We are now in a position to apply the ideas of stochastic population theory to questions of ecology and conservation (extinction times) and evolutionary theory (transitions from one peak to another on adaptive landscapes), and demography (a theory for the survival curve in the Euler–Lotka equation, which we will derive as review). These are idiosyncratic choices, based on my interests when I was teaching the material and writing the book, but I hope that you will see applications to your own interests. These applications will require the use of many, and sometimes all, of the tools that we have discussed, and will require great skill of craftsmanship. That said, the basic idea for the applications is relatively simple once one gets beyond the jargon, so I will begin with that. We will then slowly work through calculations of more and more complexity.

The basic idea: “escape from a domain of attraction”

Central to the computation of extinction times and extinction probabilities or the movement from one peak in a fitness landscape to another is the notion of “escape from a domain of attraction.” This impressive sounding phrase can be understood through a variety of simple metaphors (Figure 8.1). In the most interesting case, the basic idea is that deterministic and stochastic factors are in conflict – with the deterministic ones causing attraction towards steady state (the bottom of the bowl or the stable steady states in Figure 8.1) and the stochastic factors causing perturbations away from this steady state.