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Mathematical models are used to explore the interaction between two prey species that share a common predator. The models assume that the predator experiences density dependence via some mechanism other than prey depletion. The models also assume that the predator's functional response to each prey decreases as the density of the other prey species increases. This can occur either because of predator satiation or predator switching. The results suggest that positive indirect effects of one prey on the equilibrium density of others should occur frequently, especially when there is predator switching. Decreasing the mortality rate of one prey or adding a prey species may make it easier for additional prey species to invade the system and coexist. This occurs because the resulting decrease in the predator's functional response is greater than its positive numerical response. In many cases, different magnitudes of perturbation to one prey species will have opposite effects on the population density of the other prey species.

This paper studies some properties and implications of higher-order mutual information functions, which should serve for the analysis of general complex systems. We note that the higher-order mutual information can either be positive or negative depending on the correlation among ensembles. Two opposite types of correlations are discussed in connection with the concept of frustration. Simple examples are presented to demonstrate that our concepts are especially helpful in understanding the nature of correlations in frustrated systems. The higher-order mutual information provides an appropriate measure of the frustration effect.

We analyze simple models of predator-prey systems in which there is adaptive change in a trait of the prey that determines the rate at which it is captured by searching predators. Two models of adaptive change are explored: (1) change within a single reproducing prey population that has genetic variation for vulnerability to capture by the predator; and (2) direct competition between two independently reproducing prey populations that differ in their vulnerability. When an individual predator's consumption increases at a decreasing rate with prey availability, prey adaptation via either of these mechanisms may produce sustained cycles in both species' population densities and in the prey's mean trait value. Sufficiently rapid adaptive change (e.g., behavioral adaptation or evolution of traits with a large additive genetic variance), or sufficiently low predator birth and death rates will produce sustained cycles or chaos, even when the predator-prey dynamics with fixed prey capture rates would have been stable. Adaptive dynamics can also stabilize a system that would exhibit limit cycles if traits were fixed at their equilibrium values. When evolution fails to stabilize inherently unstable population interactions, selection decreases the prey's escape ability, which further destabilizes population dynamics. When the predator has a linear functional response, evolution of prey vulnerability always promotes stability. The relevance of these results to observed predator-prey cycles is discussed.

We analyze simple models of predator-prey systems in which there is adaptive change in a trait of the prey that determines the rate at which it is captured by searching predators. Two models of adaptive change are explored: (1) change within a single reproducing prey population that has genetic variation for vulnerability to capture by the predator; and (2) direct competition between two independently reproducing prey populations that differ in their vulnerability. When an individual predator's consumption increases at a decreasing rate with prey availability, prey adaptation via either of these mechanisms may produce sustained cycles in both species' population densities and in the prey's mean trait value. Sufficiently rapid adaptive change (e.g., behavioral adaptation or evolution of traits with a large additive genetic variance), or sufficiently low predator birth and death rates will produce sustained cycles or chaos, even when the predator-prey dynamics with fixed prey capture rates would have been stable. Adaptive dynamics can also stabilize a system that would exhibit limit cycles if traits were fixed at their equilibrium values. When evolution fails to stabilize inherently unstable population interactions, selection decreases the prey's escape ability, which further destabilizes population dynamics. When the predator has a linear functional response, evolution of prey vulnerability always promotes stability. The relevance of these results to observed predator-prey cycles is discussed.

The question what determines the structure of natural food webs has been listed among the nine most important unanswered questions in ecology 1 . It arises naturally from many problems related to ecosystem stability and resilience 2,3 . The traditional 4 view is 5 that population-dynamical stability is crucial for understanding the observed 6,7,8,9,10,11 structures. But phylogeny (evolutionary history) has also been suggested 10 as the dominant mechanism. Here we show that observed topological features of predatory food webs can be reproduced to unprecedented accuracy by a mechanism taking into account only phylogeny, size constraints, and the heredity of the trophically relevant traits of prey and predators. The analysis reveals a tendency to avoid resource competition rather than apparent competition 12 . In food webs with many parasites 13 this pattern is reversed.Empirical food-web data is notorious for its inhomogeneity 14 . In particular, the large number of species interacting in habitats has forced researchers to disregard whole subsystems or to coarsen the taxonomic resolution 11 . The representation of trophic interactions by the simple absence or presence of links in topological food webs is problematic, because it turns out that by various measures 15 weak links are more frequent than strong links in natural food webs, and network structures depend on a somewhat arbitrary thresholding among the weak links 16,17 . Furthermore, the use of different methods for determining links 14 might affect the result. Our analysis takes these difficulties into account by employing a quantitative link-strength concept, an appropriate data standardization (see Supplementary Methods), and by reflecting the inhomogeneity of empirical methodology in our food-web model and data analysis. 2Specifically, the following model ("matching model") describing the evolution of an abstract species pool is employed: The foraging and vulnerability traits of each species 18,19,20 are modeled by two sequence of ones and zeros of length n (the reader might think of oppositions such as sessile/vagile, nocturnal/diurnal, or benthic/pelagic). The strength of trophic links increases (nonlinearly) with the number m of foraging traits of the consumer that match the corresponding vulnerability traits of the resource (Figure 1). A trophic link is considered as present if the number of matched traits m exceeds some threshold m ≥ m 0 .In addition, each species is associated with a size parameter s characterizing the (logarithmic) body size of a species (0 ≤ s < 1). Consumers cannot forage on species with size parameters larger than their own by more than λ. The model parameter λ (0 ≤ λ < 1) controls the amount of trophic loops 21 in a food web.The complex processes driving evolution are modeled by speciations and extinctions that occur for each species randomly at rates r + and r − , respectively 22 . New species invade the habitat at a rate r 1 . Such continuous-time birth-death processes are well understood 23 .With r + < r − the steady-state ...

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