Pub Date : 2024-11-26DOI: 10.1016/j.tpb.2024.10.003
Ananda Shikhara Bhat
Infinitely many distinct trait values may arise in populations bearing quantitative traits, and modelling their population dynamics is thus a formidable task. While classical models assume fixed or infinite population size, models in which the total population size fluctuates due to demographic noise in births and deaths can behave qualitatively differently from constant or infinite population models due to density-dependent dynamics. In this paper, I present a stochastic field theory for the eco-evolutionary dynamics of finite populations bearing one-dimensional quantitative traits. I derive stochastic field equations that describe the evolution of population densities, trait frequencies, and the mean value of any trait in the population. These equations recover well-known results such as the replicator-mutator equation, Price equation, and gradient dynamics in the infinite population limit. For finite populations, the equations describe the intricate interplay between natural selection, noise-induced selection, eco-evolutionary feedback, and neutral genetic drift in determining evolutionary trajectories. My methods use ideas from statistical physics, calculus of variations, and SPDEs, providing alternative methods that complement the measure-theoretic martingale approach that is more common in the literature.
{"title":"A stochastic field theory for the evolution of quantitative traits in finite populations.","authors":"Ananda Shikhara Bhat","doi":"10.1016/j.tpb.2024.10.003","DOIUrl":"https://doi.org/10.1016/j.tpb.2024.10.003","url":null,"abstract":"<p><p>Infinitely many distinct trait values may arise in populations bearing quantitative traits, and modelling their population dynamics is thus a formidable task. While classical models assume fixed or infinite population size, models in which the total population size fluctuates due to demographic noise in births and deaths can behave qualitatively differently from constant or infinite population models due to density-dependent dynamics. In this paper, I present a stochastic field theory for the eco-evolutionary dynamics of finite populations bearing one-dimensional quantitative traits. I derive stochastic field equations that describe the evolution of population densities, trait frequencies, and the mean value of any trait in the population. These equations recover well-known results such as the replicator-mutator equation, Price equation, and gradient dynamics in the infinite population limit. For finite populations, the equations describe the intricate interplay between natural selection, noise-induced selection, eco-evolutionary feedback, and neutral genetic drift in determining evolutionary trajectories. My methods use ideas from statistical physics, calculus of variations, and SPDEs, providing alternative methods that complement the measure-theoretic martingale approach that is more common in the literature.</p>","PeriodicalId":49437,"journal":{"name":"Theoretical Population Biology","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142751658","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"生物学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-08DOI: 10.1016/j.tpb.2024.10.002
Mahdi Salehzadeh, John M. Stockie, Ailene MacPherson
Tree-killing bark beetle infestations are a cause of massive coniferous forest mortality impacting forest ecosystems and the ecosystem services they provide. Models predicting bark beetle outbreaks are crucial for forest management and conservation, necessitating studies of the effect of epidemiological traits on the probability and severity of outbreaks. Due to the aggregation behaviour of beetles and host tree defence, this epidemiological interaction is highly non-linear and outbreak behaviour remains poorly understood, motivating questions about when an outbreak can occur, what determines outbreak severity, and how aggregation behaviour modulates these quantities. Here, we apply the principle of distributed delays to create a novel and mathematically tractable model for beetle aggregation in an epidemiological framework. We derive the critical outbreak threshold for the beetle emergence rate, which is a quantity analogous to the basic reproductive ratio, , for epidemics. Beetle aggregation qualitatively impacts outbreak potential from depending on the emergence rate alone in the absence of aggregation to depending on both emergence rate and initial beetle density when aggregation is required. Finally, we use a stochastic model to confirm that our deterministic model predictions are robust in finite populations.
