N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

This paper deals with the problem of a ratio-dependent predator-prey model. The deterministic and stochastic behaviour of the model system around biologically feasible equilibria are studied. Conditions for which the deterministic model enter into Hopf-bifurcation are worked out. Stochastic stability of the system around positive interior equilibrium is studied. To substantiate our analytical findings numerical simulations are carried out for a hypothetical set of parameter values.

http://iopscience.iop.org/article/10.1088/0951-7715/18/2/022/pdf

Ratio-dependent predator–prey model: effect of environmental fluctuation and stability

39.Conventional theories of population and community dynamics are based on a singlecurrency such as number of individuals, biomass, carbon or energy. However,organisms are constructed of multiple elements and often require them (in particularcarbon, phosphorus and nitrogen) in different ratios than provided by their resources;this mismatch may constrain the net transfer of energy and elements through trophiclevels. Ecological stoichiometry, the study of the balance of elements in ecologicalprocesses, offers a framework for exploring ecological effects of such constraints. Wereview recent theoretical and empirical studies that have considered how stoichiometrymay affect population and community dynamics. These studies show thatstoichiometric constraints can affect several properties of populations (e.g. stability,oscillations, consumer extinction) and communities (e.g. coexistence of competitors,competitive interactions between different guilds). We highlight gaps in generalknowledge and focus on areas of population and community ecology whereincorporation of stoichiometric constraints may be particularly fruitful, such asstudies of demographic bottlenecks, spatial processes, and multi-species interactions.Finally, we suggest promising directions for new research by recommending potentialstudy systems (terrestrial insects, detritivory-based webs, soil communities) to improveour understanding of populations and communities. Our conclusion is that a betterintegration of stoichiometric principles and other theoretical approaches in ecologymay allow for a richer understanding of both population and community structure anddynamics.S. J. Moe, Norwegian Inst. for Water Research, NO-0411 Oslo, Norway (jannicke.moe@niva.no) and Centre for Ecological and Evolutionary Synthesis (CEES), Univ.ofOslo, NO-0316 Oslo, Norway.

/pdf/recent-advances-in-ecological-stoichiometry-insights-for-3turwfk5i7.pdf

Recent advances in ecological stoichiometry: insights for population and community ecology

Towards a resolution of ‘the paradox of the plankton’: A brief overview of the proposed mechanisms

Dynamic analysis of time fractional order phytoplankton–toxic phytoplankton–zooplankton system

Modelling phytoplankton allelopathy in a nutrient-plankton model with spatial heterogeneity

Dynamics of an autotroph-herbivore ecosystem with nutrient recycling

We present and analyze a three-species mathematical model of tumor progression and regression using prey-predator dynamics. The tumor cell population is supposed to be the prey species and the cytotoxic T-lymphocytes (CTL) and macrophages (which are killer cells of the immune system) as their predator. Using a ratio-dependent predation functional form, the ODE model is analyzed for stability criteria. A discrete time delay required for destruction of tumor cells is then incorporated into the basic model and the resulting delayed model is studied to explore the significance of various system parameters controlling the stability of the system. The active killer cells are shown to be the key population species controlling the system dynamics. A range of the delay parameter is estimated that ensures stability of different system components. Numerical simulations are performed to support analytical findings.

Role of cytotoxic t-lymphocytes in tumor stability: a prey-predator modeling approach

In the present paper, a mathematical model, originally proposed by Danziger and Elmergreen and describing the thyroid-pituitary homeostatic mechanism, is modified and analyzed for its physiological and clinical significance. The inuence of different system parameters on the stability behavior of the system is discussed. The transportation delays of different hormones in the bloodstream, both in the discrete and distributed forms, are considered. Delayed models are analyzed regarding the stability and bifurcation behavior. Clinical treatment of periodic catatonic schizophrenia is discussed in presence of transportation delays. Numerical simulations are presented to support analytic results.

/pdf/a-mathematical-model-describing-the-thyroid-pituitary-axis-z9vy7cbbt7.pdf

A mathematical model describing the thyroid-pituitary axis with time delays in hormone transportation

We present and analyze an epidemiological model containing Susceptible (S(t)) and Infected (I(t)) populations. The incidence rate is assumed to be nonlinear in the infected fraction (Ip(t)) as well as the susceptible fraction (Sq(t)). The dynamical behavior of the system is investigated from the point of view of stability and bifurcation. To model the recovery time of infected populations, a recovery delay, both in distributed and discrete form is introduced. In all the cases, it is shown that the infected incidence fraction p plays a vital role in controlling the dynamical behavior of the system. Numerical simulations are performed to justify the analytical findings.

R. Bhattacharyya

Papers

Modelling phytoplankton allelopathy in a nutrient-plankton model with spatial heterogeneity

Dynamics of an autotroph-herbivore ecosystem with nutrient recycling

Role of cytotoxic t-lymphocytes in tumor stability: a prey-predator modeling approach

A mathematical model describing the thyroid-pituitary axis with time delays in hormone transportation

Dynamics of a delayed epidemiological model with nonlinear incidence: the role of infected incidence fraction