This book presents the understanding of how the different forms of regulatory mechanisms, like birth and death, competition, consumption and the like, result in changes in the stability and dynamics of ecological systems. It deals with a profound and unique insight into the mathematical richness of basic ecological models. Organised into eight chapters, the book discusses the models of mathematical ecology, the dynamical models of single-species system in a polluted environment, the dynamical behaviour of different nonautonomous two species systems in a polluted environment, the influence of environmental noise in Gompertzian and logistic growth models, stability behaviour in randomly fluctuating versus deterministic environments of two interacting species, stochastic analysis of a demographic model of urbanization and stability behaviour of a social group by means of loop analysis, thermodynamic criteria of stability and stochastic criteria of stability. The book will be useful to the researchers and graduate students who wish to pursue research in mathematical ecology.

In this contribution, several probabilistic tools to study population dynamics are developed. The focus is on scaling limits of qualitatively different stochastic individual based models and the long time behavior of some classes of limiting processes. Structured population dynamics are modeled by measure-valued processes describing the individual behaviors and taking into account the demographic and mutational parameters, and possible interactions between individuals. Many quantitative parameters appear in these models and several relevant normalizations are considered, leading to infinite-dimensional deterministic or stochastic large-population approximations. Biologically relevant questions are considered, such as extinction criteria, the effect of large birth events, the impact of environmental catastrophes, the mutation-selection trade-off, recovery criteria in parasite infections, genealogical properties of a sample of individuals. These notes originated from a lecture series on Structured Population Dynamics at Ecole polytechnique (France). Vincent Bansaye and Sylvie Méléard are Professors at Ecole Polytechnique (France). They are a specialists of branching processes and random particle systems in biology. Most of their research concerns the applications of probability to biodiversity, ecology and evolution.

Survival of the fittest” is a tautology, because those that are “fit” are the ones that survive, but to survive, a species must be “fit”. Modern evolutionary theory avoids the problem by defining fitness as reproductive success, but the complexity of life that we see today could not have evolved based on selection that favors only reproductive ability. There is nothing inherent in reproductive success alone that could result in higher forms of life. Evolution from a Thermodynamic Perspective presents a non-circular definition of fitness and a thermodynamic definition of evolution. Fitness means maximization of power output, necessary to survive in a competitive world. Evolution is the “storage of entropy”. “Entropy storage” means that solar energy, instead of dissipating as heat in the Earth, is stored in the structure of living organisms and ecosystems. Part one explains this in terms comprehensible to a scientific audience beyond biophysicists and ecosystem modelers. Part two applies thermodynamic theory in non-esoteric language to sustainability of agriculture, and to conservation of endangered species. While natural systems are stabilized by feedback, agricultural systems remain in a mode of perpetual growth, pressured by balance of trade and by a swelling population. The constraints imposed by thermodynamic laws are being increasingly felt as economic expansion destabilizes resource systems on which expansion depends.

Previous research on the extinctions that occur in niche model food webs with deterministic and stochastic dynamics has shown that the structure of the food web can play an important role in extinction cascades. In this thesis, other types of synthetic food web models are considered, namely the cascade and generalized cascade models, and the extinction cascades of these food webs are compared with previous findings on the extinction cascades from the niche model. It was found that there are many similarities in the results for all three models, which prompted a closer analysis using food webs with deterministic dynamics. We developed a method to theoretically predict the survival or extinction of species in two- and three-species food webs, and compared the predictions with numerical results.