{"title":"Aggregation unveiled: A sequential modelling approach to bark beetle outbreaks","authors":"Mahdi Salehzadeh, John M. Stockie, Ailene MacPherson","doi":"10.1016/j.tpb.2024.10.002","DOIUrl":"10.1016/j.tpb.2024.10.002","url":null,"abstract":"<div><div>Tree-killing bark beetle infestations are a cause of massive coniferous forest mortality impacting forest ecosystems and the ecosystem services they provide. Models predicting bark beetle outbreaks are crucial for forest management and conservation, necessitating studies of the effect of epidemiological traits on the probability and severity of outbreaks. Due to the aggregation behaviour of beetles and host tree defence, this epidemiological interaction is highly non-linear and outbreak behaviour remains poorly understood, motivating questions about when an outbreak can occur, what determines outbreak severity, and how aggregation behaviour modulates these quantities. Here, we apply the principle of distributed delays to create a novel and mathematically tractable model for beetle aggregation in an epidemiological framework. We derive the critical outbreak threshold for the beetle emergence rate, which is a quantity analogous to the basic reproductive ratio, <span><math><msub><mrow><mi>R</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>, for epidemics. Beetle aggregation qualitatively impacts outbreak potential from depending on the emergence rate alone in the absence of aggregation to depending on both emergence rate and initial beetle density when aggregation is required. Finally, we use a stochastic model to confirm that our deterministic model predictions are robust in finite populations.</div></div>","PeriodicalId":49437,"journal":{"name":"Theoretical Population Biology","volume":"160 ","pages":"Pages 62-69"},"PeriodicalIF":1.2,"publicationDate":"2024-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142631407","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"生物学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-23DOI: 10.1016/j.tpb.2024.10.001
Alejandra Avalos-Pacheco , Mathias C. Cronjäger , Paul A. Jenkins , Jotun Hein
Motivation:
A main challenge in molecular evolution is to find computationally efficient mutation models with flexible assumptions that properly reflect genetic variation. The infinite sites model assumes that each mutation event occurs at a site never previously mutant, i.e. it does not allow recurrent mutations. This is reasonable for low mutation rates and makes statistical inference much more tractable. However, recurrent mutations are common enough to be observable from genetic variation data, even in species with low per-site mutation rates such as humans. The finite sites model on the other hand allows for recurrent mutations but is computationally unfeasible to work with in most cases. In this work, we bridge these two approaches by developing a novel molecular evolution model, the almost infinite sites model, that both admits recurrent mutations and is tractable. We provide a recursive characterization of the likelihood of our proposed model under complete linkage and outline a parsimonious approximation scheme for computing it.
Results:
We show the usefulness of our model in simulated and human mitochondrial data. Our results show that the AISM, in combination with a constraint on the total number of mutation events, can recover accurate approximations to the maximum likelihood estimator of the mutation rate.
Availability and implementation:
An implementation of our model is freely available along with code for reproducing our computational experiments at https://github.com/Cronjaeger/almost-infinite-sites-recursions.
{"title":"An almost infinite sites model","authors":"Alejandra Avalos-Pacheco , Mathias C. Cronjäger , Paul A. Jenkins , Jotun Hein","doi":"10.1016/j.tpb.2024.10.001","DOIUrl":"10.1016/j.tpb.2024.10.001","url":null,"abstract":"<div><h3>Motivation:</h3><div>A main challenge in molecular evolution is to find computationally efficient mutation models with flexible assumptions that properly reflect genetic variation. The infinite sites model assumes that each mutation event occurs at a site never previously mutant, i.e. it does not allow recurrent mutations. This is reasonable for low mutation rates and makes statistical inference much more tractable. However, recurrent mutations are common enough to be observable from genetic variation data, even in species with low per-site mutation rates such as humans. The finite sites model on the other hand allows for recurrent mutations but is computationally unfeasible to work with in most cases. In this work, we bridge these two approaches by developing a novel molecular evolution model, the almost infinite sites model, that both admits recurrent mutations and is tractable. We provide a recursive characterization of the likelihood of our proposed model under complete linkage and outline a parsimonious approximation scheme for computing it.</div></div><div><h3>Results:</h3><div>We show the usefulness of our model in simulated and human mitochondrial data. Our results show that the AISM, in combination with a constraint on the total number of mutation events, can recover accurate approximations to the maximum likelihood estimator of the mutation rate.</div></div><div><h3>Availability and implementation:</h3><div>An implementation of our model is freely available along with code for reproducing our computational experiments at <span><span>https://github.com/Cronjaeger/almost-infinite-sites-recursions</span><svg><path></path></svg></span>.</div></div>","PeriodicalId":49437,"journal":{"name":"Theoretical Population Biology","volume":"160 ","pages":"Pages 49-61"},"PeriodicalIF":1.2,"publicationDate":"2024-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142511627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"生物学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-09DOI: 10.1016/j.tpb.2024.09.001
