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Statistical inference on the location of the optima (global maxima or minima) is one of the main goals in the area of Response Surface Methodology, with many applications in engineering and science. While there exist previous methods for computing confidence regions on the location of optima, these are for linear models based on a Normal distribution assumption, and do not address specifically the difficulties associated with guaranteeing global optimality. This paper describes distribution-free methods for the computation of confidence regions on the location of the global optima of response surface models. The methods are based on bootstrapping and Tukey's data depth, and therefore their performance does not rely on distributional assumptions about the errors affecting the response. An R language implementation, the package \code{OptimaRegion}, is described. Both parametric (quadratic and cubic polynomials in up to 5 covariates) and nonparametric models (thin plate splines in 2 covariates) are supported. A coverage analysis is presented demonstrating the quality of the regions found. The package also contains an R implementation of the Gloptipoly algorithm for the global optimization of polynomial responses subject to bounds. 

Abstract Nutritional geometry has advanced our understanding of how macronutrients (e.g., proteins and carbohydrates) influence the expression of life history traits and their corresponding trade‐offs. For example, recent work has revealed that reproduction and immune function in male decorated crickets are optimized at very different protein:carbohydrate (P:C) dietary ratios. However, it is unclear how an individual's macronutrient intake interacts with its perceived infection status to determine investment in reproduction or other key life history traits. Here, we employed a fully factorial design in which calling effort and immune function were quantified for male crickets fed either diets previously demonstrated to maximize calling effort (P:C = 1:8) or immune function (P:C = 5:1), and then administered a treatment from a spectrum of increasing infection cue intensity using heat‐killed bacteria. Both diet and a simulated infection threat independently influenced the survival, immunity, and reproductive effort of males. If they called, males increased calling effort at the low infection cue dose, consistent with the terminal investment hypothesis, but interpretation of responses at the higher threat levels was hampered by the differential mortality of males across infection cue and diet treatments. A high protein, low carbohydrate diet severely reduced the health, survival, and overall fitness of male crickets. There was, however, no evidence of an interaction between diet and infection cue dose on calling effort, suggesting that the threshold for terminal investment was not contingent on diet as investigated here. 

Abstract Despite widespread variation in life span across species, three clear patterns exist: sex differences in life span are ubiquitous, life span is commonly traded against reproduction, and nutrition has a major influence on these traits and how they trade‐off. One process that potentially unites these patterns is intralocus sexual conflict over the optimal intake of nutrients for life span and reproduction. If nutrient intake has sex‐specific effects on life span and reproduction but nutrient choice is genetically linked across the sexes, intralocus sexual conflict will occur and may prevent one or both sexes from feeding to their nutritional optima.Here we determine the potential for this process to operate in the decorated cricketGryllodes sigillatus. Using the Geometric Framework for Nutrition, we restrict male and female crickets to diets varying in the ratio of protein to carbohydrates and total nutrient content to quantify the effects on life span and daily reproductive effort in the sexes. We then use inbred lines to estimate the quantitative genetic basis of nutrient choice in males and females. We combine the nutrient effects and genetic estimates to predict the magnitude of evolutionary constraint for these traits in each sex. Finally, we present male and female crickets with a much broader range of diet pairs to determine how the sexes actively regulate their intake of nutrients.We show that protein and carbohydrate intake have contrasting effects on life span and reproduction in the sexes and that there are strong positive intersexual genetic correlations for the intake of these nutrients under dietary choice. This is predicted to accelerate the evolutionary response of nutrient intake in males but constrain it in females, suggesting they are losing the conflict. Supporting this view, males and females regulate nutrient intake to a common nutrient ratio that was not perfectly optimal for life span or reproduction in either sex, especially in females.Our findings show that intralocus sexual conflict over the optimal intake of nutrients is likely to be an important process generating sex differences in life span and reproduction and may help explain why females age faster and live shorter than males inG. sigillatus. A freePlain Language Summarycan be found within the Supporting Information of this article.