Large Number Of Steady States Research Articles

Spatially heterogeneous populations with mixed negative and positive local density dependence

Identifying the steady states of a population is a key issue in theoretical ecology, that includes the study of spatially heterogeneous populations. There are several examples of real ecosystems in patchy environments where the habitats are heterogeneous in their local density dependence. We investigate a multi-patch model of a single species with spatial dispersal, where the growth of the local population is logistic in some localities (negative density dependence) while other patches exhibit a strong Allee effect (positive density dependence). When the local dynamics is logistic in each patch and the habitats are interconnected by dispersal then the total population has only the extinction steady state and a componentwise positive equilibrium, corresponding to persistence in each patch. We show that animal populations in patchy environments can have a large number of steady states if local density dependence varies over the locations. It is demonstrated that, depending on the network topology of migration routes between the patches, the interaction of spatial dispersal and local density dependence can create a variety of coexisting stable positive equilibria. We give a detailed description of the multiple ways dispersal can rescue local populations from extinction.

Effect of Spatial Segregation on Commensalistic Cultures—Series Reactors

A comprehensive analysis of static and dynamic behavior of a mixed culture in two bioreactors in series is presented considering anaerobic digestion involving acidogens (X) and methanogens (Y) as the example bioprocess. A single continuous culture may operate at up to seven steady states, including up to four coexistence steady states, with only one coexistence steady state being locally stable. The bioreactors in the two-reactor system are identical in terms of composition of and dilution rate with respect to fresh feed. A two-reactor system may admit up to forty nine steady states, which are comprised of up to forty coexistence steady states, at least at very low interaction rates (R). When physically realizable, up to five coexistence steady states are locally stable. The static and dynamic analysis of the two-reactor system is facilitated by appropriate grouping of large number of steady states arising for very low R into nine clusters and grouping of the clusters into three cluster groups. Numerical ...

Fate of commensalistic cultures in identical coupled bioreactors

A comprehensive analysis of static and dynamic behavior of a mixed culture in two identical coupled bioreactors is presented considering anaerobic digestion involving acidogens (X) and methanogens (Y) as the example bioprocess. A single continuous culture may operate at up to seven steady states, including up to four coexistence steady states, with only one coexistence steady state being locally stable. The one-way interaction between X and Y allows for compartmentalization of the system for a stand-alone bioreactor and two coupled bioreactors into two subsystems, which facilitates the analysis of steady state types and stability characteristics of these and classification of dynamic behavior. The bioreactors in the two-reactor system are identical only in terms of feed composition and reactor space time. A two-reactor system may admit up to forty nine steady states, which are comprised of up to forty coexistence steady states, at least at very low interaction rate ( R). The static and dynamic analysis of the two-reactor system is facilitated by appropriate grouping of large number of steady states arising for very low R into nine clusters. Numerical illustrations reveal the rich steady state structure of the bioprocess in coupled bioreactors. While a single bioreactor can operate at only one locally stable coexistence steady state, the coupled bioreactors can operate at up to five locally stable coexistence steady states over certain ranges of R. The two-reactor system is operationally more flexible and more robust vis-a-vis single reactor as concerns maintenance of mixed culture. Emergence of four additional steady state clusters and additional coexistence and partial washout steady states at intermediate R reveals that the coupled bioreactors are an example of a complex system.

Multiple attractors in immunology: theory and experiment

This selective survey discusses the relative merits of various modeling approaches in immunology that exhibit multiple attractors, and also assesses the ability of the different models to contribute to deeper biological understanding. The first topic is global anti-idiotypic network models, which, like Hopfield neural network models, exhibit a large number of steady states that are identified with memory. It is shown that a `reverse engineering approach' to T-cell vaccination for autoimmunity, featuring steady states corresponding respectively to `normality', `vaccination' and `disease', is able to spur new experiments, in spite of the model's deliberate neglect of almost all biological detail. Mention is made of several other T-cell models that feature bistability for Th1 or Th2 dominance, or for activation and unresponsiveness.

Emergent properties of coupled enzyme reaction systems 1. Switching and clustering behaviour

The dynamics of locally and globally coupled cells that convert a substrate to a product via an uncompetitive substrate-inhibition mechanism is studied. When the cell-cell coupling strength is below a threshold value, the coupled system exhibits a large number of steady states; however, all cells cluster to one state when coupling exceeds the threshold value. The coupled system also exhibits a buffering capacity that maintains low and almost constant intracellular and extracellular substrate levels; however, there exists a threshold value on the influx rate of extracellular substrate beyond which the system switches to higher substrate levels. This transition becomes sharper as the number of coupled cells increases. Propagation failure of concentration fronts between adjacent cells is also exhibited by the system.

Bifurcation and complex dynamics in fixed-bed catalytic reactors

Bifurcation theory and numerical continuation techniques are used to investigate the complex static and dynamic characteristics of a fixed-bed reactor modelled by a heterogeneous cell model. The model takes into account the thermal coupling between adjacent reactor cells due to radiation and conduction. A rich variety of interesting static phenomena is uncovered, including, multiplicity with a large number of steady states. A Hopf bifurcation point is identified from which a periodic branch emanates. The periodic branch terminates at a homoclinical infinite period (IP) bifurcation point. Wrong-way behaviour and sensitivity of the final steady-state profile to initial conditions are also observed.

On the steady-state behavior of the nonadiabatic autothermal tubular reactor

Singularity theory, local bifurcation analysis and numerical simulations are combined to investigate the steady-state behavior of the nonadiabatic autothermal tubular reactor in which a first-order exothermic reaction occurs. The steady-state model of the reactor consists of three first-order nonlinear ordinary differential equations with unmixed boundary conditions and contains six dimensionless groups. It is found that this simple convection-reaction model can have an arbitrarily large number of steady states. In addition, when the residence time is varied, the different possible bifurcation diagrams that exist vary from all those that occur in the classical cooled CSTR model to those found in the classical nonadiabatic tubular reactor model. We use local bifurcation techniques to make these predictions and confirm them with numerical solution of the full model. Singularity theory is used to classify the different types of bifurcation diagrams for a case where there is a maximum of five steady states.

Dynamics of commensalistic systems with self‐ and cross‐inhibition

AbstractUp to three stable steady states are possible in a simple commensalistic system, taking place in an open‐loop mixed reactor when the growth rates of the two species are inhibited by the substrates they prey on (Self‐inhibition). Two stable states are possible in a system with noncompetitive inhibition of the species by the substrate they are not preying on (cross‐inhibition). A large number of steady states as well as oscillatory states are possible when both self‐ and cross‐inhibition are strong. Multiplicity of steady states is also possible in a reactor with biomas recirculation for these kinetics. Yet, the latter is more stable than the open‐loop reactor in the sense that the domain of steady‐state multiplicity is narrower. The stability of steady states and the dynamics of the systems for each of the investigated kinetics are summarized in a qualitative phase plane. The importance of the analysis for improving the selectively and yield of the system and for predicting the response of the system to changes in the operating conditions, is discussed.