Neimark Sacker Research Articles

A general egg-limited and search-limited host–parasitoid model with proportional host refuge

Motivated by the article [Getz and Mills. Host–parasitoid coexistence and egg-limited encounter rates. Am Nat. 1996;148:301–315], in this paper, we explore a discrete model involving a host and a parasitoid with search and egg limitations and arbitrary host-escape function. We added proportional refuge for the hosts to the model. We focus on the system's behavior at the equilibrium points and the nearby regions. In addition to the topological classification of these points, we examined local behavior. In the case of the extinction equilibrium, we obtain the global result. We describe the dynamical behavior scenarios in the neighborhood of the non-isolated exclusion equilibrium point (1:1 resonant). For the unique coexisting equilibrium, we prove the emergence of the Neimark–Sacker bifurcation and calculate the first Lyapunov exponent. This bifurcation can be either super or sub-critical. We have also established the occurrence of the Chenciner bifurcation. Our findings indicate that proportional refuge may or may not stabilize the system, and the choice of host-escape function plays a crucial role in shaping the system's dynamics. We also provide numerical examples to support our theoretical results.

Dynamics of a discrete Rosenzweig–MacArthur predator–prey model with piecewise-constant arguments

This paper investigates the dynamics of a new discrete form of the classical continuous Rosenzweig–MacArthur predator–prey model and the biological implications of the dynamics. The discretization method used here is to modify the continuous model to another with piecewise-constant arguments and then to integrate the modified model, which is quite different from the traditional Euler discretization method. First, the existence and local stability of fixed points have been thoroughly discussed. Then, all codimension-1 bifurcations have been studied, including the transcritical bifurcations at the trivial fixed point and the boundary fixed point, and the Neimark–Sacker bifurcation at the unique positive fixed point. Furthermore, it is proved that there are no codimension-2 bifurcations. Finally, the control of the bifurcations is studied. Regarding the biological significance, we have considered the following four aspects: When no harvesting effort is applied to both predators and prey, it is observed that providing more resources to the prey species leads to predator extinction, causing the ecosystem to lose stability. This implies that the paradox of enrichment occurs. When predators are harvested, the system exhibits the hydra effect, i.e. increasing the harvesting effort on predators will lead to an increase in the mean population density of predators. When the prey is harvested, numerical examples indicate that increasing the harvesting effort initially reduces prey density. Upon reaching the bifurcation point, prey density stabilizes while the predator density continues to decline. When both prey and predators are harvested, the biological significance is similar to the previous case. The outcomes of this paper have significant theoretical meaning in the study of population ecology. By MATLAB, the bifurcation diagrams, maximal Lyapunov exponent diagrams, phase portraits and mean population density curves are plotted. Numerical simulations not only show the correctness of the theoretical findings but also reveal many new and interesting dynamics.

Nonlinear dynamics of a Darwinian Ricker system with strong Allee effect and immigration

In this paper, we investigate the complex dynamics of a Darwinian Ricker system through a comprehensive qualitative and dynamical analysis. Our research shows that the system exhibits Neimark–Sacker bifurcation, period-doubling bifurcation, and codimension-two bifurcations associated with 1:2, 1:3, and 1:4 resonances. These findings are derived using bifurcation and center manifold theories. We numerically illustrate all bifurcation results and chaotic features, providing a thorough understanding of the system’s behavior. This detailed examination of the Darwinian Ricker system, with a focus on the interplay between immigration and the strong Allee effect, enhances our understanding of the intricate mechanisms driving population dynamics. Furthermore, it highlights the significant implications for ecological modeling, particularly in predicting ecosystem responses to external perturbations such as climate change and species invasions.

