Systems of differential equations

This paper deals with an almost global convergence result for
Lotka-Volterra systems with predator-prey interactions. These systems can
be written as (negative) feedback systems. The subsystems of the feedback
loop are monotone control systems, possessing particular input-output
properties. We use a small-gain theorem, adapted to a context of systems
with multiple equilibrium points to obtain the desired almost global
convergence result, which provides sufficient conditions to rule out
oscillatory or more complicated behavior that is often observed in
predator-prey systems.

On predator-prey systems and small-gain theorems.

This paper aims to exploit the geometrical and dynamical properties of general type-K competitive systems. We prove that there is a defined countable family of disjoint invariant n−1 cells that attract all non-convergent persistent trajectories in the type-K competitive system. For strongly type-K competitive systems, we prove that there exists an invariant n−1 manifold attracting all nontrivial orbits if the system is dissipative. We also establish the index theory of equilibria for general Kolmogorov systems. Combining such an index theory and the global attractivity, we classify the three-dimensional type-K Lotka–Volterra systems. There are four classes. We exploit the Hopf bifurcation and the global stability for positive equilibrium.

https://iopscience.iop.org/article/10.1088/0951-7715/16/3/301/pdf

The dynamical behaviour of type-K competitive Kolmogorov systems and its application to three-dimensional type-K competitive Lotka–Volterra systems

The asymptotic behavior of dynamical systems with limited competition is investigated. We study index theory for fixed points, permanence, global stability, convergence everywhere and coexistence. It is shown that the system has a globally asymptotically stable fixed point if every fixed point is hyperbolic and locally asymptotically stable relative to the face it belongs to. A nice result is the necessary and sufficient conditions for the system to have a globally asymptotically stable positive fixed point. It can be used to establish the sufficient conditions for the system to persist uniformly and the convergence result for all orbits. Applications are made to time-periodic ordinary differential equations and reaction-diffusion equations.

/pdf/on-the-finite-dimensional-dynamical-systems-with-limited-4wqtu1mgho.pdf

On the finite-dimensional dynamical systems with limited competition

Traveling wave solutions in delayed reaction-diffusion systems with mixed monotonicity

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

This paper provides necessary and sufficient conditions for the type-K Lotka-Volterra system to have a globally asymptotically stable positive steady state. The generalization of such a result is given.

/pdf/the-necessary-and-sufficient-conditions-for-the-global-59ws5naqya.pdf

The necessary and sufficient conditions for the global stability of type- Lotka-Volterra system

The asymptotic behavior of models of cooperation with a concavity property

The Coexistence of a Community of Species with Limited Competition

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Papers

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

The necessary and sufficient conditions for the global stability of type- Lotka-Volterra system

The asymptotic behavior of models of cooperation with a concavity property

The Coexistence of a Community of Species with Limited Competition