Léonard Dekens
Specialist species thriving under specific environmental conditions in narrow geographic ranges are widely recognized as heavily threatened by climate deregulation. Many might rely on both their potential to adapt and to disperse towards a refugium to avoid extinction. It is thus crucial to understand the influence of environmental conditions on the unfolding process of adaptation. Here, I study the eco-evolutionary dynamics of a sexually reproducing specialist species in a two-patch quantitative genetic model with moving optima. Thanks to a separation of ecological and evolutionary time scales and the phase-line study of the selection gradient, I derive the critical environmental speed for persistence, which reflects how the existence of a refugium impacts extinction patterns and how it relates to the cost of dispersal. Moreover, the analysis provides key insights about the dynamics that arise on the path towards this refugium. I show that after an initial increase of population size, there exists a critical environmental speed above which the species crosses a tipping point, resulting into an abrupt habitat switch. In addition, when selection for local adaptation is strong, this habitat switch passes through an evolutionary “death valley”, leading to a phenomenon related to evolutionary rescue, which can promote extinction for lower environmental speeds than the critical one.
{"title":"Sharp habitat shifts, evolutionary tipping points and rescue: Quantifying the perilous path of a specialist species towards a refugium in a changing environment","authors":"Léonard Dekens","doi":"10.1016/j.tpb.2024.09.001","DOIUrl":"10.1016/j.tpb.2024.09.001","url":null,"abstract":"<div><div>Specialist species thriving under specific environmental conditions in narrow geographic ranges are widely recognized as heavily threatened by climate deregulation. Many might rely on both their potential to adapt and to disperse towards a refugium to avoid extinction. It is thus crucial to understand the influence of environmental conditions on the unfolding process of adaptation. Here, I study the eco-evolutionary dynamics of a sexually reproducing specialist species in a two-patch quantitative genetic model with moving optima. Thanks to a separation of ecological and evolutionary time scales and the phase-line study of the selection gradient, I derive the critical environmental speed for persistence, which reflects how the existence of a refugium impacts extinction patterns and how it relates to the cost of dispersal. Moreover, the analysis provides key insights about the dynamics that arise on the path towards this refugium. I show that after an initial increase of population size, there exists a critical environmental speed above which the species crosses a tipping point, resulting into an abrupt habitat switch. In addition, when selection for local adaptation is strong, this habitat switch passes through an evolutionary “death valley”, leading to a phenomenon related to evolutionary rescue, which can promote extinction for lower environmental speeds than the critical one.</div></div>","PeriodicalId":49437,"journal":{"name":"Theoretical Population Biology","volume":"160 ","pages":"Pages 25-48"},"PeriodicalIF":1.2,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142394689","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"生物学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-09DOI: 10.1016/j.tpb.2024.09.002
Sosuke Iwai
The evolution of microbe–microbe mutualistic symbiosis is considered to be promoted by repeated exchanges of fitness benefits, which can generate positive fitness feedbacks (‘partner fidelity feedback’) between species. However, previous evolutionary models for mutualism have not captured feedback dynamics or coupling of fitness between species. Here, a simple population model is developed to understand the evolution of mutualistic symbiosis in which two microbial species (host and symbiont) continuously grow and exchange fitness benefits to generate feedback dynamics but do not strictly control each other. The assumption that individual microbes provide constant amounts of resources, which are equally divided among interacting partner individual, enables us to reveal a simple rule for the evolution of costly mutualism with positive fitness feedbacks: the product of the benefit-to-cost ratios for each species exceeds one. When this condition holds, high cooperative investment levels are favored in both species regardless of the amount invested by each partner. The model is then extended to examine how symbiont mutation, immigration, or switching affects the spread of selfish or cooperative symbionts, which decrease and increase their investment levels, respectively. In particular, when a host associates with numerous symbionts without enforcement, neither mutation nor immigration but rather random switching would allow the spread of cooperative symbionts. Examples using symbiont switching for evolution would include large ciliates hosting numerous intracellular endosymbionts. The simple model and rules would provide a basis for understanding the evolution of microbe–microbe mutualistic symbiosis with positive fitness feedbacks and without enforcement mechanisms.
