Chaotic mating systems

Research Update

TRENDS in Ecology & Evolution Vol.17 No.11 November 2002

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Chaotic mating systems Rob P. Freckleton Why is reproduction so erratic in some plants? Mast seeding in plants is not just variable reproduction, but (in the extreme) is characterized by a bimodal distribution of population-level reproductive output among years, with huge seed production in some years, interspersed by almost zero reproduction in other years. Rees et al. have now shown that highly variable seed production by individuals of one species is explained by a simple mechanistic resourcebased model that assumes that plants must exceed a threshold level of resources to reproduce. The fitted model has chaotic (but highly periodic) dynamics. Environmental forcing then synchronizes seed production at the population level to generate the population-level masting behaviour.

that synchronous reproduction is necessary for successful reproduction. Previously, it was difficult to distinguish between these hypotheses, in spite of the resource-matching hypothesis effectively postulating that masting is a passive response to environmental variation, whilst, according to the others, masting is an evolved response. An added complication is that masting is a population-level response, whereby all individuals within a masting population reproduce simultaneously. The simplest way to achieve this synchrony is through responding to an environmental cue. Thus, environmental correlations might exist that drive synchronization across a population [6].

Rees et al. [2] have explored the ecology and evolution of masting behaviour in snow tussocks, which exhibit extreme mast seeding [3], with the variance in seed output between years being more than nine times the mean (coefficient of variation of up to 3.08). The pattern of seeding is extremely variable and erratic (Fig. 1a), and is characterized by a very distinctive relationship between seeding in successive years, with only very few plants seeding two years in a row (Fig. 1b). Work on the masting behaviour of snow tussocks before the analysis by Rees et al. lead to several important conclusions about the potential factors driving seeding behaviour. There is little effect on the numbers of seed set of varying the number

Published online: 19 September 2002

Variability in ecological systems can have two different sources. On the one hand, variability can result from extrinsic sources of stochastic variations, for example from the effects of weather or disturbance. On the other hand, the structure of many ecological systems is such that even if dynamics are completely deterministic (i.e. there is no stochastic component), then variability might result from chaotic or other nonlinear dynamics. Although it is nearly 30 years since it was recognized that chaos is possible in ecological systems [1], there are still very few examples of this kind of instability. Now, new work on mast seeding in snow tussocks Chionochloa pallens by Rees et al. [2] has yielded one of the very few examples. The phenomenon of ‘mast seeding’ is widespread in plants, occurring across a range of taxa, including trees, herbs and grasses [3,4]. There are three well-known hypotheses to explain this phenomenon. First, the theory of resource matching proposes that plants can only allocate resources to reproduction when conditions are sufficiently favourable [5]. Second, the predator satiation hypothesis suggests that evolution favours the production of huge crops to reduce the proportion of seeds that are consumed by seed predators. And third, the wind pollination hypothesis proposes http://tree.trends.com

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Fig. 1. Analysis of mast seeding behaviour in snow tussocks Chionochloa pallens [1]. (a) Long-term flowering of individual plants. (b) Phase-plane plot of flowering in successive years. (c) Relationship between the number of inflorescences produced per plant and deviations from the long-term energy budget. The fitted line is the threshold model. The red points are from years when the January temperature was<11.5°C, the green points are from years when the January temperature was >11.5°C. (d) The modelled phase plane plot of flowering trajectories in successive years, for comparison with (b). Note that the model predicts that very few small plants are able to reproduce two years in succession, as is observed in the data in (b). Reproduced, with permission, from [2].

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of flowers [7]. This effectively rules out the wind pollination hypothesis [8]. Evidence in support of the predator satiation hypothesis is that seed predation can be as low as 10% in mast years compared to >80% in non-mast years [9]. However, important questions remain: what is the mechanism for masting? And, is masting driven by stochastic or deterministic processes? In contrast to the passive, stochastic resource-matching hypothesis, simple deterministic models have been proposed to explain the evolution of mast seeding [8,10]. These simple ‘tent-map’ models assume that the ability to reproduce is constrained by the energy reserves of individual plants. The state variable of the models is the average energy store (S) of an individual in the population at time t, S(t). During the year, photosynthesis increases this store by an amount P. Reproduction only occurs if the sum of the existing and newly added resources [S(t) + P] exceeds some threshold level L. The energy reserve of the plant in the next year is thus S(t) + P if no reproduction occurs (if S(t) < L), or S(t) + P − R, if reproduction does occur (if S(t) ≥ L), where R is the amount of energy allocated to reproduction. R is a function of reserve size as well as the cost of flowers and fruits. Although simply stated, the dynamics of systems governed by these kinds of rule can be complex. In general, the dynamics of such systems consists of two phases, one of growth (when S is below the threshold) and decline (when S is above the threshold) resulting in some degree of periodic behaviour or cycling, because the system is not stable in either of these states. In the extreme, dynamics can even be chaotic, although often with a strong periodicity [8]. This framework for modelling the dynamics of masting is simple in principle, but is difficult to apply: the internal energy stores of plants cannot be measured directly without destructive sampling. However, energy expenditure can be estimated indirectly from the amount of inflorescences produced by an individual, and energy accumulation can be approximated by the number of days on which plant growth is possible (defined as growing degree days; in an alpine context this is the number of days per year on which the temperature exceeds 6°C). Given these measures, Rees et al. reconstructed the energy dynamics of individuals in two stages. http://tree.trends.com

