At the largest scale of biotic evolution there has been a directional change where simple self-replicators evolved into large organisms with high metabolic rates and long pre-reproductive periods. Associated with this increase there has been a transition from an asomatic, non-senescing, haploid, and asexually reproducing organism to a somatic, senescing, diploid, and sexually reproducing organism with male and female individuals.

Malthusian Relativity explains this major evolutionary trajectory from only two types of natural selection. It shows that selection for an increase in the energetic state of the organism puts a direction to evolution, and that selection by density dependent competitive interactions can explain the major life history transitions as a function of the energetic state of the organism. It also shows that an upper constraint on body mass can induce an additional transition into eusocial communities, and it suggests that the evolutionary trajectory is reversible when the energetic flux to the biotic system begins to decline.

The theory also deals with the evolution of body mass allometries, the evolution of ploidy levels and sex ratios in eusocial species, and the evolution of cyclic population dynamics. As a general conclusion it is suggested that the majority of complex life history traits in mobile organisms may have evolved from the ecological constraints of selection by density dependent competitive interactions.

Malthusian Relativity is a theory of directional evolution by natural selection. It was developed in response to one of the most basic paradoxes in the Darwinian theory of evolution by natural selection. This is the paradox that frequency independent natural selection favours the intrinsic population dynamic growth rate -- also known as the intrinsic Malthusian parameter -- to the degree that it cannot really explain anything but the evolutionary maintenance of simple self-replicating entities.

This potential of frequency independent natural selection suggests that evolution on Earth, and probably anywhere else in the universe, is driven by a comparable strong and contra acting force of natural selection. And being a believer in simplicity, I find that this contra acting force, which appears to be responsible for the evolution of the majority of complexities in natural organisms, necessarily must arise from only a few axiomatic principles of natural selection.

Malthusian Relativity is my proposal to solve this major evolutionary puzzle. The theory suggests that an overall evolutionary direction may arise from selection for an increase in the energetic state of the organism and that the density dependent interactions, which follow from the population growth of the self-replicating process, may be responsible for the evolution of complex life histories.

In this paper I introduces some of the many topics that are covered by the theory. I first summarise some selection principles that form a technical background covering the game theoretical concept of Continuously Stable Strategies, long-term evolutionary stability, and Fisher's fundamental theorem of natural selection. I then deal with patterns of long-term evolution describing directional evolution, major life history transitions, and the evolution of multi-level selection. The next section deals with the evolution of population dynamics. The last section deals more explicitly with life history traits covering the evolution of body mass, body mass allometries, sexual reproduction, eusocial reproduction, sexual selection, sex ratios, senescence and soma, and reproductive rates. For more detailed coverage you may consult the more specific scientific papers.

Continuously Stable Strategies

Malthusian Relativity deals primarily with the evolution of the average life history in populations with asexual, or sexually interbreeding, mobile individuals. In the classical life-history models reviewed by Roff (1992) , Stearns (1992) , Charnov (1993) , Bulmer (1994) and Charlesworth (1994) the optimisation process of natural selection is usually given by density independent r-selection or by density dependent r- and k-selection (reviewed by Mueller, 1997). The basic idea of r- and k-selection was first considered by Fisher (1930) when he proposed the fundamental theorem of natural selection (see Witting, 2000a). But the concept was formally proposed by MacArthur (1962) and MacArthur and Wilson (1967) , and developed theoretically by others (Anderson, 1971; Charlesworth, 1971, 1994; Roughgarden, 1971; Clarke, 1972). The process of r- and k-selection operates through an increase in the intrinsic population dynamic growth rate, which induces an increase in the population dynamic growth rate (r) and/or in the carrying capacity (k).

r- and k-selection is appropriate when the relative fitnesses among the different variants are constant at a given population density. When instead the relative fitnesses are frequency dependent the optimisation process of natural selection is better described by the game theoretical concepts of Evolutionary Stable Strategies (Maynard Smith and Price, 1973; Maynard Smith, 1982) and Continuously Stable Strategies (Eshel and Motro, 1981; Eshel, 1983).

The Evolutionary Stable Strategy (ESS) was defined as the strategy that, once almost fixed in the population, will be advantageous over any other mutant strategy present in a sufficiently low frequency. This concept does not consider the chances by which a population that is not situated at the ESS will evolve toward the ESS. To address the latter question Taylor (1988) and Christiansen (1991) considered the alternative requirement of Convergence Stability. This is the requirement that a small perturbation of the entire population away from the ESS will end up with selective advantage to mutations that carry the population back to the ESS. A strategy that satisfies both the requirements of the Evolutionary Stable Strategy and Convergence Stability was defined as the Continuously Stable Strategy (CSS) by Eshel and Motro (1981) . In the original version proposed by Witting (1997) , the theory of Malthusian Relativity was based on the concept of Convergence Stability. Today, the theory applies the optimisation concept Continuously Stable Strategies to a multi-dimensional life history of continuous traits.

