Mathematical Biosciences最新文献

Modeling and analysis of the transmission dynamics in metapopulation networks incorporating individual contact heterogeneity 包含个体接触异质性的元种群网络传播动力学建模与分析。

IF 1.8 4区数学 Q2 BIOLOGY

Mathematical Biosciences

Pub Date : 2025-12-12 DOI: 10.1016/j.mbs.2025.109598

Pengle Sun , Shanshan Feng , Zhen Jin

In studies of traditional metapopulation networks, researchers typically assumed that individuals within subpopulations were well-mixed, ignoring individual contact heterogeneity. However, in real life, individual contact exhibits high levels of heterogeneity. To address this, we build an SIS model that couples individual contact heterogeneity on a metapopulation network, which is characterized by constructing dynamic subnetworks within subpopulations. Theoretically, the basic reproduction number is obtained, the existence and uniqueness of equilibria, as well as their global stability, are proved. Through numerical simulations, theoretical results are validated. Additionally, the findings reveal a positive correlation between individual contact heterogeneity and the basic reproduction number, while its effect on the scale of the epidemic exhibits a dual nature, contingent upon infection rate and the degree of subpopulations. Furthermore, when the order of magnitude of the migration rate is below

10^{- 3}

, the scale of the epidemic expands while showing no dependence on the basic reproduction number; and when the order of magnitude of the migration rate exceeds

10^{- 3}

, the scale decreases and displays a negative correlation with the basic reproduction number. Based on the research findings, we propose a systematic disease control framework, which can serve as a strategic reference and practical guide for future infectious disease prevention efforts.

在传统的元种群网络研究中，研究人员通常假设亚种群内的个体混合良好，忽略了个体接触的异质性。然而，在现实生活中，个体接触表现出高度的异质性。为了解决这个问题，我们建立了一个SIS模型，该模型在元种群网络上耦合个体接触异质性，其特征是在亚种群内构建动态子网络。从理论上得到了平衡点的基本繁殖数，证明了平衡点的存在唯一性及其全局稳定性。通过数值模拟，验证了理论结果。此外，研究结果表明，个体接触异质性与基本繁殖数之间存在正相关关系，而其对流行病规模的影响表现出双重性质，取决于感染率和亚种群的程度。当迁移率数量级低于10-3时，疫情规模扩大，但不依赖于基本繁殖数；当迁移速率的数量级超过10-3时，尺度减小，且与基本繁殖数呈负相关。在此基础上，我们提出了一个系统的传染病控制框架，为今后的传染病预防工作提供战略参考和实践指导。

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引用次数: 0

Multiseasonal modeling of pesticide resistance in maize stalk borer 玉米茎秆螟虫农药抗性的多季节模拟。

IF 1.8 4区数学 Q2 BIOLOGY

Mathematical Biosciences

Pub Date : 2025-12-09 DOI: 10.1016/j.mbs.2025.109584

B.S. Tchienkou Tchiengang , I. Tankam Chedjou , I.V. Yatat Djeumen , J.J. Tewa

This study presents a comprehensive approach to modelling the infestation of maize by the maize stalk borer (Busseola fusca) using both chemical control and cultural practices consisting of post-harvest residue management. Two distinct mathematical models are developed: a semi-discrete integro-differential model and a semi-discrete differential model, each addressing different aspects of pest resistance. The integro-differential model captures the dynamics of quantitative resistance, considering resistance as a continuous variable from fully sensitive to fully resistant. The second model, on the other hand, accounts for qualitative resistance by incorporating discrete genetic mutations. Both models consider key factors such as pesticide decay rates, fitness costs associated with resistance, and the impact of integrated pest management (IPM) strategies. Our findings highlight the critical role of fitness costs in delaying resistance development and demonstrate the enhanced effectiveness of IPM techniques over conventional chemical control. This dual-model approach provides a robust framework for designing sustainable pest management practices in agriculture.

