Positive Initial Conditions Research Articles

Model Matching Problems for Positive Systems

The problem of compensating a given plant by means of a static compensator in such a way that, for any input, the output of the compensated system matches that of a given positive model, when both are initialized at 0, and its state evolves in the positive cone for positive initial conditions and inputs is considered. Under a mild structural assumption for the output-difference system between the plant and the model, a complete characterization of solvability of the problem in terms of necessary and sufficient conditions is obtained by means of structural geometric methods. Solvability conditions are practically checkable by algorithmic procedures and by solving a set of linear inequalities. The problem of asymptotic matching for any initial condition is then considered and solvability is characterized by necessary and sufficient conditions. A necessary condition that is practically checkable is given. Solvability by a dynamic compensator is also studied and a sufficient condition to characterize it is given.

Stability and Numerical Simulation of Prey-predator System with Holling Type-II Functional Responses for Adult Prey

In this paper, we analyze the local stability of the prey-predator model. This model was constructed from two prey involving stage-structure and one predator. The population of prey is divided into two subpopulations, adult prey and immature. The ratio of intrinsic growth of predator population density divided by adult prey conversion factors into a predator. Reduction of predator populations has a reciprocal relationship with the availability of favorite foods determined by the environmental resources. We also consider that Holling type II the functional response for the predator. We have investigated the local stability and the solution of the system. We have studied the existence and feasibility of various equilibrium points. The system has three positive equilibria, namely the original, the extinction of the predator and the interior point. We explore the stabilities of all nonnegative equilibrium point by the real parts of the eigenvalues of the Jacobian matrix at each equilibrium point must be negative. We also use the Routh-Hurwitz criterion to investigate the stability at the interior equilibrium point. This study aims to detect the limit cycle with its phase portrait and its time-series graphs of the prey-predator system. Numerical simulated are represented as phase portraits to draw the stability of the equilibrium point. In our Numerical outcomes confirmed the analytical results and the local behavior prey-predator model which start from their positive initial condition. The parameter value is taken mainly from the literature and assumption. By a computational Python using the fourth-order Runge-Kutta method, we solved the prey-predator system and showed some new phase portraits such as the existence of the stable or unstable equilibrium point under a suitable value of the parameter. The next research, we will explore the global stability analysis behavior of the interior equilibrium point.

Stochastic dynamics of Michaelis–Menten kinetics based tumor-immune interactions

In this paper, we investigate deterministic and stochastic dynamics of Michaelis–Menten kinetics based tumor-immune interactions. For the deterministic case, stability analysis is performed by Routh–Hurwitz criteria. Chaos is observed in bifurcation analysis and examined by the method of 0−1 test. The stochastic system is constructed by incorporating multiplicative white noise terms into the deterministic system. We establish a unique positive solution ensuring the positiveness and boundedness of solution from the positive initial condition. The sufficient condition is obtained for weak persistence in mean. We also derive the parametric restrictions for stochastic permanence and global attractivity in mean. Finally, we validate the extinction of tumor cells with the transition from co-existence domain by crossing the estimated threshold values of intensity of environmental noise.

Multiple bifurcations analysis in a delayed predator–prey system with disease in prey and stage structure for predator

In this paper, a delayed eco-epidemiological model with disease in prey, anti-predator factor for the prey and stage structure for the predator is proposed. The conditions for the stability and coexistence of the populations have been derived. Positivity and boundedness of all solutions with positive initial conditions has been proved. Sufficient conditions for the occurrence of several Hopf and Bogdanov–Takens bifurcations about the endemic steady state are derived. By taking the delay as the bifurcation parameter, it is shown that Hopf and Bogdanov–Takens bifurcations occur at positive steady state for some critical values of the delay. The normal form of the Bogdanov–Takens singularity is obtained by performing a center manifold reduction and by using the normal form theory of FDEs. Some numerical simulations of solutions in different parametric regions illustrating Hopf and homoclinic bifutcations are given to support the analytical results.

Feedback Control of Nonlinear Hyperbolic PDE Systems Inspired by Traffic Flow Models

This paper investigates and provides results, including feedback control, for a nonlinear, hyperbolic, one-dimensional partial differential equation (PDE) system on a bounded domain. The considered model consists of two first-order PDEs with a dynamic boundary condition on the one end and actuation on the other. It is shown that, for all positive initial conditions, the system admits a globally defined, unique, classical solution that remains positive and bounded for all times; these properties are important, for example for traffic flow models. Moreover, it is shown that global stabilization can be achieved for arbitrary equilibria by means of an explicit boundary feedback law. The stabilizing feedback law depends only on collocated boundary measurements. The efficiency of the proposed boundary feedback law is demonstrated by means of a numerical example of traffic density regulation.

