Pattern formation in mathematical modeling of biology has been a popular topic for a long time. In this study, we focus on possible patterns driven by Turing instability under the prey-predator model of nonlinear reaction-diffusion type in 2D space with the Allee effect and intra-species competition. We investigate pattern formation in predator-prey systems on both continuous media and network frameworks, highlighting how diffusion in continuous space and its Laplacian analogue on networks shape the emergence of Turing patterns. We derive the model from scratch, discuss three separate cases, and analyze the dynamics of both temporal and spatiotemporal dynamics of the models. Under circumstances where the Turing pattern does not exist, we present detailed proofs and use numerical simulation to demonstrate temporal Hopf bifurcation and patchy invasion. On the other hand, if the analytical conditions for the Turing pattern can be satisfied, we display patterns in our numerical simulations and give corresponding biological interpretations at the end.

Grasping how factors like the Allee effect, resource limits, and developmental delays interact is key to forecasting an ecosystem's long-term persistence and stability. Here, we present and examine a prey-dependent, three-level food-chain model that embeds an Allee effect in the prey, a finite carrying capacity for resource limitation, and a discrete gestation delay in the top predator - capturing real-world phenomena such as survival thresholds at low prey densities, restricted environmental resources, and time lags in predator reproduction. Following nondimensionalization, we carry out a thorough analytical study, establishing positivity and boundedness of solutions, exploring both local and global stability, and performing Hopf and transcritical bifurcation analyses. When delays are included, we derive stability criteria that depend on the length of the gestation period and pinpoint critical delay values that trigger oscillatory behavior. Finally, numerical simulations paired with global sensitivity analysis corroborate our theoretical results. Together, these findings shed light on how the interplay among Allee effects, carrying capacity, and time delays can drive population dynamics toward stability, sustained oscillations, or collapse - insights that are vital for effective biodiversity management and conservation planning.

This study investigates the dynamical consequences of predator-induced fear in a predator–prey system by integrating nonlinear fear effects, intraspecific competition, and Holling type II predation. We analyze both continuous- and discrete-time formulations to capture a wide spectrum of ecological dynamics. The model exhibits rich behaviors, including stable coexistence, bistability, oscillations, and chaotic dynamics, revealing how fear can fundamentally reshape population interactions. Analytical results establish well-posedness and persistence, while numerical bifurcation analysis demonstrates that variations in key ecological parameters can trigger qualitative transitions such as oscillatory outbreaks and complex irregular fluctuations. Global sensitivity analysis identifies the dominant parameters controlling long-term population densities, thereby quantifying system robustness. To account for environmental variability, a stochastic extension with multiplicative noise is examined, showing that moderate fluctuations preserve coexistence, whereas strong disturbances may destabilize populations. Discrete-time analysis further uncovers period-doubling routes to chaos and intricate bifurcation structures. Overall, the results highlight predator-induced fear as a critical regulatory mechanism that can either stabilize or destabilize ecosystems, offering important ecological insights into population management and the prediction of complex biological dynamics.

The present work examines the dynamical features of a discrete-time Rosenzweig–MacArthur predator–prey system in which the predation rate is modulated by prey density. Through a combination of theoretical derivations and computational experiments, we establish the occurrence of period-doubling (PD) and Neimark–Sacker (NS) bifurcations, both of which culminate in chaotic regimes. The Lyapunov exponent spectrum corroborates this picture: Positive values mark the onset of chaos and reveal a pronounced dependence of the trajectories on parameter perturbations. Numerical simulations further reveal chaotic attractors and invariant closed trajectories. The Ott–Grebogi–Yorke (OGY) method is employed to stabilise unstable orbits to control chaos. When extending the analysis to a coupled network, we show that chaos appears beyond a certain critical coupling threshold. These insights advance understanding of predator–prey dynamics and chaos control in ecological systems.