Long-term evolutionary stability

Apart from being based on Continuously Stable Strategies, the theory of Malthusian Relativity is based also on a concept of long-term evolution, where evolutionary equilibria are independent of short-term constraints. In A General Theory of Evolution this concept is referred to as all-dimensional optima because it aims at predictions where the life history traits (dimensions) and their covariance follow from the selection process of the model instead of being given as the assumptions of the model. The challenge with this concept is to construct life-history models where the relationships among the different traits reflect only true constraints that cannot be altered by natural selection, i.e., constraints that are evolutionarily fixed because they arise from laws that lie outside the domain of evolutionary biology. In the approach taken it is assumed that the different life-history traits evolve independently of each other unless we have a non-biological law that clearly explains why the different traits should trade-off against one another.

At the level intrinsic to the individual the long-term constraints are defined by physical laws, like those of energetic constraints. The intrinsic components of genetics and phylogeny, on the other hand, tend only to constrain short-term evolution. This is because these components are products of natural selection and, thus, they may be unstable on the longer time-scale. At the level of ecological constraints extrinsic to the individual the long-term constraints tend also to be given by laws of physics. But there are important exceptions that arise from the biological constraints associated with the origin of natural selection.

The origin of natural selection is usually considered to be define by the origin of self-replicators (Michod, 1999; but see Lifson and Lifson, 1999). Self-replicators are also referred to as exponential replicators because the self-replicating process generates an exponential increase in the abundance of self-replicating individuals. And in a limited world it follows that exponential increase induces competition among the individual self-replicators for the limited resources. Thus, through the process of density dependent competition, the origin of self-replication defines the origin of density regulation and density regulated population growth.

Density regulation can arise from exploitative and interactive (interference) competition, also referred to as scramble and contest competition respectively. The density regulation of exploitative competition is the reduction in the per capita share of resource that occurs when the population is increasing, assuming that there are no interactions among individuals. The density regulation of interactive competition arises instead from competitive interactions, where the time and energy that an individual spends interacting with other individuals is an increasing function of the number of individuals in the population. Another essential component of interactive competition is that it causes a bias in resource access in favour the individuals with the highest interactive qualityI have earlier used the term competitive quality but this term is ambiguous because it can also be understood in terms of exploitative competition, i.e., in favour of the individuals that are best at dominating other individuals. Malthusian Relativity deals primarily with the long-term evolutionary transitions to be expected when the component of interactive competition is added to the density dependent environment.

The case with no interactive competition is dealt with by the classical life-history theory (reviewed by Roff, 1992; Stearns, 1992; Charnov, 1993; Bulmer, 1994; Charlesworth, 1994). However, the classical theory is not directly comparable with Malthusian Relativity because the classical models were also developed to deal with short-term evolution. This is generally done by partitioning the life-history traits into two groups that may be referred to as respectively the fundamental and the derived traits. It is then assumed that the derived traits evolve from the short-term constraints of the fundamental traits, with short-term constraints being defined, e.g., by genetic covariance and phylogeny. In this way it is possible to use the optimisation approach of r- and k-selection to predict complex life histories from the assumptions of short-term constraints.

But on the scale of long-term evolution, where the essential constraints are those that cannot be altered by natural selection, the predictions of most classical models collapse to that of simple self-replicating entities (Witting, 1997). The life history collapses because the traits to be explained trade-off against the classical definition of fitness and, thus, these traits cannot be maintained on the longer time-scale where there are no local constraints to prevent evolution toward the long-term optimum. If instead the evolutionary equilibria are given by Continuously Stable Strategies the theory of Malthusian Relativity has shown that the density dependent constraints of interactive competition can select for complex life histories that are stable also on the time-scale of long-term evolution.

Fisher's fundamental theorem of natural selection

Selection by density dependent competitive interactions provides a solution to a long-standing problem relating to Fisher's (1930) fundamental theorem of natural selection. Defining fitness as the population dynamic growth rate (r), also known as the Malthusian parameter, Fisher introduced the fundamental theorem as

The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time.

But Price (1972a) made it clear that the fundamental theorem makes statements only about a partial increase in r caused by natural selection. Hence the population evolves toward an equilibrium where the partial increase is balanced against a partial decline caused by the populations impair of the environment. The result is an average Malthusian parameter that takes a rather stable value around zero while the population equilibrium, or carrying capacity, increases steadily, as suggested also by k-selection.

After Price the fundamental theorem regained some of its glory as a correct mathematical statement (e.g., Frank and Slatkin, 1992; Edwards, 1994; Burt, 1995). But selection by density dependent competitive interactions shows that the distinction between a partial and a total change in r does not save Fisher's idea of a partial increase in r (Witting, 2000a). Instead of a partial increase we may expect a partial decline when the level of interactive competition is sufficiently high. It is only when the organism lives in a density independent environment that the fundamental theorem seems to hold as a general principle. At this limit the theorem defines a law of hyper-exponential increase in the population abundance (Witting, 2000b); a law that includes the Malthusian law of exponential increase (Malthus, 1798) as the special case with no evolutionary potential.

Directional evolution

Long-term evolution on Earth has shown a directional trend. Although there are many exceptions it is evident that simple self-replicators have evolved into large multi-cellular organisms with high metabolic rates and complex behavioural interactions. But after Darwin's (1859) introduction of natural selection as a process of chance and history the consensus has arisen that there is no overall direction to the evolutionary process (e.g., Mayr, 1988; Gould, 1989; Williams, 1992). As concluded by Maynard Smith and Szathmary (1995) : ''On the theoretical side, there is no reason why evolution by natural selection should lead to an increase in complexity".

