The nest site lottery:
how selectively neutral density dependent growth
induces frequency dependent selection
Theoretical Population Biology 90 (2013) 82-90
[0pt] Department of Mathematics, University of Sussex,
[0pt] Brighton BN1 9QH, UK.
tel. 012 73877345
[0pt] Department of Mathematics, City University London,
[0pt] Northampton Square, London EC1V 0HB, UK.
The project is realized under grant Marie Curie grant PIEF-GA-2009-253845.
Modern developments in population dynamics emphasize the role of the turnover of individuals. In the new approaches stable population size is a dynamic equilibrium between different mortality and fecundity factors instead of an arbitrary fixed carrying capacity. The latest replicator dynamics models assume that regulation of the population size acts through feedback driven by density dependent juvenile mortality. Here, we consider a simplified model to extract the properties of this approach. We show that at the stable population size, the structure of the frequency dependent evolutionary game emerges. Turnover of individuals induces a lottery mechanism where for each nest site released by a dead adult individual a single newborn is drawn from the pool of newborn candidates. This frequency dependent selection leads toward the strategy maximizing the number of newborns per adult death. However, multiple strategies can maximize this value. Among them, the strategy with the greatest mortality (which implies the greatest instantaneous growth rate) is selected. This result is important for the discussion about universal fitness measures and which parameters are maximized by natural selection. This is related to the fitness measures and , because the number of newborns per single dead individual equals lifetime production of newborn in models without ageing. We thus have a two-stage procedure, instead of a single fitness measure, which is a combination of and . According to the nest site lottery mechanism, at stable population size, selection favours strategies with the greatest , i.e. those with the highest turnover, from those with the greatest .
Keywords: density dependence, logistic equation, frequency dependent selection, life history, evolutionary game, fitness measures
In the modern theory of evolutionary ecology (Post and Palkovacs 2009, Pelletier et al. 2009, Morris 2011, Schoener 2011) the problem of eco-evolutionary feedback is of special interest. One of the major theoretical problems in the modelling of population dynamics, and in general of evolutionary biology and ecology, is the limit of population growth and its selection consequences. This topic is very important in many disciplines such as evolutionary game theory and life history theory.
The earliest attempt to solve this problem for populations with overlapping generations is the continuous logistic equation introduced by Verhulst in the 19th century (Verhulst, 1838), which can be found in every textbook on ecology and mathematical biology. It inspired the idea of and selection (McArthur and Wilson 1967), that selection favours different strategies at low densities and near the stable population size, and is still applied in modelling (Cressman et al. 2004, Cressman and Krivan 2006, Cressman and Krivan 2010). This concept states that there is some arbitrary maximal population size at which growth is suppressed and the population remains stable. However, this approach produces some unusual predictions which provoked a wide discussion (Kozłowski 1980, Łomnicki 1988, Kuno 1991, Ginzburg 1992, Gabriel 2005, Hui 2006, Argasinski and Kozłowski 2008) presented in the next section.
The problem of the limits to growth is important not only for ecological population growth models but also for the modelling of natural selection. There is huge discussion on what is maximized by natural selection and what happens when a population reaches the limit to growth (Metz et al. 1992, Kozłowski 1993, Mylius and Diekmann 1995, Brommer 2000, Dieckmann and Metz 2006, Metz et al. 2008, Roff 2008). However in these attempts density dependence is represented by some abstract unspecified factors. Thus the proposed solutions are very general and abstract. A concrete mechanistic interpretation should be helpful in the interpretation of the mathematical notions. In the modification of the logistic equation (Kozłowski 1980, Hui 2006) which was applied to game-theoretic modelling (Argasinski and Kozłowski 2008, Argasinski and Broom 2012) there is an example of a mechanism responsible for strategically neutral density dependence called in this paper a ”nest site lottery”. The underlying assumption is that there is a limited number of nest sites in the environment and that every newborn must find a nest site to survive. Thus all newborns produced at some moment in time form a pool of candidates to be drawn from to replace the dead individuals in their nest sites. The difference is that in this case there is an arbitrary maximal population size described by a carrying capacity indicating the number of available nest sites (mechanistically interpreted as nests or holes, where individual can settle, as in Hui 2006). However, the stable population size is not the carrying capacity, as in the classical logistic equation, but the dynamic equilibrium between different factors of mortality and fecundity (Kozłowski 1980, Ginzburg 1992, Hui 2006), which can be affected by the dynamics of the population state (Argasinski and Broom 2012). The advantage of this approach is that it considers a realistic turnover of individuals (Argasinski and Kozłowski 2008). In this paper we will more rigorously analyze the properties of the nest site lottery mechanism in a simpler model than in the previous papers (Argasinski and Kozłowski 2008, Argasinski and Broom 2012).