This study constructs a discrete-time model featuring the Holling type II functional response to investigate how the Allee effect drives the appearance of both periodic and chaotic behaviors. Through rigorous algebraic analysis, it is demonstrated that Neimark-Sacker (NS) bifurcations occur when varying the Allee parameter n and the prey's intrinsic growth rate r within the positive quadrant. By applying the center manifold theorem and core results from bifurcation theory, a solid theoretical framework is established to track these qualitative changes. Extensive numerical simulations validate that the system undergoes a subcritical NS bifurcation as n and r change, marking a shift from stable equilibria into chaotic oscillations with increasing Allee intensity. Moreover, the model reveals the coexistence of periodic attractors of periods 11 and 34 over certain (n,r) pairs, underscoring how initial population levels critically shape long-term outcomes. A parallel analysis contrasts this Holling II formulation with its Holling type I analog. By varying the Allee threshold n, it becomes apparent that the type I system collapses at a lower Allee value than the type II system. This disparity suggests that predation saturation-a hallmark of Holling II-confers greater resilience under strong Allee effects. Finally, a comparative evaluation of both functional responses' advantages and limitations highlights the need for adaptive management strategies to preserve biodiversity and maintain ecological stability.

BACKGROUND:Diabetic retinopathy (DR) is a complication of diabetes that affects the eyes. This study aims to identify the genetic variants associated with DR in type 2 diabetes (T2D) patients from the UK Biobank cohort (n = 16,988). METHODS:We conducted a genome-wide association study (GWAS) of DR and integrated genomic results with multi-omics data to identify and prioritize susceptibility variants and genes. The findings are set to undergo validation in four replication cohorts. RESULTS:Here we show that the lead SNP rs6066146 in EYA2 reaches genome-wide significance (p = 4.21×10-8) and is replicated in three independent cohorts. The SNP-based heritability for DR is estimated at 14.6% (standard deviation: 0.11). Colocalization analysis at the EYA2 locus suggests moderate colocalization (PP.H4 = 0.553) alongside distinct association signals for DR and T2D, and cis-Mendelian randomization (MR) within the EYA2 region provides gene-centric evidence that T2D exerts a significant causal effect on DR. Exploratory multivariable MR identifies proinsulin as a significant mediator of T2D on DR, which may partly account for the moderate evidence for colocalization. Tissue expression, chromatin interaction, and transcriptome-wide association analyses point to the spleen, while gene set analysis identifies B-cell pathways. Together, these convergent signals suggest that splenic B-cell abundance could serve as a predictive marker for DR risk. CONCLUSIONS:Our study demonstrates a genomic risk locus in gene EYA2 associated with DR in type 2 diabetes, which offers deeper insights into broader trait architecture on DR.

Abstract We examine travelling wave solutions of a reaction–diffusion equation that incorporates a strong Allee effect in the population growth term and a Stefan-like moving boundary condition governing the advance of the population front. These two mechanisms, which have been examined separately in other model frameworks, have not previously been analysed together. We investigate the sharp-fronted travelling wave solutions that arise using both direct numerical simulation and travelling wave analysis, enabling us to obtain the relationship between the wave speed and the parameter associated with the moving boundary condition. Our results show that this model admits sharp-front travelling waves with wave speed c

We investigate the spatiotemporal dynamics of a tri-trophic food chain model incorporating a strong Allee effect on the prey and a fear effect on the middle predator. The model's well-posedness is established through the positivity and boundedness of solutions. We derive all equilibria and examine their local stability, revealing saddle-node and transcritical bifurcations under varying parameter conditions. The analysis demonstrates how shifts in the Allee threshold and fear intensity induce bistability, coexistence, or extinction. Numerical simulations highlight diffusion-driven instabilities and complex Turing patterns, including labyrinthine formations and unexpected "leaser slime" structures-resembling those observed in fungi and algae in aquatic systems. These findings reveal the crucial role of behavioral and ecological feedbacks in shaping pattern formation and species persistence.

Hip pain is a common musculoskeletal complaint that leads many people to seek medical attention. We conducted a primary genome-wide association study (GWAS) on the hip pain phenotype within the UK Biobank cohort. Sex-stratified GWAS analysis approach was also performed to explore sex specific variants associated with hip pain. We found seven different loci associated with hip pain at GWAS significance level, with the most significant single nucleotide polymorphism (SNP) being rs77641763 within the EXD3 (p value = 2.20 × 10–13). We utilized summary statistics from the FinnGen cohort and a previous GWAS meta-analysis on hip osteoarthritis as replication cohorts. Four loci (rs509345, rs73581564, rs9597759, rs2018384) were replicated with a p value less than 0.05. Sex-stratified GWAS analyses revealed a unique locus within the CUL1 gene (rs4726995, p = 2.56 × 10–9) in males, and three unique loci in females: rs1651359966 on chromosome 7 (p = 1.15 × 10–8), rs552965738 on chromosome 9 (p = 2.72 × 10–8), and rs1978969 on chromosome 13 (p = 2.87 × 10–9). This study has identified seven genetic loci associated with hip pain. Sex-stratified analysis also revealed sex specific variants associated with hip pain in males and females. This study has provided a foundation for advancing research of hip pain and hip osteoarthritis.

