Towards the Construction of a Mathematically Rigorous Framework for the Modelling of Evolutionary Fitness
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Abstract
Modelling of natural selection in selfreplicating systems has been heavily influenced by the concept of fitness which was inspired by Darwin’s original idea of the survival of the fittest. However, so far the concept of fitness in evolutionary modelling is still somewhat vague, intuitive and often subjective. Unfortunately, as a result of this, using different definitions of fitness can lead to conflicting evolutionary outcomes. Here we formalise the definition of evolutionary fitness to describe the selection of strategies in deterministic selfreplicating systems for generic modelling settings which involve an arbitrary function space of inherited strategies. Our mathematically rigorous definition of fitness is closely related to the underlying population dynamic equations which govern the selection processes. More precisely, fitness is defined based on the concept of the ranking of competing strategies which compares the longterm dynamics of measures of sets of inherited units in the space of strategies. We also formulate the variational principle of modelling selection which states that in a selfreplicating system with inheritance, selection will eventually maximise evolutionary fitness. We demonstrate how expressions for evolutionary fitness can be derived for a class of models with age structuring including systems with delay, which has previously been considered as a challenge.
Keywords
Evolutionary fitness Selection Inherited strategy Measure Age structuring1 Introduction
The concept of evolutionary fitness is one of the bestknown paradigms in theoretical approaches to the description of biological evolution and adaptation (Birch 2016; Gavrilets 2004). Charles Darwin himself stated the famous principle of the ‘survival of the fittest’, indicating the key role of selection of life traits or behavioural strategies against certain criteria which can be formalised mathematically via some fitness function (Darwin 1859). The quantitative description of evolutionary fitness came first from the seminal works of Wright (1932, 1988), in which biological evolution was visualised as a ‘hillclimbing’ process in a certain parametric landscape, through which natural selection in the population increases its fitness (Birch 2016; Davies et al. 2012; Gavrilets 2004; Roff 1992). Usually, fitness is defined as the expected number of offspring produced by an individual that will survive until reproductive age (Mangel and Clark 1988; McNamara et al. 2001). Eventually, the process of evolution under these settings should lead the population to evolve to the nearest local maximum of the fitness landscape, and then stop. For this reason, the concept of fitness is central to evolutionary modelling: given a particular fitness function, the evolutionary outcome can be predicted using an optimisation procedure to find the life traits or behaviours which maximise it.
However, the choice of fitness for a subpopulation is generally subjective and strongly depends on the personal preference of the modeller. For example, one can claim that to find optimal life traits, we should maximise (taking into account some possible constraints) the individual reproductive value (Fiksen and Carlotti 1998; Mangel and Clark 1988; McNamara et al. 2001), the growth rate (or minimise the mortality rate) (Han and Straskraba 1998, 2001), the ratio between the food intake and mortality (De Robertis 2002; Gilliam and Fraser 1987; Sainmont et al. 2015) or a certain entropy function (Levich 2000). It probably comes as no surprise that the use of different optimisation criteria can potentially produce conflicting outcomes (Kuzenkov and Ryabova 2015; Morozov and Kuzenkov 2016). In addition, more advanced gametheoretical approaches in evolutionary modelling—which take into account the strategies of the competitors—may suffer from the same drawback since the resultant evolutionarily stable strategy (ESS) generally depends on the choice of fitness—as defined by the payoff matrix (Broom and Rychtar 2013; Hofbauer and Sigmund 1998). Another shortcoming of the idea of optimising a certain fixed criterion such as the individual reproductive value is that such an approach does not take into account the possible impact of the evolution of species traits on the biotic and abiotic components of the environment (McNamara et al. 2001). For example, a successful behavioural strategy can result in the temporary proliferation of a population, but this may cause a corresponding increase of the population of their predators which can cause a decline in the original population (Morozov and Kuzenkov 2016; Sandhu et al. 2017).