本研究提出了一种综合方法来模拟玉米秸秆螟虫（Busseola fusca）的侵染，使用化学控制和包括收获后残留物管理在内的栽培实践。开发了两种不同的数学模型：半离散的积分-微分模型和半离散的微分模型，每个模型都涉及害虫抗性的不同方面。积分-微分模型捕捉了定量电阻的动态，将电阻视为一个从全敏感到全抗性的连续变量。另一方面，第二个模型通过纳入离散的基因突变来解释定性抗性。这两种模型都考虑了关键因素，如农药腐烂率、与抗性相关的适应性成本以及病虫害综合治理（IPM）策略的影响。我们的研究结果强调了适应成本在延迟抗性发展中的关键作用，并证明了IPM技术比传统的化学控制更有效。这种双模式方法为设计可持续的农业有害生物管理做法提供了一个强有力的框架。

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引用次数: 0

Quantifying the risk of long-term chikungunya persistence in Miami-Dade county 量化迈阿密-戴德县基孔肯雅热长期持续的风险。

IF 1.8 4区数学 Q2 BIOLOGY

Mathematical Biosciences

Pub Date : 2025-12-09 DOI: 10.1016/j.mbs.2025.109597

Antonio Gondim, Leonardo Schultz, Xi Huo, Shigui Ruan

Chikungunya virus (CHIKV) is a mosquito-borne arbovirus with the potential to establish sustained transmission in subtropical regions like Florida, where climatic and ecological conditions support vector proliferation. In this study, we develop a Continuous-Time Markov Chain (CTMC) model to assess the probability of long-term CHIKV establishment in Miami-Dade County following repeated introductions of external infectious individuals over a finite period of time. This work aims to identify seasonal windows of heightened endemic risk and evaluate the impact of vector control strategies on mitigating the likelihood of persistent transmission. These results generate insights into the dynamics of CHIKV and inform targeted interventions to prevent its transition from minor sporadic outbreaks to endemic circulation.

基孔肯雅病毒（CHIKV）是一种蚊媒虫媒病毒，有可能在气候和生态条件支持病媒增殖的佛罗里达州等亚热带地区形成持续传播。在这项研究中，我们建立了一个连续时间马尔可夫链（CTMC）模型，以评估在有限时间内反复引入外部感染个体后，迈阿密-戴德县长期建立CHIKV的可能性。这项工作旨在确定流行风险增加的季节窗口，并评估病媒控制战略对减轻持续传播可能性的影响。这些结果有助于深入了解CHIKV的动态，并为有针对性的干预措施提供信息，以防止其从小规模散发疫情转变为地方性流行。

引用次数: 0

Invasions in a four-species fractured cyclic ecological system 四种断裂循环生态系统中的入侵。

IF 1.8 4区数学 Q2 BIOLOGY

Mathematical Biosciences

Pub Date : 2025-12-08 DOI: 10.1016/j.mbs.2025.109590

E.Y. Siegfried , A. Bayliss , V.A. Volpert

We consider the invasion problem for a four-species cyclic ecological community. When the cyclic interspecies competition is stronger than the intraspecies competition (crowding), the system is dominated by two two-species alliances which are the competing entities. We assume that one of the alliances is fractured due to internal competition and predation. The invasion problem can then be reduced to a traveling wave problem and the two alliances will be equally matched under standstill conditions, i.e., when the speed of the traveling wave is zero. We determine the standstill condition and the role of fracturing on standstill in two regimes: (i) balanced competition, when the interspecies competition is comparable to the intraspecies competition, so that there is a significant region where the four species can live together and (ii) strong competition, where species from the two alliances cannot coexist except in a very narrow band. We employ a suitable coordinate transformation for the regime of balanced competition and a suitable linearization for the case of strong competition. In both cases we determine the role of fracturing on standstill conditions. We validate our results with numerical computations.

研究了一个四种循环生态群落的入侵问题。当种间竞争大于种内竞争（拥挤）时，系统由两个两种联盟主导，它们是竞争实体。我们假设其中一个联盟由于内部竞争和掠夺而破裂。入侵问题可以简化为行波问题，在静止条件下，即当行波速度为零时，两个联盟将相等匹配。在两种情况下，我们确定了静止状态和断裂在静止状态中的作用：(i)平衡竞争，当种间竞争与种内竞争相当时，存在四个物种可以共同生活的显著区域；（ii）强竞争，两个联盟的物种除了在非常窄的范围内不能共存。对于平衡竞争，我们采用适当的坐标变换，对于强竞争，我们采用适当的线性化。在这两种情况下，我们都确定了压裂在静止状态下的作用。我们用数值计算验证了我们的结果。