Objectives: Widespread pain is a hallmark characteristic of fibromyalgia, commonly affecting older individuals. This study aimed to identify novel genetic variants associated with widespread pain by utilizing the extensive UK Biobank dataset. Methods: We conducted a primary genome-wide association study (GWAS) using a novel definition of widespread pain, defined as pain experienced all over the body during the past month. Sex-stratified GWAS analysis approach was also performed to analyze the impact of sex on widespread pain. Results: The primary GWAS identified one novel significant genetic locus (rs34691025, p = 1.76 × 10 −8 ) on chromosome 5q13.2 within the ARHGEF28 gene and several loci that approached genome-wide significance. The sex-stratified GWAS outputs revealed biological difference widespread pain between males and females, with a novel locus identified in the female-specific analysis within the LRMDA gene on chromosome 10. Genetic Correlation analysis demonstrated significant genetic correlations between widespread pain and other phenotypes, including joint disorders and spondylosis. The PheWAS revealed associations between the significant genetic variants with hearing disorders and cardiovascular diseases. A two-sample Mendelian randomization analysis found no significant causal association between hearing loss and widespread pain. Conclusions: Our study advances the understanding of the genetic factors contributing to widespread pain, highlighting notable differences between males and females and identifying a novel genetic locus associated with this condition.

The Allee effect is crucial to population ecology and conservation biology because it clarifies how difficult it can be for tiny populations to endure and expand. Low population densities make it more difficult for individuals to survive or reproduce in a phenomenon. Therefore, it is important to study the ecological system, including the Allee effect. Accordingly, our goal in this paper was to examine equilibria's behaviors in a three-species food chain model incorporating the Allee effect. This model includes a linear type of functional response. The points of equilibrium are categorized and depicted. The behaviors of these equilibrium points are then illustrated analytically through stability and bifurcation analyses. Moreover, the numerical simulation utilizes realistic hypothetical data to confirm the analytical results and detect the influence of varying the parameters on the system's dynamics. It is observed that the system undergoes bistable behavior; otherwise, the trivial equilibrium point is globally asymptotically stable. Under specific circumstances, the food chain system experiences a transcritical bifurcation around the axial and border equilibrium points. However, under some conditions, a Hopf bifurcation happens around the border equilibrium point.

When a population is in equilibrium, both genotype and allele frequencies remain constant from one generation to the next. If a population satisfies the conditions necessary to ensure that genotypes are in Hardy–Weinberg proportions, it follows that it is also in equilibrium. Even if a population does not satisfy the Hardy–Weinberg conditions, however, it may still be in equilibrium. The frequency of recessive alleles preventing individual monkey flower plants from producing pollen, for example, is likely to represent a balance between the tendency of natural selection to eliminate the recessive allele and recurrent mutation that tends to increase its frequency. A population in which such forces are balanced might be said to be in dynamic equilibrium. Two dynamic equilibrium points in a gene regulatory network correspond to biological switch to a gene circuit.

