ESS germination strategies in randomly varying environments. I. Logistic-type models

ESS germination strategies are studied in a model of annual plant population dynamics in a randomly varying environment. The possible strategies are different values of the annual germination fraction G, either constant over time or varying in response to a "cue" correlated with upcoming environmental conditions. The model generalizes D. Cohen's model (1966, J. Theor. Biol. 12, 119-129; 1968, J. Ecol. 56, 219-228) by allowing density-dependent per capita seed yields. ESSs are characterized in terms of the resulting harmonic mean growth rate of population density. The ESS criterion cannot be solved analytically, but qualitative relationships between the value of the ESS and other population parameters are obtained, and environments in which 100% germination is an ESS are identified. Some explicit predictions of the theory are summarized and compared with ideas of M. Westoby (1981, Amer. Nat. 118, 882-885). The results of this study are compared with those of Cohen (op. cit.) in a companion paper.     

Evolutionarily stable strategies and short-term selection in Mendelian populations re-visited

This note concerns a one locus, two allele, random mating diploid population, subject to frequency-dependent viability selection. It is already known that in such a population, any evolutionarily stable strategies (ESS), if only accessible by the genotype-to-phenotype mapping, is the phenotypic image of a stable genetic equilibrium (Eshel, I. 1982. Evolutionarily stable strategies and viability selection in Mendelian populations. Theor. Popul. Biol. 22(2), 204-217; Cressman et al. 1996. Evolutionary stability in strategic models of single-locus frequency-dependent viability selection. J. Math. Biol. 34, 707-733). The opposite is not true. We find necessary and sufficient parametric conditions for global convergence to the ESS, but we also demonstrate conditions under which, although a unique, genetically accessible ESS exists, there is another, "non-phenotypic" genetically stable equilibrium.     

Simple signaling games of sexual selection (Grafen’s revisited)

We investigate several versions of a simple game of sexual selection, to explore the role of secondary sexual characters (the “handicap paradox”) with the tools of signaling theory. Our models admit closed form solutions. They are very much inspired by Grafen’s (J Theor Biol 144:517–546, 1990a; J Theor Biol 144:473–516, 1990b) seminal companion papers. By merging and simplifying his two approaches, we identify a not so minor artifact in the seminal study. We propose an alternative model to start with Grafen’s sexual selection theory, with several similarities with Getty (Anim Behav 56:127–130, 1998).     

Delayed germination of seeds: Cohen's model revisited

D. Cohen's (1966, J. Theor. Biol., 12, 110–129) model of delayed germination of seeds in a variable environment is extended to incorporate density dependence. It is shown that this has a considerable impact on the optimal germination rate, and the results are interpreted in terms of bet-hedging behavior. The effect of spatial dispersal is briefly considered.     

Evolutionary stability of plant communities and the maintenance of multiple dispersal types

S.A. Levin, D. Cohen, and A. Hastings (1984, Theor. Popul. Biol. 19, 169–200) and D. Cohen and S.A. Levin (1991, Theor. Popul. Biol. 39, 63–99) by analytic solution of the problem of invasion of a single dispersal type by any other, have provided a theory for evolutionarily stable strategies for seed dispersal in a random environment. Here the results of Cohen and Levin are extended to describe evolutionarily stable combinations of dispersal types. Such combinations of two types are coalitions that cannot be invaded by any other, although in isolation either of the types in the combination is invasible by others. These combinations appear when there is a negative correlation between the seed production of sites in successive years, or when environments are spatially heterogeneous, or presumably under other circumstances. In this work, we examine only the case of negative correlations. For this situation the configuration of evolutionarily stable strategies (ESS) and evolutionarily stable combinations (ESC) depends upon the ratio of (precompetitive) survival rates of dispersersing and nondispersing seeds, which is denoted by α. For low values of α, the purely nondispersing type is an ESS. At a somewhat higher value of α, the purely dispersing type can invade the nondispersing type, and the two types form an ESC, i.e., a combination that cannot be invaded by any other type. For still larger values of α, the purely nondispersing type is excluded by the ESC. Finally, for the largest values of α, pure dispersal is the ESS. In cases where a single dispersal type cannot exclude all others, the stationary distribution of types has a large spread. It can be adequately approximated by equations for conditional means of the proportions of various types at a site of a given quality, but these means must be conditioned upon the prior history at each site. For some purposes we have found that the history of as many as 8–10 generations is required for a good approximation. This phenomenon appears to preclude simple analytic approximations for the ESC.     