But this conclusion no longer holds. This is because the theory of Malthusian Relativity gives a strong theoretical background for the hypothesis that long-term evolution is directional. The direction arises from selection for a steady increase in the energetic state of the organism (energetic state is defined here as the net-assimilation of resource). This type of selection may be seen as an axiomatic rule of long-term evolution in stable resource rich environments. This is because the extra energy that is gained from the increase in energetic state can be invested in fitness enhancing traits like reproduction, survival, and interactive quality.

A continuous increase in the energetic state and the ability to exploit the resource has traditionally been part of the paradox of evolution toward extinction. Increased resource assimilation leads to increased reproduction and increased population growth, and thus in the long run it may lead to the over-exploitation of resources. And with a continuous increase in the exploitation rate, over-exploitation may become so server that both the population and the resource becomes extinct. But this is true only if we disregard the effects of competitive interactions. In Malthusian Relativity, with selection by density dependent competitive interactions, the increased population growth generates increased interactive competition and increased selection for interactive traits that trade-off against further population growth. The result is an abundance that stabilises at an intermediate level while the continuous increase in energetic state generates an exponential increase in traits like body mass. This type of increase has been observed, e.g., in fossil horses during the last 57 million years (MacFadden, 1986).

It has traditionally been argued that the relatively steady increase in body size that has occurred among the largest organisms on Earth is more likely to be due to evolutionary diffusion than to directional natural selection (e.g., Stanley, 1973). It is argued that this hypothesis is supported by the fact that many, if not most, organisms have been left aside from the increase and by the fact that many organisms even seem to dwarf in size. But the directional evolution of Malthusian Relativity is not in conflict with these facts. Instead the theory predicts that these patterns are what we might expect. This is because the predicted increase is based on unconstrained evolution in a stable resource-rich environment. The environment may instead impose a limit to the amount of resource that the organisms can consume, and as such limits will differ for different niches we can expect a variety of energetic levels and body masses. And if the available resource is extremely sparse the upper limit to resource consumption may be so low that the organism cannot evolve away from the low-energy self-replicator. Such low-energy self-replicators may also be unable to invade environments with more abundant resources if these are dominated by high-energy organisms that exclude the low-energy organisms by direct inter-specific interactive competition. And according to Malthusian Relativity a species may dwarf if its individuals have progressively less resource available, suggesting that dwarfing may be widespread during periods of environmental crisis.

Major evolutionary transitions

Associated with the long-term increase in body mass there has been a transition from an asomatic, non-senescing, haploid, and asexually reproducing organism to a somatic, senescing, diploid, and sexually reproducing organism. There has also been a stepwise transition in the reproducing unit, from the individual self-replicator over pair-wise sexual and co-operative reproduction to the fully evolved eusocial colony.

Malthusian Relativity is the first theory where a single selection mechanism explains these major evolutionary transitions (Witting, 2002). The essential component that induces the transitions is a step-wise equilibration of the directional increase in body mass and energetic state. Dependent upon extrinsic and intrinsic constraints the directional increase may equilibrate at four evolutionary states:

The downward constrained equilibrium

exists in low-energy organisms where the energetic state cannot evolve beyond a lower threshold value. This state has a negligible body mass and a low level of competitive interactions.

The evolutionary equilibrium

exists in high-energy organisms where the energetic state has evolved beyond the lower threshold and there are no current increase in energetic state. This state has an equilibrium level of competitive interactions and a non-negligible body mass in evolutionary equilibrium.

The evolutionary steady state

exists in high-energy organisms where the energetic state has evolved beyond the lower threshold and both the energetic state and the body mass evolves free of evolutionary constraints. This state has a higher level of competitive interactions than the evolutionary equilibrium and it is characterised by an exponential increase in energetic state and body mass.

The upward constrained equilibrium

exists in high-energy organisms where the energetic state has evolved beyond the lower threshold and the body mass is situated at an upper limit below the corresponding body masses of the evolutionary equilibrium or the evolutionary steady state. The level of competitive interactions at the upward constrained equilibrium is higher (potentially even much higher) than the level at evolutionary steady state.

The stepwise increase in the level of interactive competition associated with the transitions between the four equilibrium states implies selection for interactive traits at different levels. And as the major life history traits are linked to interactive quality it follows that the major transitions can be explained from transitions among the four equilibrium states of interactive competition. When the life history is allowed to equilibrate at the different equilibria

The downward constrained equilibrium

resembles the simple self-replicator. This organism is asomatic, non-senescing and haploid. It has a negligible body mass and a reproducing unit of a single asexually reproducing individual.

The evolutionary equilibrium

resembles a higher eukaryotic organism, with a non-negligible body mass, soma, senescence, a diploid genome, and pairwise sexual reproduction between a female and a male.

The evolutionary steady state

resembles a co-operatively breeding eukaryotic organism. Life history traits resemble those of the evolutionary equilibrium, except for the reproducing unit where the sexually reproducing pair receives help from a single offspring worker.

The upward constrained equilibrium

resembles an eusocial colony. Life history traits resemble those of the evolutionary equilibrium, except for the reproducing unit where the sexually reproducing pair receives help from up to infinitely many offspring workers.

