Modified cycling index for ecological applications

Ecological Modelling, 1985

The principle that energy does not cycle in ecosystems is examined by path and flow analyses of a small, steady state, energy model of an oyster reef community. Path numbers increase combinatoriaUy without bound as path length increases, at much faster rates with energy storage than without. Energy flows over only very short paths in the nonstorage case, but with storage present most flow is associated with long paths. Cycling results. A cycling index modified from the literature standard to include storage shows 11% cycling in the subject model. Energy does cycle in ecosystems; it differs from matter in its dissipation, not cycling, characteristics. A new result is the role of storage in diversifying path structure and increasing flows in networks. The basis for this is examined through influence theory of mathematical graphs, with the conclusion that energy storage as biomass is the root cause of ecosystem energy cycling. Issues of food chain length and embodied energy are reviewed in light of the long paths and related energy flows present in even small order models.

Ecological Modelling, 2006

We used Markov chains to assess residence time, first passage time, rate of transfers between compartments, recycling index with a general mathematical formalism. Such a description applies to any flow-balanced system that can be modelled as a series of discrete stages or compartments through which matter flows. We derived a general set of equations from a probabilistic approach and applied them to a food web and a physical system derived from the literature. We therefore analysed preferential pathways of matter and behaviour of these systems and showed how it was possible to build up and exploit indices on the basis of a transition probability matrix describing the network, and to characterize with a generic algorithm: (1) the total indirect relationships between two compartments, (2) the residence time of one compartment and (3) the general recycling pathways including the amount of matter recycling and the implication of each compartment in recycling.

Ecological Modelling, 1978

Nutrient flow through ecosystems is modeled as a discrete Markov chain whose transition probabilities are stationary or time inhomogenous according to whether a steady state or dynamic ecosystem is modeled, respectively. Expected residence time and number of intercompartmental transfers for a nutrient within a set of compartments are derived. Variances of these random variables are also considered..~ measure for ecosystem resource recycling is given as a weighted sum of probabilities.

Ecological Modelling, 2009

Ecological network analysis (ENA), predicated on systems theory and Leontiev input-output analysis, is a method widely used in ecology to reveal ecosystem properties. An important ecosystem property computed in ENA is throughflows, the amount of matter/energy leaving each compartment of the ecosystem. Throughflows are analyzed via a matrix N representing their relationships to the driving input at the boundary. Network particle tracking (NPT) builds on ENA to offer a Lagrangian particle method that describes the activity of the ecosystem at the microscopic level. This paper introduces a Lagrangian throughflow analysis methodology using NPT and shows that the NPT throughflow matrix, N, agrees with the conventional ENA throughflow matrix, N, for ecosystems at steady-state with donor-controlled flows. The matrix N is computed solely from the pathways (particles' histories) generated by NPT simulations and its average over multiple runs of the algorithm with longer simulation time agrees with the Eulerian N matrix (Law of Large Numbers). While the traditional NEA throughflow analysis is mostly used with steady-state ecosystem models, the Lagrangian throughflow analysis that we propose can be used with non-steady-state models and paves the way for the development of dynamic throughflow analysis.

Radioprotection, 2005

Model predictions of solute transport in a wetland were compared with tracer experimental data to illustrate that the number of compartments in a compartmental model should be selected according to certain rules to accurately describe the transport process. If the input pulse is short, the model structure affects the output significantly. The temporal moments of the residence time distribution was obtained from a general solution of the compartmental model in the Laplace domain derived with an arbitrary inflow pollutograph. The variance of the residence time can be used as a useful tool to analyse the effect of model structure and long-term inflow pollutograph on the response of the model predictions.

Journal of the Nigerian Society of Physical Sciences, 2020

In this research work, we investigate the concentration profiles in the single and the interconnected multiple-compartment systems with sieve partitions for the transport of chemical species with second order chemical reaction kinetics. With assumption of unidirectional transport of chemical species and constant physical properties with same equilibrium contant, the developed partial differential equations representing the two systems are spatially discretized using the Method of Lines (MOL) technique and the resulting semi-discrete system of ODEs are solved using MATLAB ode15s solver. The results show that the interconnected multiple-compartment system has lower concentration profile than the single-compartment system for different values of diffusivities.