{"title":"A simple model and rules for the evolution of microbial mutualistic symbiosis with positive fitness feedbacks","authors":"Sosuke Iwai","doi":"10.1016/j.tpb.2024.09.002","DOIUrl":"10.1016/j.tpb.2024.09.002","url":null,"abstract":"<div><div>The evolution of microbe–microbe mutualistic symbiosis is considered to be promoted by repeated exchanges of fitness benefits, which can generate positive fitness feedbacks (‘partner fidelity feedback’) between species. However, previous evolutionary models for mutualism have not captured feedback dynamics or coupling of fitness between species. Here, a simple population model is developed to understand the evolution of mutualistic symbiosis in which two microbial species (host and symbiont) continuously grow and exchange fitness benefits to generate feedback dynamics but do not strictly control each other. The assumption that individual microbes provide constant amounts of resources, which are equally divided among interacting partner individual, enables us to reveal a simple rule for the evolution of costly mutualism with positive fitness feedbacks: the product of the benefit-to-cost ratios for each species exceeds one. When this condition holds, high cooperative investment levels are favored in both species regardless of the amount invested by each partner. The model is then extended to examine how symbiont mutation, immigration, or switching affects the spread of selfish or cooperative symbionts, which decrease and increase their investment levels, respectively. In particular, when a host associates with numerous symbionts without enforcement, neither mutation nor immigration but rather random switching would allow the spread of cooperative symbionts. Examples using symbiont switching for evolution would include large ciliates hosting numerous intracellular endosymbionts. The simple model and rules would provide a basis for understanding the evolution of microbe–microbe mutualistic symbiosis with positive fitness feedbacks and without enforcement mechanisms.</div></div>","PeriodicalId":49437,"journal":{"name":"Theoretical Population Biology","volume":"160 ","pages":"Pages 14-24"},"PeriodicalIF":1.2,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142394688","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"生物学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-30DOI: 10.1016/j.tpb.2024.08.002
Marcy K. Uyenoyama
This study describes a compact method for determining joint probabilities of identity-by-state (IBS) within and between loci in populations evolving under genetic drift, crossing-over, mutation, and regular inbreeding (partial self-fertilization). Analogues of classical indices of associations among loci arise as functions of these joint identities. This coalescence-based analysis indicates that multi-locus associations reflect simultaneous coalescence events across loci. Measures of association depend on genetic diversity rather than allelic frequencies, as do linkage disequilibrium and its relatives. Scaled indices designed to show monotonic dependence on rates of crossing-over, inbreeding, and mutation may prove useful for interpreting patterns of genome-scale variation.
