Deterministic limits to stochastic spatial models of natural enemies.

Stochastic spatial models are becoming an increasingly popular tool for understanding ecological and epidemiological problems. However, due to the complexities inherent in such models, it has been difficult to obtain any analytical insights. Here, we consider individual-based, stochastic models of both the continuous-time Lotka-Volterra system and the discrete-time Nicholson-Bailey model. The stability of these two stochastic models of natural enemies is assessed by constructing moment equations. The inclusion of these moments, which mimic the effects of spatial aggregation, can produce either stabilizing or destabilizing influences on the population dynamics. Throughout, the theoretical results are compared to numerical models for the full distribution of populations, as well as stochastic simulations.

The competition-colonization trade-off is dead; long live the competition-colonization trade-off.

When applied at the individual patch level, the classic competition-colonization models of species coexistence assume that propagules of superior competitors can displace adults of inferior competitors (displacement competition). But if adults are invulnerable to displacement by propagules (as trees are to seeds), and propagules compete to replace adults that die for reasons independent of the outcome of juvenile competition (a lottery system), a competition-colonization trade-off alone is not able to produce coexistence. However, we show that coexistence is possible if patch density varies spatially, such that it becomes a niche axis. We also show how a dispersal-fecundity trade-off can partition variation in patch density. We discuss the application of these models to empirical systems. An important implication of communities coexisting via variation in patch density is that the amount of habitat loss necessarily interacts with the pattern of loss in affecting extinctions, invasions, and coexistence, in contrast to displacement competition models, for which the spatial pattern of loss is not important or is less important. Finally, with respect to mechanisms promoting coexistence, we suggest that trade-offs between different stages of colonization could be far more common in nature than a trade-off between competitive ability and colonization ability.

Reinterpreting space, time lags, and functional responses in ecological models.

Natural enemy-victim interactions are of major applied importance and of fundamental interest to ecologists. A key question is what stabilizes these interactions, allowing the long-term coexistence of the two species. Three main theoretical explanations have been proposed: behavioral responses, time-dependent factors such as delayed density dependence, and spatial heterogeneity. Here, using the powerful moment-closure technique, we show a fundamental equivalence between these three elements. Limited movement by organisms is a ubiquitous feature of ecological systems, allowing spatial structure to develop; we show that the effects of this can be naturally described in terms of time lags or within-generation functional responses.

Dynamics and evolution: evolutionarily stable attractors, invasion exponents and phenotype dynamics.

We extend the ideas of evolutionary dynamics and stability to a very broad class of biological and other dynamical systems. We simultaneously develop the general mathematical theory and a discussion of some illustrative examples. After developing an appropriate formulation for the dynamics, we define the notion of an evolutionary stable attractor (ESA) and give some samples of ESAS with simple and complex dynamics. We discuss the relationship between our theory and that for ESSS in classical linear evolutionary game theory by considering some dynamical extensions. We then introduce and develop our main mathematical tool, the invasion exponent. This allows analytical and numerical analysis of relatively complex situations, such as the coevolution of multiple species with chaotic population dynamics. Using this, we introduce the notion of differential selective pressure which for generic systems is nonlinear and characterizes internal ESAS. We use this to analytically determine the ESAS in our previous examples. Then we introduce the phenotype dynamics which describe how a population with a distribution of phenotypes changes in time with or without mutations. We discuss the relation between the asymptotic states of this and the ESAS. Finally, we use our mathematical formulation to analyse a non-reproductive form of evolution in which various learning rules compete and evolve. We give a very tentative economic application which has interesting ESAS and phenotype dynamics.

Detecting chaos in a noisy time series.

We propose a new method for detecting low-dimensional chaotic time series when there is dynamical noise present. The method identifies the sign of the largest Liapunov exponent and thus the presence or absence of chaos. It also shows when it is possible to assign a value to the exponent. This approach can work for short time series of only 500 points. We analyse several real time series including chickenpox and measles data from New York City. For model systems it correctly identifies important spatial scales at which noise and nonlinear effects are important. We propose a further technique for estimating the level of noise in real time series if it is difficult to detect by the former method.

Chaotic stochasticity: a ubiquitous source of unpredictability in epidemics.

We address the question of whether or not childhood epidemics such as measles and chickenpox are chaotic, and argue that the best explanation of the observed unpredictability is that it is a manifestation of what we call chaotic stochasticity. Such chaos is driven and made permanent by the fluctuations from the mean field encountered in epidemics, or by extrinsic stochastic noise, and is dependent upon the existence of chaotic repellors in the mean field dynamics. Its existence is also a consequence of the near extinctions in the epidemic. For such systems, chaotic stochasticity is likely to be far more ubiquitous than the presence of deterministic chaotic attractors. It is likely to be a common phenomenon in biological dynamics.