Being a member state of the European Union is necessary for having the Euro as the official currency: the set of EU member states is a superset of the set of countries with the Euro as a national currency. Generally, if a condition X is necessary for an outcome Y, then X is a superset of Y. In this chapter, we introduce the notion of necessity and discuss the analysis of necessary conditions in R. We explain what necessity is, how to analyze and visualize necessary conditions, how to interpret the results, and how to avoid common pitfalls. We outline the basic protocol for the analysis of necessity in R, both for the analysis of individual necessary conditions and for conditions combined with a logical OR, that is necessary disjunctions composed of so-called SUIN conditions.

Learning goals:

- Understanding of the basic logic of necessity.

- Familiarity with different approaches to analyzing necessity in QCA and the protocol for the analysis of necessity.

- Basic understanding of possible analytic pitfalls when analyzing necessity and ways of avoiding them.

- Ability to implement an analysis of necessity in R and visualize the results.

In the 1950s, there was a hot debate among ecologists about what regulates animal populations so as to keep them persisting for a long time. The participants of the debate were divided into two schools of thought, one insisting on the importance of climatic factors and the other emphasizing the necessity of density-dependent biotic factors. Albeit an important subject, no clear answers came out of the debate. A fundamental problem in the indecisive debate lay in the lack of clear definitions of ‘regulation’ and ‘persistence’ of populations. Consequently, the current ecological thoughts inherit many myths and ambiguities about the nature of population regulation. Here, I delve into the subject to look for unambiguous answers. In particular, I show that a climatic (density-independent) control of populations is theoretically possible but fragile in a fluctuating environment, and that density-dependent regulation is necessary but not sufficient for keeping populations in the persistent state.

In this chapter we introduce the fundamental object ofthis book: the sub-Riemannian distance. We prove itsfiniteness and continuity (Rashevskii–Chow theorem)and the existence of length-minimizers. We thenintroduce Pontryagin extremals, which are curves inthe cotangent space satisfying a first-ordernecessary condition of length-minimality.