{"title":"Joint identity among loci under mutation and regular inbreeding","authors":"Marcy K. Uyenoyama","doi":"10.1016/j.tpb.2024.08.002","DOIUrl":"10.1016/j.tpb.2024.08.002","url":null,"abstract":"<div><p>This study describes a compact method for determining joint probabilities of identity-by-state (IBS) within and between loci in populations evolving under genetic drift, crossing-over, mutation, and regular inbreeding (partial self-fertilization). Analogues of classical indices of associations among loci arise as functions of these joint identities. This coalescence-based analysis indicates that multi-locus associations reflect simultaneous coalescence events across loci. Measures of association depend on genetic diversity rather than allelic frequencies, as do linkage disequilibrium and its relatives. Scaled indices designed to show monotonic dependence on rates of crossing-over, inbreeding, and mutation may prove useful for interpreting patterns of genome-scale variation.</p></div>","PeriodicalId":49437,"journal":{"name":"Theoretical Population Biology","volume":"159 ","pages":"Pages 74-90"},"PeriodicalIF":1.2,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142114062","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"生物学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-23DOI: 10.1016/j.tpb.2024.08.001
Erik G. Noonburg , Suzanne H. Alonzo , Craig W. Osenberg , Stephen E. Swearer , Jeffrey S. Shima
Settlement is a critical transition in the life history of reef fish, and the timing of this event can have a strong effect on fitness. Key factors that influence settlement timing are predictable lunar cyclic variation in tidal currents, moonlight, and nocturnal predation risk as larvae transition from pelagic to benthic environments. However, populations typically display wide variation in the arrival of settlers over the lunar cycle. This variation is often hypothesized to result from unpredictable conditions in the pelagic environment and bet-hedging by spawning adults. Here, we consider the hypothesis that the timing of spawning and settlement is a strategic response to post-settlement competition. We use a game theoretic model to predict spawning and settlement distributions when fish face a tradeoff between minimizing density-independent predation risk while crossing the reef crest vs. avoiding high competitor density on settlement habitat. In general, we expect competition to spread spawning over time such that settlement is distributed around the lunar phase with the lowest predation risk, similar to an ideal free distribution in which competition spreads competitors across space. We examine the effects of overcompensating density dependence, age-dependent competition, and competition among daily settler cohorts. Our model predicts that even in the absence of stochastic variation in the larval environment, competition can result in qualitative divergence between spawning and settlement distributions. Furthermore, we show that if competitive strength increases with settler age, competition results in covariation between settler age and settlement date, with older larvae settling when predation risk is minimal. We predict that competition between daily cohorts delays peak settlement, with priority effects potentially selecting for a multimodal settlement distribution.
{"title":"Patterns of spawning and settlement of reef fishes as strategic responses to post-settlement competition","authors":"Erik G. Noonburg , Suzanne H. Alonzo , Craig W. Osenberg , Stephen E. Swearer , Jeffrey S. Shima","doi":"10.1016/j.tpb.2024.08.001","DOIUrl":"10.1016/j.tpb.2024.08.001","url":null,"abstract":"<div><p>Settlement is a critical transition in the life history of reef fish, and the timing of this event can have a strong effect on fitness. Key factors that influence settlement timing are predictable lunar cyclic variation in tidal currents, moonlight, and nocturnal predation risk as larvae transition from pelagic to benthic environments. However, populations typically display wide variation in the arrival of settlers over the lunar cycle. This variation is often hypothesized to result from unpredictable conditions in the pelagic environment and bet-hedging by spawning adults. Here, we consider the hypothesis that the timing of spawning and settlement is a strategic response to post-settlement competition. We use a game theoretic model to predict spawning and settlement distributions when fish face a tradeoff between minimizing density-independent predation risk while crossing the reef crest vs. avoiding high competitor density on settlement habitat. In general, we expect competition to spread spawning over time such that settlement is