Unique Positive Periodic Solution Research Articles

Oscillation and global attractivity in a nonlinear delay periodic model of respiratory dynamics

In this paper, we shall consider the nonlinear delay differential equation ▪ where m and n are positive integers, and V( t) and λ( t) are positive periodic functions of period ω. In the nondelay case, we shall show that (∗) has a unique positive periodic solution x ( t) and provides sufficient conditions for the global attractivity of x ( t). In the delay case, we shall present sufficient conditions for the oscillation of all positive solutions of (∗) about x ( t) and establish sufficient conditions for the global attractivity of x ( t).

Oscillation and global attractivity in a periodic Nicholson's blowflies model

In this paper, we shall consider the nonlinear delay differential equation N′( t) = −δ( t) N( t) + P( t) N( t − mω) e − aN( t−mω , where m is a positive integer, δ( t) and P( t) are positive periodic functions of period ω. In the nondelay case, we shall show that (∗) has a unique positive periodic solution N ( t), and provide sufficient conditions for the global attractivity of N ( t). In the delay case, we shall present sufficient conditions for the oscillation of all positive solutions of (∗) about N ( t), and establish sufficient conditions for the global attractivity of N ( t).

Oscillation and Global Attractivity in a Nonlinear Delay Periodic Model of Population Dynamics

In this article we shall consider the following nonlinear delay differential equation $$x'(t) + p(t)x(t)-\\frac {q(t)x(t)}{r + x^{n}(t-m\\omega )} = 0\\eqno (*)$$ where m and n are positive integers, p ( t ) and q ( t ) are positive periodic functions of period y . In the nondelay case we shall show that (*) has a unique positive periodic solution $ \\overline {x}(t), $ and provide sufficient conditions for the global attractivity of $ \\overline {x}(t) $ . In the delay case we shall present sufficient conditions for the oscillation of all positive solutions of (*) about $ \\overline {x}(t), $ and establish sufficient conditions for the global attractivity of $ \\overline {x}(t). $

Innovation diffusion model in patch environment

A mathematical model is proposed to describe the dynamics of users of one product in two different patches. Advertisement force, contact rate between users and non-users, population dispersal rate and returning rate from user class to non-user class are chosen as key parameters. For the product with a long life-span, it is assumed that the returning rate is proportional to the number of users. It is shown that this model has a unique positive equilibrium which is globally stable. For the product with a short life-span, time delays are introduced to represent the duration of the product in two different markets and the stability of a positive equilibrium is analyzed in one reasonable case. Periodic advertisements are also incorporated and the existence and uniqueness of positive periodic solutions are investigated.

Asymptotic behaviour of a nonautonomous cooperative system

AbstractThis paper considers a nonautonomous cooperative system, in which all the parameters are time-dependent and asymptotically approach periodic functions. We prove that under some appropriate conditions any positive solutions of the system asymptotically approach the unique positive periodic solution of the corresponding periodic system.

Optimal harvesting policy for single population with periodic coefficients.

In this paper, we examine the exploitation of single population modeled by time-dependent Logistic equation with periodic coefficients. First, it is shown that the time-dependent periodic Logistic equation has a unique positive periodic solution, which is globally asymptotically stable for positive solutions, and we obtain its explicit representation. Further, we choose the maximum annual-sustainable yield as the management objective, and investigate the optimal harvesting policies for constant harvest and periodic harvest. The optimal harvest effort that maximizes the annual-sustainable yield, the corresponding optimal population level, the corresponding harvesting time-spectrum, and the maximum annual-sustainable yield are determined, and their explicit expressions are obtained in terms of the intrinsic growth rate and the carrying capacity of the considered population. Our interesting and brief results generalize the classical results of Clark for a population described by the autonomous logistic equation in renewable resources management.

On Time Periodic Solutions of the Dirichlet Problem for Degenerate Parabolic Equations of Nondivergence Type

In this paper, we are concerned with the existence of periodic solutions of a quasilinear parabolic equation[formula]with the Dirichlet boundary condition, where Ω is a smoothly bounded domain inRNandfis a given function periodic in time defined on[formula]. Our results depend on the first eigenvalue λ1of −Δ in Ω with the Dirichlet boundary condition. If λ1>1, then there exists a unique positive periodic solution for a positivef(γ∈R). In the case of λ1<1, we construct a nonnegative periodic solution for a negativef(1≤γ<3).

Periodic optimal control for parabolic Volterra‐Lotka type equations

AbstractThis paper considers the optimal harvesting control of a biological species, whose growth is governed by the parabolic diffusive Volterra‐Lotka equation. We prove that such equation with L∞ periodic coefficients has an unique positive periodic solution. We show the existence and uniqueness of an optimal control, and under certain conditions, we characterize the optimal control in terms of a parabolic optimality system. A monotone sequence which converges to the optimal control is constructed.

Global Attractivity and ‘Level‐Crossings’ in a Periodic Logistic Integrodifferential Equation

AbstractSufficient conditions are obtained for the existence of a globally attracting positive periodic solution of the logistic integrodifferential equation where a, b are continuous positive periodic functions and K is a nonnegative integrable function defined on {0, ∞). We also derive sufficient conditions for all positive solutions to have “level crossings” about the unique positive periodic solution.

Analysis of the Periodic Lyapunov and Riccati Equations via Canonical Decomposition

A new proof is given of the following result: a periodic Riccati equation admits a unique positive semidefinite periodic solution and the solution is stabilizing if and only if the underlying system is stabilizable and detectable. The proof hinges on the decomposition of the Riccati equation induced by the system canonical decomposition. The solution structure is therefore naturally pointed out.

Periodic solutions of periodic Riccati equations

For periodically time-varying matrix Riccati equations, controllability and observability (in the usual sense) are shown to be sufficient for the existence of a unique positive definite periodic solution.

Existence of positive periodic solutions of certain ordinary differential equations

The method of ascertaining the existence and uniqueness of positive periodic solutions of certain ordinary differential equations has been presented. A nuclear reactor with negative temperature reactivity coefficient and periodic control reactivity has been considered as an illustrative example.