Evolution of multi-level selection

During evolution on Earth there has been transitions toward reproductive units of ascending inclusion, where higher-level units arise from co-operative reproduction among lower-level units (Vrba and Gould, 1986; Buss, 1987; Maynard Smith and Szathmary, 1995). At the lowest level the reproductive unit is the asexual self-replicator. At higher levels the reproductive unit can take the form of a cell, a multicellular organism, or a group of multicellular organisms, with groups including sexual pairs, co-operative breeders, and eusocial colonies.

As defined by interactors, or interacting units, natural selection is a second dimension that exists in a multi-level hierarchy that resembles the hierarchy of reproductive units (Hull, 1980, 1981; Brandon, 1988; Wilson and Sober, 1994; Lloyd, 1988; Sober and Wilson, 1998; Gould and Lloyd, 1999; Keller, 1999). At the lowest selection level interactors are given by the hereditary material. For self-replicators at the origin of life, with no distinction between genotypes and phenotypes (Michod, 1983, 1999; Szathmary and Maynard Smith, 1997), the lowest interacting level is the self-replicator itself. For higher organisms, with genotypic and phenotypic distinction, the lowest interacting level is the gene, which may act as an interactor when genes can replicate differentially within organisms. At higher selection levels the interacting unit can be formed by cells, multicellular organisms, groups, or species.

Buss (1987) proposed and Maynard Smith and Szathmary (1995) elaborated the hypothesis that transitions to higher reproductive units are associated with transitions in the rules of natural selection. A comparable hypothesis was formulated by Michod (1999) , who describes that transitions to higher-level reproductive units are likely to be induced by transitions to higher levels of selection. Transitions to higher levels of selection can induce transitions to higher reproductive units by promoting co-operative reproduction among lower-level units. Co-operative reproduction among lower-level units is typically fitness costly at that lower level, but co-operation may provide a fitness benefit at the next higher-level, and this benefit may outweigh the cost.

Although it is well established that higher-level selection may outbalance lower-level costs of co-operation it remains a relatively open question why selection at higher levels arises in the first place. But in the theory of Malthusian Relativity it is clear that it is selection by density dependent competitive interactions that selects for evolutionary transitions to higher levels of selection (Witting, 2002). Here the interacting unit that defines the level of selection is the group of individuals that co-operate in their competitive interactions with other groups. Thus, by describing selection on the number of individuals in the interacting unit the theory also describes selection on the level of selection. The higher-level benefit of co-operation is then given by the interactive advantage to interacting units with an increasing number of individuals, while the lower-level cost is given by the cost of sharing the resource among the individuals in the unit. It then follows that the level of selection is defined by this trade-off balance, which is pushed towards higher levels of selection when the number of competitive interactions increases and it becomes more important to be interactively superior. And for the four equilibrium states of the major evolutionary transitions it follows that the level of interactive competition is so high that the interacting unit coincides with respectively the asexually reproducing organism, the sexually reproducing pair, the co-operatively reproducing unit, and the fully evolved eusocial colony.

Population dynamics

Single species population dynamic models are usually based on the consensus of the 20th century; that the population dynamic mechanism intrinsic to the species is Malthusian increase with superimposed density regulation. Although it is known that these density regulated models will not explain the widespread tendency for cyclic dynamics in natural species (Turchin, 1990; Witteman et al., 1990; Turchin and Taylor, 1992; Ginzburg and Taneyhill, 1994), the models have gained acceptance in part because they are based on very plausible mechanisms. The alternative models with delayed density regulation have never been generally accepted as adequate single species models despite of the fact that these models can explain the cyclic dynamics and the fact that delayed density dependence is documented in many species (e.g., Turchin, 1990; Turchin and Taylor, 1992; Berryman, 1996). The rejection of the delayed density regulated models is scientifically sound because the delayed density dependence of those models were given by assumption instead of being deduced from mechanisms that operate within the population.

In resent years a new class of single species population dynamic models have been developed (Ginzburg, 1980, 1998; Ginzburg and Taneyhill, 1994; Inchausti and Ginzburg, 1998; Witting, 1997, 2000b). These are the models of inertial dynamics that, at least in the version proposed by Witting, resemble traditional density regulated models with superimposed density dependent changes in the intrinsic life history. The inertia models might be compared with the older models of delayed density regulation in the sense that both models can explain the cyclic dynamics in many natural species, and that both models include delayed density dependence, although the delay in inertia models is restricted to one generation. But where the delayed density dependence is given by assumption in the delayed density regulated models, the delay in the inertia models arises from plausible mechanisms that operate within the population. A second difference is that the intrinsic growth rate is a parameter in delayed, as well as direct, density regulated models, while it is an initial condition in inertia models. And a third difference is that the density independent fundament of inertial dynamics is Fisher's (1930) fundamental theorem of natural selection (Witting, 2000a) instead of the Malthusian law of exponential increase (Malthus, 1798).

Malthusian Relativity has played an essential role in the deduction of inertial dynamics. It was shown theoretically that selection by density dependent competitive interactions can induce the changes in the intrinsic life history required for the inertia models to work (Witting, 1997, 2000b). These between-generation changes in the intrinsic life history have been observed in species with cyclic dynamics (e.g., Krebs and Myers, 1974; Stenseth, 1982; Stenseth and Ims, 1993), and they may arise either from genetic responses to selection and/or from plastic phenotypic responses, where inherited environmental effects and epigenetic inheritance systems respond to the intra-specific selection pressure. Maternal effect is one example of an epigenetic inheritance system, and this particular response is the basis for the inertia models developed by Ginzburg and Taneyhill (1994) , Ginzburg (1998) , and Inchausti and Ginzburg (1998) .

