The evolution of the primary sex ratio, the proportion of male births in an individual's offspring production strategy, is a frequency‐dependent process that selects against the more common sex. Because reproduction is shaped by the entire life cycle, sex ratio theory would benefit from explicitly two‐sex models that include some form of life cycle structure. We present a demographic approach to sex ratio evolution that combines adaptive dynamics with nonlinear matrix population models. We also determine the evolutionary and convergence stability of singular strategies using matrix calculus. These methods allow the incorporation of any population structure, including multiple sexes and stages, into evolutionary projections. Using this framework, we compare how four different interpretations of sex‐biased offspring costs affect sex ratio evolution. We find that demographic differences affect evolutionary outcomes and that, contrary to prior belief, sex‐biased mortality after parental investment can bias the primary sex ratio (but not the corresponding reproductive value ratio). These results differ qualitatively from the widely held conclusions of previous models that neglect demographic structure.

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Periodic matrix models are frequently used to describe cyclic temporal variation (seasonal or interannual) and to account for the operation of multiple processes (e.g., demography and dispersal) within a single projection interval. In either case, the models take the form of periodic matrix products. The perturbation analysis of periodic models must trace the effects of parameter changes, at each phase of the cycle, on output variables that are calculated over the entire cycle. Here, we apply matrix calculus to obtain the sensitivity and elasticity of scalar-, vector-, or matrix-valued output variables. We apply the method to linear models for periodic environments (including seasonal harvest models), to vec-permutation models in which individuals are classified by multiple criteria, and to nonlinear models including both immediate and delayed density dependence. The results can be used to evaluate management strategies and to study selection gradients in periodic environments.

1. Second derivatives of the population growth rate measure the curvature of its response to demographic, physiological or environmental parameters. The second derivatives quantify the response of sensitivity results to perturbations, provide a classification of types of selection and provide one way to calculate sensitivities of the stochastic growth rate.2. Using matrix calculus, we derive the second derivatives of three population growth rate measures: the discrete-time growth rate λ, the continuous-time growth rate r = log λ and the net reproductive rate R0, which measures per-generation growth.3. We present a suite of formulae for the second derivatives of each growth rate and show how to compute these derivatives with respect to projection matrix entries and to lower-level parameters affecting those matrix entries.4. We also illustrate several ecological and evolutionary applications for these second derivative calculations with a case study for the tropical herb Calathea ovandensis.

Abstract. The population effects of harvest depend on complex interactions between density dependence, seasonality, stage structure, and management timing. Here we present a periodic nonlinear matrix population model that incorporates seasonal density dependence with stage-selective and seasonally selective harvest. To this model, we apply newly developed perturbation analyses to determine how population densities respond to changes in harvest and demographic parameters. We use the model to examine the effects of popular control strategies and demographic perturbations on the invasive weed garlic mustard (Alliaria petiolata). We find that seasonality is a major factor in harvest outcomes, because population dynamics may depend significantly on both the season of management and the season of observation. Strategies that reduce densities in one season can drive increases in another, with strategies giving positive sensitivities of density in the target seasons leading to compensatory effects that invasive species managers should avoid. Conversely, demographic parameters to which density is very elastic (e.g., seeding survival, second-year rosette spring survival, and the flowering to fruiting adult transition for maximum summer densities) may indicate promising management targets.

Mothers that experience different individual or environmental conditions may produce different proportions of male to female offspring. The Trivers‐Willard hypothesis, for instance, suggests that mothers with different qualities (size, health, etc.) will use different sex ratios if maternal quality differentially affects sex‐specific reproductive success. Condition‐dependent, or facultative, sex ratio strategies like these allow multiple sex ratios to coexist within a population. They also create complex population structure due to the presence of multiple maternal conditions. As a result, modeling facultative sex ratio evolution requires not only sex ratio strategies with multiple components, but also two‐sex population models with explicit stage structure. To this end, we combine nonlinear, frequency‐dependent matrix models and multidimensional adaptive dynamics to create a new framework for studying sex ratio evolution. We illustrate the applications of this framework with two case studies where the sex ratios depend one of two possible maternal conditions (age or quality). In these cases, we identify evolutionarily singular sex ratio strategies, find instances where one maternal condition produces exclusively male or female offspring, and show that sex ratio biases depend on the relative reproductive value ratios for each sex.

Models of sexually-reproducing populations that consider only a single sex cannot capture the effects of sex-specific demographic differences and mate availability. We present a new framework for two-sex demographic models that implements and extends the birth-matrix mating-rule approach of Pollak. The model is a continuous-time matrix model that explicitly includes the processes of mating (which is nonlinear but homogeneous), offspring production, and demographic transitions and survival. The resulting nonlinear model converges to exponential growth with an equilibrium population composition. The model can incorporate age-or stage-structured life histories and flexible mating functions. As an example, we apply the model to analyze the effects of mating strategies (polygamy or monogamy, and mated unions composed of males and females, of variable duration) on the response to sex-biased harvesting. The combination of demographic complexity with the interaction of the sexes can have major population dynamic effects and can change the outcome of evolution on sexrelated characters.

1Marine reserves are an increasingly used and potentially contentious tool in fisheries manage-2 ment. Depending upon the way that individuals move, no-take marine reserves can be necessary for 3 maximizing equilibrium rent in some simple mathematical models. The implementation of no-take 4 marine reserves often generates a redistribution of fishing effort in space. This redistribution of 5 effort, in turn, produces sharp spatial gradients in mortality rates for the targeted stock. Using 6 a two-patch model, we show that the existence of such gradients is a sufficient condition for the 7 evolution of an evolutionarily stable conditional dispersal strategy. Thus, the dispersal strategy 8 of the fish depends upon the harvesting strategy of the manager and vice versa. We find that an 9 evolutionarily stable optimal harvesting strategy (ESOHS)-one which maximizes equilibrium rent 10 given that fish disperse in an evolutionarily stable manner-never includes a no-take marine re-11 serve. This strategy is economically unstable in the short run because a manager can generate more 12 rent by disregarding the possibility of dispersal evolution. Simulations of a stochastic evolutionary 13 process suggest that such a short-run, myopic strategy performs poorly compared to the ESOHS 14 over the long run, however, as it generates rent that is lower on average and higher in variability.