Modelling Energy Dissipation Due to Soil-Moisture Hysteresis

Cyclic wetting and drying of soils due to changing weather conditions is characterised by a well-documented hysteresis in the relationship between the pressure or matric potential and the water content in the soil. Hysteresis manifests itself through a difference in drying and wetting curves. Its presence suggests that there is dissipation of energy associated with intermittent wetting and drying of the soil, which can be released in the form of heat. In this paper, we discuss a model for evaluation of the energy dissipation rate due to soil-moisture hysteresis. The model has three main ingredients: (a) a hysteresis constitutive relationship between soil-water potential and moisture content in the soil, which is modelled by the Preisach operator; (b) a lumped form of Darcy’s law assuming that the water flux is proportional to the difference of potentials in the soil and on its surface; (c) a rainfall term governing the evolution of the outer potential, which is modelled by a periodic forcing or a Poisson process. We propose a combination of analytic and numerical methods to evaluate the energy dissipation rate and how it is affected by the variation of rain parameters, such as the frequency of arrivals of rain cells and their shape, intensity and duration.     

The response of the woodpigeon (Columba palumbus) to relaxation of intraspecific competition: A hybrid modelling approach

The recent rapid growth of the woodpigeon population in the British Isles is a cause for concern for environmental managers. It is unclear what has driven their increase in abundance. Using a mathematical model, we explored two possible mechanisms, reduced intraspecific competition for food and increased reproductive success. We developed an age-structured hybrid model consisting of a system of ordinary differential equations that describes density-dependent mortality and a discrete component, which represents the birth-pulse. We investigated equilibrium population dynamics using our model. The two hypotheses predict contrasting population age profiles at equilibrium. We adapted the model to examine the impacts of control measures. We showed that an annual shooting season that follows the period of density-dependent mortality is the most effective control strategy because it simultaneously removes adult and juvenile woodpigeons. The model is a first step towards understanding the processes that influence the dynamics of woodpigeon populations.     

Convexity in projection matrices: Projection to a calibration problem

Convexity, as a fundamental property of sets and functions defined on convex sets, plays an important role in many mathematical and applied disciplines, including extremal and optimal-control problems. We prove the set of all feasible projection matrices in a general class of matrix models for stage-structured population dynamics to be convex and the dominant eigenvalue (λ₁) of any projection 2 × 2 matrix to be either a convex, or a concave function on a simplex of the matrix first-row entries (i.e., stage-specific reproduction rates). The latter is also conjectured for the general n × n case. Though looking far from practical needs of matrix population models, this mathematical result has appeared to be quite useful in solving a practical problem to calibrate the projection matrix, i.e., to estimate all the stage-specific vital rates, from empirical data. The data from monitoring of individual life histories of marked plants on permanent sample plots during successive years enable direct calculation of the stage-specific survival and ontogenetic transition rates, but the rates of reproduction do remain uncertain as far as the parent plants can hardly be determined for the (not yet marked!) recruitment.     

The monoculture vs. rotation strategies in forestry: formalization and prediction by means of Markov-chain modelling

The monoculture strategy of forest management, where the same tree species (e.g., Picea abies) is cultivated in a number of successive planting-growing-felling cycles, is generally considered to be economically efficient, yet not sustainable as it reduces biodiversity in the forest. The sound alternative suggests a long-term strategy of forest management in which different forest types rotate either with planting after clear cutting, or by natural forest succession, yet the commercial output remains dubious. We suggest an approach to formalization and modelling forest dynamics in the long-term by means of Markov chains, the monoculture strategy resulting in an absorbing chain and the rotation one in a regular chain. The approach is illustrated with a case study of Russkii Les, a managed forest located in the Moscow Region, Russia, and the nearby forest reserve having been used as a data source for undisturbed forest dynamics. Starting with conceptual schemes of transitions among certain forest types (states of the chain) in the monoculture and rotation cases, we estimated the transition probabilities by an original method based on average duration of the corresponding states and on the likelihood of alternative transitions from a state into the next one. Formal analysis of the regular chain reveals an opportunity to achieve particular management objectives within the rotation strategy, in particular, to get the distribution of forest types in accordance with an adopted hierarchy of their commercial values, i.e. more valuable types have greater shares.