Balance simplices of 3-species May-Leonard systems

We investigate the existence of a two-dimensional invariant manifold that attracts all nonzero orbits in 3 species Lotka-Volterra systems with identical linear growth rates. This manifold, which we call the balance simplex, is the common boundary of the basin of repulsion of the origin and the basin of repulsion of infinity. The balance simplex is linked to ecological models where there is 'growth when rare' and competition for finite resources. By including alternative food sources for predators we cater for predator-prey type models. In the case that the model is competitive, the balance simplex coincides with the carrying simplex which is an unordered manifold (no two points may be ordered componentwise), but for non-competitive models the balance simplex need not be unordered. The balance simplex of our models contains all limit sets and is the graph of a piecewise analytic function over the unit probability simplex

Bifurcations of piecewise smooth flows:perspectives, methodologies and open problems

In this paper, the theory of bifurcations in piecewise smooth flows is critically surveyed. The focus is on results that hold in arbitrarily (but finitely) many dimensions, highlighting significant areas where a detailed understanding is presently lacking. The clearest results to date concern equilibria undergoing bifurcations at switching boundaries, and limit cycles undergoing grazing and sliding bifurcations. After discussing fundamental concepts, such as topological equivalence of two piecewise smooth systems, discontinuity-induced bifurcations are defined for equilibria and limit cycles. Conditions for equilibria to exist in n-dimensions are given, followed by the conditions under which they generically undergo codimension-one bifurcations. The extent of knowledge of their unfoldings is also summarized. Codimension-one bifurcations of limit cycles and boundary-intersection crossing are described together with techniques for their classification. Codimension-two bifurcations are discussed with suggestions for further study

Transport of inertial particles by Lagrangian coherent structures : application to predator-prey interaction in jellyfish feeding

We use a dynamical systems approach to identify coherent structures from often chaotic motions of inertial particles in open flows. We show that particle Lagrangian coherent structures (pLCS) act as boundaries between regions in which particles have different kinematics. They provide direct geometric information about the motion of ensembles of inertial particles, which is helpful to understand their transport. As an application, we apply the methodology to a planktonic predator–prey system in which moon jellyfish Aurelia aurita uses its body motion to generate a flow that transports small plankton such as copepods to its vicinity for feeding. With the flow field generated by the jellyfish measured experimentally and the dynamics of plankton described by a modified Maxey–Riley equation, we use the pLCS to identify a capture region in which prey can be captured by the jellyfish. The properties of the pLCS and the capture region enable analysis of the effect of several physiological and mechanical parameters on the predator–prey interaction, such as prey size, escape force, predator perception, etc. The methods developed here are equally applicable to multiphase and granular flows, and can be generalized to any other particle equation of motion, e.g. equations governing the motion of reacting particles or charged particles

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

Dynamics of Waves and Patterns (hybrid meeting)

The dynamics of waves and patterns play a significant role in the sciences, especially in fluid mechanics, material science, neuroscience and ecology. The mathematical treatment interconnects several areas, ranging from evolution equations and functional analysis to dynamical systems, geometry, topology, and stochastic as well as numerical analysis. This workshop has specifically focussed on dynamic stability on extended domains, bifurcations of waves and patterns, effects of stochastic driving, and spatio-temporal inhomogenities. During the workshop, multiple new directions, collaborations, and very interesting scientific conversations arose across the entire field

Tensegrity and Recurrent Neural Networks: Towards an Ecological Model of Postural Coordination

Tensegrity systems have been proposed as both the medium of haptic perception and the functional architecture of motor coordination in animals. However, a full working model integrating those two aspects with some form of neural implementation is still lacking. A basic two-dimensional cross-tensegrity plant is designed and its mechanics simulated. The plant is coupled to a Recurrent Neural Network (RNN). The model’s task is to maintain postural balance against gravity despite the intrinsically unstable configuration of the plant. The RNN takes only proprioceptive input about the springs’ lengths and rate of length change and outputs minimum lengths for each spring which modulates their interaction with the plant’s inertial kinetics. Four artificial agents are evolved to coordinate the patterns of spring contractions in order to maintain dynamic equilibrium. A first study assesses quiet standing performance and reveals coordinative patterns between the tensegrity rods akin to humans’ strategy of anti-phase hip-ankle relative phase. The agents show a mixture of periodic and aperiodic trajectories of their Center of Mass. Moreover, the agents seem to tune to the anticipatory “time-to-balance” quantity in order to maintain their movements within a region of reversibility. A second study perturbs the systems with mechanical platform shifts and sensorimotor degradation. The agents’ response to the mechanical perturbation is robust. Dimensionality analysis of the RNNs’ unit activations reveals a pattern of degree of freedom recruitment after perturbation. In the degradation sub-study, different levels of noise are added to the RNN inputs and different levels of weakening gain are applied to the forces generated by the springs to mimic haptic degradation and muscular weakening in elderly humans. As expected, the systems perform less well, falling earlier than without the insults. However, the same systems re-evolved again under the degraded conditions see significant functional recovery. Overall, the dissertation supports the plausibility of RNN cum tensegrity models of haptics-guided postural coordination in humans