Can optimal resource allocation models explain why ectotherms grow larger in cold?

Basically all organisms can be classified as determinate growers if their growth stops or almost stops at maturation, or indeterminate growers if growth is still intense after maturation. Adult size for determinate growers is relatively well defined, whereas in indeterminate growers usually two measures are used: size at maturation and asymptotic size. The latter term is in fact not a direct measure but a parameter of a specific growth equation, most often Bertalanffy's growth curve. At a given food level, the growth rate in determinate growers depends under given food level on physiological constraints as well as on investments in repair and other mechanisms that improve future survival. The growth rate in indeterminate growers consists of two phases: juvenile and adult. The mechanisms determining the juvenile growth rate are similar to those in determinate growers, whereas allocation to reproduction (dependent on external mortality rate) seems to be the main factor limiting adult growth. Optimal resource allocation models can explain the temperature-size rule (stating that usually ectotherms grow slower in cold but attain larger size) if the exponents of functions describing the size-dependence of the resource acquisition and metabolic rates change with temperature or mortality increases with temperature. Emerging data support both assumptions. The results obtained with the aid of optimization models represent just a rule and not a law: it is possible to find the ranges of production parameters and mortality rates for which the temperature-size rule does not hold.     

Twelve fundamental life histories evolving through allocation‐dependent fecundity and survival

An organism's life history is closely interlinked with its allocation of energy between growth and reproduction at different life stages. Theoretical models have established that diminishing returns from reproductive investment promote strategies with simultaneous investment into growth and reproduction (indeterminate growth) over strategies with distinct phases of growth and reproduction (determinate growth). We extend this traditional, binary classification by showing that allocation‐dependent fecundity and mortality rates allow for a large diversity of optimal allocation schedules. By analyzing a model of organisms that allocate energy between growth and reproduction, we find twelve types of optimal allocation schedules, differing qualitatively in how reproductive allocation increases with body mass. These twelve optimal allocation schedules include types with different combinations of continuous and discontinuous increase in reproduction allocation, in which phases of continuous increase can be decelerating or accelerating. We furthermore investigate how this variation influences growth curves and the expected maximum life span and body size. Our study thus reveals new links between eco‐physiological constraints and life‐history evolution and underscores how allocation‐dependent fitness components may underlie biological diversity.     

Growth after maturity as a sub-optimal strategy

Patterns of optimal resource allocation to growth and reproduction were investigated using a numerical simulation. As in most previous analyses, cessation of growth when reproduction begins (the determinate strategy) appeared optimal. Here, it was additionally found that fitness was only slightly lower for individuals that continue to grow after maturation. Therefore, it is argued that selection for a determinate strategy may be too weak to overwhelm random processes like environmental stochasticity or genetic drift that shape patterns of growth, especially under low mortality. The consequences of an indeterminate strategy for optimal size at maturity and final size were investigated: prolonging the period in which growth and reproduction co-occurred decreased size at maturity only slightly but markedly increased the final size.     

Unexpected discontinuities in life-history evolution under size-dependent mortality

In many organisms survival depends on body size. We investigate the implications of size-selective mortality on life-history evolution by introducing and analysing a new and particularly flexible life-history model with the following key features: the lengths of growth and reproductive periods in successive reproductive cycles can vary evolutionarily, the model does not constrain evolution to patterns of either determinate or indeterminate growth, and lifetime number and sizes of broods are the outcomes of evolutionarily optimal life-history decisions. We find that small changes in environmental conditions can lead to abrupt transitions in optimal life histories when size-dependent mortality is sufficiently strong. Such discontinuous switching results from antagonistic selection pressures and occurs between strategies of early maturation with short reproductive periods and late maturation with long reproductive cycles. When mortality is size-selective and the size-independent component is not too high, selection favours prolonged juvenile growth, thereby allowing individuals to reach a mortality refuge at large body size before the onset of reproduction. When either component of mortality is then increased, the mortality refuge first becomes unattractive and eventually closes up altogether, resulting in short juvenile growth and frequent reproduction. Our results suggest a new mechanism for the evolution of life-history dimorphisms.     