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引用次数: 0

Spreading dynamics of an SVIRS model SVIRS模型的扩散动力学。

IF 1.8 4区数学 Q2 BIOLOGY

Mathematical Biosciences

Pub Date : 2025-12-04 DOI: 10.1016/j.mbs.2025.109569

Guo Lin , Jiantao Lin , Shuxia Pan

The geographic spread of a disease epidemic has long been a key focus of public attention. This article investigates the spreading properties of a reaction-diffusion system, which models the Susceptible-Vaccinated-Infected-Recovered-Susceptible (SVIRS for short) process. Assume that the habitat of the entire population expands or contracts in a wave front pattern. Then we study the corresponding initial value problems and traveling wave solutions, which model the spatial expanding ability of the disease. A constant associated with the disease’s transmission capacity is given, enabling the exploration of practical factors that influence disease spreading. For example, both the vaccination rate and vaccines’ effective protection rate can reduce the spatial transmission capacity of diseases. Moreover, we numerically find that the proportion of recovered individuals who lose their immunity does not affect the spreading ability but changes the prevalence scale.

疾病流行的地理传播长期以来一直是公众关注的焦点。本文研究了一个反应扩散系统的传播特性，该系统模拟了易感-接种-感染-恢复-易感（简称SVIRS）过程。假设整个种群的栖息地以波前模式扩张或收缩。然后研究了相应的初值问题和行波解，模拟了疾病的空间扩展能力。给出了与疾病传播能力相关的常数，从而能够探索影响疾病传播的实际因素。例如，疫苗接种率和疫苗的有效保护率都可以降低疾病的空间传播能力。数值结果表明，丧失免疫的个体所占的比例不影响病毒的传播能力，但改变了病毒的流行规模。

引用次数: 0

Stochastic environments and migrating population dynamics 随机环境与迁徙人口动态。

IF 1.8 4区数学 Q2 BIOLOGY

Mathematical Biosciences

Pub Date : 2025-12-03 DOI: 10.1016/j.mbs.2025.109589

Yogesh Trivedi, Anushaya Mohapatra

Environmental stochasticity is pivotal in shaping population dynamics by introducing random fluctuations in habitat conditions, resource availability, and survival probabilities. These fluctuations often drive critical ecological processes, influencing persistence, extinction, and adaptive strategies. Especially in the context of population migration, environmental stochasticity plays a critical role in shaping movement patterns, survival rates, and population structure. There are various forms of population migration, and among them, partial migration is a widespread phenomenon, where only a portion of the population undertakes seasonal or periodic movements while the rest remain resident in the same area year-round. In this study, we develop discrete time stochastic population models to investigate how environmental fluctuations and disturbances affects partially migrating populations. In one class of models, random fluctuations are incorporated through density-dependent fertility functions, while in another class of models, episodic disturbance events are addressed that reduce migratory populations. By deriving the stochastic growth rate through the dominant Lyapunov exponent, we establish thresholds for population persistence and extinction. Furthermore, we explore the conditions under which partial migration emerges as an evolutionarily stable strategy (ESS) in fluctuating environments and with disturbance events. As an application, we develop our framework to incorporate temperature-dependent fertility functions, analyzing the impact of climate-driven temperature fluctuations on population dynamics. Our findings reveal that environmental stochasticity can either enhance or undermine the persistence of partially migratory populations, depending on the nature of the disturbances and the distribution of environmental variability. Numerical simulations validate these theoretical insights, demonstrating how extreme events, such as climatic shocks, shape migration patterns and population structure. This study advances the understanding of partial migration dynamics, offering a robust framework for predicting population responses to environmental changes in the context of ongoing climate variability.

通过引入栖息地条件、资源可用性和生存概率的随机波动，环境随机性是塑造种群动态的关键。这些波动常常驱动关键的生态过程，影响持久性、灭绝和适应策略。特别是在人口迁移的背景下，环境随机性在形成迁移模式、存活率和人口结构方面起着关键作用。人口移徙有各种形式，其中部分移徙是一种普遍现象，即只有一部分人口进行季节性或周期性迁移，而其余人口全年居住在同一地区。在这项研究中，我们建立了离散时间随机种群模型来研究环境波动和干扰如何影响部分迁移种群。在一类模型中，通过依赖于密度的生育率函数纳入了随机波动，而在另一类模型中，处理了减少迁徙人口的偶发干扰事件。通过优势Lyapunov指数推导随机增长率，我们建立了种群持续和灭绝的阈值。此外，我们探讨了在波动环境和干扰事件中，部分迁移作为一种进化稳定策略（ESS）出现的条件。作为一个应用，我们开发了我们的框架，以纳入温度依赖的生育率函数，分析气候驱动的温度波动对人口动态的影响。我们的研究结果表明，环境随机性可以增强或破坏部分迁移种群的持久性，这取决于干扰的性质和环境变异性的分布。数值模拟验证了这些理论见解，展示了气候冲击等极端事件如何影响迁移模式和人口结构。该研究促进了对部分迁移动力学的理解，为预测持续气候变率背景下种群对环境变化的响应提供了一个强有力的框架。