Below we give the mathematical details of this approach (sections 2 and 3). Section 4 starts the development of the selection model and in section 5 the important notion of the turnover coefficient is introduced. Section 6 contains the presentation of the selection dynamics. In section 7 the relationships between the nest site lottery mechanism and the invasion fitness concept are presented. Section 8 contains the main results which are the equations (14, 15) and their analysis describing the nest site lottery mechanism (intuitively depicted in fig. 1). We see that, eventually, selection favours the strategy with the highest turnover coefficient out of those with the greatest value of lifetime reproduction. The mechanistic reasoning from section 8 is completed by Theorem 1 describing the quantitative characterization of the restpoints of the system (14, 15). The importance of the obtained results and the general ideas inspired by them is discussed in section 9 (the last subsection contains a discussion on the two-stage maximization procedure, substituting for the single-step fitness measure, obtained by our results).
2. Density dependence
The cornerstone of mathematical ecology is the Malthusian equation describing exponential population growth,
where is the birth rate and is the death rate. However, in Argasinski and Broom (2012, see Appendix 1 there for details) it was shown that with respect to the multiplicative proportionality constant (which can be removed using a change of timescale) acting as the rate of interaction occurrence, these parameters can be interpreted as demographic parameters describing the outcomes of the average interaction with elements of the environment or other individuals. Then can be interpreted as the number of newborns produced during an interaction event and as the probability of death during an interaction event. We will assume this mechanistic interpretation in our model. Thus the Malthusian coefficient can be interpreted as the balance between mortality and fertility . The above model is not realistic, because it allows for infinite population growth. The classical solution of this problem is the use of the logistic equation, which is equation (2),
However, this relies on a problematic assumption which has very serious consequences. Equation (2) produces artifacts in population growth models (Kuno 1991, Gabriel 2005) and selection models related to replicator dynamics (Argasinski and Kozłowski 2008). For example, it suppresses the selection dynamics in the replicator dynamics by setting the right hand sides of the strategy dynamics equations to 0 (Argasinski and Kozłowski 2008), the trajectory escapes to infinity for (i.e. ) and initial population size greater than (known as Levins’ paradox, (Gabriel 2005)) or the trajectory decreases with increasing rate for and initial population size slightly smaller than (Kuno 1991). This is caused by the fact that the term is multiplied by the suppression coefficient, which implies that with population growth, both mortality and fertility decrease, and mortality decreasing with increasing population size and reaching zero at equilibrium is biologically counterintuitive. Mortality should not decrease with population growth and individuals cannot be immortal at equilibrium. The above problems suggest that models should rely on clear and mechanistic assumptions (Geritz and Kisdi 2012). Thus, density dependent suppression should act only on the number of juveniles recruited to the population (Kozłowski 1980, Ginzburg 1992) and the initial population size should be smaller than the carrying capacity (Hui 2006, Argasinski and Kozłowski 2008, Argasinski and Broom 2012) leading to
where the suppression term describes newborns’ survival probability. This provides an important distinction between newborn candidates introduced to the environment (described by per capita number ) and recruited newborns, survivors of the density dependent stage (described by ).
This problem was emphasized by Kozłowski (1980) for the first time, but surprisingly this paper did not get as wide an appreciation as it deserved. This problem was also mentioned in the classical book ”Population Ecology of Individuals” (Łomnicki 1988). Then it was reinvented by Ginzburg (1992), but (3) was rejected there as it “disagrees with our intuition about unchanging equilibrium ”. Hui (2006) argued, against Ginsburg’s claim, that (3) is the proper approach and should be substituted for (2). The discussion started by Ginzburg also did not receive wide attention. Argasinski and Kozłowski (2008) then applied equation (3) to avoid the suppression of selection that occurs after the equilibrium size is reached caused by equation (2) without knowledge of this discussion, and (3) is a cornerstone of the ecologically realistic approach to dynamic evolutionary games (Argasinski and Broom 2012). Then (3) was mentioned as an example of the proper mechanistic approach (Geritz and Kisdi 2012), but not as the general alternative to (2). However, we believe that (3) deserves much stronger attention from a general audience.