How optimal life history changes with the community size-spectrum

This paper derives optimal life histories for fishes or other animals in relation to the size spectrum of the ecological community in which they are both predators and prey. Assuming log-linear size-spectra and well known scaling laws for feeding and mortality, we first construct the energetics of the individual. From these we find, using dynamic programming, the optimal allocation of energy between growth and reproduction as well as the trade-off between offspring size and numbers. Optimal strategies were found to be strongly dependent on size spectrum slope. For steep size spectra (numbers declining rapidly with size), determinate growth was optimal and allocation to somatic growth increased rapidly with increasing slope. However, restricting reproduction to a fixed mating season changed optimal allocations to give indeterminate growth approximating a von Bertalanffy trajectory. The optimal offspring size was as small as possible given other restrictions such as newborn starvation mortality. For shallow size spectra, finite optimal maturity size required a decline in fitness for large size or age. All the results are compared with observed size spectra of fish communities to show their consistency and relevance.     

Optimal growth strategies under divergent predation pressure

The conditions leading to gigantism in nine‐spined sticklebacks Pungitius pungitius were analysed by modelling fish growth with the von Bertalanffy model searching for the optimal strategy when the model's growth constant and asymptotic fish size parameters are negatively related to each other. Predator‐related mortality was modelled through the increased risk of death during active foraging. The model was parameterized with empirical growth data of fish from four different populations and analysed for optimal growth strategy at different mortality levels. The growth constant and asymptotic fish size were negatively related in most populations. Optimal fish size, fitness and life span decreased with predator‐induced mortality. At low mortality, the fitness of pond populations was higher than that of sea populations. The differences disappeared at intermediate mortalities, and sea populations had slightly higher fitness at extremely high mortalities. In the scenario where all populations mature at the same age, the pond populations perform better at low mortalities and the sea populations at high mortalities. It is concluded that a trade‐off between growth constant and asymptotic fish size, together with different mortality rates, can explain a significant proportion of body size differentiation between populations. In the present case, it is a sufficient explanation of gigantism in pond P. pungitius.     

Dietary restriction in two rotifer species: the effect of the length of food deprivation on life span and reproduction

According to resource allocation theory, animals face a trade off between the allocation of resources into reproduction and into individual growth/maintenance. This trade off is reinforced when food conditions decline. It is well established in biological research that many animals increase their life span when food is in suboptimal supply for growth and/or reproduction. Such a situation of reduced food availability is called dietary restriction. An increase in life span under dietary restricted conditions is seen as a strategy to tolerate periods of food shortage so that the animals can start reproduction again when food is in greater supply. In this study, the effect of dietary restriction on life span and reproduction in two rotifer species, Cephalodella sp. and Elosa worallii, was investigated using life table experiments. The food concentration under dietary restricted conditions was below the threshold for population growth. It was (1) tested whether the rotifers start reproduction again after food replenishment, and (2) estimated whether the time scale of dietary restricted conditions is relevant for the persistence of a population in the field. Only E. worallii responded to dietary restriction with an increase in life span at the expense of reproduction. After replenishment of food, E. worallii started to reproduce again within 1 day. With an increase in the duration of dietary restricted conditions of up to 15 days, which is longer than the median life span of E. worallii under food saturation, the life span increased and the life time reproduction decreased. These results suggest that in a temporally (or spatially) variable environment, some rotifer populations can persist even during long periods of severe food deprivation.     

Trade‐offs between life‐history traits at range‐edge and central locations

The allocation of resources to different life‐history traits should represent the best compromise in fitness investment for organisms in their local environment. When resources are limiting, the investment in a specific trait must carry a cost that is expressed in trade‐offs with other traits. In this study, the relative investment in the fitness‐related traits, growth, reproduction and defence were compared at central and range‐edge locations, using the seaweed Ascophyllum nodosum as a model system. Individual growth rates were similar at both sites, whereas edge populations showed a higher relative investment in reproduction (demonstrated by a higher reproductive allocation and extended reproductive periods) when compared to central populations that invested more in defence. These results show the capability of A. nodosum to differentially allocate resources for different traits under different habitat conditions, suggesting that reproduction and defence have different fitness values under the specific living conditions experienced at edge and central locations. However, ongoing climate change may threaten edge populations by increasing the selective pressure on specific traits, forcing these populations to lower the investment in other traits that are also potentially important for population fitness.     

