Journal of Mathematical Biology最新文献

We investigate noise-induced bimodal distributions in self-regulated gene networks with fast dimerization, where dimerized proteins enhance gene expression. Despite their fundamental role in gene regulation, analytical study of bimodal behaviour in such networks is challenging because the nonlinear interactions introduced by dimer formation render exact steady-state distributions infeasible. To address this, we reformulate the problem as a reduced self-regulated gene-expression model that approximates fast dimerization, in which the transition rate from the promoter-off to promoter-on state depends nonlinearly on protein levels. We introduce two diagnostic quantities: the promoter activity ratio, which quantifies promoter activation as a function of protein level, and the mode detection ratio, which identifies peaks of the steady-state protein distribution. Analysis of their recurrence relations reveals how promoter activity shapes the steady-state law, and how intrinsic stochasticity can generate multimodal protein distributions in self-regulated expression circuits. We further show that the corresponding mean-field ODE system admits a unique non-negative equilibrium when the protein synthesis-to-degradation ratio lies below an explicit threshold determined by the inactivation and dimer-induced activation rates. Hence, the bimodality we observe can arise purely from stochastic effects rather than deterministic bistability. Our approach provides a general framework for diagnosing noise-induced multimodality in gene networks with nonlinear promoter transitions, without relying on exact probability distributions, which are typically infeasible for nonlinear reaction rates, particularly in our case. Beyond its theoretical contribution, this work has conceptual relevance to sustainability: our mode-detection diagnostics and the distinction between deterministic multistability and noise-induced multimodality can inform assessments of resilience, early-warning indicators, and state persistence.

We introduce the boundary reproduction number, adapted from the basic reproduction number in mathematical epidemiology, to assess whether an infusion of species will persist or become exhausted in a chemical reaction system. Our main contributions are as follows: (a) we show how the concept of a siphon, prevalent in Petri nets and chemical reaction network theory, identifies sets of species that may become depleted at steady state, analogous to a disease-free boundary steady state; (b) we develop an approach for incorporating biochemically motivated conservation laws, which allows the stability of boundary steady states to be determined within specific compatibility classes; and (c) we present an effective heuristic for decomposing the Jacobian of the system that reduces the computational complexity required to compute the stability domain of a boundary steady state. The boundary reproduction number approach significantly simplifies existing parameter-dependent methods for determining the stability of boundary steady states in chemical reaction systems and has implications for the capacity of critical metabolites and substrates in metabolic pathways to become exhausted.

Several physiological and pathological processes, such as development, wound healing, and cancer invasion, depend on cell migration through fibrous extracellular matrix (ECM). In such contexts, topographical features of the ECM, including fiber alignment and pore size, strongly bias migration, a phenomenon known as topotaxis. To explore this guidance mechanism in a controlled theoretical setting, we present a minimal particle-based model of single-cell motility in two-dimensional environments abstracted as networks of elongated obstacles. This abstraction captures key geometric and topographical constraints of fibrous microenvironments while remaining computationally tractable. Our framework integrates chemotactic bias, stochastic polarity dynamics, steric repulsion from obstacles, escape strategies from mechanical trapping, and minimal remodeling of the obstacles network. Adaptive polarity perturbations mimic active cellular responses such as invadopodial protrusion or random reorientation, while a displacement-based criterion detects trapping events. Heterogeneity is incorporated by assigning variable repulsion strengths to obstacles, and remodeling is implemented by allowing local displacements induced by cell-obstacle contact. Simulation results show that active remodeling of obstacles consistently enhances migration efficiency and target acquisition, whereas escape strategies alone provide only partial improvement, and heterogeneity introduces directional variability. At long timescales, trajectories converge toward effective diffusion, but intermediate dynamics display nontrivial deviations due to confinement and obstacle interactions, highlighting a topotaxis-driven component of motility. Overall, this work positions cell migration within the theoretical context of obstacles networks, providing mechanistic insight into how confinement, anomalous transport, and remodeling interact to shape directional migration. While simplified to two dimensions and lacking entanglement effects characteristic of real three-dimensional ECMs, the model offers a tractable and extensible framework for future studies, including the incorporation of cell deformations or more realistic ECM architectures.

Coral colonies exhibit complex, self-similar branching architectures shaped by biochemical interactions and environmental constraints. To model their growth and calcification dynamics, we propose a novel reaction-diffusion framework defined over $p$ -adic ultrametric spaces. The model incorporates biologically grounded reactions involving calcium and bicarbonate ions, whose interplay drives the precipitation of calcium carbonate (CaCO₃). Nonlocal diffusion is governed by the Vladimirov operator over the $p$ -adic integers, naturally capturing the hierarchical geometry of branching coral structures. Discretization over $p$ -adic balls yields a high-dimensional nonlinear ODE system, which we solve numerically to examine how environmental and kinetic parameters-particularly CO₂ concentration-influence morphogenetic outcomes. The resulting simulations reproduce structurally diverse and biologically plausible branching patterns. This approach bridges non-Archimedean analysis with morphogenesis modeling and provides a mathematically rigorous framework for investigating hierarchical structure formation in developmental biology.

