Lake‐dwelling fish that form species pairs/flocks characterized by body size divergence are important model systems for speciation research. Although several sources of divergent selection have been identified in these systems, their importance for driving the speciation process remains elusive. A major problem is that in retrospect, we cannot distinguish selection pressures that initiated divergence from those acting later in the process. To address this issue, we studied the initial stages of speciation in European whitefish (Coregonus lavaretus) using data from 358 populations of varying age (26–10,000 years). We find that whitefish speciation is driven by a large‐growing predator, the northern pike (Esox lucius). Pike initiates divergence by causing a largely plastic differentiation into benthic giants and pelagic dwarfs: ecotypes that will subsequently develop partial reproductive isolation and heritable differences in gill raker number. Using an eco‐evolutionary model, we demonstrate how pike's habitat specificity and large gape size are critical for imposing a between‐habitat trade‐off, causing prey to mature in a safer place or at a safer size. Thereby, we propose a novel mechanism for how predators may cause dwarf/giant speciation in lake‐dwelling fish species.

Sustainable yields that are at least 80% of the maximum sustainable yield are sometimes referred to as pretty good yield (PGY). The range of PGY harvesting strategies is generally broad and thus leaves room to account for additional objectives besides high yield. Here, we analyze stage-dependent harvesting strategies that realize PGY with conservation as a second objective. We show that (1) PGY harvesting strategies can give large conservation benefits and (2) equal harvesting rates of juveniles and adults is often a good strategy. These conclusions are based on trade-off curves between yield and four measures of conservation that form in two established population models, one age-structured and one stage-structured model, when considering different harvesting rates of juveniles and adults. These conclusions hold for a broad range of parameter settings, though our investigation of robustness also reveals that (3) predictions of the age-structured model are more sensitive to variations in parameter values than those of the stage-structured model. Finally, we find that (4) measures of stability that are often quite difficult to assess in the field (e.g. basic reproduction ratio and resilience) are systematically negatively correlated with impacts on biomass and impact on size structure, so that these later quantities can provide integrative signals to detect possible collapses.Similarly, we consider impact on size-structure through the expression Impact on size-structure = which equals the relative change in the fraction of juvenile biomass following harvesting. If the impact on size-structure is positive (negative), then harvesting has increased (decreased) the fraction of juveniles in the population.Resilience, basic reproduction ratio and recovery potential as risk measures Resilience as a risk measure is increasingly used in ecology (Pimm and Lawton 1977; Loreau and Behera 1999; Petchey et al. 2002; Montoya et al. 2006; Loeuille 2010; Valdovinos et al.

With increasing fishing pressures having brought several stocks to the brink of collapse, there is a need for developing efficient harvesting methods that account for factors beyond merely yield or profit. We consider the dynamics and management of a stage-structured fish stock. Our work is based on a consumer-resource model which De Roos et al. (2008) have derived as an approximation of a physiologically-structured counterpart. First, we rigorously prove the existence of steady states in both models, that the models share the same steady states, and that there exists at most one positive steady state. Furthermore, we carry out numerical investigations which suggest that a steady state is globally stable if it is locally stable.Second, we consider multi-objective harvesting strategies which account for yield, profit, and 1 the recovery potential of the fish stock. The recovery potential is a measure of how quickly a fish stock can recover from a major disturbance and serves as an indication of the extinction risk associated with a harvesting strategy. Our analysis reveals that a small reduction in yield or profit allows for a disproportional increase in recovery potential. We also show that there exists a harvesting strategy with yield close to the maximum sustainable yield (MSY) and profit close to that associated with the maximum economic yield (MEY). In offering a good compromise between MSY and MEY, we believe that this harvesting strategy is preferable in most instances. Third, we consider the impact of harvesting on population size structure and analytically determine the most and least harmful harvesting strategies. We conclude that the most harmful harvesting strategy consists of harvesting both adults and juveniles, while harvesting only adults is the least harmful strategy. Finally, we find that a high percentage of juvenile biomass indicates elevated extinction risk and might therefore serve as an early-warning signal of impending stock collapse.

A. Kamont has discretely characterised Besov spaces on intervals. In this paper, we give a discrete characterisation of Lipschitz spaces on fractals admitting a type of regular sequence of triangulations, and for a class of post critically finite self-similar sets. This shows that on some fractals, certain discretely defined Besov spaces, introduced by R. Strichartz, coincide with Lipschitz spaces introduced by A. Jonsson and H. Wallin for low order of smoothness.

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