SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 808-830, June 2024. Abstract. We study waves on infinite one-dimensional lattices of particles that each interact with all others through power-law forces [math]. The inverse-cube case corresponds to Calogero–Moser systems which are well known to be completely integrable for any finite number of particles. The formal long-wave limit for unidirectional waves in these lattices is the Korteweg–de Vries equation if [math], but with [math] it is a nonlocal dispersive PDE that reduces to the Benjamin–Ono equation for [math]. For the infinite Calogero–Moser lattice, we find explicit formulas that describe solitary and periodic traveling waves.
{"title":"On Long Waves and Solitons in Particle Lattices with Forces of Infinite Range","authors":"Benjamin Ingimarson, Robert L. Pego","doi":"10.1137/23m1607209","DOIUrl":"https://doi.org/10.1137/23m1607209","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 808-830, June 2024. <br/> Abstract. We study waves on infinite one-dimensional lattices of particles that each interact with all others through power-law forces [math]. The inverse-cube case corresponds to Calogero–Moser systems which are well known to be completely integrable for any finite number of particles. The formal long-wave limit for unidirectional waves in these lattices is the Korteweg–de Vries equation if [math], but with [math] it is a nonlocal dispersive PDE that reduces to the Benjamin–Ono equation for [math]. For the infinite Calogero–Moser lattice, we find explicit formulas that describe solitary and periodic traveling waves.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140833362","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 783-807, June 2024. Abstract. Evidence shows that resource quality can determine the costs and benefits of the fear effect on consumer dynamics. However, mechanistic modeling and analysis are lacking. This paper formulates a tritrophic level food chain model that integrates both stoichiometric food quality and fear effect. We establish the well-posedness of the model and examine the existence and stability of equilibria. Through extensive numerical simulations, we validate our findings and visually explore the interactive effects of fear and food quality. Our results reveal that the fear effect from predators stabilizes the system. Furthermore, we demonstrate that the fear effect amplifies the influence of food quality on consumers. When food quality is favorable, the fear effect enhances consumer production efficiency, whereas, in the case of poor food quality, the fear effect exacerbates the decline in production efficiency caused by low-nutrient food.
{"title":"Stoichiometry-Dependent Fear Effect in a Food Chain Model","authors":"Tianxu Wang, Hao Wang","doi":"10.1137/23m1581613","DOIUrl":"https://doi.org/10.1137/23m1581613","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 783-807, June 2024. <br/>Abstract. Evidence shows that resource quality can determine the costs and benefits of the fear effect on consumer dynamics. However, mechanistic modeling and analysis are lacking. This paper formulates a tritrophic level food chain model that integrates both stoichiometric food quality and fear effect. We establish the well-posedness of the model and examine the existence and stability of equilibria. Through extensive numerical simulations, we validate our findings and visually explore the interactive effects of fear and food quality. Our results reveal that the fear effect from predators stabilizes the system. Furthermore, we demonstrate that the fear effect amplifies the influence of food quality on consumers. When food quality is favorable, the fear effect enhances consumer production efficiency, whereas, in the case of poor food quality, the fear effect exacerbates the decline in production efficiency caused by low-nutrient food.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140833330","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 756-781, April 2024. Abstract. We study nematic configurations within three-dimensional (3D) cuboids, with planar degenerate boundary conditions on the cuboid faces, in the Landau–de Gennes framework. There are two geometry-dependent variables: the edge length of the square cross-section, [math], and the parameter [math], which is a measure of the cuboid height. Theoretically, we prove the existence and uniqueness of the global minimizer with a small enough cuboid size. We develop a new numerical scheme for the high-index saddle dynamics to deal with the surface energies. We report on a plethora of (meta)stable states, and their dependence on [math] and [math], and in particular how the 3D states are connected with their two-dimensional counterparts on squares and rectangles. Notably, we find families of almost uniaxial stable states constructed from the topological classification of tangent unit-vector fields and study transition pathways between them. We also provide a phase diagram of competing (meta)stable states as a function of [math] and [math].