Because of the influence of environmental feedbacks on evolution, there is a general understanding in the literature that modelling the evolution of life traits or behaviour should be somehow linked to the underlying population dynamics equations. For example, the evolution of frequencies of competing strategies can be modelled using a replicator equation (Karev 2010). The individualbased modelling approach explicitly considers the reproduction, spatial movement, competition and mortality of each individual within a population or group (Hellweger et al. 2016): this approach, however, is usually computationally demanding and does not allow us to obtain analytically tractable solutions (Hamblin 2013). A popular approach which links population dynamics with evolutionary processes and allows analytical treatment of the problem is adaptive dynamics. Adaptive dynamics is based on the concept of invasion fitness, defined as the longterm per capita invasion rate of an initially rare mutant strain introduced into an environment determined by a resident strain (Geritz et al. 1997, 1998; Morozov and Adamson 2011; Parvinen et al. 2006). The relative simplicity of the adaptive dynamics framework comes at a price: the method is only applicable to an ergodic environment and assumes rare mutations which make small steps in trait space (Metz et al. 1992). Given these assumptions, and assuming that the overall amount of mutants in the system is small, pairwise comparisons between a resident and a mutant population can be made. However, these simplifications are not always biologically justified (Waxman and Gavrilets 2005): for example, often natural selection occurs via simultaneous competition among large numbers of strains with comparable population densities.
A generic approach to modelling selection in selfreproducing systems should allow us to deal with arbitrary types of life traits (either scalar or functionvalued) across a different set of initial conditions and external forcing, in the case where the environment is not necessarily ergodic, for example. One promising approach to modelling natural selection considers dynamics of the measures of sets in the underlying space of traits (Gorban 2007). The outcome of evolutionary modelling is described by the longterm dynamics of measures of sets of inherited elements, rather than the comparison of a number of ‘pure’ life traits against each other. Note that in the case of the presence of a large number of strategies, one can approximate them by a continuum framework.
In this paper, we extend the original ideas of Gorban (2007) and revisit the concept of evolutionary fitness in selfreplicating systems in the case where the space of inherited units is an arbitrary function space (encompassing scalar life trait parameters, as well as functionvalued behavioural traits and their combinations). We first introduce the ranking order of competing sets of strategies using the underlying population dynamic equations. Then we propose a mathematically rigorous definition of evolutionary fitness, which is based on the ranking order of strategies. We show that the connection between measures and densities in the space of strategies presents a number of mathematical challenges which should be taken into account when modelling selection processes using the densitybased approach. Using the new formulation of evolutionary fitness, we formulate the variational principle of modelling selection, which postulates that the longterm outcome of a selection process will correspond to the strategy or trait which maximises evolutionary fitness. Establishing variational principles in this way when modelling biological evolution has a long history with various approaches proposed (Crow 1981; Levich 2000; McNamara et al. 2001; Metz et al. 1992, 2008; Stankova et al. 2013; Wilhelm et al. 1994), but unlike the situation in mechanics, or optics, the framework for developing a unifying variational principle for modelling selection is still missing. Finally, with the help of a few insightful examples, we show how evolutionary fitness can be derived for a class of models with age structuring including delay, which has previously been considered as a mathematical challenge.
The paper is organised as follows. In Sect. 2, we introduce the generic framework of modelling natural selection in systems with inheritance. In particular, we introduce a mathematically rigorous definition of evolutionary fitness. In Sect. 3, we demonstrate how this evolutionary fitness can be found for a class of population models with age structure. In Sect. 4, we discuss the advantages of the proposed concept of fitness when modelling evolutionary dynamics and compare this concept with those from earlier approaches.
2 Defining Evolutionary Fitness
2.1 Setting the Stage: The Space of Inherited Units and Measures
Here we introduce a generic modelling concept as well as some important assumptions necessary for a mathematically rigorous definition of evolutionary fitness in a selfreplicating system with inheritance.
Assumption 1
We consider a selfreplicating system, where any set of inherited units (strategies) v belong to a compact metric (or compact Hausdorff topological) space V.
Biologically, inherited units can be genotypes, behavioural strategies, functional traits, etc. Mathematically, an element v can be a scalar, a vector, a function or a vector of functions.
In the case where the number of inherited units v is large (or infinitely large or even uncountable), it is impossible to follow the dynamics of each element separately. In other words, for an arbitrary model, it will be meaningless to compare some finite number of inherited units against each other and such a comparison will not inform us about selection in the system (Gorban 2007). Instead, we must consider the evolution of some subsets A of space V.
