Adaptive Dynamics models for the evolution of class-structured populations in stable systems

Individuals experience varying selective pressures as they pass through different classes, defined by developmental stage, age, physiological condition, or spatial location. Adaptive Dynamics models predict the long-term evolutionary dynamics of class-structured populations, but there is a trade-off when using the two forms of the selection gradient. The determinant-form - defined by the derivative of the determinant of the class transition rate matrix - has an explicit formula but lacks a biological interpretation. Alternatively, the reproductive-value-weighted-form (RVW-form) has a clear biological interpretation, but the reproductive values do not have explicit formulas outside of special cases. In this study, I show that the determinant-form is a sum of class-mediated indirect effects of selection, where for each indirect effect, selection alters the transition rate for a particular class and changes in that rate propagate through the population and indirectly affect the density of that class. I also show how to compute the reproductive values from minors of the population's transition rate matrix. Those minors represent the total effects between classes and can be visualized as pathways through a population's life cycle graph, mirroring classical results for discrete-time stage-structured models. To illustrate the utility of these results, I analyze a model of virulence evolution in an environmentally transmitted parasite, yielding novel predictions about how density-dependent host mortality and trait-dependent decay rates affect predictions related to the Curse of the Pharaoh hypothesis. I also analyze a model of the evolution of lysis and lysogeny, which helps clarify how host density and parasite life-history traits affect viral evolution.