Stationary solutions to a system of size-structured populations with nonlinear growth rate

We study stationary solutions to a system of size-structured population models with nonlinear growth rate. Several characterizations of stationary solutions are provided. It is shown that the steady-state problem can be converted into different problems such as two types of eigenvalue problems and a fixed-point problem. In the two-species case, we give an existence result of nonzero stationary solutions by using the fixed-point problem.     

Weak convergence of a sequence of stochastic difference equations to a stochastic ordinary differential equation

We consider a sequence of discrete parameter stochastic processes defined by solutions to stochastic difference equations. A condition is given that this sequence converges weakly to a continuous parameter process defined by solutions to a stochastic ordinary differential equation. Applying this result, two limit theorems related to population biology are proved. Random parameters in stochastic difference equations are autocorrelated stationary Gaussian processes in the first case. They are jump-type Markov processes in the second case. We discuss a problem of continuous time approximations for discrete time models in random environments.     

Allee Effect and Control of Lake System Invasion

We consider the model of invasion prevention in a system of lakes that are connected via traffic of recreational boats. It is shown that in presence of an Allee effect, the general optimal control problem can be reduced to a significantly simpler stationary optimization problem of optimal invasion stopping. We consider possible values of model parameters for zebra mussels. The general N-lake control problem has to be solved numerically, and we show a number of typical features of solutions: distribution of control efforts in space and optimal stopping configurations related with the clusters in lake connection structure.     

Synaptic regulation of the state of excitable membranes

Hodgkin-Huxly equations accounting for (stationary) control signals (synaptic conductivity and current) have stationary and periodic solutions. The domain of unstability of stationary solutions determined on the control signals plane (excitation and inhibition). On one part of the domain boundary a decrease of excitation and increase of inhibition suppress oscillations. On another part of the boundary they induced oscillations, which were not explained previously in accordance with the experimental data.     

Dynamics of a physiologically structured population in a time-varying environment

Physiologically structured population models have become a valuable tool to model the dynamics of populations. In a stationary environment such models can exhibit equilibrium solutions as well as periodic solutions. However, for many organisms the environment is not stationary, but varies more or less regularly. In order to understand the interaction between an external environmental forcing and the internal dynamics in a population, we examine the response of a physiologically structured population model to a periodic variation in the food resource. We explore the addition of forcing in two cases: (A) where the population dynamics is in equilibrium in a stationary environment, and (B) where the population dynamics exhibits a periodic solution in a stationary environment. When forcing is applied in case A, the solutions are mainly periodic. In case B the forcing signal interacts with the oscillations of the unforced system, and both periodic and irregular (quasi-periodic or chaotic) solutions occur. In both cases the periodic solutions include one and multiple period cycles, and each cycle can have several reproduction pulses.     

Analysis of the periodically fragmented environment model : I – Species persistence

This paper is concerned with the study of the stationary solutions of the equationwhere the diffusion matrix A and the reaction term f are periodic in x. We prove existence and uniqueness results for the stationary equation and we then analyze the behaviour of the solutions of the evolution equation for large times. These results are expressed by a condition on the sign of the first eigenvalue of the associated linearized problem with periodicity condition. We explain the biological motivation and we also interpret the results in terms of species persistence in periodic environment. The effects of various aspects of heterogeneities, such as environmental fragmentation are also discussed.     

The long-time behavior of the solution to a non-linear diffusion problem in population genetics and combustion

The Fisher (1937) or Kolmogoroff-Petrovsky-Piscounoff (1937) equation exemplifies wave-like phenomena occurring in population genetics and combustion. In an earlier paper, we proposed an extension of this equation and obtained closed form traveling wave, stationary, and “symmetric” solutions. Employing the transformation properties of the extended equation, two integral invariants for the problem are obtained and two Lyapunov functionals, which characterize the evolution of the profile to a uniformly propagating traveling wave, are constructed. A generalization of this modified Fisher equation is proposed and we obtain its integral invariants, traveling wave solutions and wave speeds, as well as the Lyapunov functionals which describe its asymptotic evolution.     

On a Nonlinear Master Equation and the Haken-Kelso-Bunz Model

A nonlinear master equation (NLME) is proposed basedon general information measures.Classical and cut-off solutions of the NLME are considered.In the former case, the NLME exhibits uniquely defined stationary distributions. In the latter case, there are multiple stationary distributions.In particular, for classical solutions, it is shown that transient solutions converge to stationary distributions that maximize information measures (H-theorem). Cut-off distributions arestudied numerically for the Haken-Kelso-Bunz model. The Haken-Kelso-Bunz modelis known to describe multistable human motor control systems. It is shownthat a stochastic Haken-Kelso-Bunz model based on a NLME can exhibit multiplestationary cut-off distributions.In doing so, we illustrate that multistability in stochastic biological systems can beestablished by means of cut-off distributions.     