Assumption 2
We assume that the system \(\Sigma \) consisting of any subsets A from the space V form a \(\sigma \)algebra known as the Borel algebra.
To accurately describe the evolution of an inherited unit (or a set of such units), it is necessary to be able to quantify its presence at any moment of time. In the simplest case, we can quantify the presence of strategies via the total population size or total biomass of all strategies \(v\in A\). However, we can also use the logarithmic scale of the biomass, or alternatively, we can characterise the presence of strategies in the population via any positive power of population size which can be strategy dependent. In this case, it is rather natural to use the mathematical concept of measure. More, accurately:
Assumption 3

A value of zero at any given moment of time indicates the absence of A in the population.

A positive value indicates the presence of the set A in the population.

It is countably additive.

It is a smooth function of time.

If it tends to zero, this signifies extinction of v.
Under the above assumptions, the function \(\mu (t)\) should be understood as a Borel measure defined on V.
Assumption 4
We assume that \(\mu (t)\) is uniformly bounded by a constant, i.e. \(\mu (t)(V) \le c\) for any t.
This is a natural assumption to make, taking into account resource limitations for the population. Using the above assumptions, we can define a selection process.
Definition 1
Remark
According to the above definition, the measure of the whole set V does not tend to zero: this would signify the extinction of the whole population, which we want to avoid.
Remark
The above limit for \( \mu (t)(A)\) may not always exist in some systems, for example, in the case of a periodical evolutionary succession of strategies.
Using the measure dynamics, we can compare selective advantages of different sets from \(\Sigma \).
Definition 2
The introduced ranking order of sets may depend on initial conditions. This can be seen from the following illustrative example.
Example 1
Note that the introduced ranking order is not a perfect comparison because a part of A may be worse than B. To cover this situation, we will introduce the following definition of strong ranking.
Definition 3
(Strong ranking order of sets) Set A is strongly better than set B if A is better than B and any \(\mu \)nonzero subset of A is better than B.
In this case, any nontrivial part of A is better than B. The strong ranking satisfies the properties of transitivity and antisymmetry, i.e. we introduce a partial order in \(\Sigma \). This gives us the opportunity to compare elements of different sets.
Definition 4
(Ranking order of strategies) Let any neighbourhood of elements v and w be \(\mu \)nonzero. Element v is better than element w (\(v\succ w\)) if there exist neighbourhoods of these elements (O(v) of v and O(w) of w) such that O(v) is strongly better than O(w).
It is obvious that this relation also satisfies the properties of transitivity and antisymmetry, i.e. it introduces the partial ranking order in V. The ranking order of strategies may also depend on initial conditions.
If the measure is uniformly bounded (\(\mu (t)(V)<c\)) and \(v\succ w\), it follows from Definition 4 that the measure \(\mu \) of some neighbourhood O(w) of w vanishes with \(t \rightarrow +\infty \).
Using the above ranking order definition, we can now postulate evolutionary fitness as follows.
Definition 5
(Evolutionary fitness) In the case where there exists a functional J(v) which preserves the ranking order of strategies it is referred to as evolutionary fitness, i.e. from \(J(v)>J(w)\) it should follow that \(v\succ w\).
Remark
One can easily see that the above definition of fitness is not unique: any function of J would be considered as an evolutionary fitness if it preserves the ranking of strategies. It is easy to see that in this case, any strategy \(v^*\) maximising the fitness will be the same.
Remark
A generic definition of evolutionary fitness would depend on what we determine as the measure of the presence.
Only strategies \(v^*\) having the highest ranking order according to Definition 4 will remain after a long time, with the others eventually vanishing. Let A be a set from \(\Sigma \), such that the closure of A does not contain any point \(v^{*}\) of the global maximum of J(v). Then \(\mu (t)(A) \rightarrow 0\) and there is selection of V / A. We emphasise that this selection does not depend on the choice of measure on V, provided that assumption 3 is satisfied.
Maximisation of evolutionary fitness J(v) provides the variational principle of modelling evolution dynamics: among all elements v, only the strategies \(v^*\) realising the global maximum of J(v) will survive in the population over time. Thus, finding the evolutionary optimal strategies will be equivalent to finding the maximum value of J across V.
