Neural fate decisions mediated by combinatorial regulation of Hes1 and miR-9

In the nervous system, Hes1 shows an oscillatory manner in neural progenitors but a persistent one in neurons. Many models involving Hes1 have been provided for the study of neural differentiation but few of them take the role of microRNA into account. It is known that a microRNA, miR-9, plays crucial roles in modulating Hes1 oscillations. However, the roles of miR-9 in controlling Hes1 oscillations and inducing transition between different cell fates still need to be further explored. Here we provide a mathematical model to show the interaction between miR-9 and Hes1, with the aim of understanding how the Hes1 oscillations are produced, how they are controlled, and further, how they are terminated. Based on the experimental findings, the model demonstrates the essential roles of Hes1 and miR-9 in regulating the dynamics of the system. In particular, the model suggests that the balance between miR-9 and Hes1 plays important roles in the choice between progenitor maintenance and neural differentiation. In addition, the synergistic (or antagonistic) effects of several important regulations are investigated so as to elucidate the effects of combinatorial regulation in neural decision-making. Our model provides a qualitative mechanism for understanding the process in neural fate decisions regulated by Hes1 and miR-9.     

On the dynamics of bursting systems

The dynamics of three-variable models of bursting are studied. It is shown that under certain conditions, the dynamics on the attractor can be essentially reduced to two dimensions. The salient dynamics on the attractor can thus be completely described by the return map of a section which is a logistic interval map. Two specific bursting models from the literature are shown to fit in the general framework which is developed. Bifurcation of the full system for one case in investigated and the dynamical behavior on the attractor is shown to depend on the position of a certain nullcline.Supported in part by N.S.F.On leave at University of Maryland     

Stability of discrete age-structured and aggregated delay-difference population models

Sufficiency conditions for local stability are derived for a class of density dependent Leslie matrix models. Four of the recruitment functions in common use in fisheries management are then considered. In two of these oscillating instability can never occur (Beverton and Holt and Cushing forms). In the other two (Deriso-Schnute and Shepherd forms) undamped oscillations are possible within the region of parameter space described here. An algorithm is developed for calculating necessary and sufficient local stability conditions for a simplified form of the general age-structured model. The complete spectrum of stability states (monotonic stability; monotonic instability; oscillating-stable; oscillating-unstable) and the bifurcation periods are given for selected examples of this model. The examples cover a large portion of the parameter space of interest in resource management. It is shown that in perfectly deterministic systems which are observed with error, oscillating instabilities may be missed, and such systems could be erroneously assumed to be stable.     

Substrate degradation by a mutualistic association of two species in the Chemostat

We discuss a system of ordinary differential equations that can be used to model the interspecies hydrogen transfer common in anaerobic degradation of organic matter. The mutualistic character of the interaction is not modeled explicitly but emerges as a consequence of the kinetics of nutrient uptake. Using monotonicity assumptions on the reaction terms, we characterise the equilibria and their stability and demonstrate two-parameter bifurcation of periodic solutions near singularities of the Bogdanov-Takens type. We have persistence and extinction results in a wide range of parameter values. Finally, we give some conditions for equivalence and non-equivalence to a cooperative system and compare to related models.     

Interactions between neural networks: a mechanism for tuning chaos and oscillations

We show that chaos and oscillations in a higher-order binary neural network can be tuned effectively using interactions between neural networks. Our results suggest that network interactions may be useful as a means of adjusting the level of dynamic activities in systems that employ chaos and oscillations for information processing, or as a means of suppressing oscillatory behaviors in systems that require stability. URL: http:// www.ntu.edu.sg/home/elpwang     

Catastrophic shifts in vertical distributions of phytoplankton The existence of a bifurcation set

A model of phytoplankton dynamics within a water column was analyzed with special consideration on the existence of a bifurcation set in the parameter space. We considered two resources, light and a limiting nutrient, for phytoplankton growth and assumed that the water column is separated into two layers by thermal and/or density stratification. It was shown that there exists a bifurcation set in the parameter space when the growth function meets several conditions that are general for growth functions of two essential resources. Specifically, these conditions include that a less abundant of the two resources limits the growth while the effect of the other is sufficiently small. Folded structure with two stable states separated by one unstable state appears in the catastrophe manifold when parameters move to a certain direction with a certain curvature from a point in the bifurcation set. These results suggest that occurrence of discontinuous transition between two alternative vertical patterns is possible nature of phytoplankton dynamics within a stratified water column.     