TRENDS in Ecology & Evolution Vol.17 No.11 November 2002

First, the cumulative reproductive output over several years can be expressed as (Eqn 1): CF(t) = D(0) + c × CPs(t) − D(t) [Eqn 1] where CF(t) is the cumulative flower production at time t, and c × CPs(t) is the cumulative contribution to flowering from cumulative photosynthesis (CPs), expressed as the conversion of energy into flowers (c). The key to the estimation lies in the parameters D(0) and D(t). These are defined as the deviations at time 0 and t, respectively, of the energy budget from its long-term average or equilibrium (S ), that is, D(t) = S(t) − S. According to Eqn 1, the time series of deviations from the equilibrium energy budget [D(t)] can be estimated as the residuals of the regression of cumulative flowering on cumulative energy input. The second stage of the reconstruction of energy budgets relates flower production in year t to the deviations from the equilibrium resource level. Flowering should only occur if the value of D(t) is greater than the difference between the mean value of S and the threshold for reproduction. By applying a threshold regression model, it becomes possible to estimate the rate at which flower production increases with D(t), termed A, as well as the threshold value of D required for reproduction. Importantly, the value of A determines the nature of flowering dynamics: if A is >2, then seed production is predicted to be chaotic. As shown in Fig. 1c, this relationship describes flowering behaviour very well, and there is a threshold response of flowering to the deviations from the energy budget. Moreover, the analysis shows that the dynamics of flowering and seed set are in the chaotic region, with A estimated as 2.34 (CI: 2.20–2.60). Figure 1c also shows that flowering only occurs when temperatures in January exceed 11.5°C. Thus, in addition to the requirement that the energy budget is above the threshold, there appears to be a sharp division between years that are suitable for reproduction and those that are unsuitable. This statistical model points to the conclusion that the threshold resource allocation model explains the pattern of reproduction in snow tussocks at the individual level, and reproduces the pattern of reproductive behaviour seen in the field remarkably well (Fig. 1d). However, two questions remain: how does periodic reproduction at the level of the

individual become synchronized at the population level? And can the resource-matching hypothesis be dismissed? The answer to the first question is that if all individuals respond to the same cues to induce flowering, reproduction rapidly becomes synchronized across the population. And, second, comparison of models suggests that the resourcematching mechanism does not account for the observed patterns of flowering: simulations show that parameters estimated from the fitted models yield levels of synchrony that are in line with those observed in the data. By contrast, models based on simple regressions that relate flowering to environmental variables do not replicate the patterns of variation in flowering observed in the data in the same way that the threshold resource models do. Moreover an important difference between the threshold and resource matching mechanisms is that owing to resource limitation, reproduction will sometimes not occur in favourable years (when the January temperature is >11.5°C). This is indeed observed in the data when the value of D(t) is below the threshold for reproduction (Fig. 1c). Chaos is an extremely influential concept in ecology, yet there are few documented examples of ecological systems that exhibit chaotic dynamics. This is in contrast to hundreds of theoretical analyses that have explored chaotic regions of model parameter space. In snow tussocks, the role of chaos is to generate periodic bursts of reproduction that predators find difficult to track. However, this does not imply that stochastic variations are not important. In this system, the population-level synchronous dynamics are the result of stochastic variations in the flowering cue. Stochastic variability thus modulates the individual-level chaotic dynamics to yield the population-level masting response. This is one of the few welldocumented examples of the application of chaos in ecology. In this system, it is the existence of a threshold that leads to endogenous chaotic dynamics. Threshold management has also been shown (theoretically) to be capable of yielding exogenously driven chaotic dynamics in weed populations [11], and it will be exciting to see whether the existence of such thresholds generates chaos in other systems. References 1 May, R., M. (1974) Biological populations with non-overlapping generations: stable points, stable cycles and chaos. Science 186, 645–647

Research Update

2 Rees, M. et al. (2002) Snow tussocks, chaos and the evolution of mast seeding. Am. Nat. 160, 44–59 3 Kelly, D. (1994) The evolutionary ecology of mast seeding. Trends Ecol. Evol. 9, 465–470 4 Silvertown, J. (1980) The evolutionary ecology of mast seeding in trees. Biol. J. Linn. Soc. 14, 235–250 5 Busgen, M. and Munch, E. (1929) The Structure and Life of Forest Trees, Chapman & Hall 6 Norton, D.A. and Kelly, D. (1988) Mast seeding over 33 years by Dacrydium cupressinum Lamb. (rimu)

