Journal of Mathematical Biology最新文献

We consider a reaction-diffusion model with free boundaries in one space dimension for a single population species with density u(t, x) and population range [g(t), h(t)]. The equations governing the evolution of the range boundary are deduced from the biological assumption that the species maintains its population density at a fixed positive level at the range boundary by advancing or retreating the fronts. Our mathematical results suggest that the Allee effects are eliminated if the species maintains its population density at suitable levels at the range boundary, namely with such a strategy at the range edge the species can invade the environment successfully with all admissible initial populations, exhibiting the dynamics of super invaders. Numerical simulations are used to help understand what happens if the population density levels at the range boundary are maintained in other ranges.

During development, precise cellular patterning is essential for the formation of functional tissues and organs. These patterns arise from conserved signaling networks that regulate communication both within and between cells. Here, we develop and present a model-independent ordinary differential equation (ODE) framework for analyzing pattern formation in a homogeneous cell array. In contrast to traditional approaches that focus on specific equations, our method relies solely on general assumptions about global intercellular communication (between cells) and qualitative properties of local intracellular biochemical signaling (within cells). Prior work has shown that global intercellular communication networks alone determine the possible emergent patterns in a generic system. We build on these results by demonstrating that additional constraints on the local intracellular signaling network lead to a single stable pattern which depends on the qualitative features of the network. Our framework enables the prediction of cell fate patterns with minimal modeling assumptions, and provides a powerful tool for inferring unknown interactions within signaling networks by analyzing tissue-level patterns.

Fire blight is a bacterial plant disease that affects apple and pear trees. We present a mathematical model for its spread in an orchard during bloom. This is a PDE-ODE coupled system, consisting of two semilinear PDEs for the pathogen, coupled to a system of three ODEs for the stationary hosts. Exploratory numerical simulations suggest the existence of travelling waves, which we subsequently prove, under some conditions on parameters, using the method of upper and lower bounds and Schauder's fixed point theorem. Our results are likely not optimal in the sense that our constraints on parameters, which can be interpreted biologically, are sufficient for the existence of travelling waves, but probably not necessary. Possible implications for fire blight biology and management are discussed.

We consider stochastic population processes that are almost surely absorbed at the origin within finite time. Our interest is in the quasistationary distribution,  $u$ , and the expected time, $\tau$ , from quasistationarity to extinction, both of which we study via WKB approximation. This approach involves solving a Hamilton-Jacobi partial differential equation specific to the model. We provide conditions under which analytical solution of the Hamilton-Jacobi equation is possible, and give the solution. This provides a first approximation to both  $u$ and  $\tau$ . We provide further conditions under which a corresponding 'transport equation' may be solved, leading to an improved approximation of  $u$ . For multitype birth and death processes, we then consider an alternative approximation for  $u$ that is valid close to the origin, provide conditions under which the elements of this alternative approximation may be found explicitly, and hence derive an improved approximation for  $\tau$ . We illustrate our results in a number of applications.

When an exotic species is introduced outside its natural range, new interspecific interactions with native species may arise. These interactions can induce phenotypic changes, which may originate from phenotypic plasticity or adaptive processes. Phenotypic change may play an important role in biological invasions, either by promoting or by preventing its success. In this work, a mathematical modeling approach is used to study a native predator-prey system exposed to an exotic species that predates on the native species and that also competes by interference with the native predator. This proposed approach allows to describe the eco-evolutionary dynamics involving the inducible defense of the prey and the inducible offense of both predators. The model is represented by a system of ordinary differential equations (ODEs), analyzed using advanced analytical and numerical methods. Specifically, we applied the qualitative theory of ODEs and developed numerical algorithms for parameter sweeps. Parameter values for the numerical experiments were based on the American mink, one of the most harmful invasive species in Europe and South America. The results show that the role of phenotypic change in invasion success depends on three components: the efficiency of the new trait values, the associated costs, and the speed of trait change. The specific conditions that lead to an unsuccessful invasion are: the prey's defense efficiency against the exotic predator is higher than its defense efficiency against the native predator. The cost imposed by the exotic predator is greater than the cost imposed by the native predator. Lastly, the speed of phenotypic change is faster in the native predator than in the exotic predator.