Optimal individual growth and reproduction in perennial species with indeterminate growth

Summary A model predicting optimal timing of growth and reproduction in perennial species with indeterminate growth living in a seasonal environment, is presented. According to the model, the optimal fraction of growing season devoted to growth decreases with increasing individual age and size, which leads to S-shaped growth curves. Winter mortality seems to be a crucial factor affecting the timing of growth and reproduction, under the same function describing the dependence of growth rate and reproductive rate on body size. When winter mortality is heavy, it is often optimal to start reproducing in the first year, and to devote a large proportion of the subsequent years to reproduction, thus leading to small adult body sizes.The model has been applied to two species of mollusc and one species of fish. The model predictions fit well to the field data for these three species.     

The effect of different light regimes on adult life span in Drosophila melanogaster is partly mediated through reproductive output

The effects of different light regimes on the fitness of organisms have typically been studied using mean or median adult life span as the sole index of physiological well-being. It is, however, known that life span is inversely related to reproductive output in many species. Moreover, the effects of a given environmental treatment on life span can be due to effects on either age-independent mortality or the "rate of aging," or a combination of both. Drawing evolutionary inferences from the effects of light regime on mean or median adult life span alone is difficult and, at best, speculative. We examined the effects of constant light (LL), alternating light-dark cycles (LD 12:12 h), and constant darkness (DD) on the life span of reproducing and virgin flies in four populations of Drosophila melanogaster and also estimated lifetime fecundity in the three light regimes. The light regime effects on life span were further dissected by examining the age-independent mortality and the Gompertz rate of aging under the three light regimes. While mean adult life span of reproducing males and females and virgin females was significantly shorter in LL compared to LD 12:12 h and DD, life-time egg production was highest in LL. Life span of virgin males was not significantly affected by light regime. The rate of aging in reproducing females was higher in LL as compared to DD, whereas age-independent mortality was higher in DD. As reproductive output, especially early in life, is a far more significant contributor to fitness than is life span, our results suggest that the earlier reported deleterious effects of LL on fitness are partly an artifact of examining life span alone, without considering other components of adult fitness that trade off with life span. Our results suggest that detailed investigation of the effects of light regime on the physiological and behavioral processes that accompany reproduction is necessary to fully understand the effects of different light regimes on adult fitness in Drosophila.     

Growth and reproduction of fungal feeding Collembola as affected by fungal species,melanin and mixed diets

Fungal feeding soil invertebrates feed on a wide spectrum of fungal species suggesting that mixed diets increase fitness. We investigated relationships between food preferences for seven saprophytic fungal species/forms and fitness parameters (mortality, growth, time to reproduction, reproduction, egg size) in two Collembola species, Folsomia candida and Protaphorura armata. The fungal species/forms studied included the wild type and a melanin-deficient form of Aspergillus fumigatus to investigate the role of melanin in collembolan nutrition. Also, three mixed diets consisting of a preferred fungal species (Cladosporium cladosporioides) and species of intermediate or low food quality were investigated. Both Collembola species preferred similar fungal species/forms as food. Food preference generally matched fitness parameters, i.e. growth and reproduction of Collembola was at a maximum when feeding on preferred fungi. This was not the case for A. fumigatus. The wild type and the melanin-deficient form ranked among the least preferred fungi. Growth and reproduction of Collembola were low when feeding on the wild type but high when feeding on the melanin-deficient form indicating that the Collembola misjudged the food quality of the latter in the preference tests. The results show for the first time that genes driving melanin syntheses (pksP) strongly affect the food quality of fungi for fungal feeding invertebrates. Feeding on mixed diets generally increased growth and reproduction of Collembola except when the diets included toxic species (Penicillium sp.). The results support the nutrient balance hypothesis and also show that the detection of toxic species in the diet is important. They indicate that the widespread generalist feeding mode of Collembola maximizes fitness if toxic fungal species are avoided. The fitness parameters growth, reproduction and time until onset of reproduction were correlated closely but egg volume, which also varied with fungal diet, correlated poorly with the other fitness parameters. Variation in egg size with fungal diet shows that the diet of Collembola may have transgenerational effects.     

