arXiv subjects

Blaine van Rensburg

Publications and source records attributed to Blaine van Rensburg.

4 recordsLinked to original sources

arXiv2026

Global Dynamics of Trait-Structured Generalised Lotka-Volterra Systems with Trait-Independent Interactions

We study the long time dynamics of a selection-mutation integro-differential Lotka-Volterra system of $N$ populations. In our model, fitness depends on a continuous phenotypic trait, but the effect of one population on another is independent of this trait. We establish that, under some usual assumptions on the interactions between populations, the long time behaviour of solutions is exactly determined by the well studied Generalised Lotka-Volterra (ordinary differential) Equation. We also show that, all else being equal, the minimal mutation rate is selected for in a static environment, whereas in an a changing environment, an intermediate mutation rate could be selected instead. The key step in our proofs is establishing that the total population sizes are asymptotically governed by a Generalised Lotka-Volterra Equation, which then allows the use of a general result on asymptotically autonomous dynamical systems.

math.AP↗

arXiv2026

Eco-evolutionary dynamics of a trait-structured predator-prey model

The coupling between evolutionary and ecological changes (eco-evolutionary dynamics) has been shown to be relevant among diverse species, and is also of interest outside of ecology, i.e. in cancer evolution. These dynamics play an important role in determining survival in response to climate change, motivating the need for mathematical models to capture this often complex interplay. Models incorporating eco-evolutionary dynamics often sacrifice analytical tractability to capture the complexity of real systems, do not explicitly consider the effect of population heterogeneity, or focus on long-term behaviour. In order to capture population heterogeneity, both transient, and long-term dynamics, while retaining tractability, we generalise a moment-based method applicable in the regime of small segregational variance to the case of time-dependent mortality and birth. These results are applied to a predator-prey model, where ecological parameters such as the contact rate between species are trait-structured. The trait-distribution of the prey species is shown to be approximately Gaussian with constant variance centered on the mean trait, which is asymptotically governed by an autonomous ODE. In this way, we make explicit the impact of eco-evolutionary dynamics on the transient behaviour and long-term fate of the prey species.

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arXiv2025

The role of mutants in the spatio-temporal progression of inflammatory bowel disease: three classes of permanent form travelling waves

Despite its high prevalence and impact on the lives of those affected, a complete understanding of the cause of inflammatory bowel disease (IBD) is lacking. In this paper, we investigate a novel mechanism which proposes that mutant epithelial cells are significant to the progression of IBD since they promote inflammation and are resistant to death. We develop a simple model encapsulating the propagation of mutant epithelial cells and immune cells which results from interactions with the intestinal barrier and bacteria. Motivated by the slow growth of mutant epithelial cells, and relatively slow response rate of the adaptive immune system, we combine geometric singular perturbation theory with matched asymptotic expansions to determine the one-dimensional slow invariant manifold that characterises the leading order dynamics at all times beyond a passive initial adjustment phase. The dynamics on this manifold are controlled by a bifurcation parameter, $σ$, which depends on the ratio of growth to decay rates of all components except mutants and determines three distinct classes of permanent-form travelling waves that describe the propagation of mutant epithelial and immune cells. These are obtained from scalar reaction-diffusion equations with the reaction being (i) a bistable nonlinearity with a cut-off, (ii) a cubic Fisher nonlinearity and (iii) a KPP or Fisher nonlinearity. Our results suggest that mutant epithelial cells are critical to the progression of IBD. However, their effect on the speed of progression is subdominant.

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arXiv2024

Adaptive Dynamics of Diverging Fitness Optima

We analyse a non-local parabolic integro-differential equation modelling the evolutionary dynamics of a phenotypically-structured population in a changing environment. Such models arise in a variety of contexts from climate change to chemotherapy to the ageing body. The main novelty is that there are two locally optimal traits, each of which shifts at a possibly different linear velocity. We determine sufficient conditions to guarantee extinction or persistence of the population in terms of associated eigenvalue problems. When it does not go extinct, we analyse the solution in the long time, small mutation limits. If the optimas have equal shift velocities, the solution concentrates on a point set of "lagged optima" which are strictly behind the true shifting optima. If the shift velocities are different, we determine that the solution in fact concentrates as a Dirac delta function on the positive lagged optimum with maximum lagged fitness, which depends on the true optimum and the rate of shift. Our results imply that for populations undergoing competition in temporally changing environments, both the true optimal fitness and the required rate of adaptation for each of the diverging optimal traits contribute to the eventual dominance of one trait.

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