Although (3) has been applied in complex selection models (Argasinski and Kozłowski 2008, Zhang and Hui 2011, Argasinski and Broom 2012), the selection consequences of this approach have not been rigorously analyzed, since previous papers (Kuno 1991, Ginzburg 1992, Gabriel 2005, Hui 2006) focused on population density dynamics and ecological aspects. This distinction between adults and newborn candidates is very important for ecological and evolutionary reasoning, because differences between juvenile and adult mortality can have serious selection consequences. For example, a lack of mortality differences means that a small fecundity advantage can favour evolution of semelparity over iteroparity (this problem is known as Coles Paradox, Cole 1954), while mortality differences can significantly change the situation (Charnov and Shaffer 1973). The selection mechanism induced by (3) is thus very interesting and will be analyzed in later sections.
3. The population in equilibrium
We can calculate equilibrium size, by setting the right hand side of Equation (3) to be equal to 0, which gives either or
Note that for positive , the condition should be satisfied. After substitution of into the logistic coefficient , we obtain the equilibrium newborn survival probability . This is reasonable; due to the turnover of individuals, in any short time interval for every newborns we have dead individuals. Thus describes the number of newborns competing for each single nest site vacated by a dead individual. Only one newborn can settle in a single place, thus each newborn can survive with probability . This newborn survival should be valid for any density dependent mortality acting on juveniles, not only for logistic suppression, because only in this case does fertility equal overall mortality.
4. The case of multiple individual strategies
Assume that there are different individual phenotypes each characterized by per capita reproduction and mortality . Thus every strategy is described by a two dimensional vector describing demographic parameters interpreted as in (1). Note that describes the number of nest sites and is the same for all phenotypes. Denoting and , we can describe the following dynamics:
The value increases with time if
and decreases in the opposite case. Thus for every strategy there is a critical population size which is a threshold between regions of growth and decline. Above the population size critical for a particular strategy, the effective fertility will be smaller than the mortality Thus the dynamics of the population size plays an important role, which is described by the equation
where and . We can easily calculate that in this case, instead of reaching the stable equilibrium, the population size converges to the stationary density manifold (Cressman et al 2001,Cressman and Garay 2003a and b)
the form of which is conditional on the strategy frequencies. Thus we introduced diversity among individual strategies to our model. In our model, in the general case, newborn survival is a phenomenological function, linear with respect to the fraction of free nest sites. Thus in this approach the recruitment probability equals the probability of finding a free nest site in a single trial. This is a very specific mechanism which will not be suitable for many species. However, similar mechanisms will work for any density dependent factor acting on births that is monotonically decreasing with respect to . Then for growth rate the critical population size will be The newly produced offspring of the carriers of the different strategies form a pool of candidates from which randomly drawn individuals will be recruited to settle in the available nest sites. This is the core of the ”nest site lottery” mechanism which will be analyzed in the following sections. Note that equations (5) and (7) suggest the importance of the factors and . This will be analyzed in the next section.
5. The turnover coefficient L
Here we will introduce an important characterization of population dynamics. We shall define the function for a single strategy . describes the number of newborns per single dead individual, which we shall refer to as the turnover coefficient (for the relationship of the turnover coefficient with lifetime reproduction, see the Discussion). Surprisingly, a similar coefficient describing the energy allocated to reproduction divided by mortality can be found in life history papers (Taylor and Williams 1984, Kozłowski 1992, Kozłowski 1996, Werner and Anholt 1993, Perrin and Sibly 1993, for an overview see Kozłowski 2006). Analogously, for a mixture of strategies where is the average strategy contained in the convex hull of the strategies (see fig. 1), we define . Thus
which is a weighted average of the s and describes the fraction of strategists among individuals dying during a small time interval (according to Appendix A, (A.2)). is thus the average among dead adult individuals. The -function can be useful in describing the multiplicative newborn survival (recruitment probability) because after substitution of the stable population size into the logistic suppression coefficient we obtain:
which can be denoted as .
If there is any variation in the s, then we have that lies strictly between the smallest and largest values of , . describes the number of newborn candidates produced per single dead individual for the strategy , during . When the strategic argument is the averaged vector, describing a population with a mixture of strategies, then the value of is the average number of newborn candidates produced per single dead individual in this population. When the population is in size equilibrium (at the stationary density manifold), then the newborn survival component can be described by the value of of the average population strategy; thus it becomes frequency dependent.
