Abstract. 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.

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We investigate the existence, uniqueness and Gaussian curvature of the invariant carrying simplices of 3 species autonomous totally competitive Lotka-Volterra systems. Explicit examples are given where the carrying simplex is convex or concave, but also where the curvature is not single-signed. Our method monitors the curvature of an evolving surface that converges uniformly to the carrying simplex, and generally relies on establishing that the Gaussian image of the evolving surface is confined to an invariant cone. We also discuss the relationship between the curvature of the carrying simplex near an interior fixed point and its Split Lyapunov stability. Finally we comment on extensions to general Lotka-Volterra systems that are not competitive.

We consider the geometry of carrying simplices of discrete-time competitive Kolmogorov systems. An existence theorem for the carrying simplex based upon the Hadamard graph transform is developed, and conditions for when the transform yields a sequence of convex or concave graphs are determined. As an application it is shown that the planar Leslie-Gower model has a carrying simplex that is convex or concave.

We show that the flow generated by the totally competitive planar Lotka-Volterra equations deforms the line connecting the two axial equilibria into convex or concave curves, and that these curves remain convex or concave for all subsequent time. We apply the observation to provide an alternative proof to that given by Tineo in 2001 that the carrying simplex, the globally attracting invariant manifold that joins the axial equilibria, is either convex, concave or a straight-line segment.

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Zeeman and Zeeman [E.C. Zeeman and M.L. Zeeman, From local to global behavior in competitive Lotka-Volterra systems, Trans. Amer. Math. Soc. 355 (2003), pp. 713-734] show that if a strongly competitive Lotka-Volterra system (i) has a unique interior fixed point p and (ii) the carrying simplex AE lies below (above) the strongly balanced tangent plane to AE at p then the system has no periodic orbits and p is a global attractor (repellor) relative to AE. Condition (ii) is then translated into the definiteness of a certain quadratic function on the tangent plane, which is equivalent to the definiteness of an (N À 1) Â (N À 1) real symmetric matrix that can be computed. Here we adapt these methods to show that the above conclusions are still true without the assumption (i). Hence, our results apply to globally attracting or repelling fixed points on the boundary, as well as in the interior, of R N þ . Moreover, the algebraic condition for global attraction also implies global asymptotic stability of the fixed point. We also show that the global attraction holds not just relative to AE, but also relative to the interior of the first quadrant.