{"title":"Multistability for Nematic Liquid Crystals in Cuboids with Degenerate Planar Boundary Conditions","authors":"Baoming Shi, Yucen Han, Apala Majumdar, Lei Zhang","doi":"10.1137/23m1604606","DOIUrl":"https://doi.org/10.1137/23m1604606","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 756-781, April 2024. <br/> Abstract. We study nematic configurations within three-dimensional (3D) cuboids, with planar degenerate boundary conditions on the cuboid faces, in the Landau–de Gennes framework. There are two geometry-dependent variables: the edge length of the square cross-section, [math], and the parameter [math], which is a measure of the cuboid height. Theoretically, we prove the existence and uniqueness of the global minimizer with a small enough cuboid size. We develop a new numerical scheme for the high-index saddle dynamics to deal with the surface energies. We report on a plethora of (meta)stable states, and their dependence on [math] and [math], and in particular how the 3D states are connected with their two-dimensional counterparts on squares and rectangles. Notably, we find families of almost uniaxial stable states constructed from the topological classification of tangent unit-vector fields and study transition pathways between them. We also provide a phase diagram of competing (meta)stable states as a function of [math] and [math].","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140805645","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 732-755, April 2024. Abstract. We study a diffusive Susceptible-Infected-Susceptible (SIS) epidemic model with the mass-action transmission mechanism and show, under appropriate assumptions on the parameters, the existence of multiple endemic equilibria (EE). Our results answer some open questions on previous studies related to disease extinction or persistence when [math] and the multiplicity of EE solutions when [math]. Interestingly, even with such a simple nonlinearity induced by the mass-action, we show that the diffusive epidemic model may have an S-shaped or backward bifurcation curve of EE solutions. This strongly highlights the impacts of environmental heterogeneity on the spread of infectious diseases as the basic reproduction number alone is insufficient as a threshold quantity to predict its extinction. Our results also shed some light on the significance of disease transmission mechanisms.
SIAM 应用数学杂志》第 84 卷第 2 期第 732-755 页，2024 年 4 月。 摘要。我们研究了一个具有大规模作用传播机制的扩散性易感-感染-易感（SIS）流行病模型，并证明在适当的参数假设下，存在多个流行病均衡（EE）。我们的结果回答了以往研究中的一些公开问题，这些问题涉及当[math]时疾病的灭绝或持续以及当[math]时 EE 解的多重性。有趣的是，即使由质量作用引起的非线性如此简单，我们仍然发现扩散流行病模型的 EE 解可能呈 S 形或向后分叉曲线。这有力地凸显了环境异质性对传染病传播的影响，因为仅凭基本繁殖数量不足以作为预测其灭绝的临界量。我们的研究结果还揭示了疾病传播机制的重要性。
{"title":"Multiplicity of Endemic Equilibria for a Diffusive SIS Epidemic Model with Mass-Action","authors":"Keoni Castellano, Rachidi B. Salako","doi":"10.1137/23m1613888","DOIUrl":"https://doi.org/10.1137/23m1613888","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 732-755, April 2024. <br/> Abstract. We study a diffusive Susceptible-Infected-Susceptible (SIS) epidemic model with the mass-action transmission mechanism and show, under appropriate assumptions on the parameters, the existence of multiple endemic equilibria (EE). Our results answer some open questions on previous studies related to disease extinction or persistence when [math] and the multiplicity of EE solutions when [math]. Interestingly, even with such a simple nonlinearity induced by the mass-action, we show that the diffusive epidemic model may have an S-shaped or backward bifurcation curve of EE solutions. This strongly highlights the impacts of environmental heterogeneity on the spread of infectious diseases as the basic reproduction number alone is insufficient as a threshold quantity to predict its extinction. Our results also shed some light on the significance of disease transmission mechanisms.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140613108","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 710-731, April 2024. Abstract. The creation of hairpin or lambda vortices, typical for the early stages of the laminar-turbulent transition process in various boundary layer flows, in some sense may be associated with blow-up solutions of the Fisher–Kolmogorov–Petrovsky–Piskunov equation. In contrast to the usual applications of this nonlinear evolution equation of the reaction-diffusion type, the solution quantity in the present context needs to stay neither bounded nor positive. We focus on the solution behavior beyond a finite-time point blow-up event, which consists of two moving singularities (representing the cores of the vortex legs) propagating in opposite directions, and their initial motion is determined with the method of matched asymptotic expansions. After resolving subtleties concerning the transition between logarithmic and algebraic expansion terms regarding asymptotic layers, we find that the internal singularity structure resembles a combination of second- and first-order poles in the form of a singular traveling wave with a time-dependent speed imprinted through the characteristics of the preceding blow-up event.