An unconventional and environmentally friendly mosquito management approach offers a sustainable solution that protects both the environment and human health. One such method is the Sterile Insect Technique (SIT), which holds promise as a mosquito control strategy by releasing sterilized male mosquitoes into the wild-type (WT) mosquito population. Since the success of SIT depends on the strategic planning of sterile mosquito releases, this paper examines a stage-structured model for mosquito populations with a density-dependent threshold for sterile male mosquito release, where releases occur only when the ratio of WT to sterile mosquito populations exceeds a critical threshold. Using intermittent releases, the proposed SIT model is designed to optimally align the release of sterile male mosquitoes with WT and sterile mosquito population densities, maintaining WT mosquito suppression at a predefined threshold and offering a more effective alternative to continuous release strategies. We employ Filippov's modelling approach to investigate how intermittent releases, represented by piecewise-smooth functions, affect the dynamics of the system, particularly when mosquito populations exceed the predefined threshold. To explore the dynamical complexities, we employ Filippov's convex method by defining the vector field in the discontinuous region as convex combinations of adjacent fields, allowing for the analysis of sliding motion and the identification of discontinuity-induced bifurcations through differential inclusions. Our findings identify the minimum release rate of sterile mosquitoes required to achieve the desired suppression level, highlighting the need to increase this rate due to increased WT mosquito immigration, reduced survival and mating fitness of sterile mosquitoes, and limitations in mosquito surveillance accuracy.

The time-elapsed model for neural assemblies is a nonlinear age-structured equation where the renewal term describes the network activity and influences the discharge rate, possibly with a delay due to the length of connections. We first solve a long standing question, namely that an inhibitory network without delay can promote desynchronization and stabilizes network activity by proving rigorously that the solution converges to a unique steady state. Our approach is based on the observation that a non-expansion property holds. However a non-degeneracy condition is needed and, besides the standard one, we introduce a new condition based on strict nonlinearity. When a delay is included, following previous works for Fokker-Planck models, we prove that the network can generate periodic solutions, both in inhibitory and excitatory networks. To this end, we introduce a new formalism to establish rigorously this property for large delays. Moreover, the fundamental contraction property can extend to other age-structured equations and systems.

In this manuscript, we develop a mobility-based Susceptible-Infectious-Recovered (SIR) model to elucidate the dynamics of pandemic propagation. While traditional SIR models within the field of epidemiology aptly characterize transitions among susceptible, infected, and recovered states, they typically neglect the inherent spatial mobility of particles. To address this limitation, we introduce a novel dynamical SIR model that incorporates nonlocal spatial motion for three distinct particle types, thereby bridging the gap between epidemiological theory and real-world mobility patterns. This paper primarily focuses on analyzing the long-term behavior of this dynamic system, with specific emphasis on the computation of first and second moments. We propose a new reproduction number $R_0^m$ and compare it with the classical reproduction number $R_0$ in the traditional SIR model. Furthermore, we rigorously examine the phenomenon of intermittency within the context of this enhanced SIR model. The results contribute to a more comprehensive understanding of pandemic spread dynamics, considering both the interplay between disease transmission and population mobility and the impact of spatial motion on the system's behavior over time.

This paper presents an extended mathematical model for tumor angiogenesis incorporating oxygen dynamics as a main regulator. We enhance a five-component PDE system describing endothelial cells, proteases, inhibitors, extracellular matrix, and oxygen concentration, with a focus on their spatiotemporal interactions. We establish existence, uniqueness, and boundedness of solutions through a mathematical analysis. A numerical scheme using method of lines and fourth-order Runge-Kutta methods is developed, with proven stability constraints and convergence properties. Numerical experiments demonstrate biologically plausible vascular formation with oxygen-mediated regulation.

We examine a generalized KPP equation with a "q-diffusion", which is a framework that unifies various standard linear diffusion regimes: Fickian diffusion ( $q=0$ ), Stratonovich diffusion ( $q=1/2$ ), Fokker-Planck diffusion ( $q=1$ ), and nonstandard diffusion regimes for general $q\in R$ . Using both analytical methods and numerical simulations, we explore how the ability of persistence (measured by some principal eigenvalue) and how the asymptotic spreading speed depend on the parameter q and on the phase shift between the growth rate r(x) and the diffusion coefficient D(x). Our results demonstrate that persistence and spreading properties generally depend on q: for example, appropriate configurations of r(x) and D(x) can be constructed such that q-diffusion either enhances or diminishes the ability of persistence and the spreading speed with respect to the traditional Fickian diffusion. We show that the spatial arrangement of r(x) with respect to D(x) has markedly different effects depending on whether $q>0$ , $q=0$ , or $q<0$ . The case where r is constant is an exception: persistence becomes independent of q, while the spreading speed displays a symmetry around $q=1/2$ . This work underscores the importance of carefully selecting diffusion models in ecological and epidemiological contexts, highlighting their potential implications for persistence, spreading, and control strategies.