Survival benefits select for group living in a social spider despite reproductive costs

The evolution of cooperation requires benefits of group living to exceed costs. Hence, some components of fitness are expected to increase with increasing group size, whereas others may decrease because of competition among group members. The social spiders provide an excellent system to investigate the costs and benefits of group living: they occur in groups of various sizes and individuals are relatively short-lived, therefore life history traits and Lifetime Reproductive Success (LRS) can be estimated as a function of group size. Sociality in spiders has originated repeatedly in phylogenetically distant families and appears to be accompanied by a transition to a system of continuous intra-colony mating and extreme inbreeding. The benefits of group living in such systems should therefore be substantial. We investigated the effect of group size on fitness components of reproduction and survival in the social spider Stegodyphus dumicola in two populations in Namibia. In both populations, the major benefit of group living was improved survival of colonies and late-instar juveniles with increasing colony size. By contrast, female fecundity, female body size and early juvenile survival decreased with increasing group size. Mean individual fitness, estimated as LRS and calculated from five components of reproduction and survival, was maximized for intermediate- to large-sized colonies. Group living in these spiders thus entails a net reproductive cost, presumably because of an increase in intra-colony competition with group size. This cost is traded off against survival benefits at the colony level, which appear to be the major factor favouring group living. In the field, many colonies occur at smaller size than expected from the fitness curve, suggesting ecological or life history constraints on colony persistence which results in a transient population of relatively small colonies.     

Life history variation in Pisidium (Bivalvia: Pisidiidae)

Intraspecific variation in the maximum shell length of Pisidium is small in comparison with variation in the life span of the clams, from 4 mo to 4 yr. The use of shell length as a measure of size is complicated by large intraspecific variation in the weight-length relationship, a possible reflection of resource availability on the weight of the soft parts. Maximum embryo length is relatively constant (1 mm) in the majority of species but litter size is variable (1 to 40), generally increasing with parent size. The time of egg-laying is determined by the size of the clam, season, and previous episode of reproduction, while the timing of the release of embryos, depending also on embryonic growth rate, is additionally affected by temperature and oxygen conditions. In brief, the maximum adult and embryo lengths are relatively conservative traits while growth rates vary by an order of magnitude.
Reproductive effort, when measured by the ratio of litter weight to parent weight, may increase or decrease with parent length and age. Both semelparous and iteroparous populations are known in several species. We suggest that if maximum adult length can be reached by the first reproductive season (in a favourable environment) the population is semelparous, otherwise (in an unfavourable environment) the population is iteroparous. Gravid clams in semelparous populations continue to grow, and thus assimilation cannot be meaningfully partitioned into reproduction, somatic growth and maintenance.
Along a depth gradient from 8 to 65 m, maximum and mature shell lengths, embryo length and litter size varies significantly in Pisidium conventus subpopulations not distinguishable electrophoretically. Large clams in deep water produce large embryos but small litters, which is contrary to the general trend of increasing litter size with parent length. Low food availability that decreases juvenile growth rate may restrict litter size in this case.     