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引用次数: 0

A compartmental model for epidemiology with human behavior and stochastic effects 具有人类行为和随机效应的流行病学分区模型。

IF 1.8 4区数学 Q2 BIOLOGY

Mathematical Biosciences

Pub Date : 2025-12-02 DOI: 10.1016/j.mbs.2025.109588

Christian Parkinson , Weinan Wang

We propose a compartmental model for epidemiology wherein the population is split into groups with either comply or refuse to comply with protocols designed to slow the spread of a disease. Parallel to the disease spread, we assume that noncompliance with protocols spreads as a social contagion. We begin by deriving the reproductive ratio for a deterministic version of the model, and use this to fully characterize the local stability of disease free equilibrium points. We then append the deterministic model with stochastic effects, specifically assuming that the transmission rate of the disease and the transmission rate of the social contagion are uncertain. We prove global existence and nonnegativity for our stochastic model. Then using suitably constructed stochastic Lyapunov functions, we analyze the behavior of the stochastic system with respect to certain disease free states. We demonstrate all of our results with numerical simulations.

我们提出了一个流行病学的分区模型，其中人群被分成遵守或拒绝遵守旨在减缓疾病传播的协议的群体。与疾病传播平行，我们假设不遵守协议作为一种社会传染病传播。我们首先推导出模型的确定性版本的生殖比率，并用它来充分表征无病平衡点的局部稳定性。然后，我们在确定性模型中加入随机效应，具体假设疾病的传播率和社会传染的传播率是不确定的。证明了随机模型的全局存在性和非负性。然后利用适当构造的随机Lyapunov函数，分析了随机系统在某些无病状态下的行为。我们用数值模拟来证明我们所有的结果。

引用次数: 0

Environmental drivers of tuberculosis transmission in Guangdong, China: Integrating generalized additive models and dynamic simulations 中国广东结核病传播的环境驱动因素：综合广义加性模型和动态模拟。

IF 1.8 4区数学 Q2 BIOLOGY

Mathematical Biosciences

Pub Date : 2025-12-01 DOI: 10.1016/j.mbs.2025.109587

Lingming Kong , Yanying Mo , Guanghu Zhu , Liang Chen , Zhen Wang

Tuberculosis (TB) remains a critical global public health challenge, particularly in high-burden regions like Guangdong Province, China. This study develops an integrated framework combining generalized additive models (GAM) and non-autonomous dynamical modeling to elucidate the synergistic effects of environmental and socioeconomic factors on TB transmission dynamics. Utilizing weekly TB case data, air quality index (AQI), absolute humidity (AH), and holiday indicators from Guangdong (2014–2019), GAM quantified nonlinear lagged effects of environmental exposures (AQI, AH) and aperiodic drivers (holidays) on incidence. Results revealed that a 10-unit increase in AQI elevated TB risk by 3.8 % (95 % CI: 1.2–6.5 %), while AH exhibited a negative regulatory effect on transmission. Holiday-related population aggregation amplified case fluctuations by 37 % (p < .01), with post-holiday rebounds up to 68 %. These time-varying parameters were incorporated into a non-autonomous SEIR model with recurrence mechanisms. The basic reproduction number R₀ was estimated at 1.9 (95 % CI: 1.2–2.6). Bifurcation analysis confirmed global stability of the disease-free equilibrium when R₀ < 1 and endemic persistence when R₀ > 1. Sensitivity analysis identified infection rate and relapse probability as dominant drivers of transmission intensity. The model predicted a declining long-term trend (-2.6 % annually) but persistent winter-spring seasonality. This hybrid approach providing a quantitative tool for optimizing intervention strategies. Key recommendations include reducing airborne pollutants, enhancing surveillance, and targeting relapse prevention to mitigate endemic persistence.