种子萌发对策:理论与实验

植物种子的萌发／休眠现象有复杂的原因和机制,综述了理论生态学家的研究结果。应用的理论基础是最优化理论和进化稳定对策(Ess)理论。当环境条件随机波动,种群受非密度依赖因素调节时,采用最优化理论的两头下注对策预测休眠一定会得到进化且萌发率与环境条件直接相关。环境条件稳定时采用进化稳定对策理论可得到在亲属竞争,种子扩散,基因冲突等等因素影响下的进化稳定休眠／萌发率,预测了休眠／萌发与它们之间的相互关系。以上各种环境条件影响种子萌发行为的方式可以表述为若种子立即萌发会遭遇到不良环境使适合度下降,那么就会推迟萌发,出现休眠,形成土壤种子库。萌发率应使种群适合度最优或具有进化稳定性。一些实验也部分验证了理论预测。     

Seed Germination in Desert Annuals: An Empirical Test of Adaptive Bet Hedging

Temporal variability in survivorship and reproduction is predicted to affect the evolution of life-history characters. Desert annual plants experience temporal variation in reproductive success that is largely caused by precipitation variability. We studied several populations of the desert annual Plantago insularis along a precipitation gradient. Whereas models of bet hedging in unpredictable environments generally predict one optimal germination fraction for a population, empirical studies have shown that environmental conditions during germination can cause a range of germination fractions to be expressed. In a 4-yr field study, we found that populations in historically more xeric environments had lower mean germination fractions, as is predicted by bet-hedging models. However, populations exhibited significant variation in germination among years. Two experimental studies measuring germination under several environment conditions were conducted to elucidate the source of this in situ variation. Germination fractions exhibited phenotypic plasticity in response to water availability and date within the season. Populations differed in their norms of reaction such that seeds from more xeric populations germinated under less restrictive conditions. A pattern of delayed germination consistent with among-year bet-hedging predictions arose in the field through the interaction of seed germinability and the distribution of environmental conditions during germination.     

The exponential model for a regulatory enzyme. An interpretation of the linear free-energy relationship.

A physical mechanism is suggested to explain the linear free-energy relationship employed in the exponential model for a regulatory enzyme [Ainsworth (1977) J. Theor. Biol. 68, 391-413]. The interpretation depends on the assumption that the structure of the enzyme changes in proportion to its saturation by substrate but at a rate that is low compared with the rates of the association-dissociation reactions of the enzyme-substrate system.     

The evolution of n-player cooperation-threshold games and ESS bifurcations

An evolutionary game of individuals cooperating to obtain a collective benefit is here modelled as an n-player Prisoner's Dilemma game. With reference to biological situations, such as group foraging, we introduce a threshold condition in the number of cooperators required to obtain the collective benefit. In the simplest version, a three-player game, complex behaviour appears as the replicator dynamics exhibits a catastrophic event separating a parameter region allowing for coexistence of cooperators and defectors and a region of pure defection. Cooperation emerges through an ESS bifurcation, and cooperators only thrive beyond a critical point in cost-benefit space. Moreover, a repelling fixed point of the dynamics acts as a barrier to the introduction of cooperation in defecting populations. The results illustrate the qualitative difference between two-player games and multiple player games and thus the limitations to the generality of conclusions from two-player games. We present a procedure to find the evolutionarily stable strategies in any n-player game with cost and benefit depending on the number of cooperators. This was previously done by Motro [1991. Co-operation and defection: playing the field and the ESS. J. Theor. Biol. 151, 145-154] in the special cases of convex and concave benefit functions and constant cost.     

Multiple switches between vegetative and reproductive growth in annual plants

A model of growth and reproduction in annual plants was developed by Cohen (1971, J. Theor. Biol.33, 299–307) to determine the allocation strategy which maximizes seed yield. The model divides the plant into vegetative and reproductive parts and predicts that yield is maximized by a strategy consisting of a switch from purely vegetative to strictly reproductive growth. We generalize Cohen's model to include vegetative and reproductive loss terms. Both growth and loss rates are allowed to vary with time. Using optimal control theory we find that seed yield is maximized by a strategy consisting of multiple switches between vegetative and reproductive growth, for certain ranges of the model parameters. In natural systems a predictable vegetative loss burst may be necessary to promote multiple switches.     