Body mass

In classical life-history theory with no competitive interactions large body masses are set to evolve from the assumption that reproduction is approximately proportional to body mass (e.g., Roff, 1981, 1992). The proportional relationship is documented in many species, and it seems that it has evolved by natural selection.

On the scale of long-term evolution there is no reason to expect that the proportional relationship can act as constraint on body mass. The proportional relationship is clearly not an absolute constraint because lifetime reproduction is independent of body mass at the scale of between-species comparison. And at the theoretical level there seems to be no physical law that will explain the relationship from constraints intrinsic to the individual. From absolute intrinsic constraints we can instead expect an inverse relation between body mass and lifetime reproduction. This is because the number of offspring that can be produced from a given amount of resource is inversely related to the body mass of the offspring. Given this energetic constraint the classical models of r- and k-selection predict the evolution of only negligible body masses.

The paradox of a body mass that is evolutionarily unstable at the time scale of long-term evolution is avoided by Malthusian Relativity (Witting, 2002). With the energetic constraint of an inverse relationship between lifetime reproduction and body mass it follows that density dependent competitive interactions selects for non-negligible body masses in high-energy organisms. This is because larger individuals can be selected to dominate smaller individuals during competitive interactions, and because the level of interactive competition is so high that the interactive advantage of a large body mass can outbalance the reproductive disadvantage. And at the evolutionary steady state with an exponentially increasing body mass the bias in resource distribution over body mass caused by competitive interactions is exactly so large that lifetime reproduction is proportional to body mass (Witting, 1997).

Body mass allometries

The inter-specific body mass allometries describe the traits of organisms as power functions of body mass, i.e., as t=w^a where t is the trait, w the body mass, and a the exponent. For life-history traits like the metabolic rate, life-span, life-time reproduction, and the population growth rate the exponents have been estimated to 3/4, 1/4, 0, and -1/4 for terrestrial organisms, while the exponents for ecological traits like the homerange area and the population density have been estimated to 1 and -3/4 (Calder, 1984; Peters, 1983).

Until recently there was no general explanation for the exponents of the body mass allometries (LaBarbera, 1986). But in Malthusian Relativity the above mentioned exponents have been deduced from the ecological constraints on the foraging process in mobile organisms (Witting, 1995). The deduction suggests that the characteristic values of ±1/4 and ±3/4 apply only to organisms that forage in two spatial dimensions, while the corresponding values for organisms that forage in three spatial dimensions are ±1/6 and ±5/6. This prediction has been confirmed empirically: the lifespan exponent is 0.25 ±0.04 among 195 species of terrestrial mammals, while it is 0.16 ±0.02 among 40 species of pelagic mammals (terrestrial mammals normally forage in two dimensions, while pelagic mammals are likely to forage in three dimensions).

More recently West et al. (1997) have proposed that the 3/4 exponent for the metabolic rate reflects the constraints associated with the branching of vascular systems. Unlike Witting's proposal the latter model does not explain the exponents of other ecological allometries, and nor does it explain why the 3/4 exponent apparently applies only to organisms that forage in two dimensions, while the metabolic exponent appears to be 5/6 among organisms that forage in three dimensions [see Gates and Gittleman (1998) and Witting (1998) for discussion].

Sexual reproduction

In Malthusian Relativity the evolution of sexual reproduction is given by the evolution of the interacting, reproducing, and sexual units. Here the interacting unit is the number of individuals that join in their interactions with other interacting units, while the reproducing and sexual units are defined by the compositions of the individuals in the interacting unit.

The reproducing unit is given by a single replicating individual and from zero to infinite non-replicating individuals. The replicating individual is the individual that produces offspring and/or macrogametes and it is usually an asexually reproducing individual or a sexually reproducing female or hermaphrodite. The non-replicating individuals can be males that produce microgametes copying their heritable code to the offspring through the sexual reproduction of the female or the hermaphrodite. The non-replicating individuals may also be offspring workers that get their heritable code copied indirectly through the relatedness with the replicating and/or male individuals.

The sexual unit is not present in asexual organisms. In sexual organisms the number of individuals in the sexual unit is usually one - a self-fertilising hermaphrodite - or two - a female and a male. But more generally there may be higher levels of sexual reproduction where females allocate heritable codes from more than a single male to the offspring. And sexually produced offspring need not necessarily obtain half of their genome from the mother and the other half from a single farther. To allow for the evolution of larger sexual units Malthusian Relativity applies a model where the sexual unit with three individuals is a unit with a single female and two males with each of the three individuals allocating one third of their heritable code to the offspring. And at the upper limit of sexuality the model allows for infinitely large sexual units composed of a single female and an infinite number of males with all individuals allocating an infinitely small fraction of their heritable code to the offspring.