2.2 Measure Dynamics and Measure Densities
However, the description based on (2) is too generic to allow us to obtain mathematically rigorous but meaningful results; in particular, the equation does not impose any restriction on mutation rates. In this paper, we will investigate the particular scenario, where inheritance is strong so that we can initially neglect the effect of mutations and assume that offspring have the same genotype as their parents.
Mathematically, it is generally difficult to explore the dynamics of measures directly from Eqs. (3) or (4). However, often we can model measure dynamics using the density distribution across the space of inherited units. To be able to use the densitybased modelling framework, we need to assume the following
Assumption 5
Theorem 1
The proof of this theorem is given in ‘Appendix A’.
The importance of the assumption about uniform convergence in this theorem is crucial, as can be seen from the following example.
Example 2
Remark
We should stress here that one should not confuse the density \(\eta (v,t)\) with the ‘true’ population density: \(\eta (v,t)\) should be considered as a particular characteristic (which is a function of the population density) for which if it tends to zero, the underlying population density also tends to zero. In this particular case, we have \(\eta (v,t)\equiv \rho (v,t)\), where \(\rho (v,t)\) is the ‘true’ population density (e.g. see Example 2).
2.3 Constructing Evolutionary Fitness
The definition of evolutionary fitness stated in the previous section is axiomatic, in the sense that it does not provide a procedure for finding J(v). In this section as well as in Sect. 3, we demonstrate several ways of constructing evolutionary fitness using the underlying density equations.
Theorem 2
Assume the existence of a longterm time average (7) of the generalised reproduction coefficient for all elements v with nonzero initial density \(\eta (v,0)\). Moreover, assume that for any point v there is a neighbourhood O(v), where convergence in limit (7) is uniform on O(v). In this case, from \(J_1(v)> J_1(w)\) it follows that \(v\succ w\). Thus, we can consider the longterm time average \(J_1(v)\) as evolutionary fitness.
The proof of the theorem is given in ‘Appendix C’.
Corollary 2.1
If the conditions of Theorem 2 hold, \(J_1(v)\) is a continuous functional and \(v^*\) is a point of the global maximum of \(J_1\) defined by (7), we then have \(J_1(v^*)=0\).
The proof of Corollary 2.1 is given in ‘Appendix C’. Biologically, this signifies that the maximal average per capita growth rate should be zero.
Remark
In the proof of Theorem 2 (see ‘Appendix C’), it is shown that the fitness functional \(J_1\) can be expressed as \(J_1(v)=\lim _{T\rightarrow \infty }(\ln (\eta (T,v))/T)\).
Remark
Remark
The function of evolutionary fitness depends on what we determine as the density \(\eta \) of measures, which can be different from a biological definition, i.e. the population density. For example, it can be a certain power of the population density: \(\eta =\rho ^{R(v)}\), where \(\rho \) is the biological population density and \(R(v)>0\) is a certain function(al) of v. The choice of the formulation of \(\eta \) may depend on the particular biological study case.
Computing the time average per capita growth rate (7) is not the only possible way of finding evolutionary fitness. In particular, one can prove that a certain combination of model parameters can be considered as a fitness if it preserves the ranking order given by Definition 2. Interestingly, in some cases it is possible to construct an evolutionary fitness which does not depend on the initial conditions. Consider the following insightful example.
Example 3
Remark
The definition of fitness in Sect. 2.1 based on ranking order and its mathematical representation via the generalised reproduction coefficient (7) are extensions of the Darwinian idea of ‘Survival of the Fittest’. This, however, contains some seeds of tautology due to a deterministic interpretation of Darwin’s evolutionary theory: the fittest strategies will be those selected by longterm evolution, and the strategies selected by evolution will be the fittest ones. This is an a posteriori interpretation of fitness (also see Sect. 4). Establishing a rigorous definition of fitness, however, is a vital step between connecting the a posteriori and a priori concepts of fitness. The next section gives examples of a priori implementation of fitness which provides tools to create links between measurable/observable characteristics of organisms such as life traits or behavioural patterns (described by model coefficients) and longterm evolutionary outcomes, thus predicting which combination of biological parameters (under some constraints) should produce a successful strategy. We should stress that obtaining a mathematical expression for evolutionary fitness as a function(al) of model coefficients can be a difficult task. We show below that for some classes of models with age structuring one can derive evolutionary fitness and predict the evolution outcome. Moreover, some numerical procedures have recently been proposed for finding evolutionary fitness for a generic population model (see Sect. 4 for more details).