Boundedness properties of the generalized Todd's difference equation

ABSTRACTWe consider the third-order difference equation with and We show that if then there exist positive initial conditions such that the solutions are unbounded except for the case and

An Algorithm for Maximizing the Biogas Production in a Chemostat

In this work, we deal with the optimal control problem of maximizing biogas production in a chemostat. The dilution rate is the controlled variable, and we study the problem over a fixed finite horizon, for positive initial conditions. We consider the single reaction model and work with a broad class of growth rate functions. With the Pontryagin maximum principle, we construct a one-parameter family of extremal controls of type bang-singular arc. The parameter of these extremal controls is the constant value of the Hamiltonian. Using the Hamilton–Jacobi–Bellman equation, we identify the optimal control as the extremal associated with the value of the Hamiltonian, which satisfies a fixed point equation. We then propose a numerical algorithm to compute the optimal control by solving this fixed point equation. We illustrate this method with the two major types of growth functions of Monod and Haldane.

Analytical Study of the Difference Equation xn+1 = xnxn-6 / xn-5 (±1 – xnxn-6)

This paper is devoted to find the form of the solutions of a rational difference equations with arbitrary positive real initial conditions. Specific form of the solutions of two special cases of this equation are given.

On a chemotaxis model with competitive terms arising in angiogenesis

In this paper we study an anti-angiogenic therapy model that deactivates the tumor angiogenic factors. The model consists of four parabolic equations and considers the chemotaxis and a logistic law for the endothelial cells and several boundary conditions, some of them are non homogeneous. We study the parabolic problem, proving the existence of a unique global positive solution for positive initial conditions, and the stationary problem, justifying the existence of one real number, an eigenvalue of a certain problem, which determines if the semi-trivial solutions are stable or unstable and the existence of a coexistence state.

Operator splitting and discontinuous Galerkin methods for advection-reaction-diffusion problem. Application to plant root growth

The time dependant advection-reaction-diffusion equation is used in the C-Root model to simulate root growth. This equation can also be applied in many others applications in life sciences. In this context the unknown is related to densities and one of the important property of the problem is that the solution is non-negative for positive initial conditions. One of the difficulty at the discrete level is to preserve the positivity of the approximated solution during the simulations. In this work we solved the model using Discontinuous Galerkin elements combined with an operator splitting technique. The DG method is briefly presented then we motivated the use of the operator splitting technique by doing some numerical experiments. Those experiments showed that the same time approximation scheme may not be suitable for all the operators of the model. We validated our implementation of the splitting technique in a simple test case. Then we performed a simulation of a plagiotropic root of Eucalyptus.

On the boundedness of solutions of a class of third-order rational difference equations

ABSTRACTWe consider the difference equation with and If we show that for all positive initial conditions the solutions of the difference equations are bounded. If then there exists positive initial conditions such that the solutions are unbounded.

Stability analysis for positive singular systems with distributed delays

This paper is concerned with the stability analysis for continuous-time positive singular systems with distributed delays. By introducing an auxiliary system, a necessary and sufficient positivity condition is proposed for singular systems with distributed lags. Based on this condition, we present a sufficient stability condition for positive singular systems with distributed delays and extend the result to systems with distributed delays over a bounded time-varying interval. In addition, this stability condition is also necessary when strictly positive initial conditions are available. Numerical simulations are utilized to illustrate the effectiveness of our results.

Consensus Driven by the Geometric Mean

Consensus networks are usually understood as arithmetic mean driven dynamical averaging systems. In applications, however, network dynamics often describe inherently non-arithmetic and nonlinear consensus processes. In this paper, we propose and study three novel consensus protocols driven by geometric mean averaging: a polynomial, an entropic, and a scaling-invariant protocol, where terminology characterizes the particular nonlinearity appearing in the respective differential protocol equation. We prove exponential convergence to consensus for positive initial conditions. For the novel protocols, we highlight connections to applied network problems: The polynomial consensus system is structured like a system of chemical kinetics on a graph. The entropic consensus system converges to the weighted geometric mean of the initial condition, which is an immediate extension of the (weighted) average consensus problem. We find that all three protocols generate gradient flows of free energy on the simplex of constant mass distribution vectors albeit in different metrics. On this basis, we propose a novel variational characterization of the geometric mean as the solution of a nonlinear constrained optimization problem involving free energy as cost function. We illustrate our findings in numerical simulations.