For organisms beyond the lowest level of reproduction there is a cost related to each of the three units. For the interacting unit the cost is the cost of sharing local resources, with the cost being proportional to the number of individuals in the unit. For the reproducing unit the cost is the cost of non-replicating individuals with the cost being proportional to the number of non-replicating individuals in the unit. This cost resembles the well-known two-fold cost of the male when the reproducing unit is composed of a single female and a single male. For the sexual unit the cost is the cost of dilution of the replicating individuals heritable code in the offspring. This cost is zero if the heritable code is provided entirely by the replicating individual. It is the two-fold cost of meiosis for the usual form of sexual reproduction. And it is close to infinity at the limit where the sexual unit contains a single female and an infinite number of males and all individuals transmit an infinitely small fraction of their genome to the offspring.

For natural organisms beyond the lowest level of reproduction we expect that the costs of the interacting, reproducing, and sexual units be balanced by advantages provided by those units. It is thus essential to explain why the apparent benefit in mobile organisms of both the interacting and the reproducing unit is zero for asexual self-replicators, two-fold for pair-wise reproduction, three-fold for co-operative reproduction among three individuals, and close to infinity for eusocial reproduction. Likewise it is essential to explain why the apparent benefit of the sexual unit is zero for the asexual self-replicator, and why it is two-fold and only two-fold for dioecious sexual organisms no matter whether the organism reproduces pairwisely, co-operatively, or eusocially.

For Malthusian Relativity it has been shown (Witting, 2002) that selection by the density dependent competitive interactions among the interacting units may outbalance the cost of interacting, reproducing, and sexual units given that the three units evolve as traits that enhance the interactive quality of the interacting unit. And this might actually be the case. It is evident that the interacting unit may evolve as an interactive trait because, other things being equal, units with more individuals should be able to evolve a higher interactive quality. And the reproducing unit may also evolve as an interactive trait because the energy and time that the non-replicating individuals save on replication may be invested in interactive quality and competitive interactions. And even the sexual unit may evolve as an interactive trait because the replicating individual may use the father/s fraction of the offspring's genome to attract interactively superior sexually reproducing males to the reproducing unit.

But sexual reproduction can evolve by this mechanism only if the level of interactive competition is sufficiently high. Interestingly it follows that it is exactly the predicted increase in interactive competition associated with the transition from low-energy organisms with negligible body masses to high-energy organisms with large body masses that will explain the evolutionary transition from asexual reproduction to the usual form of sexual reproduction with a single male per female and a two-fold cost of meiosis.

However, if offspring workers can be produced only asexually, selection by density dependent competitive interactions predicts unknown forms of sexual reproduction where n-1 males mate with a single female and the sexually produced offspring receive genes from n>3 parents. But with the possibility of sexually produced offspring workers it follows that the sexual unit in pair-wise, co-operative, and eusocially reproducing organisms is a single male per female with a two-fold cost of meiosis, instead of any other number of males per female and any other level of cost. Sexual units with more than two individuals are not expected to evolve for the case of sexually produced offspring workers because the interactive quality of the male is transferred to the sexually produced offspring workers. This generates a diminishing return where the interactive quality that can be gained by exchanging an offspring worker with a sexual male is a declining function of the number of males per female. And it is only for the initial transition from asexual to pair-wise sexual reproduction that the interactive quality of the male can outbalance the cost of sexual reproduction.

This model is the first to explain the balance between fitness and the cost of the interacting, reproducing, and sexual units in organisms ranging from simple self-replicators to eusocial colonies. The model is also quite different from the models traditionally considered in relation to the evolution of sexual reproduction. Nearly all traditional models are reflections over the hypothesis that genetic diversity is beneficial to the organism per se. The various versions of the Fisher-Muller hypothesis suggest that sex and recombination protect against a genetic deterioration caused by the accumulation of deleterious mutations (e.g., Fisher, 1930; Muller, 1932, 1964; Manning and Thompson, 1984; Wagner and Gabriel, 1990; Charlesworth et al., 1993; Lynch et al., 1993, 1995; Peck, 1994; Peck et al., 1997). Most of these studies provide only a long-term advantage to sexual populations, while they lack a short-term advantage that will explain the evolution of sexual reproduction as well as the maintenance for cases where asexual variants can arise in sexual populations. Kondrashov's (1982) synergetic-fitness theory, however, can provide a short-term advantage given a special type of deleterious mutations that act together so that each gene becomes increasingly deleterious as the number of deleterious mutations increases. And Peck et al. (1999) show that the Fisher-Muller hypothesis may provide a short-term advantage if the population is subdivided into demes with sufficiently low migration among demes.

Another class of models is based on the idea that sexual reproduction may evolve because recombination produces genetically variable offspring (Weismann, 1889), which may increase the speed and efficiency of natural selection (Kondrashov, 1993; Barton, 1995; Feldman et al., 1997). Genetically variable offspring may provide an advantage in biotic interactions (Bell, 1982; Bell and Maynard Smith, 1987), for example, in host-parasite interactions where sexual reproduction can stores genes that currently are bad but can protect against future mutant parasites (Hamilton, 1980; Hamilton et al., 1990; Ebert and Hamilton, 1996). Another example is the sib competition models of Williams (1975) and Young (1981) that assume that competition is more severe between asexual sibs, which are genetically identical, than between sexual sibs, which are genetically diverse. More recently it has been suggested that sexual reproduction may evolve or be evolutionarily maintained by non-random mating that accelerates the evolution of beneficial traits (Kodric-Brown and Brown, 1987; Davis, 1995; Jaffe, 1996, 1999), or because of interactions among the different genetic models (West et al., 1999).