Spatiotemporal complexity analysis of a discrete space-time cancer growth model with self-diffusion and cross-diffusion

We investigate spatiotemporal pattern formation in cancer growth using discrete time and space variables. We first introduce the coupled map lattices (CMLs) model and provide a dynamical analysis of its fixed points along with stability results. We then offer parameter criteria for flip, Neimark–Sacker, and Turing bifurcations. In the presence of spatial diffusion, we find that stable homogeneous solutions can experience Turing instability under certain conditions. Numerical simulations reveal a variety of spatiotemporal patterns, including patches, spirals, and numerous other regular and irregular patterns. Compared to previous literature, our discrete model captures more complex and richer nonlinear dynamical behaviors, providing new insights into the formation of complex patterns in spatially extended discrete tumor models. These findings demonstrate the model’s ability to capture complex dynamics and offer valuable insights for understanding and treating cancer growth, highlighting its potential applications in biomedical research.

Some Bifurcations of Codimensions 1 and 2 in a Discrete Predator–Prey Model with Non-Linear Harvesting

This paper delves into the dynamics of a discrete-time predator–prey system. Initially, it presents the existence and stability conditions of the fixed points. Subsequently, by employing the center manifold theorem and bifurcation theory, the conditions for the occurrence of four types of codimension 1 bifurcations (transcritical bifurcation, fold bifurcation, flip bifurcation, and Neimark–Sacker bifurcation) are examined. Then, through several variable substitutions and the introduction of new parameters, the conditions for the existence of codimension 2 bifurcations (fold–flip bifurcation, 1:2 and 1:4 strong resonances) are derived. Finally, some numerical analyses of two-parameter planes are provided. The two-parameter plane plots showcase interesting dynamical behaviors of the discrete system as the integral step size and other parameters vary. These results unveil much richer dynamics of the discrete-time model in comparison to the continuous model.

Invariant Sets, Global Dynamics, and the Neimark–Sacker Bifurcation in the Evolutionary Ricker Model

In this paper, we study the local, global, and bifurcation properties of a planar nonlinear asymmetric discrete model of Ricker type that is derived from a Darwinian evolution strategy based on evolutionary game theory. We make a change of variables to both reduce the number of parameters as well as bring symmetry to the isoclines of the mapping. With this new model, we demonstrate the existence of a forward invariant and globally attracting set where all the dynamics occur. In this set, the model possesses two symmetric fixed points: the origin, which is always a saddle fixed point, and an interior fixed point that may be globally asymptotically stable. Moreover, we observe the presence of a supercritical Neimark–Sacker bifurcation, a phenomenon that is not present in the original non-evolutionary model.

Global dynamic analyzes of the discrete SIS models with application to daily reported cases

Emerging infectious diseases, such as COVID-19, manifest in outbreaks of varying magnitudes. For large-scale epidemics, continuous models are often employed for forecasting, while discrete models are preferred for smaller outbreaks. We propose a discrete susceptible-infected-susceptible model that integrates interaction between parasitism and hosts, as well as saturation recovery mechanisms, and undertake a thorough theoretical and numerical exploration of this model. Theoretically, the model incorporating nonlinear recovery demonstrates complex behavior, including backward bifurcations and the coexistence of dual equilibria. And the sufficient conditions that guarantee the global asymptotic stability of a disease-free equilibrium have been obtained. Considering the challenges posed by saturation recovery in theoretical analysis, we then consider the case of linear recovery. Bifurcation analysis for of the linear recovery model displays a variety of bifurcations at the endemic equilibrium, such as transcritical, flip, and Neimark–Sacker bifurcations. Numerical simulations reveal complex dynamic behavior, including backward and fold bifurcations, periodic windows, period-doubling cascades, and multistability. Moreover, the proposed model could be used to fit the daily COVID-19 reported cases for various regions, not only revealing the significant advantages of discrete models in fitting, evaluating, and predicting small-scale epidemics, but also playing an important role in evaluating the effectiveness of prevention and control strategies. Furthermore, sensitivity analyses for the key parameters underscore their significant impact on the effective reproduction number during the initial months of an outbreak, advocating for better medical resource allocation and the enforcement of social distancing measures to curb disease transmission.