NATURAL GENETIC VARIATION OF LIFE SPAN,REPRODUCTION, AND JUVENILE GROWTH IN DAPHNIA

The evolutionary theory of senescence predicts that high extrinsic mortality in natural populations should select for accelerated reproductive investment and shortened life span. Here, we test the theory with natural populations of the Daphnia pulex-pulicaria species complex, a group of freshwater zooplankton that spans an environmental gradient of habitat permanence. We document substantial genetic variation in demographic life-history traits among parent and hybrid populations of this complex. Populations from temporary ponds have shorter life spans, earlier and faster increases of intrinsic mortality risk, and earlier and steeper declines in fecundity than populations from permanent lakes. We also examine the age-specific contribution to fitness, measured by reproductive value, and to expected lifetime reproduction; these traits decline faster in populations from temporary ponds. Despite having more rapid senescence, pond Daphnia exhibit faster juvenile growth and higher early fitness, measured as population growth rate (r). Among populations within this species complex we observed negative genetic correlations between r and indices of life-history timing, suggesting trade-offs between early- and late-life performance. Our results cannot be explained by a trade-off between survival and fecundity or by nonevolutionary theories of senescence. Instead, our data are consistent with the evolutionary theory of senescence because the genetic variation in life histories we observed is roughly congruent with the temporal scale of environmental change in the field.     

Evolution of body size: an optimization model

The growth of individuals is often described by bioenergetic equations which partition assimilated energy into maintenance, growth, reproduction, and so on. Such energy flows vary with body size. The bioenergetic equations in this paper, under the assumptions that organisms will adopt behavior that channels a maximum amount of energy into the production of surviving progeny, produce a model optimizing growth and reproduction. Using the Weierstrass theorem, we show that a solution of the optimization problem exists. The problem is further solved analytically using the Pontryagin maximum principle. The main conclusion of the paper is that in a given environment an optimal body size exists, one which maximizes the energy channeled to the production of progeny. This body size depends on the mortality rate, the maximum life span, and the derivative of the growth equation with respect to body size. The biological results predicted from the model are compared with ecological data for zooplankton and vertebrate species, which support the conclusions obtained.     

The Role of Life Cycle and Migration in Selection for Variance in Offspring Number

For two genotypes that have the same mean number of offspring but differ in the variance in offspring number, naturalselection will favor the genotype with lower variance. In such cases, the average growth rate is not sufficient as a measure of fitness or as a predictor of fixation probability. However, the effect of variance in offspring number on the fixationprobability of mutant strategies has been calculated under several scenarios with the general conclusion that variance in offspring number reduces fitness in proportion to the inverse of the population size [Gillespie, J., Genetics 76:601–606, 1974; Proulx, S.R., Theor. Popul. Biol. 58:33–47, 2000]. This relationship becomes more complicated under a metapopulation scenario where the “effective” population size depends on migration rate, population structure, and lifecycle. It is shown that in a life cycle where reproduction and migration (the birth-migration-regulation life cycle, or BMR)occur prior to density regulation within every deme, the fitness of a strategy depends on migration rate. When migration rates are near zero, the fitness of the strategy is determined by the size of individual demes, so that the strategy favoredin small populations tends to be fixed. As migration rate increases and approaches panmixis between demes, the fitness ofa reproductive strategy approaches what its value would be in a single, panmictic deme with a population size correspondingtothe census size of the metapopulation. Interestingly, when the life cycle is characterized by having density regulation in each deme prior to migration (the BRM life cycle) the fixation probability of a strategy is independent of migration rate. These results are found to be qualitatively consistent with the individual-based simulation results in Shpak [Theor. Biosci.124:65–85, 2005]. An erratum to this article can be found at     

Age-specific demography in Plantago: uncovering age-dependent mortality in a natural population

Accurate measures of age-dependent mortality are critical to life-history analysis and measures of fitness, yet these measures are difficult to obtain in natural populations. Age-dependent mortality patterns can be obscured not only by seasonal variation in environmental conditions and reproduction but also by changes in the heterogeneity among individuals in the population over time due to selection. This study of Plantago lanceolata uses longitudinal data from a field study with a large number of individuals to develop a model to estimate the shape of the baseline hazard function that represents the age-dependent risk of mortality. The model developed here uses both constant (genetics, spatial location) and time-varying (temperature, rainfall, reproduction, size) covariates not only to estimate the underlying mortality pattern but also to demonstrate that the risk of mortality associated with fitness components can change with time/age. Moreover, this analysis suggests that increasing size after reproductive maturity may allow this plant species to escape from demographic senescence.     