It has been said that none of the genetic hypotheses for the evolution of sexual reproduction are very convincing (Green and Noakes, 1995). The models will generally not explain why asexual reproduction in mobile organisms is more common in negligible sized organisms than in larger organisms. And nor will they explain why the sexual unit is a single male per female with the average offspring receiving half of the genes from the father and the other half from the mother. The genetical models tend to predict that only a small degree of sex is fine (Green and Noakes, 1995; Hurst and Peck, 1996; but see Peck and Waxman, 2000), suggesting that the degree of gene exchange that occurs among haploid and asexually reproducing prokaryotes may be sufficient to account for most of the genetic diversity hypotheses. The pluralist approach of West et al. (1999) may provide a more efficient framework for generating an advantage that may outbalance the two-fold cost of meiosis. But it might be more likely that the beautiful and simple phenomenon of sexual reproduction in higher organisms has evolved by a simple and clear-cut mechanism, instead of being explained by messy interactions among very different processes (Kondrashov, 1999).

Eusocial reproduction

In eusocial colonies the sexually reproducing pair is associated with a large caste of non-reproducing offspring workers. The evolution of such colonies have traditionally been explained by Hamiltonian models of inclusive fitness (Hamilton, 1963, 1964), relabelled kin selection by Maynard Smith (1964) . These models are essentially group selection models (Hamilton, 1975; Sober and Wilson, 1998) that aim at explaining why offspring choose to raise sibs instead of raising their own offspring. Assuming that the haplodiploid genome is more ancestral to hymenoptera than eusociality, Hamilton (1964) noted that hymenoptera can be predisposed to eusociality because of the high relatedness among sisters of haplodiploids. Trivers and Hare (1976) showed that sex allocation theory is central for this mechanism to work; haplodiploids are predisposed to eusociality only if the sex ratio of the sexual caste of the ancestor is female biased. Female biased sex ratios and, thus, the potential evolution of eusociality might arise through split sex ratios (Grafen, 1986), where broods produced at different times (Seger, 1983) or conditions (Frank and Crespi, 1989) have different sex ratios.

But as co-operative and eusocial reproduction are widespread also in diploid species the high relatedness among sisters in haplodiploids may not be the factor that induces the evolution of eusocial colonies. And in Malthusian Relativity the eusocial colony evolves independently of the haplodiploid genome; in fact the theory even suggests that the haplodiploid genome may evolve as a consequence of a particular mating strategy in hymenoptera-like eusocial colonies. Here the eusocial colony evolves because the interacting, reproducing, and sexual units can enhance the interactive quality of the interacting unit (Witting, 2002).

Many of the eusocial colonies in insects can be seen as being defined by a large interacting unit that contains a single reproducing unit where a sexually reproducing pair receives help from a large caste of non-reproducing offspring workers. A large interacting unit with only one large reproducing unit will evolve by density dependent competitive interactions if the body mass is upward constrained relative to the energetic level of the organism. Then, both the population density and the level of interactive competition will be high inducing selection for a high interactive quality, and this is obtained by a large interacting unit with many non-replicating individuals that allocate time and energy to competitive interactions instead of replication. But such units will not necessarily evolve into eusocial colonies. If e.g. the offspring workers can be produced only asexually, then, the sexual units will evolve to unknown levels where the sexually reproducing female mate with a large number of male and all sexual individuals transfer only a tiny fraction of their hereditary material to the offspring. In these colonies there will be no offspring workers but only a single sexual female and a large caste of sexually reproducing males.

But if offspring workers can be produced also by sexual reproduction, then, we can expect that density dependent competitive interactions select for eusocial colonies. In this case sexually produced offspring workers can evolve because the interactive quality of the male is transferred to the sexually produced offspring workers. This generates a diminishing return where the interactive quality that can be gained by exchanging an offspring worker with a sexual male is a declining function of the number of males per female. It is only for the initial transition from asexual to pair-wise sexual reproduction that the interactive quality of the male can outbalance the cost of sexual reproduction. For all potential remaining transitions, the transfer of male interactive quality to sexually produced offspring workers implies that the interactive quality that can be gained by exchanging an offspring worker with an additional sexual male cannot outbalance the extra cost to sexual reproduction associated with that transition.

The conclusion is that sexually produced offspring workers and kin selection evolve as a result of large interacting units, instead of being the traits that induce the evolution of such units. Here kin selection and the sexually produced offspring workers evolve as a balance between the cost of sexual reproduction and the overall interactive quality of the interacting unit. In result, the expected number of sexually produced offspring workers per reproducing unit becomes a function of the level of interactive competition. The optimum is zero workers for low-energy organisms and high-energy organisms with body masses in evolutionary equilibrium, while it is one and up to an infinite number of workers for high-energy organisms with body masses at respectively the evolutionary steady state and the upward constrained equilibrium.

The fully evolved eusocial colonies are known mainly from social insects where they occur in one form in ants and bees, and in another form in termites. Ants and bees have a haplodiploid genome, their workers are the daughters of the queen, and there are typically three queens per sexual male. In termites however the genome is diploid, the workers are both the daughters and sons of the queen, and there are typically one king per queen. In Malthusian Relativity these differences can evolve from the distinction that ant/bee colonies are formed only the queen while the termite colony is formed by the sexually reproducing pair.