3 Revealing Fitness in Some AgeStructured Models
Here we demonstrate how evolutionary fitness can be derived for several generic population models with age structure.
3.1 Fitness in a Population Model with Discrete Stages
This system is the partial case of (9), where \(f=y\), \(q_{11}=p_1a_1\), \(q_{1i}=b_i\), \(q_{ii1}=p_{i1}\), \(q_{ii}=p_ia_i\), \(2 \le i\le n1\), \(p_n=0\). The above results provide the function of generalised fitness (19) for the given model.
3.2 Fitness in a Generic Population Model with Delay
In the particular case of \(n=1\), we can consider Eq. (23) as the equation for the density of the measure described by Eq. (4).
Example 4
The given system is a particular case of above model (23), where \(f=y\), \(m=n1\), \(q_{ljl1}=b_j(v)\exp \Big (\sum _{k=1}^{l1}a_k(v)(\tau _{k}\tau _{k1})\Big )\), \(q_{ljl}=b_j(v)\exp \Big (\sum _{k=1}^{l}a_k(v)(\tau _{k}\tau _{k1})\Big )\), \(r_{lj}=a_l(v)\), other coefficients are equal to zero.
4 Discussion and Conclusions
Fitness is one of the most influential concepts in evolutionary modelling following the seminal idea of Sewall Wright concerning a hypothetical adaptive fitness landscape (Wright 1932, 1988). The metaphor of ‘climbing uphill’ to reach a local peak is compelling and easily graspable, which has probably helped to make the concept of fitness so popular in the literature. However, the initial idea by Wright has also met with criticism from many different backgrounds. A major point of criticism centres on the fact that the shape of the fitness landscape should not be stationary, but instead should constantly change in the course of evolution due to a permanent dynamical feedback between the strategies of individuals and the environment (Nowak and Sigmund 2004). To rectify this situation, more advanced approaches have been proposed. In particular, adaptive dynamics introduces the wellknown concept of invasion fitness, which provides the condition for the invasion of a rare mutant in an environment set by a resident type. Since the resident type is changed via consecutive replacements of successful mutants, the overall fitness landscape should be constantly being varied until we arrive at an evolutionary endpoint and the evolutionenvironment feedback loop is captured (Geritz et al. 1997, 1998; Parvinen et al. 2006).
Our revised concept of fitness is actually an extension of the idea of Wright in that we still assume that the evolutionary outcome should maximise some fitness. However, we also take into account the dynamical feedback between the population and the environment. Our idea of fitness is closely linked to the concept of the ranking order of strategies, given by the longterm limit of the ratio of their densities (6). The ranking order and thus the definition of evolutionary fitness may both depend on what we understand by the density of measures \(\eta (v)\), which should not necessarily be some ecological density \(\rho (v)\) such as the population size or biomass per volume, but could also be a function of such a density, possibly depending on the strategy v itself. For example, we might choose \(\eta (v)=\rho (v)^{R(v)}\), where R(v) is an arbitrary continuous positive functional on V, in which case changing R(v) may give us a different ranking order. The choice of measure density \(\eta (v)\) can be based on the underlying equations for the population density \(\rho (v)\) (see Example 3 in Sect. 2 and Sects. 3.1, 3.2), but this is not necessary. The main requirement is that using some other density \(\eta \) should not change the evolutionary outcome of selection, i.e. the maximum fitness strategies remain the same even if the ordering of less fit strategies changes. We should also emphasise that the ranking order (6) might strongly depend on initial conditions, i.e. the initial presence/absence of other strategies.