distributed around the lunar phase with the lowest predation risk, similar to an ideal free distribution in which competition spreads competitors across space. We examine the effects of overcompensating density dependence, age-dependent competition, and competition among daily settler cohorts. Our model predicts that even in the absence of stochastic variation in the larval environment, competition can result in qualitative divergence between spawning and settlement distributions. Furthermore, we show that if competitive strength increases with settler age, competition results in covariation between settler age and settlement date, with older larvae settling when predation risk is minimal. We predict that competition between daily cohorts delays peak settlement, with priority effects potentially selecting for a multimodal settlement distribution.</p></div>","PeriodicalId":49437,"journal":{"name":"Theoretical Population Biology","volume":"160 ","pages":"Pages 1-13"},"PeriodicalIF":1.2,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142057074","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"生物学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-21DOI: 10.1016/j.tpb.2024.07.003
Andrej Depperschmidt , Andreas Greven , Peter Pfaffelhuber
<div><p>For two Polish state spaces <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>Y</mi></mrow></msub></math></span>, and an operator <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span>, we obtain existence and uniqueness of a <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span>-martingale problem provided there is a bounded continuous duality function <span><math><mi>H</mi></math></span> on <span><math><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>×</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>Y</mi></mrow></msub></mrow></math></span> together with a dual process <span><math><mi>Y</mi></math></span> on <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>Y</mi></mrow></msub></math></span> which is the unique solution of a <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>Y</mi></mrow></msub></math></span>-martingale problem. For the corresponding solutions <span><math><msub><mrow><mrow><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>t</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span> and <span><math><msub><mrow><mrow><mo>(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>t</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span>, duality with respect to a function <span><math><mi>H</mi></math></span> in its simplest form means that the relation <span><math><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>x</mi></mrow></msub><mrow><mo>[</mo><mi>H</mi><mrow><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>]</mo></mrow><mo>=</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>y</mi></mrow></msub><mrow><mo>[</mo><mi>H</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></mrow><mo>]</mo></mrow></mrow></math></span> holds for all <span><math><mrow><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>∈</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>×</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>Y</mi></mrow></msub></mrow></math></span> and <span><math><mrow><mi>t</mi><mo>≥</mo><mn>0</mn></mrow></math></span>. While duality is well-known to imply uniqueness of the <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span>-martingale problem, we give here a set of conditions under which duality also implies existence without using approximating sequences of processes of a different kind (e.g. jump processes to approximate diffusions) which is a widespread strategy for proving existence of solutions of martingale problems. Given the process <span><math><msub><mrow><mrow><mo>(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>t</mi><mo>≥</mo><mn>0</mn></mrow></msub></mat
对于两个波兰状态空间 EX 和 EY 以及一个算子 GX，只要在 EX×EY 上存在一个有界连续对偶函数 H 以及在 EY 上存在一个对偶过程 Y，且该过程是 GY-鞅问题的唯一解，我们就能得到 GX-鞅问题的存在性和唯一性。对于相应的解[公式：见正文]和[公式：见正文]，关于函数 H 的对偶性的最简单形式是指对于所有 (x,y)∈EX×EY 且 t≥0 的关系 Ex[H(Xt,y)]=Ey[H(x,Yt)]成立。众所周知，对偶性意味着 GX-马汀厄尔问题的唯一性，我们在此给出一组条件，在这些条件下，对偶性也意味着存在性，而无需使用另一种过程的近似序列（例如近似扩散的跃迁过程），这是证明马汀厄尔问题解的存在性的一种普遍策略。给定过程[公式：见正文]和对偶函数 H，要证明[公式：见正文]的存在性，必须证明对偶关系的 r.h.s. 为每个 y 定义了 EX 上的一个度量，即存在从 EX 到 EX 的过渡核[公式：见正文]，对于所有 (x,y)∈EX×EY 和所有 t≥0，Ey[H(x,Yt)]=∫μt(x,dx')H(x',y)。作为示例，我们处理了重采样和分支模型，如弗莱明-维奥特（Fleming-Viot）度量值扩散及其空间对应模型（包括离散空间和连续空间），以及分支系统，如费勒的分支扩散。虽然我们的主要结果和所有例子都涉及（局部）紧凑状态空间，但我们讨论了将我们的结果提升到谱系值过程或历史过程的策略，从而导致非紧凑（离散和连续）状态空间。在本文的基础上，我们将在接下来的工作中讨论这类应用。