Global Dynamics of Certain Mix Monotone Difference Equation

We investigate global dynamics of the following second order rational difference equation x n + 1 = x n x n − 1 + α x n + β x n − 1 a x n x n − 1 + b x n − 1 , where the parameters α , β , a , b are positive real numbers and initial conditions x − 1 and x 0 are arbitrary positive real numbers. The map associated to the right-hand side of this equation is always decreasing in the second variable and can be either increasing or decreasing in the first variable depending on the corresponding parametric space. In most cases, we prove that local asymptotic stability of the unique equilibrium point implies global asymptotic stability.

A vicinal surface model for epitaxial growth with logarithmic free energy

We study a continuum model for solid films that arises from the modeling of one-dimensional step flows on a vicinal surface in the attachment-detachment-limited regime. The resulting nonlinear partial differential equation, $u_t = -u^2(u^3+\alpha u)_{hhhh}$, gives the evolution for the surface slope $u$ as a function of the local height $h$ in a monotone step train. Subject to periodic boundary conditions and positive initial conditions, we prove the existence, uniqueness and positivity of global strong solutions to this PDE using two Lyapunov energy functions. The long time behavior of $u$ converging to a constant that only depends on the initial data is also investigated both analytically and numerically.

Mathematical modeling of optimized SIRS epidemic model and some dynamical behavior of the solution

In this paper, a generalized mathematical model of spread of infectious disease as SIRS epidemic model is considered as a nonlinear system of differential equation. We prove that for positive initial conditions the resulting equivalence system has positive solution and under some hypothesis, this system with initial positive condition, has a positive $T$-periodic solution which is globally asymptotically stable. For numerical simulations the fourth order Runge-Kutta method is applied to the nonlinear system of differential equations.

Qualitative Analysis of an ODE Model of a Class of Enzymatic Reactions : Some Results on Global Stability of Messenger RNA-MicroRNA Interaction.

The present paper analyzes an ODE model of a certain class of (open) enzymatic reactions. This type of model is used, for instance, to describe the interactions between messenger RNAs and microRNAs. It is shown that solutions defined by positive initial conditions are well defined and bounded on [Formula: see text] and that the positive octant of [Formula: see text] is a positively invariant set. We prove further that in this positive octant there exists a unique equilibrium point, which is asymptotically stable and a global attractor for any initial state with positive components; a controllability property is emphasized. We also investigate the qualitative behavior of the QSSA system in the phase plane [Formula: see text]. For this planar system we obtain similar results regarding global stability by using Lyapunov theory, invariant regions and controllability.

Observer-based adaptive neural iterative learning control for a class of time-varying nonlinear systems

In this paper an adaptive iterative learning control scheme is presented for the output tracking of a class of nonlinear systems. An observer is designed to estimate the tracking errors. A mixed time domain and s-domain representation is constructed to derive an error model with relative degree one for our purpose. And time-varying radial basis function neural network is employed to deal with system uncertainty. A new signal is constructed by using a first-order filter, which removes the requirement of strict positive real (SPR) condition and identical initial condition of iterative learning control. Based on property of hyperbolic tangent function, the system tracing error is proved to converge to the origin as the iteration tends to infinity by constructing Lyapunov-like composite energy function, while keeping all the closed-loop signals bounded. Finally, a simulation example is presented to verify the effectiveness of the proposed approach.

A density-dependent model of competition for one resource in the chemostat

This paper deals with a two-microbial species model in competition for a single-resource in the chemostat including general intra- and interspecific density-dependent growth rates with distinct removal rates for each species. In order to understand the effects of intra- and interspecific interference, this general model is first studied by determining the conditions of existence and local stability of steady states.With the same removal rate, the model can be reduced to a planar system and then the global stability results for each steady state are derived. The bifurcations of steady states according to interspecific interference parameters are analyzed in a particular case of density-dependent growth rates which are usually used in the literature.The operating diagrams show how the model behaves by varying the operating parameters and illustrate the effect of the intra- and interspecific interference on the disappearance of coexistence region and the occurrence of bi-stability region. Concerning the small enough interspecific interference terms, we would shed light on the global convergence towards the coexistence steady state for any positive initial condition. When the interspecific interference pressure is large enough this system exhibits bi-stability where the issue of the competition depends on the initial condition.

Solution of first order system of differential equation in fuzzy environment and its application

In this paper, we solve a system of differential equation of first order in fuzzy environment using generalised Hukuhara derivative approach. Four different cases are discussed for the said differential equation: 1) coefficient is positive crisp number and initial condition is fuzzy number; 2) coefficient is negative crisp number and initial condition is fuzzy number; 3) coefficient is fuzzy number and initial condition is fuzzy number; 4) coefficient is interval number and initial condition is fuzzy number. Finally, we apply the results in Lanchaster combat model. The solution that comes in the application are defuzzified by removal area method.