The framework for modelling selection processes proposed here is based on exploring the longterm dynamics of measures in the space of strategies, as suggested in some earlier works Cressman and Hofbauer (2005), Gorban (2007). This approach is generic, so it can be equally applied to modelling selection in chemistry, sociology, turbulence theory and economics. Although realistic populations are discrete, in many cases, we have the situation where a large number of subpopulations (genotypes) with close traits simultaneously compete with each other, so that we can use a continuum approximation. This is relevant to biological reality, where in studies we usually compare competitive efficiency of some traits against each other: we technically deal with subpopulations with life traits located within some continuum ranges. However, we show here that we should implement the densitybased modelling framework with care and compare particular strategies with each other only under some mathematical restrictions (uniform convergence). In this case, we actually do not only compare pure strategies w and v per se but also their surroundings; thus, we still appeal to the overall measurebased framework.
The proposed evolutionary fitness concept using ranking order of strategies is actually an extension of Darwin’s ‘Survival of the Fittest’ idea. This implies, in particular, that the fittest strategies are those which will eventually survive after a very long time, and so their fitness J will be maximal. We should say, however, that only postulating such a definition is in a certain sense tautological as it follows from the deterministic interpretation of Darwinian evolutionary theory. Indeed, the inherited unit(s) selected by longterm evolution will be the fittest one(s) and, inversely, the fittest inherited unit(s) should be those which will eventually survive (Mills and Beatty 1979). In this case, the idea of fitness becomes an a posteriori rather than an a priori concept. However, we should still stress that the formulated definition allows us to analytically and/or computationally more easily derive which strategies are the fittest. Moreover, creating a rigorous definition is a necessary step to sort out the above tautological situation.
In Sect. 3, we show a possible way of how the above mentioned tautological aspects can be resolved and then used as an a priori framework. The main idea is that for the given environment (deterministic or stochastic) we can establish links between measurable characteristics of organisms such as life traits or behavioural patterns and longterm evolutionary outcomes. In other words, our a priori knowledge of life traits or behavioural strategies should allow us to predict longterm evolutionary outcome provided we have enough information about intraspecific and interspecies interactions as well as the environment (mathematically, such information is fixed in terms of algebraic or/and differential constraints representing the model). For example, the strategy maximising the ratio between the reproduction rate and mortality is shown to be eventually dominant in a simple system under variable predation, provided the model itself is correct (Morozov and Kuzenkov 2016). In more complex models, the expression for fitness will be obviously more complicated (Sect. 3). Therefore, the major goal is to provide a mathematical function(al) expressing the link between observable life traits or behaviours (i.e. a priori information) and the longterm evolution success. Here the definition of fitness serves as major tool for constructing such a link. Finally, we should also stress that we explore evolution and selection in the absence of rare or singular catastrophic events (e.g. a single earthquake, forest fire, atomic explosion) which might results in some paradoxical misinterpretation of fitness [see Examples in Mills and Beatty (1979)].
The fitness J introduced here allows us to formulate the variational principle of modelling evolutionary dynamics: the outcome of longterm selection should correspond to the maximum of the fitness functional across the strategies initially present. Ideas of optimisation in evolution have been suggested in earlier works, for example in the adaptive dynamics framework, where it was found that evolution will optimise the invasion fitness of a mutant introduced in the resident population in the case where the environment affects fitness in an effectively monotone onedimensional manner (Metz et al. 1992, 2008). However, generic classes of models for which this property holds are still poorly understood (Gyllenberg et al. 2011). For example, it has been shown that the optimisation principle (i.e. the existence of a certain function(al) whose maximisation would provide the evolutionary attractor) should require the absence of socalled rock–scissors–paper cycles of invasion of mutants into the environment set by a resident (Gyllenberg and Service 2011). We should stress that the meaning of evolutionary optimisation in the current paper is somewhat different to the one considered in the adaptive dynamics framework (Gyllenberg and Service 2011; Metz et al. 1992, 2008), since it does not implement the invasionreplacement paradigm of adaptive dynamics. In particular, in our case, the strategy which maximises the fitness J may depend on the initial conditions, as in Example 1.
