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This chapter focuses on mechanistic theoretical approaches to explain the emergence of ratio dependence at a global scale from various behavioral models of species interaction. It explains how refuges and spatial heterogeneity as well as forms of temporal and biological heterogeneities can lead to a ratio-dependent functional response. It provides a two-patch model that shows by what method donor control can emerge from being a prey refuge and being subject to intense predation. It also discusses how ratio dependence emerges in realistic models of predator-prey interactions where spatial distribution of both the predator and prey is explicitly followed. It highlights application of these models to the explanation of successful biological control of insect pests.

This chapter examines the controversy surrounding the topic of ratio-dependence model of predator-prey interactions. It considers the practice of many ecologists in using the Lotka-Volterra model and other similar prey-dependent models when building complex food web models, either for theoretical purposes or for applied studies. It looks into the issue of concealing evidence of ratio dependence in collected works, and highlights the continuing acceptance of prey-dependent views with the paradox of enrichment and the cascading response of trophic chains. It tackles some of the arguments raised against ratio-dependency such as the theory's lack of a mechanical basis, and discusses the agreement of a “middle opinion” that predator dependence has to be included in the theory of predator-prey interactions.

Language does not necessarily support all human cognition. Comparative research continues to uncover the links between the cognitive abilities of humans and other animal species that lack language. Representing and manipulating numbers are important for most aspects of human life, and much of this involves abstract symbolic representations. Given this perspective, the authors illustrate how the comparative approach has been used in the domain of numbers to identify the origins of mathematical thinking. The authors in this chapter found out that a system for representing numbers nonverbally is shared by both humans and many nonhuman animal species. A host of behavioral parallels between human and animal numerical cognition such as ratio dependence, semantic congruity, cross-modal matching, and nonverbal arithmetic are revealed in this chapter. The homologous brain structures appear to support numerical representations in humans and macaque monkeys.

This chapter studies the methods of direct measurements of prey captures, either in the laboratory or in the field. It reviews experimental or observational studies that lead to direct quantitative estimates of the numbers of captures and of the way they depend on consumer densities; and short-scale behavioral studies of the influence of consumer density performed in the laboratory with insects and similarly small animals. It provides a detailed analysis of the wasps and chrysomelids field experiment by Dominique Schenk and Sven Bacher, which is the only experiment ever set up to purposely test the ratio dependence hypothesis in the field. It considers long-scale demographic studies such as the field observation of wolves and moose on Isle Royale.

This chapter explains the role of ratio-dependence as an invariance in ecology. It analyzes the importance of using scaling invariance in mathematics and proposes that all models of interacting species must be fundamentally invariant to a proportional change in the system. It uses Newton's law of inertia as a metaphor for the Malthusian law of population growth and extends this to include interacting species. It looks into Andrey Kolmogorov's insight on the possibility of ratio-dependent interaction, and examines H. Reşit Akçakaya's famous field experiment of the Canadian lynx and hare cycles. It compares Reşit's number of unsupported parameters with other research to measure overfitting. It also discusses idealizations or limit myths of theories and applies this to the ratio-independence model.