Global Stability Of Positive Solutions Research Articles

The Solution of a System of Higher-Order Difference Equations in Terms of Balancing Numbers

In this paper, we are interested in the closed-form solution of the following system of nonlinear difference equations of higher order,un+1 = 1/34-vn-m , vn+1 = 1/34-un-m, n, m ∈ N0,and the initial values u-j and v-j , j∈{0, 1, ..., m} are real numbers do not equal 34. We show that the solutions of this system are associated with Balancing numbers. As consequence, these solutions are also associated with Pell numbers, Pell-Lucas numbers, and Lucas-Balancing numbers. It is shown that the global stability of positive solutions of this system holds. Our results are illustrated via numerical examples.

Dynamic behaviors of a discrete Lotka-Volterra competitive system with the effect of toxic substances and feedback controls

By noting the fact that the intrinsic growth rate are not positive everywhere, we revisit Lotka-Volterra competitive system with the effect of toxic substances and feedback controls. The corresponding results about permanence and extinction for the species given in (Chen and Chen in Int. J. Biomath. 8(1):1550012, 2015) are extended. Furthermore, a very important fact is found in our results, that is, the feedback controls and toxic substances have no effect on the permanence and extinction of species. Moreover, we also derive sufficient conditions for the global stability of positive solutions. Finally, some numerical simulations show the feasibility of our main results.

Existence, uniqueness, stochastic persistence and global stability of positive solutions of the logistic equation with random perturbation

AbstractThis paper discusses a randomized logistic equation equation image with initial value x(0)=x0>0, where B(t) is a standard one‐dimension Brownian motion, and θ∈(0, 0.5). We show that the positive solution of the stochastic differential equation does not explode at any finite time under certain conditions. In addition, we study the existence, uniqueness, boundedness, stochastic persistence and global stability of the positive solution. Copyright © 2006 John Wiley & Sons, Ltd.

Existence, uniqueness, and global stability of positive solutions to the food-limited population model with random perturbation

This paper discusses a randomized food-limited population model Ṅ( t) = ( r + αḂ( t)). N( t)[( K − N( t))/( K + CN( t))] with initial value N(0) = N 0, and N 0 is a random variable satisfying 0 < N 0 < K. The existence, uniqueness, and global stability of positive solutions of the equation are studied.

Permanence and global stability of positive solutions of a nonautonomous discrete ratio‐dependent predator‐prey model

We first give sufficient conditions for the permanence of nonautonomous discrete ratio‐dependent predator‐prey model. By linearization of the model at positive solutions and construction of Lyapunov function, we also obtain some conditions which ensure that a positive solution of the model is stable and attracts all positive solutions.