On existence and uniqueness of the carrying simplex for competitive dynamical systems

Certain dynamical models of competition have a unique invariant hypersurface to whichevery nonzero tractory is asymptotic, having simple geometry and topology.Comment: Submitted to Journal of Biological Dynamics. 13 page

Rank of Stably Dissipative Graphs

For the class of stably dissipative Lotka-Volterra systems we prove that the rank of its defining matrix, which is the dimension of the associated invariant foliation, is completely determined by the system's graph

The Jungle Universe

In this paper, we exploit the fact that the dynamics of homogeneous and isotropic Friedmann-Lemaitre universes is a special case of generalized Lotka-Volterra system where the competitive species are the barotropic fluids filling the Universe. Without coupling between those fluids, Lotka-Volterra formulation offers a pedagogical and simple way to interpret usual Friedmann-Lemaitre cosmological dynamics. A natural and physical coupling between cosmological fluids is proposed which preserve the structure of the dynamical equations. Using the standard tools of Lotka-Volterra dynamics, we obtain the general Lyapunov function of the system when one of the fluids is coupled to dark energy. This provides in a rigorous form a generic asymptotic behavior for cosmic expansion in presence of coupled species, beyond the standard de Sitter, Einstein-de Sitter and Milne cosmologies. Finally, we conjecture that chaos can appear for at least four interacting fluids.Comment: 26 pages, 4 figure

Balance manifolds in Lotka-Volterra systems

The Lotka-Volterra equations are a dynamical system in the form of an autonomous ODE. The aim of this thesis is to explore the carrying simplex for non-competitive Lotka-Volterra systems for the case of 2- and 3-species, where it is referred to as a balance simplex. Carrying simplices were developed by M.W. Hirsch in a series of papers. They are hypersurfaces which asymptotically attract all non-zero solutions in the phase portrait. This essentially means that all the non-trivial dynamics occur on the carrying simplex, which is one dimension less than the system itself. Many of its properties have been studied by various authors, for example: E.C. Zeeman, M.L. Zeeman, S. Baigent, J. Mierczyński. The first few chapters of this thesis explores the 2-species scaled Lotka-Volterra system, where all intrinsic growth rates and intraspecific interaction rates are set to the value 1. This simplification of the model allows for an explicit, analytic form of the balance simplex to be found. This is done by transforming the system to polar co-ordinates and explicitly integrating the new system. The balance simplex for this 2-species model is precisely composed of the heteroclinic orbits connecting non-zero steady states, along with these states themselves. The later chapters of this thesis focuses on the 3-species case. The existence of the balance simplex in particular parameter cases is proven and it is shown to be piecewise analytic (when the interaction matrix containing the parameters is strictly copositive). These chapters also work towards plotting the balance simplex so it can be visualised for the 3-species system. In another chapter, more general planar Kolmogorov models are considered. Conditions sufficient for the balance simplex to exist are given, and it is again composed of heteroclinic orbits between non-zero steady states

Chaos to Permanence - Through Control Theory

Work by Cushing et al. [18] and Kot et al. [60] demonstrate that chaotic behavior does occur in biological systems. We demonstrate that chaotic behavior can enable the survival/thriving of the species involved in a system. We adopt the concepts of persistence/permanence as measures of survival/thriving of the species [35]. We utilize present chaotic behavior and a control algorithm based on [66, 72] to push a non-permanent system into permanence. The algorithm uses the chaotic orbits present in the system to obtain the desired state. We apply the algorithm to a Lotka-Volterra type two-prey, one-predator model from [30], a ratio-dependent one-prey, two-predator model from [35] and a simple prey-specialist predator-generalist predator (for ex: plant-insect pest-spider) interaction model [67] and demonstrate its effectiveness in taking advantage of chaotic behavior to achieve a desirable state for all species involved

Chaotic attractors in Atkinson-Allen model of four competing species

We study the occurrence of chaos in the Atkinson-Allen model of four competing species, which plays the role as a discrete-time Lotka-Volterra-type model. We show that in this model chaos can be generated by a cascade of quasiperiod-doubling bifurcations starting from a supercritical Neimark-Sacker bifurcation of the unique positive fixed point. The chaotic attractor is contained in a globally attracting invariant manifold of codimension one, known as the carrying simplex. Biologically, our study implies that the invasion attempts by an invader into a trimorphic population under Atkinson-Allen dynamics can lead to chaos.Peer reviewe

Chaos to Permanence-Through Control Theory

Work by Cushing et al. \cite{Cushing} and Kot et al. \cite{Kot} demonstrate that chaotic behavior does occur in biological systems. We demonstrate that chaotic behavior can enable the survival/thriving of the species involved in a system. We adopt the concepts of persistence/permanence as measures of survival/thriving of the species \cite{EVG}. We utilize present chaotic behavior and a control algorithm based on \cite{Vincent97,Vincent2001} to push a non-permanent system into permanence. The algorithm uses the chaotic orbits present in the system to obtain the desired state. We apply the algorithm to a Lotka-Volterra type two-prey, one-predator model from \cite{Harvesting}, a ratio-dependent one-prey, two-predator model from \cite{EVG} and a simple prey-specialist predator-generalist predator (for ex: plant-insect pest-spider) interaction model \cite{Upad} and demonstrate its effectiveness in taking advantage of chaotic behavior to achieve a desirable state for all species involved