## Final Report Summary - MATHECOEVO (Between evolutionary games and life history theory)

This is a report summarising the outcomes of project MATHECOEVO. The project can be regarded as successful. The goal of building a connection allowing for the integration of evolutionary game theory and life history evolution models was achieved. The results obtained provide sufficient background for the development of the modelling framework of age structured populations diversified by individual traits (strategies) with dynamics driven by interactions between individuals. This is significant progress in the mathematical methods applied in evolutionary biology and ecology. The methods developed during the fellowship will be useful for the formulation of quantitative mechanistic and falsifiable models for the wider range of complex biological problems. Below is the descriptive summary of achievements.

Obtained research objectives:

(1) Development of methodology allowing the derivation of dynamic game theoretic models from basic demographic parameters

The classical approach to game theoretic modelling and strategic analyses in evolutionary biology relies on the analysis of some evolutionary 'costs' versus 'benefits' expressed in terms of 'Darwinian fitness' or 'reproductive success'. Thus the basic element of every evolutionary game is the 'fitness function'. However, the following question arises: what is the 'cost' and 'benefit', and how can 'Darwinian fitness' be defined? During the project we developed a conceptual framework which allows us to describe such abstract 'costs' and 'benefits' in terms of the outcomes of basic demographic events, which shape the population dynamics, i.e. births and deaths. This allows for a clear mechanistic interpretation of the outcomes of the model. In effect the models are expressed in terms of clear, unambiguously measurable parameters.

Benefits:

The new conceptual framework is very useful for general evolutionary theory because it explains evolutionary effects as an aggregated outcome of the basic ecological events. Thus, this explanation scheme cannot be described as 'tautological'. The fact that in the new approach, models can be parameterised by demographic data and tested, implies that the new framework is fully falsifiable.

(2) Explicit calculation of births and deaths as the basic events shaping evolutionary dynamics

Instead of a single fitness function describing the excess from the average population growth rate, as in the classical approach, the new approach uses two types of payoffs: mortality and fecundity. This makes the new methodology more realistic and 'ergonomic', in the sense that the development of the models is simpler. Explicit consideration of mortality and fecundity as major opposite forces shaping the population dynamics allows for a full description of the causal chains shaping the evolutionary process. Different events or different types of interactions can lead to the successful raising of offspring or death of the individual. The new theory contains several frameworks to calculate how the overall fitness is shaped by different mortality and fecundity factors and the interplay between them.

Benefits:

The new methodology allows for explicit expression of trade-offs between mortality and fecundity. This is the first step toward the integration of game theoretic methods with life history optimisation. It is also the crucial step for an increase in the ecological realism of game theoretic models. The example application of the new approach (analytical results and numerical simulations) to the classical problem of selection toward non-aggressive behaviour (known as the Hawk-Dove game) shows how over-idealised and in effect naive the classical approach was. This leads to the next important outcome of the project, below.

(3) An extended version of the classical Hawk-Dove game model

The Hawk-Dove game is the basic example in every textbook for evolutionary game theory and evolutionary biology. It describes the mechanism of selection toward reduced aggressiveness in the population. Application of the methods developed in points 1 and 2 shows that the classical approach is oversimplified and does not contain crucial elements necessary for biological realism. Disadvantages of the classical approach are hidden by unclear terminology operating on abstract 'costs' and 'benefits'. For example, the new approach clearly shows that classical results hold only under the assumption that Dove strategists are immortal.

Benefits:

The new model shows that environmental feedback plays a very important role, which affects the value of the 'benefit', in the new approach described as the number of successful matings. Decrease in newborn survival caused by limited habitat availability can seriously change the rules of the game, in a sense that strategies profitable at low densities and unlimited growth can be unprofitable at high densities. This mechanism is so important that additional work was devoted to the analysis of its properties, which is the subject of the next objective.

(4) Explanation of the role of the turnover of individuals and density dependence in the processes underlying population dynamics and natural selection

Separation of the birth and death processes out of the fitness function in the classical approach, allowed for deeper mechanistic insight into how limitation of the population growth affects the process of selection. It was shown that when a population reaches its equilibrium size, a frequency dependent mechanism is induced. This can be thought of as a lottery, leading toward the maximisation of the so called 'turnover coefficient' L=b/d (the number of newborns produced per single dead adult individual, b-fertility and b-mortality). This mechanism can seriously alter the outcomes of the model, as was shown using a Hawk-Dove example, where the emergence of an invasion barrier for Hawks was observed. This mechanism is important for results from objective 2, since juvenile mortality affects the expected 'benefits' measured in the outcomes of successful matings. Thus the rules of the game are not constant, and are determined by strictly ecological factors.