Gout is a prevalent and painful form of inflammatory arthritis associated with hyperuricemia, which leads to monosodium urate crystal deposition in joints and surrounding tissues, triggering acute inflammatory responses. This disease is also closely linked to serious comorbidities, including cardiovascular diseases, chronic kidney diseases, diabetes, and increased mortality risk, significantly impacting global health. In this study, we conducted a comprehensive genome-wide association study (GWAS) based on the UK Biobank pain questionnaire 2019, comprising 10 474 gout cases and 140 068 controls, identifying 13 loci associated with gout. These findings were further explored in the FinnGen cohort, with 10 loci being replicated significantly. Sex-stratified analyses revealed notable differences, with 16 loci identified in males and two loci identified in females, reflecting both shared and sex -stratified genetic influences on gout susceptibility. In addition, genetic correlation analyses demonstrated strong associations between gout and traits related to urate levels, specific medication use, and metabolic functions. Transcriptome-wide association studies highlighted several genes, such as SLC16A9 and ASAH2B, which showed significant expression patterns across various tissues, implicating metabolic and immune pathways in gout. Phenome-wide association studies of significant single nucleotide polymorphisms revealed links to metabolic, immunological, and skeletal traits, underscoring the multi-faceted nature of gout. These results contribute valuable insights into the genetic architecture and biological mechanisms underlying gout, suggesting potential avenues for tailored interventions.

This study aims to thoroughly investigate the dynamics of a predator-prey model with a Beddington-De Angelis functional response. We assume that the prey refuge is proportional to both species. We establish the standard properties of boundedness, permanence, and local stability. We show that under certain parameter conditions, transcritical bifurcation and Hopf bifurcation occur. To understand the nature of the limit cycle, we determine the direction of the Hopf bifurcation. We focus on the significant ranges of the predators' Prey capturing rate and examine how the level of prey fear and the predator's mutual interference affect the system's stability. Through numerical analysis, we study the behavior of the Lyapunov exponent and observe multiple self-repeating shrimp-shaped patterns that indicate periodic attractors in discrete-time predator-prey system. These structures appear across a broad region associated with chaotic dynamics. Additionally, if the intensity of white noise is kept below a specific threshold, the deterministic control approach is equally effective in environmental fluctuation. Numerical simulations support these findings..

Purpose: Diabetic retinopathy (DR), a complication affecting the eyes, is associated with diabetes. This study aims to identify genetic variants associated with DR in patients with type 1 diabetes in the UK Biobank cohort (n = 1,004). Methods: A genome-wide association study (GWAS) was conducted to identify significant genetic variants of DR in type 1 diabetes. The findings are set to undergo validation during the replication and meta-analysis stages by using six cohorts: African American, European, FinnGen, GoSHARE, GoDARTS and Caucasian Australians. Results: In a locus, top single nucleotide polymorphism (SNP) rs184619214 in CCDC7 reached a GWAS significance level (p = 6.38 x 10-9) and rs79853754 in ITGB1 (p = 3.24 x 10-8), with both genes being adjacent to each other. The SNP-based heritability was estimated to be 31.09%. Rs184619214 was replicated and reached statistical significance (p < 5.0 x 10-8) in the meta-analysis stage. Pathway analysis revealed that ITGB1 is involved in the generation of biomolecules that impact the progression of DR. PheWAS analysis revealed that osteoarthritis (OA) of the hip was significantly associated with most of the SNPs of the locus. Mendelian Randomization further confirmed an association between OA and DR. Conclusions: Our study has identified a novel genomic risk locus associated with DR in type 1 diabetes, located in the intergenic region between the CCDC7 and ITGB1 genes, providing insights for DR researchers. Keywords: Diabetic retinopathy; genome-wide association study; meta-analysis; Phenome-Wide Association Study; type 1 diabetes ### Competing Interest Statement The authors have declared no competing interest. ### Funding Statement This study was mainly funded by the Pioneer and Leading Goose R&D Program of Zhejiang Province 2023 with reference number 2023C04049 and Ningbo International Collaboration Program 2023 with reference number 2023H025. ### Author Declarations I confirm all relevant ethical guidelines have been followed, and any necessary IRB and/or ethics committee approvals have been obtained. Yes The details of the IRB/oversight body that provided approval or exemption for the research described are given below: This study adheres to all ethical guidelines and data protection protocols of the UK Biobank.This study was approved by the Ethics Committee of the University of Nottingham, Ningbo, China. The research received the necessary ethical approval from the National Health Service National Research Ethics Service (reference 11/NW/0382) I confirm that all necessary patient/participant consent has been obtained and the appropriate institutional forms have been archived, and that any patient/participant/sample identifiers included were not known to anyone (e.g., hospital staff, patients or participants themselves) outside the research group so cannot be used to identify individuals. Yes I understand that all clinical trials and any other prospective interventional studies must be registered with an ICMJE-approved registry, such as ClinicalTrials.gov. I confirm that any such study reported in the manuscript has been registered and the trial registration ID is provided (note: if posting a prospective study registered retrospectively, please provide a statement in the trial ID field explaining why the study was not registered in advance). Yes I have followed all appropriate research reporting guidelines, such as any relevant EQUATOR Network research reporting checklist(s) and other pertinent material, if applicable. Yes Detailed summary statistics data about DR within the UKB are retrievable from https://figshare.com/s/68c2e17003c694c6f453. Should any additional data pertinent to this study need to be presented within this paper or its other files, the authors can provide such data upon reasonable request.

