Teaching the Switch Point Theorem

Carlos A. de la Rosa, Ecology & Evolutionary Biology, UCLA, Los Angeles, CA 90095

Patricia Adair Gowaty, Ecology & Evolutionary Biology, UCLA, Los Angeles, CA 90095

The Switch Point Theorem (SPT, Eq. 1) is a mathematical model (Gowaty and Hubbell 2009) that predicts an individual’s reproductive decisions (who to accept or reject as a mate) given real time variation in the individual’s unique ecological and social opportunities and constraints - without necessary reference to their sex/gender. Therefore, the SPT is a quantitative alternative to the qualitative idea that females choose among males as mates based on phenotypic variation in males, such as elaborate or colorful plumage.

The SPT was deduced from a Markov model in The Theory of Mating (Gowaty and Hubbell in progress) of the time an individual spends in a series of states essential to mating and reproduction. The model is a Markov model having four states that an individual can enter. Starting at sexual maturity it can enter a receptive-to-mating state, from which it might then enter a state of encountering a potential mate during which she or he may mate with a potential mate on encounter (as if “indiscriminate”) or wait for a better mate (as if “choosy”). If the individual does mate, it then may enter a state of “post-mating time out” also called a “post-mating latency”, which can vary in duration from zero time units to infinity. From any other state an individual can also enter an “absorbing state” of death (“absorbing” because once an individual is dead it cannot enter any of the previous living states: dead is dead). What determines the movement of an individual from state to state are two probabilities: the individual’s survival probability, s, and its probability of encountering potential mates, e. Using these two probabilities alone, allows one to calculate an individual’s expected lifetime mating success.

The SPT adds some complexity to the basics of mating theory in order to predict the effects on expected lifetime fitness from mating with any of the n (number of) potential mates in a population. The added complexities include the assumption that an individual uniquely ranks all the potential mates in a population from 1 (best for the individual) to n (worst for the individual). The switch point is the dividing line for an individual that indicates the sets of potential mates “acceptable” or “unacceptable” to that individual. What determines the switch point dividing line along the axis of ranked potential mates is the high point of the curve of fitness reward for adhering to a given switch point. To find that high point of earned fitness for each possible switch point, the model calculates the switch point that maximizes fitness by comparison of all possible switch points along the axis of ranked potential mates (Eq. 1). Thus, for example, for n = 100, the switch point that maximizes fitness, f*, could be between potential mates ranked, say, 1 and 2, or say, 20 and 21, or say, 50 and 51, or even 99 and 100.

To predict f* and thus an individual’s expected lifetime mating success and their mating behavior (“mating on encounter” or “waiting for a better mate”), one uses instantaneous values of its five assumptions: an individual’s survival probability ( s ), the probability of encountering potential mates ( e ), and, if the individual is not a virgin, the individual’s post-mating latency to further receptivity to mating ( l ), the number of potential mates in the population ( n ), and the distribution of fitnesses those potential mates would confer on the individual ( w -distribution). As the SPT parameters change, so does the equation’s solution, f* , the theoretical lowest ranking of potential mates with whom an individual should mate on encounter (to enhance lifetime fitness), along with all others of equal or better rank, if she or he is to achieve maximum potential fitness over the course of their lifetime. Changes in each parameter result in unique and sometimes surprising changes to f* , which is the value that toggles the dividing line between the individual’s ranked acceptable and rejectable potential mates.

The stochastic nature of interacting model parameters, as well as their number, make teaching the SPT challenging. The six plots of Figure 4 (Gowaty and Hubbell, 2009) efficiently showcase the model’s largest movements by displaying 5 values each of s, e, and l , for two w -distributions. However, like still photos from a movie, static plots are only snapshots of a dynamic, continuous process, and many details are necessarily lost in the interest of space and readability.

C. de la Rosa created the web application “Teaching the Switch Point Theorem” (TSPT) as a learning tool for undergraduate students in the evolutionary theory course UCLA EEB 175, “The Evolutionary Dynamics of Sex,” using the app creation tool Shiny from R Studio. The TSPT interface consists of sliding bars (for survival probability, encounter probability, post-mating latency, and population size of potential mates) and radio buttons (representing different initial w -distributions) to induce instantaneous changes in f and f* in an embedded SPT graph (Table 1). The user controls parameter values, within preset limits, that when “played with” show smooth movement of the output variables. In the context of students playing with the SPT toy it is important for students to remember that l = 0 for virgins because virgins by definition have never mated and therefore it is impossible for any duration of latency above 0 to affect a virgin’s decisions to accept or reject potential mates.

The SPT toy is intuitive, simple, and student friendly. It allows students and others to get a very good idea of how real time dynamic changes in s or e may affect an individual’s reproductive decision to accept or reject potential mates.