arXiv subjects

Jane Shaw MacDonald

Publications and source records attributed to Jane Shaw MacDonald.

4 recordsLinked to original sources

arXiv2026

Persistence of n-Species Lotka-Volterra Models with Periodic Pulses

Periodic impulsive interventions arise naturally in the management of biological populations, including chemotherapy, pesticide application, and infectious-disease treatment. We develop general conditions for permanence in n-species population models subject to periodic multiplicative pulse disturbances. Our main result provides a sufficient condition for permanence in terms of weighted long-term growth rates on a Morse decomposition of the extinction set, explicitly separating the contributions of continuous population dynamics from those of the periodic pulse. To establish this result, we transform the impulsive system into an associated autonomous continuous-time dynamical system and use this correspondence to extend classical permanence theory to periodically pulsed models. We further show that the same conditions imply robust permanence under sufficiently small perturbations to the continuous dynamics, pulse period, and pulse effects. We illustrate the framework with two Lotka-Volterra models motivated by biological control: competition between chemotherapy-sensitive and chemotherapy-resistant cancer cells, and integrated control of an agricultural pest using pesticides and parasitoids. These examples demonstrate how intervention frequency and intensity interact with underlying ecological interactions to determine whether populations coexist or are excluded. Our results provide a general framework for analyzing persistence in ecological systems subject to repeated discrete disturbances.

q-bio.PE↗

arXiv2025

Unique steady states for population models in a heterogeneous environment

We revisit a model proposed by Freedman etal in \cite{freedman} which describes the dynamics of a population diffusing in a patchy environment. From their work it is known that positive steady states exist for this model, but not whether they are unique. Here, we provide sufficient conditions guaranteeing that steady states are unique. These conditions are satisfied when the reaction rates are generalized logistic growth rates. Our proofs critically exploit Chicone's ideas in \cite{chicone}, which were used to establish that the period map associated to a continuous family of periodic solutions of certain planar Hamiltonian systems is monotone.

math.DS↗

arXiv2024

A mathematical model for Nordic skiing

Nordic skiing provides fascinating opportunities for mathematical modelling studies that exploit methods and insights from physics, applied mathematics, data analysis, scientific computing and sports science. A typical ski course winds over varied terrain with frequent changes in elevation and direction, and so its geometry is naturally described by a three-dimensional space curve. The skier travels along a course under the influence of various forces, and their dynamics can be described using a nonlinear system of ordinary differential equations (ODEs) that are derived from Newton's laws of motion. We develop an algorithm for solving the governing equations that combines Hermite spline interpolation, numerical quadrature and a high-order ODE solver. Numerical simulations are compared with measurements of skiers on actual courses to demonstrate the effectiveness of the model. Throughout, we aim to illustrate how elementary concepts from undergraduate courses in calculus and scientific computing can be applied to study real problems in sport, which we hope will provide stimulating examples for both instructors and students. At the same time, we demonstrate how these concepts are capable of providing novel insights into skiing that should also be of interest to sport scientists.

physics.class-ph↗

arXiv2023

A hybrid finite element method for moving-habitat models in two spatial dimensions

Moving-habitat models track the density of a population whose suitable habitat shifts as a consequence of climate change. Whereas most previous studies in this area consider 1-dimensional space, we derive and study a spatially 2-dimensional moving-habitat model via reaction-diffusion equations. The population inhabits the whole space. The suitable habitat is a bounded region where population growth is positive; the unbounded complement of its closure is unsuitable with negative growth. The interface between the two habitat types moves, depicting the movement of the suitable habitat poleward. Detailed modelling of individual movement behaviour induces a nonstandard discontinuity in the density across the interface. For the corresponding semi-discretised system we prove well-posedness for a constant shifting velocity before constructing an implicit-explicit hybrid finite element method. In this method, a Lagrange multiplier weakly imposes the jump discontinuity across the interface. For a stationary interface, we derive optimal a priori error estimates over a conformal mesh with nonconformal discretisation. We demonstrate with numerical convergence tests that these results hold for the moving interface. Finally, we demonstrate the strength of our hybrid finite element method with two biologically motivated cases, one for a domain with a curved boundary and the other for non-constant shifting velocity.

math.NA↗