Ecological Modelling, 2003

When modelling real ecosystems, a number of techniques need steady state condition to proceed in their analysis. In ecosystem network models this means that energy entering the system exactly balances the output. Steady state, however, is not a straightforward outcome of network construction and, to have this condition satisfied, network analysis uses balancing procedure. This operation leads to restructuring the weighted network, changing the values of some network flows; this can affect drastically the results of the analysis. Presently, two algorithms are used for balancing ecosystem networks: input-based approach and output-based approach. In the former input flows are kept constant while outputs and transfer coefficients are manipulated; the latter requires that inputs and intercompartmental flows are modified. This paper discusses the effects of these algorithms on some products of network analysis, in particular system level indices such as total system throughput (TST) and ascendency. Also it suggests four new procedures that, while balancing the networks, can minimise changes on measured flows and distortion on results of the analysis. (S. Allesina). compartment are equal, so that there is no increase or decrease in mass: dx i dt = 0, ∀x i ∈ S 0304-3800/03/$ -see front matter

Mathematical Biosciences, 1984

The occupancy principle and the mean-transit-time theorem are derived for the passage of a tracer through a system that can be described by a general pool model. It is proved, using matrix theory, that if (and only if) tracer entering the system labels equally all tracee fluxes into the system, then the integral of the tracer concentration is the same in all the pools. It is also proved that if, in addition, all flow out of the system is through the observation point, the first moment of the tracer concentration at the observation point can be used to calculate the total amount of trace in the system. The necessity of this condition is analyzed. Examples are given of models in which the occupancy principle and the mean-transit-time theorem hold or do not hold.

Journal of Contaminant Hydrology, 2015

Travel-time based models simplify the description of reactive transport by replacing the spatial coordinates with the groundwater travel time, posing a quasi one-dimensional (1-D) problem and potentially rendering the determination of multidimensional parameter fields unnecessary. While the approach is exact for strictly advective transport in steady-state flow if the reactive properties of the porous medium are uniform, its validity is unclear when local-scale mixing affects the reactive behavior. We compare a two-dimensional (2-D), spatially explicit, bioreactive, advective-dispersive transport model, considered as "virtual truth", with three 1-D travel-time based models which differ in the conceptualization of longitudinal dispersion: (i) neglecting dispersive mixing altogether, (ii) introducing a local-scale longitudinal dispersivity constant in time and space, and (iii) using an effective longitudinal dispersivity that increases linearly with distance. The reactive system considers biodegradation of dissolved organic carbon, which is introduced into a hydraulically heterogeneous domain together with oxygen and nitrate. Aerobic and denitrifying bacteria use the energy of the microbial transformations for growth. We analyze six scenarios differing in the variance of log-hydraulic conductivity and in the inflow boundary conditions (constant versus time-varying concentration). The concentrations of the 1-D models are mapped to the 2-D domain by means of the kinematic (for case i), and mean groundwater age (for cases ii & iii), respectively. The comparison between concentrations of the "virtual truth" and the 1-D approaches indicates extremely good agreement when using an effective, linearly increasing longitudinal dispersivity in the majority of the scenarios, while the other two 1-D approaches reproduce at least the concentration tendencies well. At late times, all 1-D models give valid approximations of two-dimensional transport. We conclude that the conceptualization of nonlinear bioreactive transport in complex multidimensional domains by quasi 1-D travel-time models is valid for steady-state flow fields if the reactants are introduced over a wide crosssection, flow is at quasi steady state, and dispersive mixing is adequately parametrized.

Biosystems, 2001

In this contribution, we present the symbolic time course equations corresponding to a general model of a linear compartmental system, closed or open, with or without traps and with zero input. The steady state equations are obtained easily from the transient phase equations by setting the time. Special attention has been given to the open systems, for which an exhaustive kinetic analysis has been developed to obtain important properties. Besides, the results have been particularized to open systems without traps and an alternative expression for the distribution function of exit times has been provided. We have implemented a versatile computer program, that is easy to use and with a user-friendly format of the input of data and the output of results. This computer program allows the user to obtain all the information necessary to derive the symbolic time course equations for closed or open systems as well as for the derivation of the distribution function of exit times.