{"title":"Moving Singularities of the Forced Fisher–KPP Equation: An Asymptotic Approach","authors":"Markus Kaczvinszki, Stefan Braun","doi":"10.1137/23m1552905","DOIUrl":"https://doi.org/10.1137/23m1552905","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 710-731, April 2024. <br/> Abstract. The creation of hairpin or lambda vortices, typical for the early stages of the laminar-turbulent transition process in various boundary layer flows, in some sense may be associated with blow-up solutions of the Fisher–Kolmogorov–Petrovsky–Piskunov equation. In contrast to the usual applications of this nonlinear evolution equation of the reaction-diffusion type, the solution quantity in the present context needs to stay neither bounded nor positive. We focus on the solution behavior beyond a finite-time point blow-up event, which consists of two moving singularities (representing the cores of the vortex legs) propagating in opposite directions, and their initial motion is determined with the method of matched asymptotic expansions. After resolving subtleties concerning the transition between logarithmic and algebraic expansion terms regarding asymptotic layers, we find that the internal singularity structure resembles a combination of second- and first-order poles in the form of a singular traveling wave with a time-dependent speed imprinted through the characteristics of the preceding blow-up event.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140593502","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 687-709, April 2024. Abstract. This paper is concerned with the stability estimates for inverse source problems of the stochastic Helmholtz equation driven by white noise. The well-posedness is established for the direct source problems, which ensures the existence and uniqueness of solutions. The stability estimates are deduced for the inverse source problems, which aim to determine the strength of the random source. To enhance the stability of the inverse source problems, we incorporate a priori information regarding the regularity and support of the strength. In the case of homogeneous media, a Hölder stability estimate is established, providing a quantitative measure of the stability for reconstructing the source strength. For the more challenging scenario of inhomogeneous media, a logarithmic stability estimate is presented, capturing the intricate interactions between the source and the varying medium properties.
{"title":"Stability for Inverse Source Problems of the Stochastic Helmholtz Equation with a White Noise","authors":"Peijun Li, Ying Liang","doi":"10.1137/23m1586331","DOIUrl":"https://doi.org/10.1137/23m1586331","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 687-709, April 2024. <br/> Abstract. This paper is concerned with the stability estimates for inverse source problems of the stochastic Helmholtz equation driven by white noise. The well-posedness is established for the direct source problems, which ensures the existence and uniqueness of solutions. The stability estimates are deduced for the inverse source problems, which aim to determine the strength of the random source. To enhance the stability of the inverse source problems, we incorporate a priori information regarding the regularity and support of the strength. In the case of homogeneous media, a Hölder stability estimate is established, providing a quantitative measure of the stability for reconstructing the source strength. For the more challenging scenario of inhomogeneous media, a logarithmic stability estimate is presented, capturing the intricate interactions between the source and the varying medium properties.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140593599","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 621-631, April 2024. Abstract. DEM simulations by Chialvo et al. observed three distinct flow regimes for homogeneous simple shear of soft, frictional, noncohesive spheres in different domains of shear rate and density. This paper shows that all three regimes can be accommodated in a continuum description, using the CIDR formalism.
{"title":"Constitutive Relations for Granular Flow That Include the Three Flow Regimes of Chialvo et al.","authors":"David G. Schaeffer, Yuhao Hu","doi":"10.1137/23m1578097","DOIUrl":"https://doi.org/10.1137/23m1578097","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 621-631, April 2024. <br/> Abstract. DEM simulations by Chialvo et al. observed three distinct flow regimes for homogeneous simple shear of soft, frictional, noncohesive spheres in different domains of shear rate and density. This paper shows that all three regimes can be accommodated in a continuum description, using the CIDR formalism.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140593479","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Panagiotis Kaklamanos, Andrea Pugliese, Mattia Sensi, Sara Sottile
SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 661-686, April 2024. Abstract. We propose a compartmental model for a disease with temporary immunity and secondary infections. From our assumptions on the parameters involved in the model, the system naturally evolves in three time scales. We characterize the equilibria of the system and analyze their stability. We find conditions for the existence of two endemic equilibria for some cases in which [math]. Then, we unravel the interplay of the three time scales, providing conditions to foresee whether the system evolves in all three scales, or only in the fast and the intermediate ones. We conclude with numerical simulations and bifurcation analysis to complement our analytical results.
{"title":"A Geometric Analysis of the SIRS Model with Secondary Infections","authors":"Panagiotis Kaklamanos, Andrea Pugliese, Mattia Sensi, Sara Sottile","doi":"10.1137/23m1565632","DOIUrl":"https://doi.org/10.1137/23m1565632","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 661-686, April 2024. <br/> Abstract. We propose a compartmental model for a disease with temporary immunity and secondary infections. From our assumptions on the parameters involved in the model, the system naturally evolves in three time scales. We characterize the equilibria of the system and analyze their stability. We find conditions for the existence of two endemic equilibria for some cases in which [math]. Then, we unravel the interplay of the three time scales, providing conditions to foresee whether the system evolves in all three scales, or only in the fast and the intermediate ones. We conclude with numerical simulations and bifurcation analysis to complement our analytical results.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140593192","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 632-660, April 2024. Abstract. In this paper, we are interested in the spatiotemporal pattern formations and bifurcations for a nondegenerate reaction-diffusion rabies SIR model which was used to explain the epidemiological patterns observed in Europe. First, by using the iteration methods, we are able to show the global existence and boundedness of in-time solutions of the parabolic system. Second, for the ODEs, we analytically prove the phenomena observed by Anderson et al. [Nature, 289 (1981), pp. 765–771]: if the carrying capacity [math] is smaller than some positive [math], then rabies eventually dies out; if [math] is larger than [math], then the rabies prevails. Moreover, if [math] for some positive [math], then the endemic equilibrium solution is (locally asymptotically) stable, while it is unstable if [math]. In particular, at [math], the loss of the stability of the endemic equilibrium leads to a Hopf bifurcation. Finally, for the PDEs, we derive sufficient conditions on the diffusion rates so that under these conditions, Turing instability of both the endemic equilibrium solution and the Hopf bifurcating spatially homogeneous periodic solutions can occur. Once Turing instability of the solution (equilibrium or periodic solution) occurs, it is observed numerically that the system might have new spatiotemporal patterns.