6. Selection dynamics
The behaviour of equation (5) suggests frequency dependent self-regulation of the population state. Equation (7) shows attraction to the stable size manifold (8), which suggests that the dynamics on this manifold should be analyzed. To describe the frequency dependent selection associated with the system we have presented, tools appropriate to game dynamics are required. Thus we should describe the population in terms of the strategy frequencies and the population size . However, at the stable size manifold the population size is given by (10). We can assume that the strategies are close enough to each other that a separation of timescales between fast dynamics and dynamics occurs. Then we can assume that selection occurs on the stationary size manifold. Now we can describe the selection process realized by the ”nest site lottery” mechanism. Thus using (10), we can write the selection dynamics from (5) as
Because the average growth rate on the stable size manifold is zero then the equation (11) can be replaced by the replicator dynamics (see Appendix A)
Therefore the growth rate of the -th strategy becomes a function of the strategy frequencies (frequency dependent):
Note that the growth rate function describes a mixture of all mortality and fecundity components, not only the density independent mortality and fecundity as in the Malthusian parameter . The growth rate is positive when , which implies that
Thus there is a threshold between regions of growth (strategies with reproductive surplus) and reduction (strategies with death rate exceeding birth rate) which has the linear form (see Figure 1). The threshold describes the set of strategies for which the growth rate equals 0. Frequencies of strategies with a greater value of than the average strategy will increase under the replicator dynamics. In effect the averaged strategy shifts towards those strategies because it is a linear combination of the strategies present in the population. This implies an increase of of the average strategy (see Figure 1). However, among growing strategies, the greatest growth rate is by the strategy with the greatest coefficient . Because in this case the dynamics is on the stationary density manifold, the current population size is very close to (8) and (13) is equivalent to satisfying inequality (6) (passing the critical population size). Thus at the stationary size manifold the threshold between the growth and decline of the strategy frequency is equivalent to the threshold between the growth and decline of the number of carriers of that strategy (this may not be satisfied far from the stationary size manifold). Frequency dependence induces an increase of the slope of the threshold which eventually leads to the selection of the strategy with the greatest , which confirms the result of Mylius and Diekmann (1995). Note that their second result, that density dependent adult mortality leads simply to maximization as in unlimited growth models, directly comes from the independence of the replicator dynamics from background fitness.
7. The monomorphic resident-mutant case
We can simplify the above model by assuming a monomorphic population invaded by a rare mutant; thus this resembles the classical ESS approach (Maynard Smith 1982) in the context of life history evolution (Charlesworth & Leon 1976, Mylius and Diekmann 1995). In the limiting case where the strategy trait tends to zero, we approach the method known as invasion analysis which is the cornerstone of adaptive dynamics (Dieckmann and Law 1996, Metz et al. 1996, Geritz et al. 1998, Dercole and Rinaldi 2008). Using Equation (12), the resident growth rate is zero and the rare mutant growth rate function is which must be positive to invade the population. Thus the equilibrium population size increases.
To be an ESS itself, the “mutant” population should be stable against the previous resident, and thus:
which is the same condition. Thus in both cases we obtain
Note that, when we consider only the death component of the Malthusian equation then we obtain the equation ; thus this is an exponential decay with decay constant , and so the average lifetime of the individual is . In any short time interval of length , for every newborns we have dead individuals. We can change the timescale to set as the new time unit. Then initial rates, and thus respective births and deaths numbers, should be multiplied by some timescale specific constant. However this constant cancels out in . Thus is the lifetime reproduction . Therefore we have obtained for the ”nest site lottery” mechanism, the classical result that under limited growth only lifetime reproduction is maximized and there is no selection pressure on the lifespan. However this occurs only in a monomorphic resident-mutant model. The case of a population composed of an arbitrary number of individual strategies is more interesting.