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(Podocarpaceae) in New Zealand: the importance of economies of scale. Funct. Ecol. 2, 399–408 7 Kelly, D. et al. (2001) Evaluating the wind pollination benefits of mast seeding. Ecology 82, 117–126 8 Satake, A. and Iwasa, Y. (2000) Pollen coupling of forest trees: forming synchronized and periodic reproduction out of chaos. J. Theor. Biol. 203, 63–84 9 Kelly, D. and Sullivan, J.J. (1997) Quantifying the benefits of mast seeding on predator satiation and wind pollination in Chionochloa pallens (Poaceae). Oikos 78, 143–150

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10 Isagi, Y. et al. (1997) How does masting happen and synchronize? J. Theor. Biol. 187, 231–239 11 Wallinga, J. and van Oijen, M. (1997) Level of threshold weed density does not affect the long-term frequency of weed control Crop Protection 16, 273-278

Rob P. Freckleton Dept of Zoology, University of Oxford, South Parks Road, Oxford, UK OX1 3PS. e-mail: [email protected]

2002: the year of the ‘diversity–ecosystem function’ debate Tom Cameron The main conclusion of a detailed review on biodiversity and ecosystem functioning in 2001 was to recommend continuing research into general patterns between species loss and ecosystem processes, in relation to habitat management. Four key papers have emerged from an explosion of recent experimental tests addressing this question. Studies by Pfisterer and Schmid, and by Schaffers provide empirical evidence for the lack of a unique relationship between increased ecosystem diversity and ecosystem function or stability. Paine and, in a separate report, Willms et al. describe the problems associated with ignoring trophic links in the experimental testing of the ‘diversity – production’ debate. These findings highlight the importance of rigorous testing of general ecological theory before recommending it for use by habitat or population managers. Published online: 03 September 2002

There has never been so much attention focused on the implications of biodiversity loss for ecosystem functioning or primary productivity. Two pairs of reports published this year highlight the importance of careful testing of ecological paradigms and the thorough evaluation of the focus of biodiversity research [1–4]. In a two-part report, Schaffers [1] investigates the roles of soil, biomass and management as factors that control plant species diversity. The author identified interesting hump-shaped species-richness relationships with biomass and productivity; a normal distribution, as identified in Mediterranean-type ecosystems [5]. However, he points out that these relationships fail if soil or management http://tree.trends.com

variables are considered. What Schaffers is saying is that the biomass or productivity – diversity relationships that are found in plant systems can arise from covariation of biomass and /or productivity with other abiotic or management factors, which are typically ignored in such studies. More importantly he found that management practices, such as mowing and hay removal (i.e. grazing) promoted a positive relationship with species evenness and the number of rare or endangered species present. This suggests that vulnerable species might be threatened by competitive exclusion in increased biomass environments, and that a tradeoff between predation and competition exists that promotes the coexistence of rare and common species and not simply a productivity relationship. Complementing these findings, Pfisterer and Schmid [2] report on an experimental grassland system where increased biodiversity increases biomass production. However, they stress the importance of a link between such systems and an inverse relationship between the diversity and the stability of ecosystem functioning [6]. The authors found that biomass production in species-poor ecosystems was reduced less following perturbation, and returned to pre-perturbation levels faster, than did species-rich systems. Artificially increasing spatial heterogeneity to accommodate increased diversity and production might therefore prove catastrophic if these habitats contain rare or endangered populations. This is in direct contrast to the conclusions of a recent review by Loreau et al. [7]. Instead, the indirect interactions discovered by Pfisterer and Schmid could promote increased extinctions in increasingly variable environments.

These two reports [1,2] have questioned the use of previous experimental results to look for generalities in the ‘diversity begets productivity’ debate because confounding variables, such as soil microbe diversity, soil quality and ecosystem resistance, have been ignored [7,8]. Such problems could be overcome with more complete experimental null hypotheses. Where have all the rare species gone?

Part of the burning interest in biodiversity and ecosystem function relationships is born from the concern over declining species numbers in recent years and the consequences of this decline to ecosystem services and life on Earth [9]. This interest has generated a large portfolio of studies that describe the importance of ecosystem function for diversity, and vice versa [7,10,11]. But is it ecosystem function that is important? Has this generalization been experimentally tested and, if so, how rigorously? Recently, Paine [3] tested the hypothesis that trophic links cloud the relationship between species richness and ecosystem output as measured by production (see also [10,11]). Collecting data over seven years in the low intertidal zone at Tatoosh Island, WA, USA, he found that experimental plots with restricted grazer access had higher species richness and primary production. However, this increase in production was accounted for by the enemy release of only one or two highly productive algal species. Moreover, the increased richness (presence of rare species) was accounted for by the lack of grazers rather than the increased production. These key results clearly demonstrate a weakness in the ‘diversity begets production’ debate. Paine’s report is

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