{"title":"Dynamics and Bifurcations in a Nondegenerate Homogeneous Diffusive SIR Rabies Model","authors":"Gaoyang She, Fengqi Yi","doi":"10.1137/23m159055x","DOIUrl":"https://doi.org/10.1137/23m159055x","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 632-660, April 2024. <br/> Abstract. In this paper, we are interested in the spatiotemporal pattern formations and bifurcations for a nondegenerate reaction-diffusion rabies SIR model which was used to explain the epidemiological patterns observed in Europe. First, by using the iteration methods, we are able to show the global existence and boundedness of in-time solutions of the parabolic system. Second, for the ODEs, we analytically prove the phenomena observed by Anderson et al. [Nature, 289 (1981), pp. 765–771]: if the carrying capacity [math] is smaller than some positive [math], then rabies eventually dies out; if [math] is larger than [math], then the rabies prevails. Moreover, if [math] for some positive [math], then the endemic equilibrium solution is (locally asymptotically) stable, while it is unstable if [math]. In particular, at [math], the loss of the stability of the endemic equilibrium leads to a Hopf bifurcation. Finally, for the PDEs, we derive sufficient conditions on the diffusion rates so that under these conditions, Turing instability of both the endemic equilibrium solution and the Hopf bifurcating spatially homogeneous periodic solutions can occur. Once Turing instability of the solution (equilibrium or periodic solution) occurs, it is observed numerically that the system might have new spatiotemporal patterns.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140593808","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 602-620, April 2024. Abstract. Cover times measure the speed of exhaustive searches which require the exploration of an entire spatial region(s). Applications include the immune system hunting pathogens, animals collecting food, robotic demining or cleaning, and computer search algorithms. Mathematically, a cover time is the first time a random searcher(s) comes within a specified “detection radius” of every point in the target region (often the entire spatial domain). Due to their many applications and their fundamental probabilistic importance, cover times have been extensively studied in the physics and probability literatures. This prior work has generally studied cover times of a single searcher with a vanishing detection radius or a large target region. This prior work has further claimed that cover times for multiple searchers can be estimated by a simple rescaling of the cover time of a single searcher. In this paper, we study cover times of many diffusive or subdiffusive searchers and show that prior estimates break down as the number of searchers grows. We prove a rather universal formula for all the moments of such cover times in the many searcher limit that depends only on (i) the searcher’s characteristic (sub)diffusivity and (ii) a certain geodesic distance between the searcher starting location(s) and the farthest point in the target. This formula is otherwise independent of the detection radius, space dimension, target size, and domain size. We illustrate our results in several examples and compare them to detailed stochastic simulations.
{"title":"Cover Times of Many Diffusive or Subdiffusive Searchers","authors":"Hyunjoong Kim, Sean D. Lawley","doi":"10.1137/23m1576645","DOIUrl":"https://doi.org/10.1137/23m1576645","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 602-620, April 2024. <br/> Abstract. Cover times measure the speed of exhaustive searches which require the exploration of an entire spatial region(s). Applications include the immune system hunting pathogens, animals collecting food, robotic demining or cleaning, and computer search algorithms. Mathematically, a cover time is the first time a random searcher(s) comes within a specified “detection radius” of every point in the target region (often the entire spatial domain). Due to their many applications and their fundamental probabilistic importance, cover times have been extensively studied in the physics and probability literatures. This prior work has generally studied cover times of a single searcher with a vanishing detection radius or a large target region. This prior work has further claimed that cover times for multiple searchers can be estimated by a simple rescaling of the cover time of a single searcher. In this paper, we study cover times of many diffusive or subdiffusive searchers and show that prior estimates break down as the number of searchers grows. We prove a rather universal formula for all the moments of such cover times in the many searcher limit that depends only on (i) the searcher’s characteristic (sub)diffusivity and (ii) a certain geodesic distance between the searcher starting location(s) and the farthest point in the target. This formula is otherwise independent of the detection radius, space dimension, target size, and domain size. We illustrate our results in several examples and compare them to detailed stochastic simulations.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140593465","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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