Propagation of Turing patterns in a plankton model

The paper is devoted to a reaction-diffusion system of equations describing phytoplankton and zooplankton distributions. Linear stability analysis of the model is carried out. Turing and Hopf stability boundaries are found. Emergence of two-dimensional spatial structures is illustrated by numerical simulations. Travelling waves between various stationary solutions are investigated. Transitions between homogeneous in space stationary solutions and Turing structures are studied.     

Propagation of Turing patterns in a plankton model

The paper is devoted to a reaction–diffusion system of equations describing phytoplankton and zooplankton distributions. Linear stability analysis of the model is carried out. Turing and Hopf stability boundaries are found. Emergence of two-dimensional spatial structures is illustrated by numerical simulations. Travelling waves between various stationary solutions are investigated. Transitions between homogeneous in space stationary solutions and Turing structures are studied.     

Some exact solutions of a generalized Fisher equation related to the problem of biological invasion

The problem of biological invasion in a model single-species community is considered, the spatiotemporal dynamics of the system being described by a modified Fisher equation. For a special case, we obtain an exact solution describing self-similar growth of the initially inhabited domain. By comparison with numerical solutions, we show that this exact solution may be applicable to describe an early stage of a biological invasion preceding the propagation of the stationary travelling wave. Also, the exact solution is applied to the problem of critical aggregation to derive sufficient conditions of population extinction. Finally, we show that the solution we obtain is in agreement with some data from field observations.     

On the stationary state analysis of reaction-diffusion mechanisms for biological pattern formation

We present a biologically plausible two-variable reaction-diffusion model for the developing vertebrate limb, for which we postulate the existence of a stationary solution. A consequence of this assumption is that the stationary state depends on only a single concentration-variable. Under these circumstances, features of potential biological significance, such as the dependence of the steady-state concentration profile of this variable on parameters such as tissue size and shape, can be studied without detailed information about the rate functions. As the existence and stability of stationary solutions, which must be assumed for any biochemical system governing morphogenesis, cannot be investigated without such information, an analysis is made of the minimal requirements for stable, stationary non-uniform solutions in a general class of reaction-diffusion systems. We discuss the strategy of studying stationary-state properties of systems that are incompletely specified. Where abrupt transitions between successive compartment-sizes occur, as in the developing limb, we argue that it is reasonable to model pattern reorganization as a sequence of independent stationary states.     

Large time behavior in a nonlinear age-dependent population dynamics problem with spatial diffusion

In this work we analyze the large time behavior in a nonlinear model of population dynamics with age-dependence and spatial diffusion. We show that when t+ either the solution of our problem goes to 0 or it stabilizes to a nontrivial stationary solution. We give two typical examples where the stationary solutions can be evaluated upon solving very simple partial differential equations. As a by-product of the extinction case we find a necessary condition for a nontrivial periodic solution to exist. Numerical computations not described below show a rapid stabilization.This work was partially supported by the Centre National de la Recherche Scientifique through ATP 95939900     

Global stability of stationary solutions to a nonlinear diffusion equation in phytoplankton dynamics

We consider a nonlinear diffusion equation proposed by Shigesada and Okubo which describes phytoplankton growth dynamics with a selfs-hading effect.We show that the following alternative holds: Either (i) the trivial stationary solution which vanishes everywhere is a unique stationary solution and is globally stable, or (ii) the trivial solution is unstable and there exists a unique positive stationary solution which is globally stable. A criterion for the existence of positive stationary solutions is stated in terms of three parameters included in the equation.     

Effects of transients on pulsatile flow in arteries

Analytical solutions to the model problem of unsteady Newtonian fluid flow in straight, elastic-walled vessels can provide: theoretical insights into the flow of blood in arteries; a theoretical basis for clinical measurements in diagnoses of arterial flow rates; and guidance for boundary conditions in numerical simulations of flow in finite computational domains. However, while Womersley's analyses of blood flow assume solution forms that treat the flow as periodic and continuously unsteady, many flow variables in the smaller arteries are not continuously unsteady at all. They are characterized more accurately as rapid transient motions followed by a period of recovery to a stationary state, repeated in successive cycles. These flows are not continually unsteady ones described by Womersley's solutions but unsteady flows restarted from rest in each cycle, characterized as initial-boundary value problems. In this paper, we compare the Womersley and initial-boundary value solutions for model transients that stop then restart, explain these previously unreported limitations of Womersley's solutions, and demonstrate how the initial-boundary value solutions provide excellent agreement with measurements of blood flow in the anterior tibial and popliteal arteries of patients. Some consequences of these findings for understanding and interpreting measurements of blood flow, and for prescribing boundary conditions in computer simulations of arterial blood flow are discussed.     

