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In angiosperm plants the parental investment consists mainly of the endosperm, a nutritive tissue formed by genetic elements from both parents and used upon germination. We analyze a one-locus model of the evolution of the endosperm assuming that the alleles expressed in the endosperm determine the seed provisioning. In the model, we assume that large endosperm increases the probability of survival when young, but decreases seed set when the plant has reached the reproductive stage. We show that there is an evolutionarily stable strategy (ESS) of endosperm amount which is influenced by mating system, ploidy of the endosperm, paternity and genomically imprinted genes with parent-specific expression. The ESS value is higher than the value which maximizes the reproductive success of the plant. When the maternal genetic contribution is higher than the paternal (e.g. Polygonum plants), the ESS is almost always lower than when the parents have equal influence over the endosperm as in Oenothera plants. In both types, increasing the number of pollen donors to a seed crop selects for higher levels of endosperm. Accordingly, genes expressed only when inherited from the father are selected for higher endosperm amounts than genes expressed only when inherited from the mother, except when all seed have the same pollen parent. ESS values for imprinted genes do not differ between Polygonum and Oenothera types of plants. The ESS values are shown to be both locally and globally stable. The results are discussed in relation to evolutionary conflicts between the sexes.

The presence of an overcompensatory response to damage in some plant species has recently created a debate concerning whether this trait is an adaptation to herbivory, or simply a physiological consequence of adaptations to competition for light. According to the latter hypothesis, competition for light favors fast vertical growth and strong apical dominance. The removal of apical dominance by damaging the primary shoot allows the growth of secondary shoots and hence increases productivity. We compare predictions of these two hypotheses in a model-system where plants are exposed to both a risk of damage and a risk of competition. Compensatory seed production is assumed to depend on the number of dormant buds that can be activated by damage, and on the seed production of surviving shoots. In accordance with earlier theoretical analyses, we expect that intensive herbivory can favor overcompensatory seed production. In contrast, competition for light should at best lead to exact compensation when the competitive environment remains unchanged. Competition acts against overcompensation for two reasons. First, competitive plants should have poor resource reserves to support compensatory growth. Second, competition for light is assumed to favor unbranched architecture and thus. activation of many secondary shoots should not increase the seed yield. However, we cannot exclude the possibility that plants adapted to competition may overcompensate when grown singly. In spite of this caveat, it is likely that overcompensation requires damage related adaptations that may evolve only under intensive and relatively predictable risk of damage.

The regular cyclic population dynamics of herbivores has frustrated naturalists since the middle of the 16th century. Olaus Magnus, archbishop of Uppsala, Sweden, concluded already 1555 that the lemming populations showed oscillatory population changes (Olaus Magnus 1555) Furthermore, he suggested that the cyclic outbreaks of stoats and weasels could be caused by the 3-4-yr lemming cycle. Since then, several ecologists have speculated about the causes of these spectacular phenomena. For cyclic microtine populations, the hypotheses presented, may, according to Stenseth and Ims (1993) be divided into four main categories: (1) abiotic hypotheses (e.g. Elton 1924, Moran 1953a, b), (2) intrinsic factor hypotheses (e.g. Chitty 1967, Charnov and Finerty 1980), (3) prey-predator/pathogen interactions (e.g. Stenseth 1980, Ydenberg 1987, Anderson and May 1991), and (4) vegetation-herbivore interactions (e.g. Schultz 1969, Haukioja and Hakala 1975, Haukioja 1980). Obviously, this is a general classification that could be valid for all kinds of cyclic herbivore populations. Among the vegetation-herbivore hypotheses, we find the ones concerning food quality especially interesting. Green and Ryan (1972) suggested that plants could respond to herbivory by the production of proteinase inhibitors with adverse effects on herbivore performance (growth, survival and reproduction). These defence responses may, according to Haukioja and Hakala (1975), be a driving force of fluctuating herbivore populations. In a recent contribution, Seldal et al. (1994) propose that an inducible defence mechanism can be the cause of the population cycles exhibited by the lemming in the tundras of America, Fennoscandia and Russia. Their main findings are: (1) Physical damage to those plants that constitute the bulk of the lemmings' diet will induce the synthesis of large amounts of proteinase inhibitors, proteins that form irreversible complexes with trypsin and other proteolytic enzymes. This does not only effectively inhibit the protein digestion in the gut of most mammals, birds and insects, but also drain their reserves of essential amino acids that are excreted into the faeces (reviewed by e.g. Gallaher and Schneeman 1986, Ryan 1990). (2) Experiments have shown that proteinase inhibitors, when introduced with artificial food, have adverse effects on the growth and can eventually cause the death of mammals, birds and insects (reviewed by e.g. Gallaher and Schneeman 1986, Ryan 1990). Similar effects could be observed when young lemmings were given food collected in their natural habitat, if the samples were taken during the decline and trough phases of the lemming population cycle. The phytochemical induction by herbivory has been studied extensively by other workers. In the case of proteinase inhibitors, the general conclusion that can be drawn from these studies is that a plant can exhibit different levels of induction, depending on the size and number of damages (Green and Ryan 1972). A single damage event will trigger the production of inhibitors, but without further damages, the synthesis will cease and the concentration of these substances will decline in the plant tissue (Seldal et al. 1994). Repeated damages do, however, cause high levels of proteinase inhibitors that will last for extended periods (Green and Ryan 1972, Gustafson and Ryan 1976). Seldal et al. (1994) argue that this effect might cause a delay in the lemming population regulation, and that this could be sufficient for causing cyclic population dynamics. The more a plant has been damaged through grazing, the higher will its defence level be, and the longer will it remain in this induced state. In this note we propose a fairly simple mathematical model that encapsulate these properties, in order to investigate the hypothesis that inducible defences can cause oscillations in herbivore populations.