8. Multiple strategies with
We have seen that evolution leads to the fixation of the strategy with the largest value of , . What if there is more than one such strategy? The following question arises: is there selection between strategies with the same ? We can show this by applying a multipopulation game-theoretic approach (Appendix A and Argasinski 2006) and divide strategies present in the population among subpopulations with the same , but different s (Appendix B). Assume than we have such classes with different strategies in the -th -class (then the lower strategy index descxribes the number within the particular -class and the upper strategy index describes the -class). Then for all strategies (for all ) from the same -class . When we assume that the dynamics is on the stable density manifold, we obtain the following equations
describing the changes of the proportion of the -th strategy within the -th -class (Equation 14) described by and related frequencies between -classes (15) described by . Thus selection between -classes is driven by the first bracketed term from Equation (14) and affects both intra- and inter-group dynamics. However, there is selection inside each -class toward greater . When suboptimal -classes are outcompeted, the intrinsic selection driven by the bracket is also suppressed. The form of Equation (15) shows that among growing -classes, those with smaller can grow faster than those with larger, due to a greater , until they fall under the -selection threshold.
We are interested in analyzing which strategy (strategies) will dominate the population in the long term. In particular, in Equation (12) is always positive if strategy has , whenever there is variation in the values in the population. Thus the proportions of such strategies increase; but also, following Equation (14), the strategies out of these with the largest values of increase the fastest. Thus if there is either repeated small mutations involving strategies with or a constant low level of mutation involving a mix of strategies making , the population will evolve to the strategy out of those with such that takes the largest value. Thus repeated mutations or invasions of suboptimal strategies induce selection towards maximal among strategists. It is easy to show that in the absence of density dependent suppression, this strategy has the greatest but only among strategies, since . The strategies from other -classes can have even greater , but they will be outcompeted by the mechanism described by the bracketed term in equation (12). In the case when the population consists only of the individuals, the same outcome can be caused by repeated ecological catastrophes leading to a decrease of the population size. Then the strategy with the greatest will have the greatest growth rate during the growth phase of the population.
However, let us focus on the evolution of the system under the replicator dynamics in a single particular “turn”, during which no mutation occurs. Suppose that there are precisely strategies in the -class, and assume that the initial state of the -class is described by vector and initial relative size .
The replicator dynamics converges to the vector
where is a constant that satisfies the equation .
For the proof see Appendix C.
Theorem 1 shows that the restpoint describing the frequencies among strategists is fully determined by the initial state of the -class and its initial relative size . Despite frequency dependence, this occurs independently of the initial frequencies of the other strategies (note that in our model there are no direct interactions between individuals). Thus calculation of reduces to the finding of the appropriate value of . Note that the rest point can be interpreted as the state of the whole population in general coordinates and the final state of the -class (then ). Note that if initially , then obviously . The parameter can be described as the inflation coefficient because it inflates the frequencies to sum them to one and compensate the impact of .
9.1. How the nest site lottery works?
We started from the basic population growth equation which is the cornerstone of the framework underlying evolutionary game theory and replicator dynamics (Maynard Smith 1982, Cressman 1992, Hofbauer and Sigmund 1988 and 1998) and its more ecologically realistic extensions (Cressman and Garay 2003a,b, Argasinski 2006, Argasinski and Kozłowski 2008, Argasinski and Broom 2012). We presented an analysis of the dynamics of the mechanism inducing frequency dependent selection toward the strategy maximizing the turnover coefficient .
This phenomenon can be explained mechanistically. All newborns introduced into the population at the same moment in time form a pool of candidates. Each newborn has equal probability to survive (find a nest place), thus the strategy maximizing the number of newborns (trials) maximizes the fraction in the pool of candidates and in effect the amount of survivors. However, every dead adult can be substituted by an individual with any other strategy, thus each death is an additional free place in the lottery. Thus it is profitable for the strategy carried by some subpopulation to maximize the number of trials (newborns) per single offered place (dead adult). In addition, we have shown that among strategies with the largest value of the turnover coefficient there is a selection pressure toward the strategy with the greatest . This is intuitive from equation (12), because for strategies with the maximal number of newborn candidates produced per dead adult (i.e. maximizing the bracketed term in equation 12), the growth rate will increase with the number of dead adults (described by the fraction in equation 12) since each of them will be exchanged for newborns in the pool of candidates. Note that this provides a gene centered mechanistic explanation of the phenomenon which can be naively interpreted in terms of group selection and an altruistic ”sacrifice” of adults, to release the nest sites for juveniles.However, our model shows that it is an outcome of ”selfish” fitness maximization at the individual level. In addition our model suggests a possible tradeoff in resource allocation between maximization of the number of candidates in the nest site lottery and survival of the parental individual.
