Journal of Mathematical Biology最新文献

Natural populations experience a complex interplay of continuous and discrete processes: continuous growth and interactions are punctuated by discrete reproduction events, dispersal, and external disturbances. These dynamics can be modeled by impulsive or flow-kick systems, where continuous flows alternate with instantaneous discrete changes. To study species persistence in these systems, an invasion growth rate theory is developed for flow-kick models with state-dependent timing of kicks and auxiliary variables that can represent stage structure, trait evolution, or environmental forcing. The invasion growth rates correspond to Lyapunov exponents that characterize the average per-capita growth of species when rare. Two theorems are proven that use invasion growth rates to characterize permanence, a form of robust coexistence where populations remain bounded away from extinction. The first theorem uses Morse decompositions of the extinction set and requires that there exists a species with a positive invasion growth rate for every invariant measure supported on a component of the Morse decomposition. The second theorem uses invasion growth rates to define invasion graphs whose vertices correspond to communities and directed edges to potential invasions. Provided the invasion graph is acyclic, permanence and extinction are fully characterized by the signs of the invasion growth rates. Invasion growth rates are also used to identify the existence of extinction-bound trajectories and attractors that lie on the extinction set. To demonstrate the framework's utility, these results are applied to three ecological systems: (i) a microbial serial transfer model where state-dependent timing enables coexistence through a storage effect, (ii) a spatially structured consumer-resource model showing intermediate reproductive delays can maximize persistence, and (iii) an empirically parameterized Lotka-Volterra model demonstrating how disturbance can lead to extinction by disrupting facilitation. Mathematical challenges, particularly for systems with cyclic invasion graphs, and promising biological applications are discussed. These results reveal how the interplay between continuous and discrete dynamics creates ecological outcomes not found in purely continuous or discrete systems, providing a foundation for predicting population persistence and species coexistence in natural communities subject to gradual and sudden changes.

We present a model of infectious disease dynamics where individuals can transition between different compartments, which may have distinct epidemiological characteristics. Within each compartment, epidemic dynamics are represented by a staged progression epidemic model. Individual transitions between compartments occur on a faster time scale, allowing the initial model to be reduced for analysis. In the reduced model, disease eradication and endemicity are characterized by the basic reproduction number. The relationship between this basic reproduction number and those associated with each compartment is analyzed by considering each compartment in isolation. This allows the study of the role of transitions in epidemic dynamics. Endemicity is represented by uniform persistence relative to the total number of infected individuals. It is verified that, for a sufficiently large ratio between time scales, the initial model shares the uniform persistence of the reduced model. The influence of transitions on disease eradication/endemicity is illustrated by different results. In particular, the conditions for transition rates are determined so that endemicity (eradication) in each isolated compartment results in global eradication (endemicity). These results can provide some tools for managing epidemics in the context of individuals transiting between compartments with different epidemiological properties.

An adult human body is made up of some 30 to 40 trillion cells, all of which stem from a single fertilized egg cell. The process by which the right cells appear to arrive in their right numbers at the right time at the right place - development - is only understood in the roughest of outlines. This process does not happen in isolation: the egg, the embryo, the developing foetus, and the adult organism all interact intricately with their changing environments. Conceptual and, increasingly, mathematical approaches to modelling development have centred around Waddington's concept of an epigenetic landscape. This perspective enables us to talk about the molecular and cellular factors that contribute to cells reaching their terminally differentiated state: their fate. The landscape metaphor is however only a simplification of the complex process of development; it for instance does not consider environmental influences, a context which we argue needs to be explicitly taken into account and from the outset. When delving into the literature, it also quickly becomes clear that there is a lack of consistency and agreement on even fundamental concepts; for example, the precise meaning of what we refer to when talking about a 'cell type' or 'cell state.' Here we engage with previous theoretical and mathematical approaches to modelling cell fate - focused on trees, networks, and landscape descriptions - and argue that they require a level of simplification that can be problematic. We introduce random dynamical systems as one natural alternative. These provide a flexible conceptual and mathematical framework that is free of extraneous assumptions. We develop some of the basic concepts and discuss them in relation to now 'classical' depictions of cell fate dynamics, in particular Waddington's landscape. This paper belongs to the special issue "Problems, Progress and Perspectives in Mathematical and Computational Biology".

Predator-mediated apparent competition is an indirect negative interaction between two prey species mediated by a shared predator, which can lead to changes in population dynamics, competition outcomes and community structures. This paper is devoted to investigating the effects and biological consequences of the predator-mediated apparent competition based on a two prey species (one is native and the other is invasive) and one predator model with Holling type I and II functional responses. Through the analytical results and case studies alongside numerical simulations, we find that the initial mass of the invasive prey species, capture rates of prey species, and the predator mortality rate are all important factors determining the success/failure of invasions and the species coexistence/extinction. The global dynamics can be completely classified for the Holling type I functional response, but can only be partially determined for the Holling type II functional response. For the Holling type I functional response, we find that whether the invasive prey species can successfully invade to induce the predator-mediated apparent competition is entirely determined by the capture rates of prey species. For the Holling type II functional response, the dynamics are more complicated. First, if two prey species have the same ecological characteristics, then the initial mass of the invasive prey species is the key factor determining the success/failure of the invasion and hence the effect of the predator-mediated apparent competition. Whereas if two prey species have different ecological characteristics, say different capture rates, then the success of the invasion no longer depends on the initial mass of the invasive prey species, but on the capture rates. In all cases, if the invasion succeeds, then the predator-mediated apparent competition's effectiveness essentially depends on the predator mortality rate. Precisely we show that the native prey species will die out (resp. persist) if the predator has a low (resp. moderate) mortality rate, while the predator will go extinct if it has a large mortality rate. Our study reveals that predator-mediated apparent competition is a complicated ecological process, and its effects and biological consequences depend upon many possible factors.