More on recombination and selection in the modifier theory of sex-ratio distortion

G. Maffi and S.D. Jayakar suggested a model for the two-locus control of sex determination in the mosquito Aedes aegypti (1981, Theor. Pop. Biol. 19, 19-36). This model was extended to multiple alleles and analyzed in mathematical detail by S. Lessard (1987, Theor. Pop. Biol. 31, 339-358). The model supposes that males are "Mm" and females "mm" but the transmission from males is controlled by a second gene with alleles Ai. We show that in addition to the equilibrium in which mAi in females, MAi from males and mAi from males all have the same frequencies, a second class of polymorphic equilibria exists and can be stable. The former class was shown by Lessard to be stable for intermediate and/or loose linkage. The new class of equilibria may be stable for tight linkage under the conditions that preclude stability of the former. We also develop the theory of linkage modification from the neighborhood of the new equilibrium. Successful modifiers of recombination may either reduce or increase the recombination fraction with the outcome depending on the linkage of the modifier to the major genes.     

Dynamics and stability in coevolutionary ecological systems. I. Community stability and coevolutionarily stable states

An extension of J. Roughgarden's [1979, Theor. Pop. Biol. 9, 388; 1979, "An Introduction to Evolutionary Ecology and Population Genetic Theory," Macmillan, New York] formalism for investigating the effects of coevolution on community structure is presented. The extension assumes the result that a coevolved community is asymptotically stable when coevolution takes place at a genetically noninvasible boundary. This is proved for the general case of n interacting species. From this a community persistence function, phi (P), is defined that allows measuring the domain of attraction for the community as well as the resilience time, that is, the time taken for a perturbation to decay to 1-1/e (63%) of its initial value.     

Protein composition of cotyledons and aleurone grains inLupinus luteus L. Seeds during Germination

The change in protein composition of whole cotyledons and cotyledon aleurone grains ofLupinus luteus L. during seed germination was studied. SDS-polyacrylamide gel electro-phoresis showed a clear change in composition of cotyledon proteins as well as in composition of the aleurone grains during 5 days of seed germination. At this time, both in whole cotyledons as well as in aleurone grains, two subunits of β-conglutin with mol. m. 53 000 and 39 000 were rapidly hydrolyzed. After 5 days of germination traces of α-conglutin subunits could be detected in the cotyledons, whereas in aleurone grains this globulin fraction disappeared. In whole cotyledons and in cotyledon aleurone grains the γ-conglutin subunits with mol. m. 28 000 and 17 000 were not mobilized during the study period. These results indicate that the protein components with lower mol. m. were degraded later than those withhigher mol. m. during seed germination.     

The rise and fall of handicap principle: a commentary on the “Modelling and the fall and rise of the handicap principle”

The honesty of animal communication is in the spot lights in the last 30 years. During most of this time the field was dominated by one explanation: Zahavi’s handicap principle (Zahavi, J Theor Biol 67:603–605, 1975; Grafen, J Theor Biol 144:517–546, 1990). Grose (Biol Philos 2011) embarks to explain both the success of the theory and the empirical difficulties that exist despite this success. While I wholeheartedly agree with the criticism offered by Grose and with almost all the claims he makes, the treatment of the issue is far from complete and it still leaves much to be explained. Accordingly, my commentary consist of six sections: in the first section I clear up some technical issues left unexposed, most importantly the role of strategic cost in handicap signalling; in the second section I relate this to empirical testing; in the next section I comment on the historical development of the handicap principle; in the fourth section I review the biological models that came up with conclusions that contradict the handicap principle; in the fifth section I discuss the reasons behind the success of the handicap theory; finally, in the last section I discuss the application of the handicap theory to anthropology and human sciences.     

EXPERIMENTAL STUDIES OF THE EVOLUTIONARY SIGNIFICANCE OF SEXUAL REPRODUCTION. III. MATERNAL AND PATERNAL EFFECTS DURING SEEDLING ESTABLISHMENT

To determine whether genetic differences in fitness components exist among seeds and seedlings in a natural population, weighed propagules of six parents of Anthoxanthum odoratum from a reciprocal diallel cross were planted into the parental source population, a mown field. Seed families of maternal genotypes differed in germination success, while paternal families showed no detectable differences. Differential germination success could not be attributed to propagule weight. Seed families ranked differently in germination percentage in different blocks. No survivorship differences among parental seed families could be detected. There were significant cross × block × germination and cross × block × survivorship interactions; different crosses performed better or worse in different blocks. In some cases, crosses sired by different fathers within a maternal seed family differed in germination or survivorship, suggesting that natural selection may be capable of discriminating among juvenile genotypes within a maternal family despite the presence of large overall maternal effects. These results indicate that seedling establishment may differ according to genotype and that microsite heterogeneity may maintain genetic variation in juvenile traits in natural plant populations.     

