The ecological dynamics between elk and wolves in Northern Yellowstone have been a focal point of long-term research, particularly following the reintroduction of wolves to the region. Although numerous studies have explored this prey–predator interaction from ecological and behavioral perspectives, there remains a lack of comprehensive analysis using mathematical modeling approaches capable of uncovering underlying dynamical patterns. In this study, we investigate the prey–predator dynamics of the elk–wolf system in Northern Yellowstone National Park, using a data-driven modeling approach. We used yearly population data for elk and wolves from 1995 to 2022 to construct a mathematical model using a sparse regression modeling framework. To the best of our knowledge, no previous work has applied this framework to capture elk–wolf interactions over this time period. Our modeling pipeline integrates gaussian process regression for data smoothing, sparse identification of nonlinear dynamics for model discovery, and model selection techniques to identify the most suitable mathematical representation. Stability and bifurcation analyzes are then performed to understand the system’s qualitative behavior. A saddle–node bifurcation identifies the parameter range in which both species can coexist, while values outside this range lead to the extinction of one or both species. Hopf and saddle–node bifurcations define regions of stable co-existence, periodic oscillations, and extinction. Bifurcations such as Bogdanov–Takens and cusp are examined by varying two parameters simultaneously. Ecologically, these bifurcations reflect the interplay between wolf pressure and elk defence strategies. They suggested that small changes in parameter can trigger sudden shifts between co-existence, oscillations, or extinction.

The study of hunting cooperation and fear effects is emerging as important ecological factors in population dynamics. These two features are analyzed independently in the literature by several researchers in detail. It is observed that both effects are important but poorly understood mechanisms that mediate the way predators organize ecosystems. The literature suggests that the outcomes of predator–prey interactions and their impact on ecosystems can be influenced together by these two factors. Therefore, we review the expanding body of research that integrates hunting cooperation and/or the effect of fear phenomena into the ecology of predator–prey. Our aim is to provide a framework for examining how the increasing type of functional response is affected by fear factor. The temporal dynamics, including the stability and bifurcation analysis of the system, is discussed briefly. Various parametric planes are analyzed to identify the regions of stability, instability, and bistability, along with some invariant manifolds in the phase plane that divide the basins of attraction. The temporal model is extended to the spatiotemporal framework to capture the movements of populations, and the conditions for Turing instability are derived, revealing spatial dynamics that produce various Turing patterns (spots, stripes, and mixed type) in response to the changes in fear effect and diffusion coefficients. Extensive numerical simulations are also performed to illustrate the dynamics of the model in temporal and spatio-temporal contexts.

Rubella is a contagious viral infection that has serious public health consequences, especially for pregnant women and their children. This paper proposes a deterministic compartmental model for understanding rubella transmission dynamics, which includes critical factors such as vaccination rate, vaccine waning rate, recovery rate, and progression rate from infection to severe cases. We apply the proposed model to actual rubella data from Nigeria to analyze the epidemic dynamics within the Nigerian population. We utilized the rubella case reports from Nigeria covering January to December 2020 from the Nigeria Centre for Disease Control to obtain the best model fit; this help to determine the accuracy of the proposed model’s representation to the real-world data. To discover key characteristics that influence disease propagation, sensitivity analysis is performed using the Partial Rank Correlation Coefficient approach. When R0<1, the disease-free equilibrium remains globally asymptotically stable, while for R0>1, the system transitions to an endemic equilibrium. Numerical simulations show that achieving at least 70% vaccination coverage minimizes cumulative new rubella cases. However, vaccine waning increases the disease burden, highlighting the importance of high immunization rates and prompt booster vaccinations. These findings highlight the significance of strategic immunization plans for effective rubella control.

This study develops and analyzes an SIRS epidemic model with convex incidence and saturated treatment under both autonomous and non-autonomous frameworks. For the autonomous system we characterize the disease-free and endemic equilibria and perform bifurcation analysis, showing backward and saddle-node bifurcations, Hopf bifurcations that generate endemic bubbles. Also, bifurcation analyses reveal a codimension-two double-zero bifurcation arising from the intersection of saddle-node and Hopf bifurcations. The non-autonomous extension incorporates seasonal variations in transmission and recovery rates, capturing realistic periodic forcing observed in diseases such as influenza. Using data from the Democratic Republic of the Congo, we confirm December as the peak influenza season. Analytical results establish conditions for the existence and global stability of a positive periodic solution, while numerical simulations show that seasonality induces complex behaviors, including multi-periodic and chaotic oscillations. Low seasonal intensity sustains coexistence, whereas high-intensity forcing leads to population extinction. The emergence of quasi-periodic (torus) and chaotic (strange) attractors demonstrates how seasonal forcing can transform regular epidemic cycles into irregular outbreaks, offering new insights into the role of seasonality in infectious disease dynamics and control.

In a two-dimensional prey–predator model with a Holling type-II functional response and a food-limited prey growth rate, this study explores the consequences of predator harvesting. There are utmost three coexisting equilibrium points in the system. It has been shown that the prey population’s half-saturation constant has a significant role in boosting the complicated bifurcation structure. To do this, we thoroughly examine the suggested system for codimension-1 bifurcations, including Hopf and saddle-node bifurcations. Calculating the conditions of Sotomayor’s theorem the presence of saddle-node bifurcation is ensured and the signs of first Lyapunov number are computed for the stability of periodic solutions via Hopf bifurcation. With the implementation of saddle-node bifurcation, an unforeseen scenario of species extinction is carried out in the case of three interior equilibria with respect to the half-saturation constant, which gives results apart from the traditional one. Additionally, we executed the continuation of codimension-1 bifurcations for the emergence of codimension-2 bifurcations like generalized-Hopf, cusp, and Bogdanov–Takens bifurcations to understand the role of harvesting the predator population. The work becomes more appealing because it displays topologically different phase diagrams for suitable parameters. Ecologically, the generalized-Hopf bifurcation shows that the system’s behavior is quite sensitive to the prey saturation constant and predator harvesting. In addition, we compare the results of proposed system with the model for a saturated harvesting and linear functional response. Extensive numerical simulations are performed to validate the conclusions for stability and bifurcations.

In this article, we propose a novel nonlinear cross-diffusion framework to model the distribution of susceptible and infected individuals within their habitat using a reduced SIR model that incorporates saturated incidence and treatment rates. The study investigates solution boundedness through the theory of parabolic partial differential equations, thereby validating the proposed spatio-temporal model. Through the implementation of the suggested cross-diffusion mechanism, the model reveals at least one non-constant positive equilibrium state within the susceptible–infected (SI) system. This work demonstrates the potential coexistence of susceptible and infected populations through cross-diffusion and unveils Turing instability within the system. By analyzing codimension-2 Turing–Hopf bifurcation, the study identifies the Turing space within the spatial context. In addition, we explore the results for Turing–Bogdanov–Takens bifurcation. To account for seasonal disease variations, novel perturbations are introduced. Comprehensive numerical simulations illustrate diverse emerging patterns in the Turing space, including holes, strips, and their mixtures. Additionally, the study identifies non-Turing and Turing–Bogdanov–Takens patterns for specific parameter selections. Spatial series and surfaces are graphed to enhance the clarity of the pattern results. This research provides theoretical insights into the implications of cross-diffusion in epidemic modeling, particularly in contexts characterized by localized mobility, clinically evident infections, and community-driven isolation behaviors.

In this study, we present a comprehensive mathematical model to investigate the dynamics of Susceptible-Infectious-Recovered (SIR) type infectious diseases. Our model incorporates various factors that play crucial roles in disease transmission and control, including saturated incidence, vertical transmission, inherited passive immunity in newborns of susceptible and recovered individuals, and treatment. We explore the interplay between these factors and their impact on disease spread and control measures using a reduced SI model. After ensuring the existence of the model, we obtain the stability of the system around the disease-free and endemic equilibria. Also, for comprehending the long-term dynamics of the system, we obtain the results of persistence and global stability. By varying the parameters, we observe various bifurcations in the reduced SI system such as backward, transcritical, saddle-node, and Hopf bifurcations. To understand the behavior of the medication rate, vertical transmission, and the measure of passive immunity of newborns in the S and R compartments, we conduct numerical simulations that investigate the existence of periodic solutions through Hopf bifurcation, elucidating multiple endemic bubbles in the bifurcation diagram. We also validate the proposed system for the zoonotic disease by analyzing the 2022 monkeypox outbreak in the USA using daily new infection data obtained from the official website of the Centers for Disease Control and Prevention (CDC). By demonstrating that our simulations of the reduced SI model is closely aligned with officially reported data and average daily cases, policymakers can have confidence in its predictive capabilities, enhancing their ability to anticipate and respond to future outbreaks effectively. We have followed the dual approach of validating analytical results both theoretically as well as empirically using real data. This dual approach distinguishes our work from existing studies, which typically focus on either modeling a specific disease or exploring hypothetical dynamics exclusively.

This manuscript accounts for an investigation of the complex dynamics of a spatial model for interacting populations. We discuss the existence and boundedness of solutions for the proposed spatio-temporal system. The global stability of the co-existing steady state of the proposed system is analyzed with the help of a suitable Lyapunov function. We provide results on the existence and non-existence of positive non-constant solutions of the model. The priori estimate for the positive steady state is obtained for the nonexistence of the non-constant positive steady state by using the maximum principle. The existence of a non-constant positive steady state is studied with the help of Leray–Schauder degree theory. The stability and Hopf bifurcation are briefly revisited for the co-existing steady state in the corresponding temporal model, where a bubble-like structure is observed. The onset of Hopf bifurcation has been analyzed, and different conditions for the formation of the Turing pattern have been established through diffusion-driven instability analysis. Numerical simulations are performed in detail to figure out the effects of saturated harvesting on Turing patterns. The Turing as well as non-Turing patterns in their respective domains are also examined. Finally, the criteria of Turing–Hopf bifurcation is briefly demonstrated with relevant numerical examples and corresponding plots that give a better illustration of this work.

In the realm of disease transmission dynamics, assigning blame to every susceptible individual for propagating an illness proves untenable. This limitation plagues prevailing compartmental models, including SI, SIS, SIR, SEIR, and others, impeding accurate disease-spread prognostication. This study innovatively partitions the susceptible population into two distinct classes: the unaware and the aware. This conceptual leap facilitates the evolution of the SIR model into a more realistic UAIR epidemic model, characterized by a bilinear incidence rate in the unaware class and a saturated incidence rate in the aware class. The study rigorously establishes key insights, encompassing the stability of the endemic equilibrium state, transcritical and Hopf bifurcations, and stability of bifurcated periodic solutions in the reduced UAI epidemic model. Notably, the analysis uncovers potential disease-induced chaos, the turbulence that heightened community awareness can potentially harness. Delving deeper, the research explores a self-diffusive spatio-temporal UAI epidemic model with zero-flux boundary conditions, unveiling instability insights that preclude the emergence of Turing patterns and associated Turing bifurcation. Rigorous numerical simulations validate the analytical framework, unveiling captivating non-Turing patterns within a 2-dimensional domain. This study proffers a predictive exploration, poised to unearth diverse medically significant observations, further enriching our understanding of disease dynamics.

The current study presents complex dynamics of an SIR epidemic model that incorporates a saturated type incidence rate as well as treatment. We provide here rigorous results for asymptotic stability of equilibrium states of the proposed system. Several bifurcations including Hopf, Generalized Hopf, saddle–node, transcritical and Bogdanov–Takens are also discussed. The stability of bifurcated periodic solutions is verified with the help of first Lyapunov number. Extensive numerical simulations are performed to validate these results. In a numerical example it is observed that if the saturation factor increases slowly, then the unique endemic equilibrium state is asymptotically stable for a certain range. The further increase in the value of saturation parameter, the endemic equilibrium state loses its stability and periodic solutions appear through Hopf bifurcation. It is also observed that the increase in saturation parameter beyond Hopf bifurcation threshold, results in regaining the stability of the endemic equilibrium state, which forms an interesting dynamical phenomenon in the bifurcation diagram named as an endemic bubble. It is pointed out that in the case of two endemic equilibrium states, one of these two is always saddle, whereas, the other one becomes unstable through Hopf bifurcation. In this scenario, the periodic solution is initially stable and it becomes unstable through generalized Hopf bifurcation. In numerical example for Bogdanov–Takens bifurcation two pairs of feasible bifurcation thresholds exist for the same set of parameters value. The bifurcation diagrams and equilibrium surfaces are also plotted to observe the combined effects of medication and saturation parameters.