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In this paper we provide a systematic investigation of a family of composed aggregation functions which generalize the Bonferroni mean. Such extensions of the Bonferroni mean are capable of modeling the concepts of hard and soft partial conjunction and disjunction, as well as that of k-tolerance and k-intolerance. There are several interesting special cases with quite an intuitive interpretation for application.

Aim Extreme climatic events and large wildfires are predicted to increase as the world's climate warms. Understanding how they shape species’ distributions will be critical for conserving biodiversity. We used a 7‐year dataset of mammals collected during and after south‐east Australia's Millennium Drought to assess the roles of fire history, climatic extremes and their interactions in shaping mammal distributions. Location Grampians National Park, south‐eastern Australia. Methods We surveyed mammals at 36 sites along a ~50‐year post‐fire chronosequence in each of the 7 years. We modelled ten mammal species in relation to fire history, productivity and recent rainfall. Next, we examined the consistency of species’ fire response curves across each of three climatic phases relating to the Millennium Drought. Finally, we identified the optimal distribution of fire ages for small and medium‐sized mammal conservation in each of the three climatic phases. Results The majority of species were influenced by fire history, and all native species were negatively associated with recently burned vegetation. Seven of ten species responded positively to the end of the Millennium Drought, but six of these declined quickly thereafter. Species’ responses to fire history differed depending on the climatic conditions. However, the optimal distribution of fire‐age classes consistently emphasized the importance of older age classes, regardless of climatic phase. This distribution is in stark contrast to the current distribution of fire ages across the study region. Main conclusions Mammals in the study region face an uncertain future. The negative impact of drought, the short‐lived nature of post‐drought recovery and, now, the possibility of a new drought beginning forewarn of further declines. The stark contrast between the optimal and current fire‐age distributions means that reducing the incidence of further fires is critical to enhance the capacity of native mammal communities to weather an increasingly turbulent climate.

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Clinical trials with multiple endpoints incur increased familywise type I errors. The Bonferroni correction is a common method used to modify the p-values to account for multiple significance testing. For independent endpoints the Bonferroni method is slightly conservative whereas with high correlation the conservatism is extreme, as demonstrated by Pocock et al. This paper presents a procedure which allows for the correlation present, whilst adjusting the multiple p-values. The method is based on an approximation derived for multinormal probabilities.

Conservation of biodiversity in urban environments depends on species’ responses to the intensity of urban development. “Land sharing” and “land sparing” represent alternate ends of a gradient that conceptualises a trade‐off between the human population and biodiversity. We used a linear optimisation procedure to (a) identify the optimal allocation of land for people and nature, (b) assess whether the optimal allocation is closer to land sparing or land sharing and (c) examine how this might change under scenarios of human population growth. We surveyed birds in 28 landscapes, each 25 ha in size, along a gradient of human population density (zero to c. 1,600 persons/25 ha) in the Greater Melbourne region, Australia. Species’ responses to population density were estimated using generalised additive models (GAMs). These relationships were then used to determine the optimal allocation of land among different categories of population density based on maximising a community index, the geometric mean of relative abundance (G) of bird species. Human population density was an important driver of the reporting rate for 28 species. Response curves differed among “urban avoider,” “urban adapter” and “urban exploiter” species. For the current human population in the study area, the optimal allocation of land included elements of both land sharing and land sparing. However, for scenarios of increased population size, optimal allocation converged upon a land sparing design. Synthesis and applications. Urban areas represent a mosaic of land uses that offer habitats of differing quality. Land sharing, based on sustaining biota among residential areas, performed poorly under all scenarios due to its inability to support species that depend on natural or seminatural habitat. To sustain more than a homogenised avifauna in urban regions, large tracts of natural vegetation are needed within, or adjacent to, the urban environment. Protecting natural areas on urban fringes will be critical to the safeguarding of nature in the future as urban populations and land‐use inevitably expand.

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