{"title":"Duality and the well-posedness of a martingale problem","authors":"Andrej Depperschmidt , Andreas Greven , Peter Pfaffelhuber","doi":"10.1016/j.tpb.2024.07.003","DOIUrl":"10.1016/j.tpb.2024.07.003","url":null,"abstract":"<div><p>For two Polish state spaces <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>Y</mi></mrow></msub></math></span>, and an operator <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span>, we obtain existence and uniqueness of a <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span>-martingale problem provided there is a bounded continuous duality function <span><math><mi>H</mi></math></span> on <span><math><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>×</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>Y</mi></mrow></msub></mrow></math></span> together with a dual process <span><math><mi>Y</mi></math></span> on <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>Y</mi></mrow></msub></math></span> which is the unique solution of a <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>Y</mi></mrow></msub></math></span>-martingale problem. For the corresponding solutions <span><math><msub><mrow><mrow><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>t</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span> and <span><math><msub><mrow><mrow><mo>(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>t</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span>, duality with respect to a function <span><math><mi>H</mi></math></span> in its simplest form means that the relation <span><math><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>x</mi></mrow></msub><mrow><mo>[</mo><mi>H</mi><mrow><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>]</mo></mrow><mo>=</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>y</mi></mrow></msub><mrow><mo>[</mo><mi>H</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></mrow><mo>]</mo></mrow></mrow></math></span> holds for all <span><math><mrow><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>∈</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>×</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>Y</mi></mrow></msub></mrow></math></span> and <span><math><mrow><mi>t</mi><mo>≥</mo><mn>0</mn></mrow></math></span>. While duality is well-known to imply uniqueness of the <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span>-martingale problem, we give here a set of conditions under which duality also implies existence without using approximating sequences of processes of a different kind (e.g. jump processes to approximate diffusions) which is a widespread strategy for proving existence of solutions of martingale problems. Given the process <span><math><msub><mrow><mrow><mo>(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>t</mi><mo>≥</mo><mn>0</mn></mrow></msub></mat","PeriodicalId":49437,"journal":{"name":"Theoretical Population Biology","volume":"159 ","pages":"Pages 59-73"},"PeriodicalIF":1.2,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0040580924000765/pdfft?md5=3a0d0ba95ef090a854236fc78278e994&pid=1-s2.0-S0040580924000765-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142001150","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"生物学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-31DOI: 10.1016/j.tpb.2024.07.005
Bjarke Frost Nielsen , Christian Berrig , Bryan T. Grenfell , Viggo Andreasen
Leveraging the simplicity of nucleotide mismatch distributions, we provide an intuitive window into the evolution of the human influenza A ‘nonstructural’ (NS) gene segment. In an analysis suggested by the eminent Danish biologist Freddy B. Christiansen, we illustrate the existence of a continuous genetic “backbone” of influenza A NS sequences, steadily increasing in nucleotide distance to the 1918 root over more than a century. The 2009 influenza A/H1N1 pandemic represents a clear departure from this enduring genetic backbone. Utilizing nucleotide distance maps and phylogenetic analyses, we illustrate remaining uncertainties regarding the origin of the 2009 pandemic, highlighting the complexity of influenza evolution. The NS segment is interesting precisely because it experiences less pervasive positive selection, and departs less strongly from neutral evolution than e.g. the HA antigen. Consequently, sudden deviations from neutral diversification can indicate changes in other genes via the hitchhiking effect. Our approach employs two measures based on nucleotide mismatch counts to analyze the evolutionary dynamics of the NS gene segment. The rooted Hamming map of distances between a reference sequence and all other sequences over time, and the unrooted temporal Hamming distribution which captures the distribution of genotypic distances between simultaneously circulating viruses, thereby revealing patterns of nucleotide diversity and epi-evolutionary dynamics.