Cyanobacterial blooms (CBs) pose significant global challenges due to their harmful toxins and socio-economic impacts, with nutrient availability playing a key role in their growth, as described by ecological stoichiometry (ES). However, real-world ecosystems exhibit spatial heterogeneity, limiting the applicability of simpler, spatially uniform models. To address this, we develop a spatially explicit partial differential equation model based on ES to study cyanobacteria in the epilimnion of freshwater systems. We establish the well-posedness of the model and perform a stability analysis, showing that it admits two linearly stable steady states, leading to either extinction or a spatially uniform positive equilibrium where cyanobacterial biomass stabilizes at its carrying capacity. Further, we discuss the possibility of long-term spatially nonuniform solution with small diffusion and space-dependent parameters. We use the finite elements method (FEM) to numerically solve our system on a real lake domain derived from Geographic Information System (GIS) data and realistic wind conditions extrapolated from ERA5-Land. Additionally, we use a cyanobacteria estimation (CE) obtained from Sentinel-2 to set initial conditions, and we achieve strong model validation metrics. Our numerical results highlight the importance of lake shape and size in bloom monitoring, while global sensitivity analysis using Sobol Indices identifies light attenuation and intensity as primary drivers of bloom variation, with water movement influencing early bloom stages and nutrient input becoming critical over time. This model supports continuous water-quality monitoring, informing agricultural, recreational, economic, and public health strategies for mitigating CBs.

The ideal free distribution in ecology was introduced by Fretwell and Lucas to model the habitat selection of animal populations. In this paper, we revisit the concept via a mean field game system with local coupling, which models a dynamic version of the habitat selection game in ecology. We establish the existence of classical solution of the ergodic mean field game system, including the case of heterogeneous diffusion when the underlying domain is one-dimensional and further show that the population density of agents converges to the ideal free distribution of the underlying habitat selection game, as the cost of control tends to zero. Our analysis provides a derivation of ideal free distribution in a dynamical context.

We revisit the problem of characterising the mean-field limit of a network of Hopfield-like neurons. Building on the previous works of Ben Arous and Guionnet we establish for a large class of networks of Hopfield-like neurons, i.e. rate neurons, the mean-field equations on a time interval $[0,T]$ , $T>0$ , of the thermodynamic limit of these networks, i.e. the limit when the number of neurons goes to infinity. Here, we do not assume that the synaptic weights describing the connections between the neurons are i.i.d. as zero-mean Gaussians. The limit equations are stochastic and very simply described in terms of two functions, a "correlation" function noted $K_Q(t,s)$ and a "mean" function noted $m_Q\left(t\right)$ . The "noise" part of the equations is a linear function of the Brownian motion, which is obtained by solving a Volterra equation of the second kind whose resolving kernel is expressed as a function of $K_Q$ . We give a constructive proof of the uniqueness of the limit equations. We use the corresponding algorithm for an effective computation of the functions $K_Q$ and $m_Q$ , given the weights distribution. Several numerical experiments are reported.

In this paper, a class of reaction-diffusion equations for Multiple Sclerosis is presented. These models are derived by means of a diffusive limit starting from a proper kinetic description, taking account of the underlying microscopic interactions among cells. At the macroscopic level, we discuss the necessary conditions for Turing instability phenomena and the formation of two-dimensional patterns, whose shape and stability are investigated by means of a weakly nonlinear analysis. Some numerical simulations, confirming and extending theoretical results, are proposed for a specific scenario.

We consider a mathematical model to explore the effects of human behavioural changes on the transmission of two respiratory viruses, where co-infection is possible. The model includes an index to describe the human choices induced by information and rumours regarding the diseases. We first consider the case in which the public health authorities rely only on non-pharmaceutical containment measures and perform a qualitative analysis of the model through bifurcation theory, in order to analyse the existence and stability of both endemic and co-endemic equilibria. We also show the impact of the most relevant information-related parameters on the system dynamics. Then, we extend the model by assuming that a vaccine is available for each of the two viruses. We show how adherence to social distancing may be affected by information and rumours regarding the vaccination coverage in the community. Finally, we investigate the effects of seasonality by introducing a two-state switch function to represent a reduction in both vaccination and transmission rates during the summer season. We found that seasonality causes an increase in the prevalence peaks, suggesting that the detrimental effects due to the reduction of vaccination rates prevail over the beneficial ones due to the reduction of transmission.

In this paper, we investigate a two-species Lotka-Volterra competition patch model in a Y-shaped river network, where the two species are assumed to be identical except for their random and directed movements. We show that competitive exclusion can occur under certain conditions, i.e., one of the semi-trivial equilibria is globally asymptotically stable. Specifically, if the random dispersal rates of the two species are equal, the species with a smaller drift rate will drive the other species to extinction, which suggests that smaller drift rates are favored.