9.2. Importance of the nest site lottery mechanism
The model presented in this paper is as simple as possible, to emphasize the mechanistic aspects of the analyzed phenomenon. For example, there are no direct interactions between individuals as in game-theoretic models. Our model has an extremely simplified age structure consisting only of juveniles and adults (more on the limitations of pure age-dependent models can be found in Metz and Diekmann 1986). However, it was shown that the impact of density dependent factors (thus also the mechanism described in this paper and its generalizations) can significantly affect and alter the outcomes of game-theoretic models (Argasinski and Kozłowski 2008, Argasinski and Broom 2012). This is caused by the feedback driven by the fact that when population size is on the stable size manifold every newborn should find a new nest site vacated by a dead adult. Our model is the simple case example of a one-dimensional monotone density dependence acting on the effective birth rate (more on this and other general cases can be found in Metz et al 2008). However, as was shown at the end of section 4, our results can be extrapolated to other factors that are monotonically decreasing with respect to the population size, acting like juvenile survival. It is possible that other environmental feedback loops of the same type to those in our model may induce similar selection mechanisms.
The mechanism shown in this paper supports the intuition underlying and selection theory (McArthur and Wilson 1967), that natural selection favours different strategies in growing populations than in populations with suppressed growth. The theoretical and methodological aspects of this approach were criticized (Barbault 1987, Getz 1993, Stearns 1977), however as an intuition it still seems to be relevant (for modern approaches see for example Metz et al. 2008). An alternative to the and approach is life history theory (Roff 1992, Stearns 1992), where the problem of different selection mechanisms in limited and unlimited populations also exists. Maybe phenomena similar to those revealed by our simple model can be found in other, more general or different specific models related to general population dynamics, life history evolution, adaptive dynamics or population genetics. This can be the subject of future research.
9.3. What is maximized by natural selection, and when?
The exact meaning of “fitness” is a subject of endless discussion (Metz et al. 1992, Kozłowski 1993, Mylius and Diekmann 1995, Brommer 2000, Dieckmann and Metz 2006, Metz et al. 2008, Roff 2008). Basically ”fitness” can be defined as the instantaneous growth rate or invasion exponent (Metz 2008). However, if eco-evolutionary feedback is of a particulary simple kind, the optimization approach can be applied (Metz et al 2008, Gyllenberg et al 2011) where some ”fitness measures” or ”proxies” are maximized. There is a widely known fact in life history theory that in a population with unlimited growth, the Malthusian growth rate is a proper fitness measure, while on the stable size manifold, lifetime production of newborns (before juvenile mortality selection) is the correct measure. In Mylius and Diekmann (1995) there is a statement that the invasion fitness method (Metz et al. 1992) suggests that and are necessarily both maximal at the ESS (although this statement is unclear since some strategies can maximize and others can maximize ). Our results support this claim, and show that an analogous mechanism can act in population dynamic and game-theoretic models.
Here an important claim is that of Brommer and Kokko (2002), who say that is a rate independent reproductive measure which does not depend on the timing of reproductive events. This is because is described on the lifespan timescale, not the population dynamic timescale like . Despite its simplified form our model can be a useful illustrative example for this problem. How can we use a lifespan perspective in our approach? At first, assume that population growth is unlimited (every newborn candidate can find a nest site). Then demographic parameters and describing strategy , are constant and the average lifetime of the individual is (as in section 7). Then and is the lifetime reproduction (or the average among individuals dying during in a population described by ). Therefore, the formula shows how the growth rate is affected by lifetime reproduction in the case of a non-age structured population with unlimited growth. It shows that an individual should basically replace itself, but for the strategy growth rate to be positive requires a reproductive surplus during a lifetime. Now introduce the limitation of the nest sites causing density dependent selection. Our results show that under density dependence, the growth rates of the strategies are affected by the frequency dependent surplus reducing mechanism, described by newborn survival which is the function of the other strategies present in the population. In effect is replaced by density dependent growth rate and . Thus the strategies maximizing the lifetime production of newborn candidates will maximize and the bracketed term of . However among maximizers, the strategy with smallest will have greatest .
Our simple model suggests an insight into the mechanistic nature of selection under limited growth and has serious interpretational consequences. It clearly shows that this problem should not be formulated as the alternative: evolution maximizes OR . In our simple model, when the population reaches a stable size manifold, then a mechanism that modifies the ’s of competing strategies, which are no longer constants, emerges to select the strategy with maximal , or with maximal , among multiple strategies with maximal . Thus, our model suggests the existence of another fitness measure which is the combination of and if our reasoning holds in age structured and other more complex models. However, it will be not a function which should be maximized, but a two staged procedure. The first stage should identify the strategies maximizing the turnover coefficient, while the second stage should find strategies with the greatest from strategies chosen in the first stage. We note that our analysis is a simplification, and whereas in the models without age structure as presented in this paper, this is not necessarily satisfied in age structured models. This should be the subject of future research.
