelled phenomenon. The classical theory is phenomenological at the level of determining payoff values expressed in terms of excess over the average growth rate. There are no rules describing how the value of deviation from the average growth rate is determined by strategy. We can always pose the question: why is it not greater (or lower)? In our new approach it can be determined by mortality risk (the probability of death during an interaction with an opponent) and the number of successful reproduction events (for example successful matings). Therefore our new approach can be called “event based modelling”.

### 5.2 Ecological results

Modern ecological modelling uses very sophisticated mathematical tools such as Lyapunov exponents (Metz et al. 1992). This may cause problems with the understanding of the results obtained by readers with lesser mathematical skills. The framework presented in this paper is an attempt at a solution of this problem, by providing a simple interpretation of parameters. Empirical testing of models formulated under the new framework will be much easier because mortalities (fractions of dead individuals) and fertilities (per capita number of newborns) can be easily measured unlike abstract “costs” and “benefits”. Also costs and benefits are not separated quantities. The Hawk–Dove example shows that “cost” is a function of expected “benefit” defined as a number of offspring. In effect, the general but abstract condition that “cost” should exceed “benefit” collapses to specific but clear and empirically testable statements such as: “if the expected number of offspring is 2 than over half of Hawks should die during a fight for there to be a mixed stable equilibrium”.

Density dependence may affect the structure of the game by increasing newborn mortality; in effect some strategies can be profitable in low densities but not profitable in high densities. The opposite situation can be obtained by density dependent background fertility. Therefore, costs and benefits are not constants. The assumption of constant payoff coefficients was criticized by Dieckmann and Metz (2006) as not realistic. Our model supports their arguments. It shows that even in the application of classical methods, such as logistic suppression, we obtain a varying value of benefit.

The system in ecological equilibrium is not equivalent to the purely frequency dependent case. The equation on frequencies is more complicated, additional rest points may exist (for example in the Hawk–Dove game, a pure Hawk population may become evolutionarily stable). Trajectories of selection determine trajectories of stable population size.

Our new approach shows that the rules of the evolutionary game are not constant but written by ecological conditions and are very sensitive to changes of those conditions. We believe that abstract reasonings related to unspecified “costs” and “benefits” are insufficient. A strong bond with reality via empirically measurable parameters with a clear mechanistic meaning is necessary to produce realistic models. It was clearly shown using the example of the Battle of the Sexes model (Mylius 1999) that more realistic details included in the model (such as realistic pair-formation and the mechanistic interpretation of “costs” as time delays during courtship) can seriously affect predictions, which thus can differ from the “abstract” game model. Therefore, when developing an abstract game theoretic model such a mechanistic analysis should be considered. Our work supports those conclusions.

## Acknowledgments

The project is realized under grant Marie Curie Grant PIEF-GA-2009-253845. We want to thank Jan Kozłowski, John McNamara, Jan Rychtar and Franjo Weissing.

### Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

## Appendix 1: Discrete and continuous models

For a population changing under continuous growth, how do we estimate the parameters from real data? Here we show that we can estimate the real parameters of the model if we record the number of births and deaths within an interval, provided that this interval is sufficiently short, effectively using a straight line approximation to a curve. Thus while we really have a continuously interacting population, we can still get a handle on reality using the method of counting births and deaths as in classical discrete models by comparing our continuous population to a discrete model with a small number of interactions in each time step.

For the continuous system to be well-approximated by the first order Taylor expansion, we require the remainder term \(o(\Delta t)\) to be small, which occurs for sufficiently small \(\Delta t\), when the number of births (respectively deaths) in a time period of length \(\Delta t\) is approximated by \(\tau n_{i}(t)W_{i}\Delta t\) (respectively \(\tau n_{i}(t)d_{i}\Delta t\)).

Thus whilst \(W_{i}\) and \(d_{i}\) are not small in our model, the number of births and deaths within a sufficiently small interval of length \(\Delta t\) will be small. By a change of timescale \(\tau \Delta t\) can be set to 1 and removed. In effect we obtain equation (5). There is thus a natural link between the number of births (deaths) observed in any time interval and the birth (death) rate from a continuous model and we can use real data on births and deaths to inform the model parameters of our continuous model.

## Appendix 2: Derivation of equations for the Hawk–Dove example game

## Appendix 3: Calculation and stability of the rest points in the Hawk–Dove example game

## Appendix 4: Proof of Theorem 1

To compare both approaches we should remove from the new model factors not considered in the classical approach. In the Maynard Smith model there is no density dependence and no explicit analysis of background fitness. Therefore we should assume that background survivability \(b\) is equal to 1, background fertility \(W_{b}\) is 0 and there is no suppression coefficient.

## Appendix 5: Proof of Theorem 2

###
*Case 1*

###
*Case 2*

###
*Case 3*

###
*Case 4*

In the special case where \(W_{b}=0,L=1<b+(1-b)/d\) so there cannot be a unique mixed stable equilibrium, and so that there is always a pure Hawk stable equilibrium. Using (86) we obtain a mixed stable equilibrium as well if \(b>0.5\) and \(d>4b(1-b)\). \(\square \)