Levels of aminoacyl-tRNA synthetases, tRNA nucleotidyltransferase and ATP in germinating lupin seeds

Transfer RNAs in dry lupin seeds are aminoacylated to a low extent (Kedzierski, W. and Pawe?kiewicz, J. (1977) Phytochemistry 16, 503-504) and are partly degraded at the acceptor terminus (Dziegielewski, T. and Pawe?kiewicz, J. (1977) Bull. Acad. Polon. Sci. Ser. Biol. 7, 4oo-435). Increase in the levels of tRNA aminoacylation and disappearance of defective tRNA molecules during seed germination are not accompanied by significant changes in the levels of phenylalanyl-, arginyl-, valyl-tRNA synthetases and tRNA nucleotidyltransferase. Additionally, no inhibitor of aminoacylation of valine tRNA has been detected in dry seeds. However, dry seeds contain very low ATP amounts, which increase dramatically during germination. The above results suggest that a very low ATP level is a factor limiting the aminoacylation and reparation of tRNA molecules at early stages of seed germination.     

Resistance to inactivation by EGTA of the enzyme-substrate and enzyme-phosphate complexes of alkaline phosphatase.

Bovine intestinal mucosal alkaline phosphatase is inactivated by the chelating agent EGTA. Several concentrations of the enzyme were incubated with EGTA and a range of concentrations of the substrate p-nitrophenyl phosphate to determine the substrate concentration as a function of time. As predicted by a recently developed theory [Duggleby (1986) J. Theor. Biol. 123, 67-80], catalysis ceases before all substrate is exhausted. An analysis of these final substrate concentrations according to the theory revealed that, whereas the free enzyme is unstable, the effect of EGTA is counteracted when either the substrate or product (phosphate) is bound. Comparison of the results with those obtained by direct stability measurements and steady-state kinetic experiments gave a qualitatively and quantitatively consistent body of evidence in support of this interpretation.     

The stochastic modelling of kleptoparasitism using a Markov process

Kleptoparasitism, the stealing of food items from other animals, is a common behaviour observed across a huge variety of species, and has been subjected to significant modelling effort. Most such modelling has been deterministic, effectively assuming an infinite population, although recently some important stochastic models have been developed. In particular the model of Yates and Broom (Stochastic models of kleptoparasitism. J. Theor. Biol. 248 (2007), 480-489) introduced a stochastic version following the original model of Ruxton and Moody (The ideal free distribution with kleptoparasitism. J. Theor. Biol. 186 (1997), 449-458), and whilst they generated results of interest, they did not solve the model explicitly. In this paper, building on methods used already by van der Meer and Smallegange (A stochastic version of the Beddington-DeAngelis functional response: Modelling interference for a finite number of predators. J. Animal Ecol. 78 (2009) 134-142) we give an exact solution to the distribution of the population over the states for the Yates and Broom model and investigate the effects of some key biological parameters, especially for small populations where stochastic models can be expected to differ most from their deterministic equivalents.     

Alternate plant life history strategies and coexistence in randomly varying environments

Environmental fluctuations can in theory allow the coexistence ofecologically similar species by time-sharing a niche, as envisioned by Hutchinson. The evolution of this situation is studied in a competition model, using as an example the evolution of seed germination strategies. Coexistence occurs via the evolution of low-risk and high-risk strategies for dealing with the variability by different species. Coexistence is promoted by intermediate levels of variability or disturbance, and by a trade-off between seed yield and seed survivorship. These results may be applicable also to other low vs. high risk life history options in unpredictably varying environments, such as: stress resistance vs. potentially rapid growth, high adult survivorship vs. high reproductive output. The model's predictions differ from those obtained without consideration of life history evolution in response to environmental variability, and are consistent with some recent studies of plant strategies in intermittently stressed communities.I thank A. Shmida for many discussions on this topic, D. Cohen and I. Noy-Meir for comments after a seminar presentation of this paper, and H. de Kroon, H. During, and E. Van der Maarel for decreasing my ignorance of the empirical literature.Research conducted while the author was recipient of a Sir Charles Clore Postdoctoral Fellowship in the Department of Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel.