Discrete-time Model Research Articles

Predator–Prey Interaction with Fear Effects: Stability, Bifurcation and Two-Parameter Analysis Incorporating Complex and Fractal Behavior

This study investigates the dynamics of predator–prey interactions with non-overlapping generations under the influence of fear effects, a crucial factor in ecological research. We propose a novel discrete-time model that addresses limitations of previous models by explicitly incorporating fear. Our primary question is: How does fear influence the stability of predator–prey populations and the potential for chaotic dynamics? We analyze the model to identify biologically relevant equilibria (fixed points) and determine the conditions for their stability. Bifurcation analysis reveals how changes in fear levels and predation rates can lead to population crashes (transcritical bifurcation) and complex population fluctuations (period-doubling and Neimark–Sacker bifurcations). Furthermore, we explore the potential for controlling chaotic behavior using established methods. Finally, two-parameter analysis employing Lyapunov exponents, spectrum, and Kaplan–Yorke dimension quantifies the chaotic dynamics of the proposed system across a range of fear and predation levels. Numerical simulations support the theoretical findings. This study offers valuable insights into the impact of fear on predator–prey dynamics and paves the way for further exploration of chaos control in ecological models.

A discrete approximation method for modeling interval-censored multistate data.

Many longitudinal studies are designed to monitor participants for major events related to the progression of diseases. Data arising from such longitudinal studies are usually subject to interval censoring since the events are only known to occur between two monitoring visits. In this work, we propose a new method to handle interval-censored multistate data within a proportional hazards model framework where the hazard rate of events is modeled by a nonparametric function of time and the covariates affect the hazard rate proportionally. The main idea of this method is to simplify the likelihood functions of a discrete-time multistate model through an approximation and the application of data augmentation techniques, where the assumed presence of censored information facilitates a simpler parameterization. Then the expectation-maximization algorithm is used to estimate the parameters in the model. The performance of the proposed method is evaluated by numerical studies. Finally, the method is employed to analyze a dataset on tracking the advancement of coronary allograft vasculopathy following heart transplantation.

Infection-induced increases to population size during cycles in a discrete-time epidemic model

One-dimensional discrete-time population models, such as those that involve Logistic or Ricker growth, can exhibit periodic and chaotic dynamics. Expanding the system by one dimension to incorporate epidemiological interactions causes an interesting complexity of new behaviors. Here, we examine a discrete-time two-dimensional susceptible-infectious (SI) model with Ricker growth and show that the introduction of infection can not only produce a distinctly different bifurcation structure than that of the underlying disease-free system but also lead to counter-intuitive increases in population size. We use numerical bifurcation analysis to determine the influence of infection on the location and types of bifurcations. In addition, we examine the appearance and extent of a phenomenon known as the ‘hydra effect,’ i.e., increases in total population size when factors, such as mortality, that act negatively on a population, are increased. Previous work, primarily focused on dynamics at fixed points, showed that the introduction of infection that reduces fecundity to the SI model can lead to a so-called ‘infection-induced hydra effect.’ Our work shows that even in such a simple two-dimensional SI model, the introduction of infection that alters fecundity or mortality can produce dynamics can lead to the appearance of a hydra effect, particularly when the disease-free population is at a cycle.

Almost automorphic behaviours of nonlocal stochastic fuzzy Cohen–Grossberg lattice neural networks

Discrete-time Cohen–Grossberg neural networks (CGNNs) play important role in feedback neural networks. The existing literatures of CGNNs only regarded integral order discrete-time models without other variables. This paper builds the lattice model for nonlocal stochastic fuzzy CGNNs with reaction diffusions by employing the finite difference and Mittag–Leffler time Euler difference techniques. Further, the existence of a unique bounded almost automorphic sequence solution in distribution and global exponential convergence in the mean-square sense to the achieved difference model are investigated. At last, by using the tool of Matlab and time Euler difference for the Brownian motions, an illustrative example with simulations is used to show the feasible of the works of the current paper. We think that this work can contribute to achieve some effective real tasks, e.g. content-addressable memory, auto-association, dynamic reconstruction of a chaotic process, etc.

Open Problems and Conjectures in the Evolutionary Periodic Ricker Competition Model

In this paper, we present a survey about the latest results in global stability concerning the discrete-time evolutionary Ricker competition model with n species, in both, autonomous and periodic models. The main purpose is to convey some arguments and new ideas concerning the techniques for showing global asymptotic stability of fixed points or periodic cycles in these kind of discrete-time models. In order to achieve this, some open problems and conjectures related to the evolutionary Ricker competition model are presented, which may be a starting point to study global stability, not only in other competition models, but in predator–prey models and Leslie–Gower-type models as well.

Chaotic Dynamics of a Three-Species Food Chain Model

Abstract An analysis of a discrete-time food chain model is presented in this paper to investigate the effects of strong prey pressure on the chain. A bifurcation and stability theory is used to analyze equilibrium formation in system. It is shown by numerical simulations that chaos occurs as a result of the bifurcation of invariant curves following Neimark-Sacker (NS) bifurcation rules. The OGY method on stable periodic orbits of period 1 further controls chaos. Stability is achieved through iterations for chaotic motion controlled by different regulator poles. Numerical simulations are performed to examine the effects of the control method along with the results of our theoretical analysis.

Perceived discrimination and incident dementia among older adults in the U.S.: The buffering role of social relationships.

Recent studies have found that perceived discrimination as a chronic stressor predicts poorer cognitive health. However, little research has investigated how social relationships as potential intervening mechanisms may mitigate or exacerbate this association. Using a nationally representative sample of U.S. older adults, this study examined how the existence and quality of four types of relationships-with a partner, children, other family members, and friends-may modify the impact of perceived discrimination on incident dementia. We analyzed data from the 2006-2016 Health and Retirement Study (N=12,236) using discrete-time event history models with competing risks. We used perceived discrimination, social relationships, and their interactions at the baseline to predict the risk of incident dementia in the follow-ups. Perceived discrimination predicted a higher risk of incident dementia in the follow-ups. Although having a partner or not did not modify this association, partnership support attenuated the negative effects of discrimination on incident dementia. Neither the existence nor quality of relationships with children, other family members, or friends modified the association. Our findings imply that intimate partnership plays a critical role in coping with discrimination and, consequently, influencing the cognitive health of older adults. While perceived discrimination is a significant risk factor for the incidence of dementia, better partnership quality may attenuate this association. Policies that eliminate discrimination and interventions that strengthen intimate partnership may facilitate better cognitive health in late life.

Hadamard fractional discrete-time relaxation equation’s solutions and asymptotic stability

This paper suggests two numerical schemes for the famous fractional logistic equations by use of the time scale theory. First, the Hadamard discrete fractional calculus is introduced. Leibniz integral law is used to obtain the fractional sum equation. Then, Hadamard fractional derivative and integral’s numerical schemes are obtained and employed for numerical solutions. It can be concluded the fractional differential equation on time scales is very suitable for both continuous and discrete–time modeling.

Hypertension, Cardiovascular Risk Factors, and Uterine Fibroid Diagnosis in Midlife.

Fibroids are benign neoplasms associated with severe gynecologic morbidity. There are no strategies to prevent fibroid development. To examine associations of hypertension, antihypertensive treatment, anthropometry, and blood biomarkers with incidence of reported fibroid diagnosis in midlife. The Study of Women's Health Across the Nation is a prospective, multisite cohort study in the US. Participants were followed-up from enrollment (1996-1997) through 13 semiannual visits (1998-2013). Participants had a menstrual period in the last 3 months, were not pregnant or lactating, were aged 42 to 52 years, were not using hormones, and had a uterus and at least 1 ovary. Participants with prior fibroid diagnoses were excluded. Data analysis was performed from November 2022 to February 2024. Blood pressure, anthropometry, biomarkers (cholesterol, triglycerides, and C-reactive protein), and self-reported antihypertensive treatment at baseline and follow-up visits were measured. Hypertension status (new-onset, preexisting, or never [reference]) and hypertension treatment (untreated, treated, or no hypertension [reference]) were categorized. Participants reported fibroid diagnosis at each visit. Discrete-time survival models estimated hazard ratios (HRs) and 95% CIs for associations of time-varying hypertension status, antihypertensive treatment, anthropometry, and biomarkers with incident reported fibroid diagnoses. Among 2570 participants without a history of diagnosed fibroids (median [IQR] age at screening, 45 [43-48] years; 1079 [42.1%] college educated), 526 (20%) reported a new fibroid diagnosis during follow-up. Risk varied by category of hypertension treatment: compared with those with no hypertension, participants with untreated hypertension had a 19% greater risk of newly diagnosed fibroids (HR, 1.19; 95% CI, 0.91-1.57), whereas those with treated hypertension had a 20% lower risk (HR, 0.80; 95% CI, 0.56-1.15). Among eligible participants with hypertension, those taking antihypertensive treatment had a 37% lower risk of newly diagnosed fibroids (HR, 0.63; 95% CI, 0.38-1.05). Risk also varied by hypertension status: compared with never-hypertensive participants, participants with new-onset hypertension had 45% greater risk of newly diagnosed fibroids (HR, 1.45; 95% CI, 0.96-2.20). Anthropometric factors and blood biomarkers were not associated with fibroid risk. Participants with untreated and new-onset hypertension had increased risk of newly diagnosed fibroids, whereas those taking antihypertensive treatment had lower risk, suggesting that blood pressure control may provide new strategies for fibroid prevention.

Bifurcations and complex dynamics in a banking duopoly model with macroprudential policy

We consider a banking duopoly model with a macroprudential policy in Indonesia called loan-to-deposit ratio-based reserve requirement (LDR-RR). The objective of the policy is to control the banking loans growth using a LDR-based penalty scheme that requires banks to save more money in the central bank to maintain their liquidity ratio. Following recent studies on banking models, we analyse a piecewise discrete-time model with two banks. We assume that the dynamics of the deposits follows the discrete logistic growth. Moreover, our model has two borders, hence we examine the resulting border-collision bifurcations. From the local stability analysis we find that, according to the parameter values, only the border-collision bifurcation or the flip bifurcation occurs. Finally, we perform several numerical simulations to confirm the stability analysis’ results. Our discrete dynamical system offers various possibilities of development for future research perspectives.

Economic Cycles and Entry into Parenthood: Is the Association Changing and Does it Affect Macro-Level Trends? Micro-Level Hazard and Simulation Models of Belgian Fertility Trends, 1960–2010

The association between economic cycles—typically measured in terms of GDP growth or swings in unemployment—and macro-level fertility trends has received ample attention in the literature. Compared to studies that consider macro-level fertility, individual-based models can address the association between economic cycles and specific stages of family formation (e.g. entry into parenthood) more precisely while allowing for structural factors that contribute to fertility postponement. Using population-wide longitudinal microdata from the Belgian censuses we combine discrete-time hazard models of entry into parenthood for the period 1960 to 2010 with microsimulation models to assess whether economic cycles in tandem with educational expansion can account for year-to-year variation in the proportion of women entering parenthood and variation in the pace of fertility postponement at the macro-level. Results indicate that educational expansion has been a structural driver of fertility postponement, whereas the procyclical effect of economic cycles accounts for accelerations and decelerations of fertility postponement throughout the period considered. Microsimulation of macro-level fertility trends indicates that individual-based models predict the annual proportion of women entering parenthood and the mean age at first birth with average errors of prediction below 1 per cent and 3 months, respectively, while also showing strong correlations between first differences of observed and simulated time-series. Because the extended observation window encompasses several severe recessions, we test whether the association between economic cycles and entry into parenthood has changed over time and how this affected macro-level trends, discussing several mechanisms that may account for such temporal variation.

Distributed Adaptive Tracking Control of Hidden Leader-Follower Multi-Agent Systems with Unknown Parameters

The distributed leader-follower control of multi-agent systems is discussed. Each agent is expressed in a discrete-time and non-linear dynamic model with an unknown parameter and can be affected by its neighbors’ history information. For each agent, to identify the parameter, one switching set of the parameter estimates is constructed and the optimal parameter estimate is chosen based on the index switching function. Using the given desired reference signal, the leader agent’s control law is designed, and relying on the neighbors’ history information, each follower agent’s local control law is designed. With the designed distributed tracking adaptive control laws, the whole system tracks the given desired reference signal, and in the face of strong couplings the closed-loop system ultimately reaches an agreement. Finally, by comparing simulations of the control strategy with a normal projection algorithm, the results indicate that the adaptive control method with a switching set of the parameter estimates is effective in improving the control performance.

Option Pricing Using a Skew Random Walk Binary Tree

We develop a binary tree pricing model with underlying asset price dynamics following Itô–McKean skew Brownian motion. Our work was motivated by the Corns–Satchell, continuous-time, option pricing model. However, the Corns–Satchell market model is incomplete, while our discrete-time market model is defined in the natural world, extended to the risk-neutral world under the no-arbitrage condition where derivatives are priced under uniquely determined risk-neutral probabilities, and is complete. The skewness introduced in the natural world is preserved in the risk-neutral world. Furthermore, we show that the model preserves skewness under the continuous-time limit. We provide empirical applications of our model to the valuation of European put and call options on exchange-traded funds tracking the S&P Global 1200 index.

A Canadian Simulation Model for Major Depressive Disorder: Study Protocol.

Major depressive disorder (MDD) is a common, often recurrent condition and a significant driver of healthcare costs. People with MDD often receive pharmacological therapy as the first-line treatment, but the majority of people require more than one medication trial to find one that relieves symptoms without causing intolerable side effects. There is an acute need for more effective interventions to improve patients' remission and quality of life and reduce the condition's economic burden on the healthcare system. Pharmacogenomic (PGx) testing could deliver these objectives, using genomic information to guide prescribing decisions. With an already complex and multifaceted care pathway for MDD, future evaluations of new treatment options require a flexible analytic infrastructure encompassing the entire care pathway. Individual-level simulation models are ideally suited for this purpose. We sought to develop an economic simulation model to assess the effectiveness and cost effectiveness of PGx testing for individuals with major depression. Additionally, the model serves as an analytic infrastructure, simulating the entire patient pathway for those with MDD. Key stakeholders, including patient partners, clinical experts, researchers, and modelers, designed and developed a discrete-time microsimulation model of the clinical pathways of adults with MDD in British Columbia (BC), including all publicly-funded treatment options and multiple treatment steps. The Simulation Model of Major Depression (SiMMDep) was coded with a modular approach to enhance flexibility. The model was populated using multiple original data analyses conducted with BC administrative data, a systematic review, and an expert panel. The model accommodates newly diagnosed and prevalent adult patients with MDD in BC, with and without PGx-guided treatment. SiMMDep comprises over 1500 parameters in eight modules: entry cohort, demographics, disease progression, treatment, adverse events, hospitalization, costs and quality-adjusted life-years (payoff), and mortality. The model predicts health outcomes and estimates costs from a health system perspective. In addition, the model can incorporate interactive decision nodes to address different implementation strategies for PGx testing (or other interventions) along the clinical pathway. We conducted various forms of model validation (face, internal, and cross-validity) to ensure the correct functioning and expected results of SiMMDep. SiMMDep is Canada's first medication-specific, discrete-time microsimulation model for the treatment of MDD. With patientpartner collaboration guiding its development, it incorporates realistic care journeys. SiMMDep synthesizes existing information and incorporates provincially-specific data to predict the benefits and costs associated with PGx testing. These predictions estimate the effectiveness, cost-effectiveness, resource utilization, and health gains of PGx testing compared with the current standard of care. However, the flexible analytic infrastructure can be adapted to support other policy questions and facilitate the rapid synthesis of new data for a broader search for efficiency improvements in the clinical field of depression.

Leaving home across the recent cohorts in Italy: does economic vulnerability due to labour market status matter?

We focus on the process of leaving home among young Italians and aim to discern: (a) the extent to which the potential economic vulnerability associated with certain individual labour market conditions affects the pathway of leaving the parental home; (b) whether the impact of such conditions has changed over generations of young adults. Data come from the pooling of two cross-sectional rounds of the ‘Families and Social Subjects’ survey conducted by the Italian National Institute of Statistics in 2009 and 2016. Specifically, we consider young adults who at the time of interview were between the ages of 20 and 44, for a total of 22,218 respondents. We employ discrete-time event history models, including a competing risks approach. Our findings suggest that labour market status differs in its influence according to reason for exit and gender. While we observe a clearly negative impact of being out of the labour market for both union and non-union related reasons for men, this negative impact is observed only for non-union-related motives for women. Meanwhile, economic uncertainty connected with precarious jobs does not necessarily have a negative impact on the risk of leaving parental home, both for union and non-union related reasons. Furthermore, we see signs of differences across cohorts relative to this latter set of motives for men. The results highlight the importance of considering reasons for exit in analyses of departure from the parental home.

The necessity of the sausage-string structure for mode-locking regions of piecewise-linear maps

Piecewise-smooth maps are used as discrete-time models of dynamical systems whose evolution is governed by different equations under different conditions (e.g. switched control systems). By assigning a symbol to each region of phase space where the map is smooth, any period-p solution of the map can be associated to an itinerary of p symbols. As parameters of the map are varied, changes to this itinerary occur at border-collision bifurcations (BCBs) where one point of the periodic solution collides with a region boundary. It is well known that BCBs conform broadly to two cases: persistence, where the symbolic itinerary of a periodic solution changes by one symbol, and a nonsmooth-fold, where two solutions differing by one symbol collide and annihilate. This paper derives new properties of periodic solutions of piecewise-linear continuous maps on Rn to show that under mild conditions BCBs of mode-locked solutions on invariant circles must be nonsmooth-folds. This explains why Arnold tongues of piecewise-linear maps exhibit a sausage-string structure whereby changes to symbolic itineraries occur at codimension-two pinch points instead of codimension-one persistence-type BCBs. But the main result is based on the combinatorical properties of the itineraries, so the impossibility of persistence-type BCBs also holds when the periodic solution is unstable or there is no invariant circle.

Aperiodic intermittent sampling for nonlinear discrete-time event-based model predictive control

In this article, an aperiodic intermittent sampling method is designed in event-based model predictive control for nonlinear discrete systems subject to external disturbances. The proposed aperiodic intermittent sampling is applied to find the appropriate triggering instant satisfying the desired sub-optimal performance. In the proposed event-based MPC, selection of the sampling instants is based on the prediction of the worst case induced by disturbances and the mapping principle with updated state feedback. With the proposed event-based MPC, the measurement frequency and the triggering rate can be reduced by sampling the system states only at certain time instants. Recursive feasibility and robust stability are established under sufficient conditions.

Partnership Dynamics of LGB People and Heterosexuals: Patterns of First Partnership Formation and First Cohabitation

In this study we examine partnership dynamics among people with different sexual orientations in Germany. More specifically, we explore the process of first partnership formation and first cohabitation among men and women who self-identify as heterosexual, homosexual or bisexual. Given the various discriminations against same-sex lifestyles, and the limited opportunities to meet potential partners, we assume that lesbian, gay and bisexual (LGB) people form partnerships later in life and less frequently than heterosexuals. We further expect that the constantly improving social and legal climate for sexual minorities will lead to a reduction in differences in partnership behaviour by sexual orientation. We use retrospectively reported partnership biographies from the German Socio-Economic Panel, which was supplemented in 2019 with a boost sample of sexual and gender minority households. Using discrete-time event history models, we analyse nearly 15,000 episodes of being single and nearly 20,000 episodes of living without a partner in the household. Around 4.5% of these episodes are from people who self-identify as LGB. The results clearly show that patterns of partnership and coresidential union formation differ by sexual orientation. People with a homosexual orientation—and to a lesser extent people with a bisexual orientation—are less likely to enter into a first partnership and a first cohabitation than people with a heterosexual orientation. Significant changes occur across cohorts: LGB people from younger birth cohorts enter (cohabiting) partnerships much earlier and more frequently than those from older cohorts. Thus, the union formation patterns of LGB and straight people have converged slightly.

Gender Differences in the Protective Role of Grandparenting in Dementia Risk.

This study provides one of the first national longitudinal studies of the association between caring for grandchildren (i.e., grandparenting) and the risk of dementia in the U.S., with a focus on gender-specific variations. We estimated discrete-time event history models, drawing upon data from the Health and Retirement Study (2000-2016). The analytic sample included 10,217 community-dwelling White and Black grandparents aged 52 years and older at baseline. Noncoresident grandparenting was associated with a lower risk of dementia for both women and men compared to grandparents who did not take care of grandchildren. However, the cognitive advantage showed different patterns based on gender and the combination of care intensity and family structure. Grandmothers had a lower risk of dementia than noncaregiving grandmothers when providing a light level of noncoresident grandparenting, while grandfathers who provided intensive noncoresident grandparenting had a reduced risk of dementia compared to their noncaregiving counterparts. Grandparenting experiences within multigenerational households and skipped-generation households were not associated with dementia risk. Intergenerational caregiving plays a pivotal role in shaping cognitive health during later life; however, the impact is nuanced, depending on factors such as gender, care intensity and family structure.

Asymptotics for ruin probabilities in a bidimensional discrete-time risk model with dependent and consistently varying tailed net losses

Consider an insurer who operates two lines of businesses and hence receives two types of insurance net losses. In the bidimensional discrete-time risk model with a constant interest rate, the net loss vectors from the two business lines form a sequence of independent and identically distributed real-valued random vectors. We propose a bidimensional extended regularly varying structure in terms of survival copulas to model the dependence between the two components of each net loss vector, which includes not only asymptotic independence but also asymptotic dependence. Under the framework of consistent variation, we obtain several (uniformly) asymptotic formulas for various ruin probabilities. In particular, the investigation of random-time ruin probabilities unifies and extends the results on both finite-time and infinite-time ruin probabilities. Some numerical studies are also performed to check the accuracy of our obtained asymptotic results.