结核病（TB）仍然是一个重大的全球公共卫生挑战，特别是在中国广东省等高负担地区。本研究建立了一个结合广义加性模型（GAM）和非自主动力学模型的集成框架，以阐明环境和社会经济因素对结核病传播动力学的协同效应。利用2014-2019年广东省每周结核病病例数据、空气质量指数（AQI）、绝对湿度（AH）和节假日指标，GAM量化了环境暴露（AQI、AH）和非周期性驱动因素（节假日）对发病率的非线性滞后效应。结果显示，AQI每增加10个单位，结核病风险增加3.8% (95% CI: 1.2-6.5%)，而AH对传播表现出负调节作用。假日相关的人群聚集将病例波动放大了37% (p 0估计为1.9 （95% CI: 1.2-2.6）。分岔分析证实了R0 0 > 1时无病平衡的全局稳定性。敏感性分析确定感染率和复发率是传播强度的主要驱动因素。该模型预测了下降的长期趋势（每年-2.6%），但持续的冬春季节。这种混合方法为优化干预策略提供了定量工具。主要建议包括减少空气污染物、加强监测和以预防复发为目标，以减轻流行病的持久性。

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引用次数: 0

New results on traveling wave solutions for a Keller-Segel system with nonlinear chemical gradient 具有非线性化学梯度的Keller-Segel体系行波解的新结果。

IF 1.8 4区数学 Q2 BIOLOGY

Mathematical Biosciences

Pub Date : 2025-11-30 DOI: 10.1016/j.mbs.2025.109586

Shangbing Ai , Jianhe Shen

<div><div>We study a Keller-Segel system with nonlinear chemical gradient and two parameters <em>c</em> > 0 and ε > 0. The system has a family of equilibria <span><math><mrow><mo>(</mo><msup><mi>u</mi><mo>*</mo></msup><mo>,</mo><msubsup><mi>v</mi><mrow><mo>±</mo></mrow><mo>*</mo></msubsup><mo>)</mo></mrow></math></span> with <span><math><mrow><msubsup><mi>u</mi><mn>1</mn><mo>*</mo></msubsup><mo><</mo><msup><mi>u</mi><mo>*</mo></msup><mo><</mo><msubsup><mi>u</mi><mn>2</mn><mo>*</mo></msubsup></mrow></math></span> where <span><math><msubsup><mi>u</mi><mn>1</mn><mo>*</mo></msubsup></math></span> and <span><math><msubsup><mi>u</mi><mn>2</mn><mo>*</mo></msubsup></math></span> are explicitly defined. We investigate the existence of traveling wave solutions <span><math><mrow><mo>(</mo><msubsup><mi>u</mi><mi>s</mi><mrow><mi>c</mi><mo>,</mo><mrow><mi>ε</mi></mrow></mrow></msubsup><mrow><mo>(</mo><mi>x</mi><mo>−</mo><mi>s</mi><mi>t</mi><mo>)</mo></mrow><mo>,</mo><msubsup><mi>u</mi><mi>s</mi><mrow><mi>c</mi><mo>,</mo><mrow><mi>ε</mi></mrow></mrow></msubsup><mrow><mo>(</mo><mi>x</mi><mo>−</mo><mi>s</mi><mi>t</mi><mo>)</mo></mrow><mo>)</mo></mrow></math></span> of this system that connect a pair of equilibria <span><math><mrow><mo>(</mo><msup><mi>u</mi><mo>*</mo></msup><mo>,</mo><msubsup><mi>v</mi><mrow><mo>±</mo></mrow><mo>*</mo></msubsup><mo>)</mo></mrow></math></span> and establish the following result: there exists <span><math><msubsup><mi>u</mi><mn>0</mn><mo>*</mo></msubsup></math></span> with <span><math><mrow><msubsup><mi>u</mi><mn>1</mn><mo>*</mo></msubsup><mo><</mo><msubsup><mi>u</mi><mn>0</mn><mo>*</mo></msubsup><mo><</mo><msubsup><mi>u</mi><mn>2</mn><mo>*</mo></msubsup></mrow></math></span> such that if <span><math><mrow><msubsup><mi>u</mi><mn>1</mn><mo>*</mo></msubsup><mo><</mo><msup><mi>u</mi><mo>*</mo></msup><mo>≤</mo><msubsup><mi>u</mi><mn>0</mn><mo>*</mo></msubsup></mrow></math></span>, then <span><math><mrow><mo>(</mo><msubsup><mi>u</mi><mi>s</mi><mrow><mi>c</mi><mo>,</mo><mrow><mi>ε</mi></mrow></mrow></msubsup><mo>,</mo><msubsup><mi>v</mi><mi>s</mi><mrow><mi>c</mi><mo>,</mo><mrow><mi>ε</mi></mrow></mrow></msubsup><mo>)</mo></mrow></math></span> exists for all <em>c</em> > 0 and <em>s</em> > 0 and sufficiently small ε > 0; if <span><math><mrow><msubsup><mi>u</mi><mn>0</mn><mo>*</mo></msubsup><mo><</mo><msup><mi>u</mi><mo>*</mo></msup><mo><</mo><msubsup><mi>u</mi><mn>2</mn><mo>*</mo></msubsup></mrow></math></span>, then there exists <em>c</em>* > 0 such that for <em>c</em> > <em>c</em>*, such a solution exists for all <em>s</em> > 0 and sufficiently small ε, while for 0 < <em>c</em> ≤ <em>c</em>*, such a solution exists only for <em>s</em> in a disconnected set of (0, ∞) which includes two connected components <span><math><mrow><mo>(</mo><mn>0</mn><mo>,</mo><msubsup><mi>s</mi><mi>c</mi><mo>*</mo></msubsup><mo>)</mo></mrow></math></span> and <span><math><mrow><mo>(</mo><msub><mover><mi>s</mi><mo>^

