Modeling migratory grazing of zooplankton on toxic and non-toxic phytoplankton

Migratory grazing of zooplankton between non-toxic phytoplankton (NTP) and toxic phytoplankton (TPP) is a realistic phenomena unexplored so far. The present article is a first step in this direction. A mathematical model of NTP–TPP-zooplankton with constant and variable zooplankton migration is proposed and analyzed. The asymptotic dynamics of the model system around the biologically feasible equilibria is explored through local stability analysis. The dynamics of the proposed system is explored and displayed for different combination of migratory parameters and toxin inhibition parameters. Our analysis suggests that the migratory grazing of zooplankton has a significant role in determining the dynamic stability and oscillation of phytoplankton zooplankton systems.

Studying reactive processes with classical dynamics: rebinding dynamics in MbNO

A new surface-crossing algorithm suitable for describing bond-breaking and bond-forming processes in molecular dynamics simulations is presented. The method is formulated for two intersecting potential energy manifolds which dissociate to different adiabatic states. During simulations, crossings are detected by monitoring an energy criterion. If fulfilled, the two manifolds are mixed over a finite number of time steps, after which the system is propagated on the second adiabat and the crossing is carried out with probability one. The algorithm is extensively tested (almost 0.5 mu s of total simulation time) for the rebinding of NO to myoglobin. The unbound surface ((FeNO)-N-...) is represented using a standard force field, whereas the bound surface (Fe-NO) is described by an ab initio potential energy surface. The rebinding is found to be nonexponential in time, in agreement with experimental studies, and can be described using two time constants. Depending on the asymptotic energy separation between the manifolds, the short rebinding timescale is between 1 and 9 ps, whereas the longer timescale is about an order of magnitude larger. NO molecules which do not rebind within 1 ns are typically found in the Xenon-4 pocket, indicating the high affinity of NO to this region in the protein.

On Hamiltonian balanced dynamics and the slowest invariant manifold

The concept of a slowest invariant manifold is investigated for the five-component model of Lorenz under conservative dynamics. It is shown that Lorenz's model is a two-degree-of-freedom canonical Hamiltonian system, consisting of a nonlinear vorticity-triad oscillator coupled to a linear gravity wave oscillator, whose solutions consist of regular and chaotic orbits. When either the Rossby number or the rotational Froude number is small, there is a formal separation of timescales, and one can speak of fast and slow motion. In the same regime, the coupling is weak, and the Kolmogorov–Arnold-Moser theorem is shown to apply. The chaotic orbits are inherently unbalanced and are confined to regions sandwiched between invariant tori consisting of quasi-periodic regular orbits. The regular orbits generally contain free fast motion, but a slowest invariant manifold may be geometrically defined as the set of all slow cores of invariant tori (defined by zero fast action) that are smoothly related to such cores in the uncoupled system. This slowest invariant manifold is not global; in fact, its structure is fractal; but it is of nearly full measure in the limit of weak coupling. It is also nonlinearly stable. As the coupling increases, the slowest invariant manifold shrinks until it disappears altogether. The results clarify previous definitions of a slowest invariant manifold and highlight the ambiguity in the definition of “slowness.” An asymptotic procedure, analogous to standard initialization techniques, is found to yield nonzero free fast motion even when the core solutions contain none. A hierarchy of Hamiltonian balanced models preserving the symmetries in the original low-order model is formulated; these models are compared with classic balanced models, asymptotically initialized solutions of the full system and the slowest invariant manifold defined by the core solutions. The analysis suggests that for sufficiently small Rossby or rotational Froude numbers, a stable slowest invariant manifold can be defined for this system, which has zero free gravity wave activity, but it cannot be defined everywhere. The implications of the results for more complex systems are discussed.

Rossby number expansions, slaving principles, and balance dynamics

We consider the problem of constructing balance dynamics for rapidly rotating fluid systems. It is argued that the conventional Rossby number expansion—namely expanding all variables in a series in Rossby number—is secular for all but the simplest flows. In particular, the higher-order terms in the expansion grow exponentially on average, and for moderate values of the Rossby number the expansion is, at best, useful only for times of the order of the doubling times of the instabilities of the underlying quasi-geostrophic dynamics. Similar arguments apply in a wide class of problems involving a small parameter and sufficiently complex zeroth-order dynamics. A modified procedure is proposed which involves expanding only the fast modes of the system; this is equivalent to an asymptotic approximation of the slaving relation that relates the fast modes to the slow modes. The procedure is systematic and thus capable, at least in principle, of being carried to any order—unlike procedures based on truncations. We apply the procedure to construct higher-order balance approximations of the shallow-water equations. At the lowest order quasi-geostrophy emerges. At the next order the system incorporates gradient-wind balance, although the balance relations themselves involve only linear inversions and hence are easily applied. There is a large class of reduced systems associated with various choices for the slow variables, but the simplest ones appear to be those based on potential vorticity.

The population dynamics of disease on short and long time-scales

Observational evidence is scarce concerning the distribution of plant pathogen population sizes or densities as a function of time-scale or spatial scale. For wild pathosystems we can only get indirect evidence from evolutionary patterns and the consequences of biological invasions.We have little or no evidence bearing on extermination of hosts by pathogens, or successful escape of a host from a pathogen. Evidence over the last couple of centuries from crops suggest that the abundance of particular pathogens in the spectrum affecting a given host can vary hugely on decadal timescales. However, this may be an artefact of domestication and intensive cultivation. Host-pathogen dynamics can be formulated mathematically fairly easily–for example as SIR-type differential equation or difference equation models, and this has been the (successful) focus of recent work in crops. “Long-term” is then discussed in terms of the time taken to relax from a perturbation to the asymptotic state. However, both host and pathogen dynamics are driven by environmental factors as well as their mutual interactions, and both host and pathogen co-evolve, and evolve in response to external factors. We have virtually no information about the importance and natural role of higher trophic levels (hyperpathogens) and competitors, but they could also induce long-scale fluctuations in the abundance of pathogens on particular hosts. In wild pathosystems the host distribution cannot be modelled as either a uniform density or even a uniform distribution of fields (which could then be treated as individuals). Patterns of short term density-dependence and the detail of host distribution are therefore critical to long-term dynamics. Host density distributions are not usually scale-free, but are rarely uniform or clearly structured on a single scale. In a (multiply structured) metapopulation with coevolution and external disturbances it could well be the case that the time required to attain equilibrium (if it exists) based on conditions stable over a specified time-scale is longer than that time-scale. Alternatively, local equilibria may be reached fairly rapidly following perturbations but the meta-population equilibrium be attained very slowly. In either case, meta-stability on various time-scales is a more relevant than equilibrium concepts in explaining observed patterns.