Convex geometry of the carrying simplex for the May-Leonard map

We study the convex geometry of certain invariant manifolds, known as carrying simplices, for 3-species competitive Kolmogorov-type maps. We show that if all planes whose normal bundles are contained in a fixed closed and solid convex cone are rendered convex (concave) surfaces by the map, then, if there is a carrying simplex, it is a convex (concave) surface. We apply our results to the May-Leonard map

P matrix properties, injectivity, and stability in chemical reaction systems

In this paper we examine matrices which arise naturally as Jacobians in chemical dynamics. We are particularly interested in when these Jacobians are P matrices (up to a sign change), ensuring certain bounds on their eigenvalues, precluding certain behaviour such as multiple equilibria, and sometimes implying stability. We first explore reaction systems and derive results which provide a deep connection between system structure and the P matrix property. We then examine a class of systems consisting of reactions coupled to an external rate-dependent negative feedback process, and characterise conditions which ensure the P matrix property survives the negative feedback. The techniques presented are applied to examples published in the mathematical and biological literature

Invariant manifolds of Competitive Selection–Recombination dynamics

We study the two-locus-two-allele (TLTA) Selection-Recombination model from population genetics and establish explicit bounds on the TLTA model parameters for an invariant manifold to exist. Our method for proving existence of the invariant manifold relies on two key ingredients: (i) monotone systems theory (backwards in time) and (ii) a phase space volume that decreases under the model dynamics. To demonstrate our results we consider the effect of a modifier gene β on a primary locus α and derive easily testable conditions for the existence of the invariant manifold

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

Carrying simplicies of competitive maps

The carrying simplex is a finite-dimensional, attracting Lipschitz invariant manifold that is commonly found in both continuous and discrete-time competition models from Ecology. It can be studied using the graph transform and cone conditions often applied to study attractors in continuous-time finite and infinitedimensional models from applied mathematics, including chemical reaction networks and reaction diffusion equations. Here we show that the carrying simplex can also be studied from the point of view of the graph transform and cone conditions. However, unlike many of the models mentioned above, we do not use - at least directly - a gap condition that is often used to establish existence of a globally and exponentially attracting manifold. Instead we use contraction of phase volume to ‘suck’ hypersurfaces together uniformly, and ultimately onto the carrying simplex. We give a proof of the existence of the carrying simplex for a class of competitive maps, viewed here as also normally monotone maps. The result is not new, but is carried out in the framework of the graph transform to indicate how the carrying simplex relates to other well-known classes of invariant manifolds. We also discuss the relation between hypersurfaces with positive normals, unordered hypersurfaces and also the type of maps that preserve these types of hypersurfaces. Finally we review several examples from models in Ecology where the carrying simplex is known to exist

Mathematical Model of Ammonia Handling in the Rat Renal Medulla

The kidney is one of the main organs that produces ammonia and release it into the circulation. Under normal conditions, between 30 and 50% of the ammonia produced in the kidney is excreted in the urine, the rest being absorbed into the systemic circulation via the renal vein. In acidosis and in some pathological conditions, the proportion of urinary excretion can increase to 70% of the ammonia produced in the kidney. Mechanisms regulating the balance between urinary excretion and renal vein release are not fully understood. We developed a mathematical model that reflects current thinking about renal ammonia handling in order to investigate the role of each tubular segment and identify some of the components which might control this balance. The model treats the movements of water, sodium chloride, urea, NH3 and [Formula: see text], and non-reabsorbable solute in an idealized renal medulla of the rat at steady state. A parameter study was performed to identify the transport parameters and microenvironmental conditions that most affect the rate of urinary ammonia excretion. Our results suggest that urinary ammonia excretion is mainly determined by those parameters that affect ammonia recycling in the loops of Henle. In particular, our results suggest a critical role for interstitial pH in the outer medulla and for luminal pH along the inner medullary collecting ducts

A global picture for the planar Ricker map: convergence to fixed points and identification of the stable/unstable manifolds

A quadratic Lyapunov function is demonstrated for the non-invertible planar Ricker map (Formula presented.) which shows that for (Formula presented.), and (Formula presented.) all orbits of the planar Ricker map converge to a fixed point. We establish that for 0<r, s<2, whenever a positive equilibrium exists and is locally asymptotically stable, it is globally asymptotically stable (i.e. attracts all of (Formula presented.)). Our approach bypasses and improves on methods that rely on monotonicity, which require (Formula presented.). We also use the Lyapunov function to identify the one-dimensional stable and unstable manifolds when the positive fixed point exists and is a hyperbolic saddle

The balance simplex in non-competitive 2-species scaled Lotka–Volterra systems

Explicit expressions in terms of Gaussian Hypergeometric functions are found for a ‘balance’ manifold that connects the non-zero steady states of a 2-species, non-competitive, scaled Lotka–Volterra system by the unique heteroclinic orbits. In this model, the parameters are the interspecific interaction coefficients which affects the form of the solution used. Similar to the carrying simplex of the competitive model, this balance simplex is the common boundary of the basin of repulsion of the origin and infinity, and is smooth except possibly at steady states

What role does the new interactive educational tool NearPod have in small group anatomy teaching at medical school?

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The value of source data verification in a cancer clinical trial

Background Source data verification (SDV) is a resource intensive method of quality assurance frequently used in clinical trials. There is no empirical evidence to suggest that SDV would impact on comparative treatment effect results from a clinical trial. Methods Data discrepancies and comparative treatment effects obtained following 100% SDV were compared to those based on data without SDV. Overall survival (OS) and Progression-free survival (PFS) were compared using Kaplan-Meier curves, log-rank tests and Cox models. Tumour response classifications and comparative treatment Odds Ratios (ORs) for the outcome objective response rate, and number of Serious Adverse Events (SAEs) were compared. OS estimates based on SDV data were compared against estimates obtained from centrally monitored data. Findings Data discrepancies were identified between different monitoring procedures for the majority of variables examined, with some variation in discrepancy rates. There were no systematic patterns to discrepancies and their impact was negligible on OS, the primary outcome of the trial (HR (95% CI): 1.18(0.99 to 1.41), p = 0.064 with 100% SDV; 1.18(0.99 to 1.42), p = 0.068 without SDV; 1.18(0.99 to 1.40), p = 0.073 with central monitoring). Results were similar for PFS. More extreme discrepancies were found for the subjective outcome overall objective response (OR (95% CI): 1.67(1.04 to 2.68), p = 0.03 with 100% SDV; 2.45(1.49 to 4.04), p = 0.0003 without any SDV) which was mostly due to differing CT scans. Interpretation Quality assurance methods used in clinical trials should be informed by empirical evidence. In this empirical comparison, SDV was expensive and identified random errors that made little impact on results and clinical conclusions of the trial. Central monitoring using an external data source was a more efficient approach for the primary outcome of OS. For the subjective outcome objective response, an independent blinded review committee and tracking system to monitor missing scan data could be more efficient than SDV