Stability criteria for a nonlinear nonautonomous system with delays

Abstract

a b s t r a c tThis paper further develops a method, originally introduced by Mori et al., for proving local stability of steady states in linear systems of delay differential equations. A nonlinear nonautonomous system of delay differential equations with several delays is considered. Explicit delay-independent sufficient conditions for global attractivity of the solutions with an extremely simple form are provided. The above-mentioned conditions make the stability test quite practical. We illustrate application of this test to the Hopfield neural network models. The results obtained were also applied to a new marine protected areas model with delay that describes the ecological linkage between the reserve and fishing ground.

Global dynamics of Nicholson-type delay systems with applications

“…The qualitative theory of various delay differential systems was studied in monographs [10,11,2] and the most recent papers [12][13][14][15][16][17][18][19][20][21][22]. Numerous applications of delay differential systems can be found in [9,[23][24][25][26][27][28].…”

Stability of Hahnfeldt angiogenesis models with time lags

“…where x ∈ R n , A ∈ R n×n and F : R + × R n × R n → R n is a nonlinear and continuous vector function. Stability analysis of the delay differential equation (6) is a well-trodden area, however, some existing results rely on restrictive conditions, e.g., strict monotonicity and boundedness of the functions and operators involved, continuity of the parameters [5], [12], [21], [22], [26], [28] and [30]. For example, models under study possess non-Lipchitz nonlinearity, thus stability analysis for models (3)-(5) requires new tools and approaches.…”

On Stability of Nonautonomous Perturbed Semilinear Fractional Differential Systems of Order α∈(1,2)

“…To illustrate the biological applications [18], we consider two habitat areas, with a fish population dispersing between the two areas, whilst fishing takes place only in region 2, with region 1 established as a no-fishing zone. To describe the ecological linkage between the reserve and fishing ground we propose the fractional alternative model of the autonomous linear system 0, 1 ( ) = − ( 1 + 1 ) 1 ( ) + 2 2 ( ) , 0, 2 ( ) = − ( 2 + 2 + ) 2 ( ) + 1 1 ( ) .…”

“…The marine protected area model [18] can describe the ecological linkage between the reserve and fishing ground by the autonomous linear system…”