Dengue fever is a major public health problem and has been extensively modeled. Understanding the role of explicit vector dynamics in vector-borne diseases such as dengue fever is essential for accurately capturing transmission patterns and improving control strategies. In this study, we extend the minimalistic two-infection host-host SIRSIR model by introducing the SIRSIR-UV model, which explicitly incorporates vector population dynamics. Our aim is to investigate how these explicit vector dynamics influence the behavior of the system. In doing so, we extend previous models that assumed implicit vector effects in addition to immunity and disease enhancement factors. Using tools from nonlinear dynamics and bifurcation theory, we derive analytical conditions for transcritical and tangent bifurcations, formalize backward bifurcation using center manifold theory, and compute Hopf and global homoclinic bifurcation curves. We also show that seasonal influences in the vector populations, mimicking the seasonality of mosquitoes, contribute to the occurrence of chaotic behavior in disease transmission, reflecting the current patterns observed in epidemiological data. We thoroughly characterize the dynamics of the SIRSIR-UV model and explore the implications of including explicit vector dynamics. Finally, we discuss our results with the previous SIRSIR model and conclude that the bifurcation structures observed in the SIRSIR-UV model are consistent with those of the minimalistic SIRSIR model. This unexpected result has important implications for the modeling of vector-borne diseases. It suggests that simplifying assumptions, such as the use of implicit vector dynamics, can effectively capture important aspects of disease transmission while reducing the complexity of the mathematical analysis.

We consider the following question: how close to the ancestral root of a phylogenetic tree is the most recent common ancestor of k species randomly sampled from the tips of the tree? For trees having shapes predicted by the Yule-Harding model, it is known that the most recent common ancestor is likely to be close to (or equal to) the root of the full tree, even as n becomes large (for k fixed). However, this result does not extend to models of tree shape that more closely describe phylogenies encountered in evolutionary biology. We investigate the impact of tree shape (via the Aldous $\beta -$ splitting model) to predict the number of edges that separate the most recent common ancestor of a random sample of k tip species and the root of the parent tree they are sampled from. Both exact and asymptotic results are presented. We also briefly consider a variation of the process in which a random number of tip species are sampled.

Infectious diseases remain a significant threat to global public health, often causing substantial economic burdens. Effective disease management requires an integrated approach involving healthcare facilities, particularly hospital bed capacity, and vaccination campaigns. A four-dimensional mathematical model is investigated to study the dynamics of an emerging infectious disease, considering both vaccination efforts and the limitations of healthcare resources. The model undergoes a series of local bifurcations, including transcritical (both forward and backward), saddle-node, Hopf (supercritical, subcritical, and Bautin), and Bogdanov-Takens bifurcations, revealing the complex dynamics that govern disease transmission and control. To derive optimal control strategies, we apply a multiobjective optimal control approach, transforming the problem into a multiobjective optimization problem and solving it using the $ϵ$ -constraint method. The analysis of Pareto optimal fronts provides valuable insights into the relative effectiveness of varying vaccination and hospitalization strategies under different transmission rates. The numerical results validate the analytical findings and provide comprehensive insight into the best strategies to minimize the infected individuals and associated cost. One such result reveals that the use of saturation-type cost functions offers a cost-efficient approach for managing intervention resources, while more comprehensive cost models may incur higher implementation costs.

Mating behaviors significantly influence the dynamics of frog populations. In this study, we formulate a stage-structured model with periodic time delay that reflects the complexities of frog populations, accounting for seasonal changes, two-sex division, mating interactions, and adult competition. The model tracks the fluctuations of female and male populations in both active and hibernation phases. To analyze the global dynamics of this system, we explore fundamental properties in the natural phase space and a new phase space, in the quotient space sense, to establish the strong monotonicity of the solution periodic semiflow. Numerical simulations evaluate the effects of maturity mortality rates and mating pair numbers on population trajectories over single and multiple life cycles. The results indicate that the populations decline markedly prior to hibernation, but an increased number of mating pairs correlates with larger stable population sizes during the active phase.

ODE-based models for gene regulatory networks (GRNs) can often be formulated as smooth singular perturbation problems with multiple small parameters, some of which are related to time-scale separation, whereas others are related to 'switching', (proximity to a non-smooth singular limit). This motivates the study of reduced models obtained after (i) quasi-steady state reduction (QSSR), which utilises the time-scale separation, and (ii) piecewise-smooth approximations, which reduce the nonlinearity of the model by viewing highly nonlinear sigmoidal terms as singular perturbations of step functions. We investigate the interplay between the reduction methods (i)-(ii), in the context of a 4-dimensional GRN which has been used as a low-dimensional representative of an important class of (generally high-dimensional) GRN models in the literature. We begin by identifying a region in the small parameter plane for which this problem can be formulated as a smooth singularly perturbed system on a blown-up space, uniformly in the switching parameter. This allows us to apply Fenichel's coordinate-free theorems and obtain a rigorous reduction to a 2-dimensional system, that is a perturbation of the QSSR. Finally, we show that the reduced system features a Hopf bifurcation which does not appear in the QSSR system, due to the influence of higher order terms. Taken together, our findings suggest that the relative size of the small parameters is important for the validity of QSS reductions and the determination of qualitative dynamics in GRN models more generally. Although the focus is on the 4-dimensional GRN, our approach is applicable to higher dimensions.