Knee pain is a widespread musculoskeletal condition affecting millions globally, with significant socio-economic implications. This study endeavors to identify genetic variants associated with knee pain through a comprehensive genome-wide association study (GWAS) using data from 441,757 individuals in the UK Biobank. The primary GWAS identified ten significant loci, including eight novel loci, with the most significant single nucleotide polymorphism (SNP) being rs143384 near the GDF5 gene on chromosome 20 ( p = 4.68 x 10-19). In the replication study, seven loci (rs143384, rs919642, rs55760279, rs56076919, rs3892354, rs687878, rs368636424) were found to be significant in the FinnGen cohort. Further, sex-specific analyses revealed distinct genetic associations, identifying three loci (rs143384 with p = 1.70x10-15, rs56076919 with p = 1.60x10-9, rs919642 with p = 1.45x10-8) in females and four loci ( rs2899611 with p = 2.77 x 10-11, rs891720 with p = 5.55 x 10-11, rs2742313 with p = 4.19 x 10-9, rs2019689 with p = 6.51 x 10-9) in males. The phenome-wide association analysis and Mendelian randomization analysis revealed significant links between several phenotypes and knee pain such as leg pain on walking. These findings enhance our understanding of the genetic factors of knee pain, offering potential pathways for therapeutic interventions and personalized medical strategies. ### Competing Interest Statement The authors have declared no competing interest. ### Funding Statement This research was primarily supported by the Pioneer and Leading Goose R&D Program of Zhejiang Province 2023 (Grant No. 2023C04049) and the Ningbo International Collaboration Program 2023 (Grant No. 2023H025). ### Author Declarations I confirm all relevant ethical guidelines have been followed, and any necessary IRB and/or ethics committee approvals have been obtained. Yes The details of the IRB/oversight body that provided approval or exemption for the research described are given below: This study fully complies with the ethical standards and data protection regulations of the UK Biobank. The research was conducted using data from the UK Biobank under Application Number 89386. I confirm that all necessary patient/participant consent has been obtained and the appropriate institutional forms have been archived, and that any patient/participant/sample identifiers included were not known to anyone (e.g., hospital staff, patients or participants themselves) outside the research group so cannot be used to identify individuals. Yes I understand that all clinical trials and any other prospective interventional studies must be registered with an ICMJE-approved registry, such as ClinicalTrials.gov. I confirm that any such study reported in the manuscript has been registered and the trial registration ID is provided (note: if posting a prospective study registered retrospectively, please provide a statement in the trial ID field explaining why the study was not registered in advance). Yes I have followed all appropriate research reporting guidelines, such as any relevant EQUATOR Network research reporting checklist(s) and other pertinent material, if applicable. Yes The summary statistics from the UK Biobank regarding knee pain will be made available upon publication. Additional data relevant to this study, not included in the article or supplementary materials, can be obtained from the authors upon reasonable request.

Propagation of a disease through a spatially varying population poses complex questions about disease spread and population survival. We consider a spatio-temporal predator–prey model in which a disease only affects the predator. Diffusion-driven instability conditions are analytically derived for the spatio-temporal model. We perform numerical simulation using experimental data given in previous studies and demonstrate that travelling waves, periodicity and chaotic patterns are possible. We show that the introduction of disease in the predator species makes the standard Rosenzweig–MacArthur model capable of producing Turing patterns, which is not possible without disease. However, in the absence of infection, both species can coexist in spiral non-Turing patterns. It follows that disease persistence may be predictable, while eradication may not be.