研究了一类具有非线性化学梯度的Keller-Segel体系。该系统包含两个参数c > 0和ε > 0，并具有一个具有u1**2*的平衡族（u*,v±*），其中u1*和u2*是显式定义的。研究了连接平衡对（u*,v±*）的行波解(usc,ε(x-st),usc,ε(x-st))，得到了以下存在性结果：存在u0*与u1*0*2*，使得当u1**≤u0*时，则（usc,ε,vsc,ε）对所有c > 0和s > 0均存在，且ε > 0足够小；如果u0**2*，则存在c* > 0，使得对于c ≥ c*，对于所有s > 0和足够小的ε存在这样的解，而对于0 c*)和（s^c,∞），其中0c*cc*在c中增加，s^c在c中减少，[公式：见文]和[公式：见文]。这个结果概括了参考文献[J]中的主要结果。数学。分析的。达成。[j], 545(2025), 129128]，其中研究了系统中c=ε的特殊情况。

引用次数: 0

Early-stage invasion and spreading speed in a resource-dependent dispersal model 资源依赖扩散模型中的早期入侵和扩散速度。

IF 1.8 4区数学 Q2 BIOLOGY

Mathematical Biosciences

Pub Date : 2025-11-28 DOI: 10.1016/j.mbs.2025.109585

Jean-Baptiste Burie , Arnaud Ducrot , Ousmane Seydi

In this paper, we study the dynamics of biological invasion through complementary modeling frameworks in the context of nonlocal resource-driven dispersal. During the very early stage of invasion, when only a few individuals are present, demographic variability is crucial: extinction may occur even under favorable average conditions. To capture this, we use a branching-process approximation that provides explicit formulas for extinction probabilities, survival conditions, and mean extinction times. At larger scales and higher densities, invasion is described by a deterministic system of nonlinear integro-differential equations. For this system, we establish well-posedness and derive lower and upper bounds on the asymptotic spreading speed. A unifying threshold parameter

T_{0}

, defined as the spectral radius of a next-generation operator, characterizes invasion outcomes: if

T_{0} \leq 1

, extinction occurs; if

T_{0} > 1

, the invader persists and spreads. Importantly, the threshold derived from the early-stage approximation coincides with that of the deterministic model, thus providing a consistent criterion for invasion success. Finally, numerical simulations illustrate the transition between extinction and persistence and highlight how resource-driven dispersal shapes invasion speed.

在本文中，我们通过互补建模框架研究了非局部资源驱动扩散背景下的生物入侵动力学。在入侵的早期阶段，只有少数个体存在，人口变化是至关重要的：即使在有利的平均条件下，灭绝也可能发生。为了捕捉这一点，我们使用了一个分支过程近似，该近似提供了灭绝概率、生存条件和平均灭绝时间的明确公式。在更大的尺度和更高的密度下，入侵用非线性积分微分方程的确定性系统来描述。对于该系统，我们建立了系统的适定性，并推导了系统渐近扩散速度的下界和上界。一个统一的阈值参数T0，定义为下一代算子的频谱半径，表征入侵结果：如果T0≤1，发生灭绝；如果为0，则入侵者持续存在并扩散。重要的是，从早期近似得到的阈值与确定性模型的阈值一致，从而为入侵成功提供了一致的标准。最后，数值模拟说明了灭绝和持久之间的过渡，并强调了资源驱动的扩散如何影响入侵速度。

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引用次数: 0