We study ensembles of similar systems under load of environmental factors. The phenomenon of adaptation has similar properties for systems of different nature. Typically, when the load increases above some threshold, then the adapting systems become more different (variance increases), but the correlation increases too. If the stress continues to increase then the second threshold appears: the correlation achieves maximal value, and start to decrease, but the variance continue to increase. In many applications this second threshold is a signal of approaching of fatal outcome.
This effect is supported by many experiments and observation of groups of humans, mice, trees, grassy plants, and on financial time series. A general approach to explanation of the effect through dynamics of adaptation is developed. H. Selye introduced “adaptation energy" for explanation of adaptation phenomena. We formalize this approach in factors – resource models and develop hierarchy of models of adaptation. Different organization of interaction between factors (Liebig's versus synergistic systems) lead to different adaptation dynamics. This gives an explanation to qualitatively different dynamics of correlation under different types of load and to some deviation from the typical reaction to stress.
In addition to the “quasistatic" optimization factor – resource models, dynamical models of adaptation are developed, and a simple model (three variables) for adaptation to one factor load is formulated explicitly.

The G-function formalism has been widely used in the context of evolutionary games for identifying evolutionarily stable strategies (ESS). This formalism was developed for and applied to point processes. Here, we examine the G-function formalism in the settings of spatial evolutionary games and strategy dynamics, based on reaction-diffusion models. We start by extending the point process maximum principle to reaction-diffusion models with homogeneous, locally stable surfaces. We then develop the strategy dynamics for such surfaces. When the surfaces are locally stable, but not homogenous, the standard definitions of ESS and the maximum principle fall apart. Yet, we show by examples that strategy dynamics leads to convergent stable inhomogeneous strategies that are possibly ESS, in the sense that for many scenarios which we simulated, invaders could not coexist with the exisiting strategies.

Since Rosenzweig showed the destabilisation of exploited ecosystems, the so called Paradox of enrichment, several mechanisms have been proposed to resolve this paradox. In this paper we will show that a feeding threshold in the functional response for predators feeding on a prey population stabilizes the system and that there exists a minimum threshold value above which the predator-prey system is unconditionally stable with respect to enrichment. Two models are analysed, the first being the classical Rosenzweig-MacArthur (RM) model with an adapted Holling type-II functional response to include a feeding threshold. This mathematical model can be studied using analytical tools, which gives insight into the mathematical properties of the two dimensional ode system and reveals underlying stabilisation mechanisms. The second model is a mass-balance (MB) model for a predator-prey-nutrient system with complete recycling of the nutrient in a closed environment. In this model a feeding threshold is also taken into account for the predator-prey trophic interaction. Numerical bifurcation analysis is performed on both models. Analysis results are compared between models and are discussed in relation to the analytical analysis of the classical RM model. Experimental data from the literature of a closed system with ciliates, algae and a limiting nutrient are used to estimate parameters for the MB model. This microbial system forms the bottom trophic levels of aquatic ecosystems and therefore a complete overview of its dynamics is essential for understanding aquatic ecosystem dynamics.

We construct a stochastic model of bacteriophage parasitism of a host bacteria that accounts for demographic stochasticity of host and parasite and allows for multiple bacteriophage adsorption to host. We analyze the associated deterministic model, identifying the basic reproductive number for phage proliferation, showing that host and phage persist when it exceeds unity, and establishing that the distribution of adsorbed phage on a host is binomial with slowly evolving mean. Not surprisingly, extinction of the parasite or both host and parasite can occur for the stochastic model.

The seasonality hypothesis states that climates characterized by large annual cycles select for large body sizes. In order to study the effects of seasonality on the evolution of body size, we use a model that is based on physiological rules and first principles. At the ecological time scale, our model results show that both larger productivity and seasonality may lead to larger body sizes. Our model is the first dynamic and process-based model to support the seasonality hypothesis and hence demonstrates the importance of basing models on physiological processes. We focus not only on variability at the ecological time scale, but also on the temporal variations in seasonality existing at geological time scales. A particularly strong forcing of seasonality exists on the scale of 20,000-400,000 years, the scale of Milankovitch cycles. Therefore, we simulated the evolutionary response of body size to a Milankovitch-type of forcing of climate and food density. Results illustrate that for a given level of investment in reserves body size may track climatic cycles, and that below a certain seasonality threshold the body size will decrease rapidly, leading to extinction.

We develop a stage-structured model that describes the dynamics of two competing species each of which have sexual and clonal reproduction. This is typical of many plants including irises. We first analyze the dynamical behavior of a single species model. We show that when the inherent net reproductive number is smaller than one then the population will go to extinction and if it is larger than one then an interior equilibrium exists and it is globally asymptotically stable. Then we analyze the two-species model and establish conditions on the reproduction and survivorship rates that lead to competitive exclusion. We show that the winner species is the one that attains higher density at which its net reproductive number equals unity. Numerical results corroborating the theoretical ones are also presented.