Sexual selection

In Malthusian Relativity there is sexual selection on the level of sexual reproduction. This form of sexual selection arises from the density dependent competitive interactions that selects for a female choice for interactively superior males and a male choice for females that allocate the largest fraction of the male genome to the offspring. This causal relationship, where sexual selection arises from secondary sex traits of interactive quality and selects directly on the level of sexual reproduction, is opposite to the causal relation originally proposed by Darwin (1859, 1871) . According to the original proposal and apparently all subsequent studies (Andersson, 1994), sexual selection is an intermediate form of selection that arises from sexual reproduction and selects on secondary sex traits by means of mate choice or intrasexual contest for the opposite sex.

The models that support Darwin's sexual selection hypothesis are usually divided into the Fisherian run away process (Fisher, 1930) and indicator mechanisms (e.g., Fisher, 1915; Williams, 1966b; Maynard Smith, 1976; Zahavi, 1975; Hamilton, 1982; Hamilton and Zuk, 1982; Heywood, 1989; Hoelzer, 1989; Grafen, 1990, reviewed Andersson, 1994), also known as the good gene or handicap hypothesis. Several authors have shown that these two sexual selection mechanisms may lead to the evolution of energetically costly secondary sex traits. But most of the predictions are evolutionarily unstable in the way that the mechanisms fail to explain the evolution of energetically costly secondary sex traits at the expense of energetically non-costly secondary sex traits. If we allow for genetic variation in the energetic cost of the secondary sex trait, and costly and non-costly traits are potentially equally suited as indicators for female choice, then the models predict the evolution of non-costly secondary sex traits. This contrast to many species that have an energetic cost associated with secondary sexual characters (Andersson, 1994). Energetically costly secondary sex traits that evolve by density dependent competitive interactions for other resources than mates, however, tend to be evolutionarily stable. The requirement is that the individuals that invest less energy in the secondary sex traits tend to lose during competitive interactions with individuals investing more energy. This condition is fulfilled when the secondary sex traits are direct measures of interactive ability.

Sex ratios

The sex ratio is the number of males per female. The classical sex ratio theory that arose from Fisher (1930) suggests that the sex ratio evolves by differences in the rates whereby male and female offspring copy genes into future generations through sexual reproduction. According to this process an even sex ratio should evolve because the rare sex has an advantage in sexual reproduction over the common sex and, thus, the success of the two sexes in sexual reproduction is even when the sex ratio is even. This process depends on the mating pattern, where random mating results in an even sex ratio (Fisher, 1930), while local mating results in a female biased sex ratio (Hamilton, 1967).

The classical sex ratio theory seems to hold on a local scale where the organism has a fixed mating pattern and a fixed degree of sexual reproduction. But on a longer time-scale, where the mating pattern and degree of sexual reproduction evolve by natural selection, the theory predicts the evolution of asexual variants with no males and no sex ratios. This failure is connected to the two-fold cost of the male and the two-fold cost of meiosis, where the classical sex ratio theory provides no benefit that will outweigh the costs. In Malthusian Relativity the paradox of an evolutionary unstable sex ratio theory is avoided. This is because the density dependent competitive interactions of high-energy organisms provides a fitness advantage that can outweigh both the two-fold cost of the male and the two-fold cost of meiosis.

Senescence and soma

Classical life-history theory suggests that senescence evolves from a somatic tissue from which no part is passed on in either sexual or asexual reproduction (Williams, 1957; reviewed by Rose, 1991). This hypothesis coincides with the presence of senescence in the large-bodied eukaryotes that have a soma, and its absence in the small-bodied prokaryotes that have no soma. But the hypothesis does not explain why the higher eukaryotes have a soma when prokaryotes do not.

Malthusian Relativity incorporates an alternative hypothesis where the soma is the trait through which senescence is expressed. When this hypothesis is combined with selection by density dependent competitive interactions it predicts that senescence and soma are unlikely to evolve in small-bodied organisms like prokaryotes, while senescence and soma is expected to evolve in large-bodied organisms like the higher eukaryotes.

Reproductive rates

The evolution of reproductive rates is usually modelled by variants of Lack's (1947) hypothesis on the evolution of an optimal clutch size in birds. This hypothesis suggests that the survival of parents and offspring will decline with the number of offspring produced so that number of offspring that will survive and reproduce is optimal at an intermediate reproductive rate.

Lack's hypothesis seems to holds at a local scale, where the organism has a fixed trade-off between the rate of reproduction and the rate of offspring and/or parent survival. But on a longer time scale, where the trade-off evolves by natural selection, the hypothesis predicts a continuous decline in the trade-off and a continuous increase in reproductive rate. This paradox of a continuously increasing reproductive rate may be solved by selection by density dependent competitive interactions. The latter model predicts a limited rate of reproduction that is balanced against the level of extrinsic mortality. The hypothesis also suggests that the trade-off between reproduction and survival is evolutionarily dependent upon the optimum to the population dynamic growth rate. This goes contrary to Lack's hypothesis that suggests that it is instead the optimal growth rate that is determined by the trade-off between reproduction and survival.

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