Benefits:

The analysis of the 'lottery' mechanism allows us to revisit an important discussion on the correct universal fitness measure. It shows that maximisation of the growth rate r is a general rule, but density dependence induces a frequency dependent mechanism which modifies the r’s of different strategies to select the strategy with the greatest lifetime reproduction R0, which is an outcome of 'turnover coefficient' maximisation. Thus the results obtained are useful in integrating separate fields of theoretical biology. This leads to the final goal.

(5) Development of an age structured frequency-density dependent evolutionary game framework

In the classical approach to dynamic evolutionary games, individuals are characterised by their behavioural strategies, but their properties cannot change during their lifetime. The results from previous objectives allow creating an ecologically realistic modelling framework combining strategic properties of individuals (behaviour) with their life cycle (life history). By application of this method allowing the decomposition of the initial population into arbitrary subgroups, the new approach was extended to a population with age structure. This leads to the sets of ODEs which are a combination of replicator dynamics and the Leslie Matrix structure.

Benefits:

This result opens up a brand new modelling perspective which is the combination of the game theoretic approach with life history modelling. This will overcome the disadvantages of both theories and allow us to investigate problems requiring an integration of the two theories, such as the coevolution of behavioural patterns and life history traits.

(6) Open problems and new research perspectives shown by the project

The results obtained from this project are the first step toward the synthesis of game theoretic models of the evolution of behaviour with life history optimisation. The age structured replicator dynamics that has been developed can be a very useful tool, for example it can be used to justify fitness measures. The next step, a continuation of the results obtained during the project, will be the derivation of a corresponding static game structure, useful for the strategic interpretation of the trajectories of replicator equations. The natural approach to use is a stage based game theoretic framework. This will allow for the expression of 'costs' and 'benefits' in the evolutionary game in terms of the reproductive value of the individual, i.e. the expected future reproductive success during a lifetime. At the current stage of development, dynamic game theoretic models are limited to the outcomes of a single interaction, and do not deal with the whole life cycle of an individual, and the new models will overcome this limitation.

Obtained research objectives:

(1) Development of methodology allowing the derivation of dynamic game theoretic models from basic demographic parameters

The classical approach to game theoretic modelling and strategic analyses in evolutionary biology relies on the analysis of some evolutionary 'costs' versus 'benefits' expressed in terms of 'Darwinian fitness' or 'reproductive success'. Thus the basic element of every evolutionary game is the 'fitness function'. However, the following question arises: what is the 'cost' and 'benefit', and how can 'Darwinian fitness' be defined? During the project we developed a conceptual framework which allows us to describe such abstract 'costs' and 'benefits' in terms of the outcomes of basic demographic events, which shape the population dynamics, i.e. births and deaths. This allows for a clear mechanistic interpretation of the outcomes of the model. In effect the models are expressed in terms of clear, unambiguously measurable parameters.

Benefits:

The new conceptual framework is very useful for general evolutionary theory because it explains evolutionary effects as an aggregated outcome of the basic ecological events. Thus, this explanation scheme cannot be described as 'tautological'. The fact that in the new approach, models can be parameterised by demographic data and tested, implies that the new framework is fully falsifiable.

(2) Explicit calculation of births and deaths as the basic events shaping evolutionary dynamics

Instead of a single fitness function describing the excess from the average population growth rate, as in the classical approach, the new approach uses two types of payoffs: mortality and fecundity. This makes the new methodology more realistic and 'ergonomic', in the sense that the development of the models is simpler. Explicit consideration of mortality and fecundity as major opposite forces shaping the population dynamics allows for a full description of the causal chains shaping the evolutionary process. Different events or different types of interactions can lead to the successful raising of offspring or death of the individual. The new theory contains several frameworks to calculate how the overall fitness is shaped by different mortality and fecundity factors and the interplay between them.

