Herbivory, inducible defence and population oscillations : a preliminary theoretical analysis

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.