The global lockdown to mitigate COVID-19 pandemic health risks has altered human interactions with nature. Here, we report immediate impacts of changes in human activities on wildlife and environmental threats during the early lockdown months of 2020, based on 877 qualitative reports and 332 quantitative assessments from different studies. Hundreds of reports of unusual species observations from around the world suggest that animals quickly responded to the reductions in human presence. However, negative effects of lockdown on conservation also emerged, as confinement resulted in some park officials being unable to perform conservation, restoration and enforcement tasks, resulting in local increases in illegal activities such as hunting. Overall, there is a complex mixture of positive and negative effects of the pandemic lockdown on nature, all of which have the potential to lead to cascading responses which in turn impact wildlife and nature conservation. While the net effect of the lockdown will need to be assessed over years as data becomes available and persistent effects emerge, immediate responses were detected across the world. Thus, initial qualitative and quantitative data arising from this serendipitous global quasi-experimental perturbation highlights the dual role that humans play in threatening and protecting species and ecosystems. Pathways to favorably tilt this delicate balance include reducing impacts and increasing conservation effectiveness.

The recent context of global food emergency and ecological crisis has increased the relevance of people's struggle for food sovereignty (FSv), which promotes the transformation of the dominant food system and claims 'the right of peoples to healthy and culturally appropriate food produced through ecologically sound and sustainable methods, and their right to define their own food and agriculture systems'. Revisiting two Spanish and Catalan articles developing FSv indicators, this article aims at discussing the need and utility of developing FSv indicators at different territorial levels. Confronting these two territorial scales, the paper also identifies common steps that can facilitate other future processes of building FSv indicators. As a conclusion, the paper suggests that these processes of building indicators can contribute to providing political direction at different geographical scales for the implementation of the FSv proposal. At the same time, they favor the movement's self-reflexivity in its practices while supporting the collective shaping of future actions.

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Is there a limit to the amount of fish that can be taken from the sea? This question echoes the concern of the broader environmental movement in asking: are there ‘limits to growth’? If the answer is ‘yes’, then what must be done to remain within sustainable limits? Fifty years after the publication of the landmark report Limits to Growth, new theories about limits highlight the importance of collective self-limitation, also in the context of fisheries management, in place of external, top-down determination and imposition of limits. This paper considers the shift in fisheries governance from regulating and establishing Maximum Sustainable Yields to collectively co-managing territories and ecosystems as symptomatic of a general turn from externally-imposed to self-imposed limitations. We show how perceptions and practices of limits are changing based on an ethnographic study of six small-scale fisheries co-management plans located off the Catalan coast in the Northwestern Mediterranean. The study evidences the challenges fishers face in attempting to define the limits of their agency to manage external forces that are often beyond their control. It concludes by arguing for the adoption of an ethos of collective self-limitation in fisheries governance to protect and benefit local communities and their environments.

We prove that product-free sets of the free group over a finite alphabet have maximum density 1/2 with respect to the natural measure that assigns total weight one to each set of irreducible words of a given size. This confirms a conjecture of Leader, Letzter, Narayanan and Walters. In more general terms, we actually prove that strongly k-product-free sets have maximum density 1/k in terms of the said measure.

We prove that product-free sets of the free group over a finite alphabet have maximum density $1/2$ with respect to the natural measure that assigns total weight one to each set of irreducible words of a given length. This confirms a conjecture of Leader, Letzter, Narayanan and Walters. In more general terms, we actually prove that strongly $k$-product-free sets have maximum density $1/k$ in terms of the said measure. The bounds are tight.

The western Mediterranean basin is a high marine biodiversity area under severe pressure by changing climate and intense human activities. Beyond national jurisdictions, international institutions such as the General Fisheries Commission for the Mediterranean (GFCM) work towards canalizing a regional consensus that fishing practices should evolve to better support the 2030 Agenda for Sustainable Development. In this context, Fisheries Restricted Areas (FRA) are proposed as effective management measures to contribute towards increasing fisheries sustainability in the region that can be considered, under some conditions, as Nature-based Solutions (NbS); however, how to operationalize their framework remains unclear. In this study, based on combined ecological and fisheries criteria, we identify and prioritize six potential priority areas for management (PAMs) in the western Mediterranean Sea. They are specifically aimed at the protection and recovery of Essential Fish Habitats and the conservation of Vulnerable Marine Ecosystems, whilst requiring limited adaptation of fisheries practices due to their relative low fishing pressure. We compare the identified areas to those that are currently under protection, and to areas that have been proposed for protection at the GFCM. Our results show that the FRAs and other spatial management measures introduced in the last years marginally contribute to the protection PAMs in the western Mediterranean region. However, the adoption of FRAs that are currently under discussion at the GFCM could contribute significantly to improve the situation. FRAs could also contribute to operationalize NbS in the western Mediterranean Sea when properly designed and implemented.HighlightsBased on combined ecological and fisheries criteria, six priority areas for management (PAMs) in the western Mediterranean Sea have being identified, with multiple ecological values and relative low trawling.Current spatial management measures implemented have little contribution on PAMs protection.Fisheries Restricted Areas currently under discussion at the GFCM can significantly increase the protection level of high priority PAMs.

Generalising results of Erdős-Freud and Lindström, we prove that the largest Sidon subset of a bounded interval of integers is equidistributed in Bohr neighbourhoods. We establish this by showing that extremal Sidon sets are Fourier-pseudorandom, in that they have no large non-trivial Fourier coefficients. As a further application we deduce that, for any partition regular equation in five or more variables, every finite colouring of an extremal Sidon set has a monochromatic solution. Contents1. Introduction 1 2. Fourier uniformity 6 3. Equidistribution in progressions 8 4. Equidistribution in Bohr neighbourhoods 9 5. Partition regularity 11 6. Improving Fourier uniformity 13 Appendix A. The size of a Sidon set 14 Appendix B. Trigonometric approximation 14 Appendix C. A dense model lemma 15 References 18