Since 2018 Springer has started publishing this book series Mathematics of Planet Earth (MPE), and there are already six volumes by the end of 2019. The present book is volume 5 and is somewhat representative for the idea behind the whole book series.

The idea of the series is to describe and model several global problems that can be in any area, like climate change, epidemics, biodiversity, food security, urban societies, the Internet of Things, and many more. The list of topics is very broad and often require a multidisciplinary approach. A major objective in creating this book series is to raise interest and stimulate interdisciplinary collaboration. Therefore the papers of the books in this series are intended to be written and to be read by a broad range of pure or applied mathematicians, engineers, scientists, and other researchers. They are at a level that is not elementary but neither too technical so that it can be understood even by a beginning researcher from a different discipline. Solutions need not, or are only partially provided. The chapters are both introductions to the topic discussed and invitations to work on it.

While other volumes in the series were dealing with a specific topic like modeling of infectious diseases, tropical climate dynamics, energy transfer in atmosphere and oceans, and dynamics of biological systems, this volume has the generic MPE title of the series. Its subtitle Protecting Our Planet, Learning from the Past, Safeguarding for the Future can refer to anything and is not really enlightening the contents either. The reason is that this volume covers several different topics. There are thirteen papers divided over four parts: (1) Geo- and Physical sciences, (2) Life Sciences, (3) Ecology and Evolution, and (4) Socio-Economics and Infrastructure. Each part of almost a hundred pages consists of three or four papers. All of the papers were invited and refereed and can be read independently as an introduction to the problem discussed. Discussing each paper in some detail would be too much for this review, so what follows is just a birds eye view of the topics.

Learning from the past is applicable to the first paper in part 1. It takes data from glacial cycles in the Pleistocene Epoch to investigate the influence of CO2 by proposing different models that take care of the eccentricity of the Earth's orbit, precession, and inclination of its axis. This allows to predict the corresponding exposure to sunshine, the melting of ice, the volume of Northern Atlantic Deep Water, and other parameters. The resulting dynamical systems with bifurcation and limit cycles is investigated to make predictions about our future. The second paper is somewhat related and investigates the effect of melting of polar ice over the last 700 Kyears which changes the volume of solid land and the volume of water on the surface. Finding patterns in the evolution of precipitation of the third paper is more difficult since this has much more variable dynamics, both in time and in space.

In the Life Science part, we find a paper on the relation between malaria and climate change. The latter influences air-water temperature differences which influences the survival of the mosquito, and hence the spreading of malaria. Another paper models the spread of Buruli ulcer disease in Ghana and how the human-human and human-environment interactions influence that. These two papers obviously focus on problems of the African continent. A much more general view is taken in a paper that analyses the massive amount of data in the health sector that is becoming available. The purpose is to use this to obtain probabilistic data-driven models for some phenomena like food poisoning, transmission of avian influenza etc.

Ecology and Environment is represented by three papers. It is known that ecosystems are highly nonlinear and have several stable and unstable states to which it evolves under different circumstances. Extreme climate events or human intervention can result in a malfunctioning ecosystem. If we have proper models and can find the regions in parameter space in which some of these local states are embedded, perhaps we can we make these changes more gradual or partially prevent them. Dryland ecosystems are used as a case study. A paper about biodiversity is stressing the importance of having parameters for the number of different species, but also for how evenly they are distributed. In another paper several models for population dynamics at different scales are proposed to calculate an extinction risk.

The last part has contributions about Socio-economic Systems like the modelling of food systems, which, at a global scale, involves production, distribution, and consumption with complex network interactions. Bad management may lead to malnutrition and overweight existing simultaneously in different regions. A second paper models how the ecosystem can benefit from natural capital. Optimal management can be obtained from a dynamical optimization problem. For example what should be the optimal fishing quota for a steady state equilibrium? The Infrastructure is the subject of the remaining two papers. Network and scheduling analysis is used to obtain quantitative models to manage the restoration of infrastructure after extreme events. With the Internet of Things a massive network is created that should be managed to optimize urban societies. Transportation and municipal water services are used as case studies.

This quick survey illustrates the general set-up. It definitely illustrates the diversity of problems and methods that we have to deal with to cope with the enormous challenges that result from an increasing globalisation. Problems become very large scale, involve large networks, and become highly nonlinear. The parameters are changing drastically faster and the systems become very sensitive to small changes when they approach unstable states. If we ever want to control and manage all these effects, it will need substantial scientific research. And time is pressing. Here we meet several examples of research in action, trying to solve practical problems. Let's hope this is a wake-up call and that this will result in joining forces worldwide and across specialisations, instead of driving researchers away from each others into their own abstract unworldly niches with the only purpose to publishing papers.