Benefits:

The new methodology allows for explicit expression of trade-offs between mortality and fecundity. This is the first step toward the integration of game theoretic methods with life history optimisation. It is also the crucial step for an increase in the ecological realism of game theoretic models. The example application of the new approach (analytical results and numerical simulations) to the classical problem of selection toward non-aggressive behaviour (known as the Hawk-Dove game) shows how over-idealised and in effect naive the classical approach was. This leads to the next important outcome of the project, below.

(3) An extended version of the classical Hawk-Dove game model

The Hawk-Dove game is the basic example in every textbook for evolutionary game theory and evolutionary biology. It describes the mechanism of selection toward reduced aggressiveness in the population. Application of the methods developed in points 1 and 2 shows that the classical approach is oversimplified and does not contain crucial elements necessary for biological realism. Disadvantages of the classical approach are hidden by unclear terminology operating on abstract 'costs' and 'benefits'. For example, the new approach clearly shows that classical results hold only under the assumption that Dove strategists are immortal.

Benefits:

The new model shows that environmental feedback plays a very important role, which affects the value of the 'benefit', in the new approach described as the number of successful matings. Decrease in newborn survival caused by limited habitat availability can seriously change the rules of the game, in a sense that strategies profitable at low densities and unlimited growth can be unprofitable at high densities. This mechanism is so important that additional work was devoted to the analysis of its properties, which is the subject of the next objective.

(4) Explanation of the role of the turnover of individuals and density dependence in the processes underlying population dynamics and natural selection

Separation of the birth and death processes out of the fitness function in the classical approach, allowed for deeper mechanistic insight into how limitation of the population growth affects the process of selection. It was shown that when a population reaches its equilibrium size, a frequency dependent mechanism is induced. This can be thought of as a lottery, leading toward the maximisation of the so called 'turnover coefficient' L=b/d (the number of newborns produced per single dead adult individual, b-fertility and b-mortality). This mechanism can seriously alter the outcomes of the model, as was shown using a Hawk-Dove example, where the emergence of an invasion barrier for Hawks was observed. This mechanism is important for results from objective 2, since juvenile mortality affects the expected 'benefits' measured in the outcomes of successful matings. Thus the rules of the game are not constant, and are determined by strictly ecological factors.

Benefits:

The analysis of the 'lottery' mechanism allows us to revisit an important discussion on the correct universal fitness measure. It shows that maximisation of the growth rate r is a general rule, but density dependence induces a frequency dependent mechanism which modifies the r’s of different strategies to select the strategy with the greatest lifetime reproduction R0, which is an outcome of 'turnover coefficient' maximisation. Thus the results obtained are useful in integrating separate fields of theoretical biology. This leads to the final goal.

(5) Development of an age structured frequency-density dependent evolutionary game framework

In the classical approach to dynamic evolutionary games, individuals are characterised by their behavioural strategies, but their properties cannot change during their lifetime. The results from previous objectives allow creating an ecologically realistic modelling framework combining strategic properties of individuals (behaviour) with their life cycle (life history). By application of this method allowing the decomposition of the initial population into arbitrary subgroups, the new approach was extended to a population with age structure. This leads to the sets of ODEs which are a combination of replicator dynamics and the Leslie Matrix structure.

Benefits:

This result opens up a brand new modelling perspective which is the combination of the game theoretic approach with life history modelling. This will overcome the disadvantages of both theories and allow us to investigate problems requiring an integration of the two theories, such as the coevolution of behavioural patterns and life history traits.

(6) Open problems and new research perspectives shown by the project

The results obtained from this project are the first step toward the synthesis of game theoretic models of the evolution of behaviour with life history optimisation. The age structured replicator dynamics that has been developed can be a very useful tool, for example it can be used to justify fitness measures. The next step, a continuation of the results obtained during the project, will be the derivation of a corresponding static game structure, useful for the strategic interpretation of the trajectories of replicator equations. The natural approach to use is a stage based game theoretic framework. This will allow for the expression of 'costs' and 'benefits' in the evolutionary game in terms of the reproductive value of the individual, i.e. the expected future reproductive success during a lifetime. At the current stage of development, dynamic game theoretic models are limited to the outcomes of a single interaction, and do not deal with the whole life cycle of an individual, and the new models will overcome this limitation.