We study ensembles of similar systemsunder load of environmental factors. The phenomenon of adaptationhas similar properties for systems of different nature. Typically,when the load increases above some threshold, then the adaptingsystems become more different (variance increases), but thecorrelation increases too. If the stress continues to increasethen the second threshold appears: the correlation achievesmaximal value, and start to decrease, but the variance continue toincrease. In many applications this second threshold is a signalof approaching of fatal outcome.
This effect is supported by many experiments and observation ofgroups of humans, mice, trees, grassy plants, and on financialtime series. A general approach to explanation of the effectthrough dynamics of adaptation is developed. H. Selye introduced“adaptation energy" for explanation of adaptation phenomena. Weformalize this approach in factors – resource models anddevelop hierarchy of models of adaptation. Different organizationof interaction between factors (Liebig's versus synergisticsystems) lead to different adaptation dynamics. This gives anexplanation to qualitatively different dynamics of correlationunder different types of load and to some deviation from thetypical reaction to stress.
In addition to the “quasistatic" optimization factor – resourcemodels, dynamical models of adaptation are developed, and asimple model (three variables) for adaptation to one factor loadis formulated explicitly.

The G-function formalism has beenwidely used in the context of evolutionary games for identifyingevolutionarily stable strategies (ESS). This formalism wasdeveloped for and applied to point processes. Here, we examine the G-functionformalism in the settings of spatial evolutionarygames and strategy dynamics, based on reaction-diffusion models. We startbyextending the point process maximum principle to reaction-diffusion modelswith homogeneous, locally stable surfaces. We then develop the strategy dynamics forsuch surfaces. When the surfaces are locally stable, but nothomogenous, the standard definitions of ESS and the maximumprinciple fall apart. Yet, we show by examples that strategy dynamics leads toconvergent stable inhomogeneous strategies that are possibly ESS, in the sensethat 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, severalmechanisms 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 minimumthreshold value above which the predator-prey system is unconditionally stable with respect to enrichment. Two models areanalysed, 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 ofthe 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 ofbacteriophage parasitism of a host bacteria that accounts fordemographic stochasticity of host and parasite and allows formultiple bacteriophage adsorption to host. We analyze the associateddeterministic model, identifying the basic reproductive number forphage proliferation, showing that host and phage persist when itexceeds unity, and establishing that the distribution of adsorbedphage on a host is binomial with slowly evolving mean. Notsurprisingly, extinction of the parasite or both host and parasitecan 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.