The project is realized under grant Marie Curie grant PIEF-GA-2009-253845. We want to thank Jan Kozłowski, John McNamara, and Franjo Weissing for their support of the project and helpful suggestions. In addition we want to thank two anonymous reviewers for detailed comments and helpful suggestions.
Appendix A Multipopulation replicator dynamics.
Assume that we have individual strategies. Standard replicator dynamics can be derived by rescaling the growth equation to the related frequencies which leads to the equation (where ). This equation describes the evolution of strategy frequencies in the unstructured population. However, we might be interested in the modelling of the structured population divided into subpopulations such as different sexes, species etc. Assume that we want to decompose an entire population into subpopulations. Define
as a vector of indices of strategies exhibited by individuals from the -th subpopulation (, and is the number of strategies in the -th subpopulation). For example the notation means that, in the second subpopulation there are (only) individuals with strategies and. Every strategy should belong to a single unique subpopulation. Then according to Argasinski (2006), by the following change of coordinates
we obtain a distribution of relative frequencies of strategies in the -th subpopulation. The distribution of proportions between subpopulations has the form
where is the proportion of the -th subpopulation. Every decomposition into subpopulations can be reduced again to a single population model by the opposite change of coordinates where
When we apply the above transformations to the replicator equations, we obtain a set of equations that describes the dynamics inside the subpopulations (intraspecific dynamics). When the set of strategies in each subpopulation is characterized by the vector of indices , then the system of replicator equations will be:
where is the mean fitness in the -th subpopulation and . In practical applications of this method to the modelling of biological problems, the replicator equations can be defined on the decomposed population. This will simplify the formulation of the model, because when strategies are initially assigned to subpopulations, there is then no need to change their indices. The choice of subpopulations is arbitrary and depends on the biological assumptions underlying the analyzed problem. For example, the entire population may be divided into two competing subpopulations of hosts and parasites or prey and predators. On the other hand, it may be divided into two subpopulations of males and females, when interspecific dynamics will describe the evolution of the secondary sex ratio, and intraspecific dynamics will describe changes of the frequencies of strategies inside the male and female subpopulations. The subpopulations can be divided into subsubpopulations, and the entire population may be transformed into a complex multilevel cluster structure. However, all these structures are equivalent to a single population replicator dynamics model.
Appendix B Derivation of equations (14) and (15) describing selection strategies inside -classes and change of sizes of -classes
Let us assume than we have such classes with different strategies in the -th -class. In addition, assume that the dynamics is on the stable size manifold. Then the initial system of the replicator equations can be transformed into two sets of differential equations. Firstly, the within -class dynamics (according to (A.5)):
where is the proportion of the -th strategy in the -th -class and . Secondly, the between -class dynamics (according to (A.6)):
Appendix C: Proof of Theorem 1
From (12) we have that
Consider any pair of strategies and from the same -class (i.e. ). Using ((C.1)) we obtain
Considering in equation ((C.2)) we obtain
Equation ((C.4)) holds for any pair from the same -class, so that
for some -class specific . It is clear from equation (12) and the fact that is increasing whenever there is heterogeneity of values within the population that for the -class the corresponding value is always increasing and for any other class it is either always decreasing, or starts by increasing and then eventually switches to decreasing, when the population size passes the corresponding threshold (6). Since is bounded above and below, and a monotonic function (decreasing or increasing) then it converges. Letting gives
We know that , thus for at least one -class the corresponding should not converge to . The system (14),(15) shows that it will be -class. However, the above reasoning used coordinates describing the strategy frequencies in the whole population (a metasimplex coordinates, Argasinski 2006). According to ((A.4)), can be described in the coordinates of the system (14) and (15) and after change of the indices where is the index of the -class and is the index of the strategy within this -class, we have . Thus the rest-point will contain only the -maximizing strategies, so that the state of the -class will be equivalent to the state of the whole population (i.e according to ((A.4)) and ), but frequencies will not sum to 1. However, from ((A.4)) we have . Then ((C.5)) for the -class can be presented as:
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