Monotone dynamical systems Stability and convergence Competitive and cooperative differential equations Irreducible cooperative systems Cooperative systems of delay differential equations Nonquasimonotone delay differential equations Quasimonotone systems of parabolic equations A competition model Appendix Bibliography.

Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems

1 Strongly Order-Preserving Semiflows 6 1.1 Definitions and Basic Results . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Nonordering of Omega Limit Sets . . . . . . . . . . . . . . . . . . . . . . 12 1.3 Local Semiflows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4 The Limit Set Dichotomy . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5 Q is plentiful . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.6 Stability in Normally Ordered Spaces . . . . . . . . . . . . . . . . . . . . 21 1.7 Stable Equilibria in Strongly Ordered Banach Spaces . . . . . . . . . . . 24 1.8 The Search for Stable Equilibria . . . . . . . . . . . . . . . . . . . . . . . 27

https://link-springer-com-443.webvpn.jmu.edu.cn/content/pdf/10.1007%2F978-1-4419-9863-7_525.pdf

Monotone Dynamical Systems

Monotone subsystems have appealing properties as components of larger networks, since they exhibit robust dynamical stability and predictability of responses to perturbations. This suggests that natural biological systems may have evolved to be, if not monotone, at least close to monotone in the sense of being decomposable into a “small” number of monotone components, In addition, recent research has shown that much insight can be attained from decomposing networks into monotone subsystems and the analysis of the resulting interconnections using tools from control theory. This paper provides an expository introduction to monotone systems and their interconnections, describing the basic concepts and some of the main mathematical results in a largely informal fashion.

/pdf/monotone-and-near-monotone-biochemical-networks-1z7sxqiil5.pdf

Monotone and near-monotone biochemical networks.

A reaction-diffusion model for the evolution of dispersal rates is considered in which there is both spatial heterogeneity and temporal periodicity. The model is restricted to two phenotypes because of technical difficulties, but a wide range of mathematical techniques and computational effort are needed to obtain useful answers. We find that the question of selection is a great deal richer than in the autonomous case, where the phenotype with the lowest diffusion is selected for. In the current model either the lower or higher diffuser rate may be selected, or there may be coexistence of phenotypes. The paper raises several open questions and suggests in particular that a mutation-selection multi-phenotypic model would repay study.

The evolution of dispersal rates in a heterogeneous time-periodic environment

Publisher Summary This chapter discusses a restricted but useful class of dynamical systems—,namely, those enjoying a comparison principle with respect to a closed order relation on the state space. Such systems, variously called monotone, order-preserving or increasing, occur in many biological, chemical, physical, and economic models. The chapter introduces the basic definitions and develops the main tools of monotone dynamics. Several results on density of quasiconvergent points are proved and used to establish existence of stable equilibria.

/pdf/chapter-4-monotone-dynamical-systems-1m0rmqsitr.pdf

Chapter 4 Monotone Dynamical Systems

The author studies the periodic time-dependent type-K monotone system \[ \dot x_i = F_i \left( {t,x_1 , \ldots ,x_n } \right)\quad (i = 1, \ldots ,n) \] in the interior of the nonnegative orthant in n-space satisfying the following conditions: (i) if $x \ne y$, $x_i = y_i $ and $x_j \leq y_j $ for $j \ne i$, then $F_i (t,x) \leq F_i (t,y)$; (ii) $F(t,x)$ is periodic in t of period $\tau > 0$; (iii) F possesses an invariant function with positive gradient. It is proved that every solution to such a system either converges to a periodic solution or eventually leaves any compact set. This result gives an affirmative answer to the conjecture recently proposed by B. R. Tang, Y. Kuang, and H. Smith in [SIAM J. Math. Anal., 24 (1993), pp. 1331–1339] for periodic type-K monotone systems.

Periodic monotone systems with an invariant function

Stable Cycles for Attractors of Strongly Monotone Discrete-Time Dynamical Systems

The global stability of a class of second order differential equations

Attractors in strongly monotone flows

In this paper, the convergence of solutions of type K monotone systems is studied. The basic assumption is that the Jacobian matrix is stable for every point in $R_+^n$. The main results are the following. If the system has a positive steady state, then it is globally asymptotically stable in $\mbox{Int} R_+^n$. A sufficient and necessary condition for a nonnegative steady state of the system to be globally asymptotically stable is presented. Moreover, we provide sufficient conditions for type K monotone systems to be permanent and for existence and uniqueness of a positive steady state.

Global stability and permanence for a class of type K monotone systems

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