Purpose To identify the genetic variants associated with diabetic retinopathy in type 2 patients from the UK Biobank cohort (n = 17,015) and supporting replication cohorts GODARTS (n = 5,013), GOSHARE (n = 1,754), Caucasian Australians (n = 518), FinnGen (n = 206,664) and Chinese (n = 1,007). Methods Totally eleven genome-wide association studies were applied to search for significant genetic variants. Results We found 5 different loci associated with type 2 diabetic retinopathy in or nearest gene EYA2, MPDZ, NTNG1, CTAGE14P and MREGP1. In the primary GWAS, a significant SNP rs6066146 located in gene EYA2 showed a p value of 4.21 x 10-8 and may play a role in the development of the disease, with "spleen" reaching a significant level produced by tissue expression analysis. Corresponding heritability of DR was estimated to be 26.73% by SumHer. Among five genes, we found that genes EYA2, MPDZ, NTNG1 had genetic interactions and may affect the complex development of retinal blood vessels. Conclusion Diabetic retinopathy is a complication of diabetes that affects the eyes. It is highly likely to occur when high blood sugar damages the retinal blood vessels. There is limited awareness regarding the pathogenesis of DR. Our study identified multiple loci associated with diabetic retinopathy, which may lead to personalized treatments to reduce the burden of the disease. Keywords : Diabetic retinopathy; UK Biobank; genome-wide association study; tissue expression analysis; heritability. ### Competing Interest Statement The authors have declared no competing interest. ### Funding Statement This study was mainly funded by the Pioneer and Leading Goose R&D Program of Zhejiang Province 2023 with reference number 2023C04049 and Ningbo International Collaboration Program 2023 with reference number 2023H025. ### Author Declarations I confirm all relevant ethical guidelines have been followed, and any necessary IRB and/or ethics committee approvals have been obtained. Yes The details of the IRB/oversight body that provided approval or exemption for the research described are given below: This study was approved by the Ethics Committee of the University of Nottingham, Ningbo, China. I confirm that all necessary patient/participant consent has been obtained and the appropriate institutional forms have been archived, and that any patient/participant/sample identifiers included were not known to anyone (e.g., hospital staff, patients or participants themselves) outside the research group so cannot be used to identify individuals. Yes I understand that all clinical trials and any other prospective interventional studies must be registered with an ICMJE-approved registry, such as ClinicalTrials.gov. I confirm that any such study reported in the manuscript has been registered and the trial registration ID is provided (note: if posting a prospective study registered retrospectively, please provide a statement in the trial ID field explaining why the study was not registered in advance). Yes I have followed all appropriate research reporting guidelines, such as any relevant EQUATOR Network research reporting checklist(s) and other pertinent material, if applicable. Yes This research complies with all ethical standards and data privacy measures set by the UK Biobank. Detailed summary statistics data pertaining to Diabetic Retinopathy within the UK Biobank are retrievable from https://figshare.com/s/9d55aa774fb19307f7c8. Should there be any additional data pertinent to this study that are not presented within this paper or its additional files, the authors can provide such data upon reasonable request. Data is available upon publication.

Clonorchiasis is an important food-borne parasitic disease that has been largely understudied. We examine the dynamics of clonorchiasis with human treatment and fish vaccination versus snail control. Four mathematical models are proposed and analysed, both analytically and numerically. We show that fish vaccination is a valid method of control and derive the maximal period of molluscicide application in order to control the snail populations. It follows that a variety of methods may be necessary to control clonorchiasis.

Prey refuge from predators can play an important role in stabilising an ecological system by reducing interactions between species, while Allee effects can arise from a range of biological phenomena, such as anti-predator vigilance, genetic trends and feeding deficiencies. We develop a predator–prey model that combines these phenomena, considering variable prey refuge with additive Allee effect on the prey species, with a Holling type II response function for the prey growth function. We use the predator and prey nullclines to determine the existence and stability of interior equilibria. We also investigate all possible local and global bifurcations that the system could undergo, showing that prey refuge and a strong Allee effect can lead to saddle-node bifurcations, Hopf bifurcations or Bogdanov–Takens bifurcation. We have investigated the appearance of Hopf bifurcations in a neighborhood of the unique interior equilibrium point of the dynamical system. The rich behaviour of the dynamics suggests that both prey refuge and a strong Allee affect are important factors in ecological complexity.