{"title":"One hundred years of influenza A evolution","authors":"Bjarke Frost Nielsen , Christian Berrig , Bryan T. Grenfell , Viggo Andreasen","doi":"10.1016/j.tpb.2024.07.005","DOIUrl":"10.1016/j.tpb.2024.07.005","url":null,"abstract":"<div><p>Leveraging the simplicity of nucleotide mismatch distributions, we provide an intuitive window into the evolution of the human influenza A ‘nonstructural’ (NS) gene segment. In an analysis suggested by the eminent Danish biologist Freddy B. Christiansen, we illustrate the existence of a continuous genetic “backbone” of influenza A NS sequences, steadily increasing in nucleotide distance to the 1918 root over more than a century. The 2009 influenza A/H1N1 pandemic represents a clear departure from this enduring genetic backbone. Utilizing nucleotide distance maps and phylogenetic analyses, we illustrate remaining uncertainties regarding the origin of the 2009 pandemic, highlighting the complexity of influenza evolution. The NS segment is interesting precisely because it experiences less pervasive positive selection, and departs less strongly from neutral evolution than e.g. the HA antigen. Consequently, sudden deviations from neutral diversification can indicate changes in other genes via the hitchhiking effect. Our approach employs two measures based on nucleotide mismatch counts to analyze the evolutionary dynamics of the NS gene segment. The <em>rooted Hamming map</em> of distances between a reference sequence and all other sequences over time, and the unrooted temporal Hamming distribution which captures the distribution of genotypic distances between simultaneously circulating viruses, thereby revealing patterns of nucleotide diversity and epi-evolutionary dynamics.</p></div>","PeriodicalId":49437,"journal":{"name":"Theoretical Population Biology","volume":"159 ","pages":"Pages 25-34"},"PeriodicalIF":1.2,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141879655","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"生物学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-15DOI: 10.1016/j.tpb.2024.07.004
Ingemar Kaj , Carina F. Mugal , Rebekka Müller-Widmann
We introduce a multi-allele Wright–Fisher model with mutation and selection such that allele frequencies at a single locus are traced by the path of a hybrid jump–diffusion process. The state space of the process is given by the vertices and edges of a topological graph, i.e. edges are unit intervals. Vertices represent monomorphic population states and positions on the edges mark the biallelic proportions of ancestral and derived alleles during polymorphic segments. In this setting, mutations can only occur at monomorphic loci. We derive the stationary distribution in mutation–selection–drift equilibrium and obtain the expected allele frequency spectrum under large population size scaling. For the extended model with multiple independent loci we derive rigorous upper bounds for a wide class of associated measures of genetic variation. Within this framework we present mathematically precise arguments to conclude that the presence of directional selection reduces the magnitude of genetic variation, as constrained by the bounds for neutral evolution.
{"title":"A Wright–Fisher graph model and the impact of directional selection on genetic variation","authors":"Ingemar Kaj , Carina F. Mugal , Rebekka Müller-Widmann","doi":"10.1016/j.tpb.2024.07.004","DOIUrl":"10.1016/j.tpb.2024.07.004","url":null,"abstract":"<div><p>We introduce a multi-allele Wright–Fisher model with mutation and selection such that allele frequencies at a single locus are traced by the path of a hybrid jump–diffusion process. The state space of the process is given by the vertices and edges of a topological graph, i.e. edges are unit intervals. Vertices represent monomorphic population states and positions on the edges mark the biallelic proportions of ancestral and derived alleles during polymorphic segments. In this setting, mutations can only occur at monomorphic loci. We derive the stationary distribution in mutation–selection–drift equilibrium and obtain the expected allele frequency spectrum under large population size scaling. For the extended model with multiple independent loci we derive rigorous upper bounds for a wide class of associated measures of genetic variation. Within this framework we present mathematically precise arguments to conclude that the presence of directional selection reduces the magnitude of genetic variation, as constrained by the bounds for neutral evolution.</p></div>","PeriodicalId":49437,"journal":{"name":"Theoretical Population Biology","volume":"159 ","pages":"Pages 13-24"},"PeriodicalIF":1.2,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0040580924000777/pdfft?md5=c7ea20aafbe501d42760b57841f9e368&pid=1-s2.0-S0040580924000777-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141635177","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"生物学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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