Indeterminate growth is selected by a trade-off between high fecundity and risk avoidance in stochastic environments

 We developed a stage-structured model to describe optimal energy allocation among growth, reproduction, and survival. Our model includes stochastic fluctuations in survival rate at age 0 but constant survival rate at older ages. Many mammals and birds cease to grow after maturity (i.e., determinate growth), whereas organisms in a number of other taxa grow beyond maturation (i.e., indeterminate growth). We discuss the conditions under which each of the following strategies is optimal: (I) semelparity, (II) iteroparity with determinate growth, and (III) iteroparity with indeterminate growth. Our model demonstrates that iteroparity with indeterminate growth is selected for when a nonlinear relationship exists between weight and energy production; this strategy is also often selected for in stochastic environments, even with a linear relationship between weight and energy production. The optimal strategy in stochastic environments is to maximize the long-term population growth rate, which does not correspond with maximization of total fecundity. The optimal life history is determined by a balance between spreading a risk and increasing the number of offspring. Our model suggests that optimal life history strategy depends on the magnitude of environmental fluctuations, the advantage of investing in growth, the cost of survival, and the nonlinearity between weight and energy production. Received: February 20, 2002 / Accepted: September 20, 2002 Acknowledgments We thank Drs. Y. Matsumiya, K. Morita, K. Shirakihara, and Y. Watanabe for encouragement and helpful advice. We also thank the responsible editor and anonymous reviewers for helpful comments. This work was supported by a Japan Society for the Promotion of Science grant to H.M. Correspondence to:Y. Katsukawa     

Heuristic optimization of the general life history problem: A novel approach

Summary The general life history problem concerns the optimal allocation of resources to growth, survival and reproduction. We analysed this problem for a perennial model organism that decides once each year to switch from growth to reproduction. As a fitness measure we used the Malthusian parameterr, which we calculated from the Euler-Lotka equation. Trade-offs were incorporated by assuming that fecundity is size dependent, so that increased fecundity could only be gained by devoting more time to growth and less time to reproduction. To calculate numerically the optimalr for different growth dynamics and mortality regimes, we used a simplified version of the simulated annealing method. The major differences among optimal life histories resulted from different accumulation patterns of intrinsic mortalities resulting from reproductive costs. If these mortalities were accumulated throughout life, i.e. if they were senescent, a bangbang strategy was optimal, in which there was a single switch from growth to reproduction: after the age at maturity all resources were allocated to reproduction. If reproductive costs did not carry over from year to year, i.e. if they were not senescent, the optimal resource allocation resulted in a graded switch strategy and growth became indeterminate. Our numerical approach brings two major advantages for solving optimization problems in life history theory. First, its implementation is very simple, even for complex models that are analytically intractable. Such intractability emerged in our model when we introduced reproductive costs representing an intrinsic mortality. Second, it is not a backward algorithm. This means that lifespan does not have to be fixed at the begining of the computation. Instead, lifespan itself is a trait that can evolve. We suggest that heuristic algorithms are good tools for solving complex optimality problems in life history theory, in particular questions concerning the evolution of lifespan and senescence.     

Optimal growth strategies when mortality and production rates are size-dependent

Summary Pontryagin's maximum principle from optimal control theory is used to find the optimal allocation of energy between growth and reproduction when lifespan may be finite and the trade-off between growth and reproduction is linear. Analyses of the optimal allocation problem to date have generally yielded bang-bang solutions, i.e. determinate growth: life-histories in which growth is followed by reproduction, with no intermediate phase of simultaneous reproduction and growth. Here we show that an intermediate strategy (indeterminate growth) can be selected for if the rates of production and mortality either both increase or both decrease with increasing body size, this arises as a singular solution to the problem. Our conclusion is that indeterminate growth is optimal in more cases than was previously realized. The relevance of our results to natural situations is discussed.