Here we have considered selection processes in deterministic systems, but similar definitions can be formulated in systems with environmental or demographic noise, where we should consider dynamics of expected measures. In particular, the introduced ranking order can then be defined as the ratio of expected measures. However, adding noise would make the derivation of fitness function(al)s more complicated. For example, we would need to include information about the covariances of species growth rates: even for simple densityindependent dynamics, computing the geometric mean of the growth rate (known as ‘Malthusian fitness’) will not provide a correct fitness function (Lande 2007). Considering more complicated dynamics via a generalised logistic equation with noise would make the evolutionary outcome even more complicated: one would need to maximise the expected value of the densitydependent component of the growth rate (Lande et al. 2009). However, even this maximisation principle still does not seem to be generic enough to cover other more complicated models. Another issue related to stochasticity is that our current Definitions in Sect. 2 do not allow for the possibility of stochastic extinction of a subpopulation when its measure (e.g. the population size) drops to a very low value. However, we can incorporate this as well by setting a certain threshold \(\epsilon \) for the measures/densities and assuming that reaching this threshold would signify the extinction of species. Note that in this case, the overall axiomatic approach towards modelling selection will be similar to the current one.
It would be interesting to compare predictions of longterm selection using the current approach with some other well established approaches, such as adaptive dynamics, for example. Strictly speaking, a rigorous direct comparison between our approach and that of adaptive dynamics is hardly possible at all, since in the current settings we assume the absence of ongoing mutations. Also, a fair comparison would require us to have an ergodic environment, as is required for adaptive dynamics (Metz et al. 1992, 2008), so adaptive dynamics would not be applicable to Example 2. However, one can suggest that we already include some mutations in our settings when we consider arbitrary nonzero initial strategies in the space of strategies so that various ‘mutations’ are already present in the system at the start and the fittest mutation will grow and outcompete the other strategies. As such, one can consider the scenario where initial nonzero densities (‘mutations’) are distributed in the vicinity of the strategy \(v^*\) maximising the fitness J given by Definition 3 (applied to the scenario where initial nonzero densities are located in a small vicinity of \(v^*\)). In principle, we can also consider a polymorphic case of several strategies which maximise the fitness: \(J(v^*)=J(u^*)=\cdots =J(w^*)\) and consider initial mutations in the vicinity of each of them. To our understanding, it seems that \(v^*\) will also be an evolutionary attractor in the sense of adaptive dynamics (although a strict mathematical proof of this still needs to be done). Indeed, adaptive dynamics requires that an evolutionary attractor should be convergence stable, noninvasible and, in the case of protected polymorphism, mutation strategies closer to \(v^*\) should be able to invade (Geritz et al. 1997, 1998). Our definition of fitness covers all of these properties by requiring uniform convergence in (6). For example, uniform convergence in the vicinity of \(v^*\) in our definition of fitness rules out the possibility of a ‘Garden of Eden’ situation, in which the best strategy \(v^*\) outcompetes all other strategies, but taking out this strategy from the initial nonzero set would drastically change the evolutionary outcome (Broom and Rychtar 2013). The question about the possibility of achieving the global maximum of J via small mutations in adaptive dynamics remains an open one since the evolutionary trajectory can get stuck at a local attractor.
In this paper, using a specific change of variables we derived evolutionary fitness for some classes of population models (Sect. 3). The question of how to reveal a population fitness for some more complicated models remains. The development of efficient analytical methods to deal with other classes of population models will be the priority for our next work, but the outcome of selection in selfreproducing systems with high complexity can also be obtained via numerical methods which take into account the theoretical framework of this paper, in particular, the requirement of uniform convergence in the space of strategies. For example, in recent work Sandhu et al. (2017), it was shown how the best strategy \(v^*\) can be found using a straightforward computational algorithm. The method can be used for both scalar and functionvalued traits and can also be implemented in situations where we do not know the underlying dynamical equations. However, the proposed method is based on the assumption that the fitness does not depend on initial conditions, and so it should be modified to deal with more realistic situations.
Notes
Acknowledgements
We thank the two anonymous reviewers and the Handling Editor for providing valuable comments which substantially contributed to the improvement of the manuscript. The research collaboration was made possible via an LMS Scheme 2 Grant (the Grant holder is A. Morozov). O. Kuzenkov was supported by the Ministry of Education and Science of the Russian Federation (Project No. 14.Y26.31.0022).
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