Parametric Analysis of a Predator–prey System Stabilized by a Top Predator

We present a complete parametric analysis of a predator–prey system influenced by a top predator. We study ecosystems with abundant nutrient supply for the prey where the prey multiplication can be considered as proportional to its density. The main questions we examine are the following: (1) Can the top predator stabilize such a system at low densities of prey? (2) What possible dynamic behaviors can occur? (3) Under which conditions can the top predation result in the system stabilization? We use a system of two nonlinear ordinary differential equations with the density of the top predator as a parameter. The model is investigated with methods of qualitative theory of ODEs and the theory of bifurcations. The existence of 12 qualitatively different types of dynamics and complex structure of the parametric space are demonstrated. Our studies of phase portraits and parametric diagrams show that a top predator can be an important factor leading to stabilization of the predator-prey system with abundant nutrient supply. Although the model here is applied to the plankton communities with fish (or carnivorous zooplankton) as the top trophic level, the general form of the equations allows applications of our results to other ecological systems.     

The dynamics of a Lotka–Volterra predator–prey model with state dependent impulsive harvest for predator

According to the economic and biological aspects of renewable resources management, we propose a Lotka–Volterra predator–prey model with state dependent impulsive harvest. By using the Poincaré map, some conditions for the existence and stability of positive periodic solution are obtained. Moreover, we show that there is no periodic solution with order larger than or equal to three under some conditions. Numerical results are carried out to illustrate the feasibility of our main results. The bifurcation diagrams of periodic solutions are obtained by using the numerical simulations, and it is shown that a chaotic solution is generated via a cascade of period-doubling bifurcations, which implies that the presence of pulses makes the dynamic behavior more complex.     

Three stage semelparous Leslie models

Nonlinear Leslie matrix models have a long history of use for modeling the dynamics of semelparous species. Semelparous models, as do nonlinear matrix models in general, undergo a transcritical equilibrium bifurcation at inherent net reproductive number R ₀ = 1 where the extinction equilibrium loses stability. Semelparous models however do not fall under the purview of the general theory because this bifurcation is of higher co-dimension. This mathematical fact has biological implications that relate to a dichotomy of dynamic possibilities, namely, an equilibration with over lapping age classes as opposed to an oscillation in which age classes are periodically missing. The latter possibility makes these models of particular interest, for example, in application to the well known outbreaks of periodical insects. While the nature of the bifurcation at R ₀ = 1 is known for two-dimensional semelparous Leslie models, only limited results are available for higher dimensional models. In this paper I give a thorough accounting of the bifurcation at R ₀ = 1 in the three-dimensional case, under some monotonicity assumptions on the nonlinearities. In addition to the bifurcation of positive equilibria, there occurs a bifurcation of invariant loops that lie on the boundary of the positive cone. I describe the geometry of these loops, classify them into three distinct types, and show that they consist of either one or two three-cycles and heteroclinic orbits connecting (the phases of) these cycles. Furthermore, I determine stability and instability properties of these loops, in terms of model parameters, as well as those of the positive equilibria. The analysis also provides the global dynamics on the boundary of the cone. The stability and instability conditions are expressed in terms of certain measures of the strength and the symmetry/asymmetry of the inter-age class competitive interactions. Roughly speaking, strong inter-age class competitive interactions promote oscillations (not necessarily periodic) with separated life-cycle stages, while weak interactions promote stable equilibration with overlapping life-cycle stages. Methods used include the theory of planar monotone maps, average Lyapunov functions, and bifurcation theory techniques.      

Effects of density dependent sex allocation on the dynamics of a simultaneous hermaphroditic population: Modelling and analysis

In this work we present a mathematical model describing the dynamics of a population where sex allocation remains flexible throughout adult life and so can be adjusted to current environmental conditions. We consider that the fractions of immature individuals acquiring male and female sexual roles are density dependent through nonlinear functions of a weighted total population size. The main goal of this work is to understand the role of life-history parameters on the stabilization or destabilization of the population dynamics.The model turns out to be a nonlinear discrete model which is analysed by studying the existence of fixed points as well as their stability conditions in terms of model parameters. The existence of more complex asymptotic behaviours of system solutions is shown by means of numerical simulations.Females have larger fertility rate than males. On the other hand, increasing population density favours immature individuals adopting the male role. A positive equilibrium of the system exists whenever fertility and survival rates of one of the sexual roles, if shared by all adults, allow population growing while the opposite happens with the other sexual role. In terms of the female inherent net reproductive number, η_F, it is shown that the positive equilibria are stable when η_F is larger and closed to 1 while for larger values of η_F a certain asymptotic assumption on the investment rate in the female function implies that the population density is permanent. Depending on the other parameters values, the asymptotic behaviour of solutions becomes more complex, even chaotic. In this setting the stabilization/destabilization effects of the abruptness rate in density dependence, of the survival rates and of the competition coefficients are analysed.