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.
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