Monotone Dynamical Systems with Applications in Biology

Monotone Dynamical Systems with Applications in Biology

  • Monotone Dynamical Systems with Applications in Biology


(1) Review on General Dynamical System, topological property of w-limit set, alpha limit set, 2-d Poincare Bendixson Theorem , Stable Manifold Theorem, Hopf Bifurcation, Chaotic Dynamical Systems.
(2) Monotone Dynamical System
Definitions, Monotone, Strongly monotone, Strongly order preserving (SOP)
Hirsch’s Convergence Theorem, No attracting periodic orbit in monotone system. Non-ordering of limit sets, limit set dichotomy , generic quasi-convergence.
(3) Competitive and Cooperative Differential Equations
Kamke Condition, Positively invariant sets and Monotone Solutions, Non-oscillation principle (Ito’s Lemma), Topological Equivalence of flow of competitive or cooperative n-d systems to a flow on (n-1)-d system of Lipschitz differential equations, Poincare-Bendixson Theorem of 3-d competitive systems, Alternative Cones, The Field-Noyes Model.
(4) Irreducible Cooperative Systems
Strong Monotonicity, A Biochemical Control Circuit, Stability and the Perron-Frobenius Theorem, Competition and Migration, Smale’s Construction
(5) Quasimonotone Systems of Parabilic Equations
Parabolic Systems: The Basic Setup, Maximum Principles, Positively Invariant Sets , Comparison and Monotonicity, The Strong Order Preserving Property, The Biochemical Control Circuit with Diffusion.
(6) Two species competition in the flow reactor model in a river system.  
(7) Sublinearity of Monotone Dynamical System, Butler McGhee Lemma, Uniform Persistence Theory.
(8) Jiang’s global stability theorem for cooperative systems
(9) PDE, Reaction-Diffusion system, Principle eigenvalue
(10) Krein-Rutman Theorem , Systems with coupled PDE and ODE, Generalized Krein-Rutman’s Theorem.
(11) Abstract theory of two species competition in ordered Banach Spaces, Dancer-Hess Lemma. 
Slower Diffuser vs Fast Diffuser, Two species Competition in an unstirred chemostat.
(12) Poincare Bendixson Theorem for Monotone Cyclic Feedback System.



  • Hal L. Smith. Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems (Mathematical Surveys & Monographs),Amer Mathematical Society, 1995.
  • SZE-BI HSU, Ordinary Differential Equations with Applications (Applied Mathematics), 2nd Ed. World Scientific Publishing Company, 2013.
  • Xiao-Qiang Zhao, Dynamical Systems in Population Biology, 2nd Ed, Springer, 2017.