Linear Augmentation for Stabilizing Stationary Solutions: Potential Pitfalls and Their Application

Linear augmentation has recently been shown to be effective in targeting desired stationary solutions, suppressing bistablity, in regulating the dynamics of drive response systems and in controlling the dynamics of hidden attractors. The simplicity of the procedure is the main highlight of this scheme but questions related to its general applicability still need to be addressed. Focusing on the issue of targeting stationary solutions, this work demonstrates instances where the scheme fails to stabilize the required solutions and leads to other complicated dynamical scenarios. Examples from conservative as well as dissipative systems are presented in this regard and important applications in dissipative predator—prey systems are discussed, which include preventative measures to avoid potentially catastrophic dynamical transitions in these systems.     

The accuracy of Bartlett's small-fluctuation approximation for stochastic-difference-equation population models

A heuristic approximation procedure devised by Bartlett has often been used to estimate the stationary first- and second-order moments of difference-equation population models perturbed by “small” noise. Here, the approximation is proved to be valid under quite general assumptions: the exact and approximate moments differ by an amount of order σ³ as σ → 0, where σ² is the mean-square norm of the noise process. The existence of stationary solutions to the perturbed difference equation is also considered. If the noise is Markovian, stationary solutions satisfying the assumptions of the error analysis are proved to exist if the noise is “small” with probability 1. The results are applied to a population model with two age classes and variable recruitment.     

Stochastic Computations in Cortical Microcircuit Models

Experimental data from neuroscience suggest that a substantial amount of knowledge is stored in the brain in the form of probability distributions over network states and trajectories of network states. We provide a theoretical foundation for this hypothesis by showing that even very detailed models for cortical microcircuits, with data-based diverse nonlinear neurons and synapses, have a stationary distribution of network states and trajectories of network states to which they converge exponentially fast from any initial state. We demonstrate that this convergence holds in spite of the non-reversibility of the stochastic dynamics of cortical microcircuits. We further show that, in the presence of background network oscillations, separate stationary distributions emerge for different phases of the oscillation, in accordance with experimentally reported phase-specific codes. We complement these theoretical results by computer simulations that investigate resulting computation times for typical probabilistic inference tasks on these internally stored distributions, such as marginalization or marginal maximum-a-posteriori estimation. Furthermore, we show that the inherent stochastic dynamics of generic cortical microcircuits enables them to quickly generate approximate solutions to difficult constraint satisfaction problems, where stored knowledge and current inputs jointly constrain possible solutions. This provides a powerful new computing paradigm for networks of spiking neurons, that also throws new light on how networks of neurons in the brain could carry out complex computational tasks such as prediction, imagination, memory recall and problem solving.     

Isolated periodic solutions of the Hodgkin-Huxley equations

Periodic solutions of the current clamped Hodgkin-Huxley equations (Hodgkin & Huxley, 1952 J. Physiol. 117, 500) that arise by degenerate Hopf bifurcation were studied recently by Labouriau (1985 SIAM J. Math. Anal. 16, 1121, 1987 Degenerate Hopf Bifurcation and Nerve Impulse (Part II), in press). Two parameters, temperature T and sodium conductance gNa were varied from the original values obtained by Hodgkin & Huxley. Labouriau's work proved the existence of small amplitude periodic solution branches that do not connect locally to the stationary solution branch, and had not been previously computed. In this paper we compute these solution branches globally. We find families of isolas of periodic solutions (i.e. branches not connected to the stationary branch). For values of gNa in the range measured by Hodgkin & Huxley, and for physically reasonable temperatures, there are isolas containing orbitally asymptotically stable solutions. The presence of isolas of periodic solutions suggests that in certain current space clamped membrane experiments, action potentials could be observed even though the stationary state is stable for all current stimuli. Once produced, such action potentials will disappear suddenly if the current stimulus is either increased or decreased past certain values. Under some conditions, "jumping" between action potentials of different amplitudes might be observed.     

一类时滞反应扩散方程的稳态解和行波解

本文讨论了一类造血生物模型在Dirichlet边值条件下稳态解的全局吸引性,并利用上、下解技术和单调迭代方法讨论了行波解的存在性.