Journal of Mathematical Biology最新文献

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.

Algebraic techniques in phylogenetics have historically been successful at proving identifiability results and have also led to novel reconstruction algorithms. In this paper, we study the ideal of phylogenetic invariants of the Cavender-Farris-Neyman (CFN) model on a phylogenetic network with the goal of providing a description of the invariants which is useful for network inference. It was previously shown that to characterize the invariants of any level-1 network, it suffices to understand all sunlet networks, which are those consisting of a single cycle with a leaf adjacent to each cycle vertex. We show that the parameterization of an affine open patch of the CFN sunlet model, which intersects the probability simplex, factors through the space of skew-symmetric matrices via Pfaffians. We then show that this affine patch is isomorphic to a determinantal variety and give an explicit Gröbner basis for the associated ideal, which involves only $\left(\begin{array}{c}n\\ 2\end{array}\right)$ coordinates rather than $2^n$ . Lastly, we show that sunlet networks with at least 6 leaves are identifiable using only these polynomials and run extensive simulations, which show that these polynomials can be used to accurately infer the correct network from DNA sequence data.

How does the movement of individuals influence the persistence of a single population? A surprising phenomenon known as dispersal-induced growth (DIG) occurs when the population would become extinct if isolated or well mixed, but migration, at an appropriate rate, can induce the persistence of the population. In this paper, we investigate this phenomenon based on a time-periodic two-patch model incorporating asymmetric migration matrices. Through comprehensive analysis of the qualitative properties of the associated principal eigenvalue, including monotonicity, asymptotic behaviors, and the topological structure of the level sets as a function of the migration rate and frequency, we characterize the important factors driving the occurrence of DIG under fixed environmental oscillation frequencies. Our results provide new insights into how the interplay between spatial connectivity and temporal environmental variation enables the population persistence.

Different from the existing studies on the influence of self-diffusion or cross-diffusion on Turing instability, this paper originally focuses on the effect of nonlocal competition and host-taxis on Turing instability in a more realistic two-dimensional space, and novelly applies it to study the control of pine wilt disease. It turns out that the incorporation of host-taxis is not conducive to the generation of Turing instability, whereas nonlocal competition can promote the formation of pattern structure by facilitating the occurrence of it. The results not only reveal the new mechanism for the emergence of spatial heterogeneity patterns, but also provide an alternative theoretical explanation for the actually observed multi-point aggregation and multiple outbreaks of pine wilt disease. The various spatial patterns induced by nonlocal competition and host-taxis are numerically illustrated. We find that the Turing patterns can preserve the symmetry of the initial distribution, and contrary to the taxis diffusion, the self-diffusion of D. helophoroides promotes the pattern formation. Furthermore, the high consistency between the simulated and actual distribution patterns of pine wilt disease strongly validates the practical reference value of the paper. The most interesting finding is that we obtain the circular aggregation distribution pattern from simulations, which is consistent with the actual spread trend of pine wilt disease, and our study theoretically reveals the intrinsic evolution mechanism behind its occurrence.

We consider a two-component reaction-diffusion system that has previously been developed to model invasion of cells into a resident cell population. The system is an idealised version of models of tumour growth in which tumour cells degrade the surrounding tissue by increasing the acidity of the local environment. By numerically computing families of travelling wave solutions to this problem, we observe that a general initial condition with either compact support, or sufficiently large exponential decay in the far field, tends to the travelling wave solution that has the largest possible decay at its front. Initial conditions with sufficiently slow exponential decay tend to those travelling wave solutions that have the same exponential decay as their initial conditions. We also show that in the limit that the (nondimensional) degradation rate of resident cells is large, the system has similar asymptotic structure as previously observed in perturbed Fisher-KPP models. The asymptotic analysis in this limit explains the formation of an interstitial gap (a region between the invading and receding fronts, in which both cell populations are small), the width of which is logarithmically large in the limit of large degradation rate. These results show that the general mechanism behind the formation of the interstitial gap in reaction-diffusion tumour models is connected to perturbations of the Fisher-KPP system. Biologically, this implies that order of magnitude difference in degradation rate is required to produce appreciably different gap sizes, while the velocity of the invading front is largely determined by the Fisher-KPP velocity, and only very weakly affected by the presence of the interstitial gap.

The epithelial-mesenchymal transition (EMT), a key stage in tumor metastasis and invasion, depends on many micro-environmental factors, including oxygen levels. In this article, we use a continuum partial differential equations (PDEs) framework comprising coupled equations for the epithelial, mesenchymal, and necrotic cell densities and oxygen concentration to unravel the mysteries of how oxygen heterogeneity affects EMT. A distinguishing feature of the model is that the rates of EMT and MET (mesenchymal-epithelial transition) depend on the oxygen concentration. We assume EMT occurs when oxygen concentration drops below a critical level, and MET occurs when it rises above the critical value. We begin by studying EMT dynamics in an in vitro scenario, where oxygen levels are assumed to be spatially uniform and fluctuate over time between normoxia and hypoxia. This setup mimics aspects of the in vivo phenomenon of cyclic hypoxia. Numerical simulations indicate that the tumor cells adopt a single phenotype based on the oxygen levels: under normoxic conditions, the cells exhibit an epithelial phenotype, while under hypoxic conditions, they transition to a mesenchymal phenotype. We also observe that temporal changes in oxygen levels affect the tumor's overall growth rate. Specifically, our investigations reveal that as the timescale of the oxygen fluctuations decreases, the tumor's growth rate increases. We then use the model to study an in vivo scenario in which we account for oxygen diffusion in order to investigate the effect of spatial heterogeneity in oxygen levels on the EMT dynamics. Simulation results indicate that spatial oxygen heterogeneity generates a heterogeneous population within the tumor, with epithelial cells localized on the outer rim of the tumor and mesenchymal cells concentrated at the tumor center. We perform additional simulations which show further that an increase in mesenchymal diffusivity increases the density of the epithelial cells and the tumor's volume.

Starvation driven diffusion (SDD) describes a cognitive strategy that starvation of a species leads to its stronger movement. In this paper, to better understand the effects of SDD, we propose and analyze a type of predator-prey systems with predator and prey both obeying SDD. By analyzing the linearized eigenvalue problem, we investigate the stability and instability of a semi-trivial steady state, which depends on the conversion efficiency of prey to predator as well as on the predator's minimum motility rate when conversion efficiency is properly large. Predator and prey coexist if the unique semi-trivial steady state is unstable. Utilizing Crandall-Rabinowitz bifurcation theorem, we investigate the local existence, stability, and structure of a bifurcating nontrivial steady state. There exists only one critical conversion efficiency guaranteeing the occurrence of steady-state bifurcation at the unique semi-trivial steady state. The global existence and structure of a bifurcating nontrivial steady state are proven by the global bifurcation theorem. One nontrivial steady state always exists for sufficiently large conversion efficiency. As examples, we apply theoretical results to predator-prey models with Holling type II/IV functional response involving SDD, and verify them via numerical simulations. We numerically observe spatially inhomogeneous periodic solutions, which should arise from nontrivial steady states via Hopf bifurcation, or even via homoclinic bifurcation in the case of Holling type IV functional response. Notably, these solutions consistently mirror resource distribution patterns, aligning conceptually with the ideal free distribution.