BIFURCATION ANALYSIS AND CHAOS CONTROL OF THE DISSOLVED OXYGEN-PHYTOPLANKTON DYNAMICAL MODEL

The production of oxygen through phytoplankton photosynthesis is a crucial phenomenon in the dynamics of marine ecosystems. A generic oxygen-phytoplankton interaction model is considered to comprehend its underlying mechanism. This paper investigates the discrete-time dynamics of oxygen and phytoplankton in aquatic ecosystems, incorporating factors that cause phytoplankton mortality due to external influences. We explore the conditions for the local stability of steady states concerning the oxygen content in dissolved water and phytoplankton density. The analysis reveals that the model undergoes a co-dimension one bifurcation, encompassing flip and Neimark–Sacker bifurcations, utilizing the center manifold theorem and bifurcation theory. To manage the chaos resulting from the Neimark–Sacker bifurcation, we apply the OGY feedback control method and a hybrid control methodology. Finally, we present numerical simulations to validate the theoretical discussion.

Complex Dynamics of a Discretized Predator–Prey System with Prey Refuge Using a Piecewise Constant Argument Method

The objective of this work is to investigate the complex dynamics of a discrete predator–prey system using the method of piecewise constant argument for discretization. An analysis is conducted to examine the presence and stability of fixed points. Furthermore, the system is shown to undergo period-doubling (PD) and Neimark–Sacker (NS) bifurcations by the use of center manifold and bifurcation theories. The feedback and hybrid control strategies are used to regulate the system’s bifurcating and chaotic behaviors. Both strategies seem to be effective in managing bifurcation and chaos inside the system. Finally, the main results are validated by numerical evidence. Parameters of the system are varied to produce time graphs, phase portraits, bifurcation diagrams, and maximum Lyapunov exponent (MLE) graphs. The discrete model displays rich dynamics, as seen in the numerical simulations and graphs, indicating a complex and chaotic system.

A Generalized Beddington Host–Parasitoid Model with an Arbitrary Parasitism Escape Function

This research delves into the generalized Beddington host–parasitoid model, which includes an arbitrary parasitism escape function. Our analysis reveals three types of equilibria: extinction, boundary, and interior. Upon examining the parameters, we discover that the first two equilibria can be globally asymptotically stable. The boundary equilibrium undergoes period-doubling bifurcation with a stable two-cycle and a transcritical bifurcation, creating a threshold for parasitoids to invade. Furthermore, we determine the interior equilibrium’s local stability and analytically demonstrate the period-doubling and Neimark–Sacker bifurcations. We also prove the permanence of the system within a specific parameter space. The numerical simulations we conduct reveal a diverse range of dynamics for the system. Our research extends the results in [ Kapçak et al. , 2013 ] and applies to a broad class of the generalized Beddington host–parasitoid model.

Neimark–Sacker bifurcation of discrete Pielou model with delay

In this paper, we investigate the existence of the Neimark–Sacker bifurcation of discrete delay Pielou Model with positive parameters. It is shown that this model undergoes a Neimark–Sacker bifurcation when the parameter passes a critical value. Furthermore, based on the computational algorithm developed by Murakami in [The invariant curve caused by Neimark-Sacker bifurcation, Dyn. Contin. Discrete Impuls. Syst. Ser. A 9 (2002), pp. 121–132], the explicit algorithm for determining the direction and stability of the Neimark–Sacker bifurcations is derived. Also, an explicit approximate expression of the invariant curve caused by Neimark–Sacker bifurcation is given. Some numerical simulations are carried out to illustrate the analytical results found.

Controlling chaos and codimension-two bifurcation in a discrete fractional-order Brusselator model

This paper explores the qualitative behavior of a discrete fractional–order Brusselator model. We analyze the local dynamics of the model around its fixed point and determine its topological classification. We perform the bifurcation analysis for both codimension-one and codimension-two cases to examine the system behavior near critical parameter values. Using normal form theory and center manifold theorem (CMT), we prove that the model exhibits period-doubling bifurcation around its interior fixed point. We also study the existence and direction of Neimark–Sacker bifurcation using normal form theory. For codimension-two bifurcation, we show that the model undergoes 1:2, 1:3, and 1:4 resonances by applying normal form theory and suitable affine transformations. The system displays a rich variety of bifurcations, including quasi–periodicity, periodic orbits, chaotic behavior, and resonance bifurcation. Furthermore, the existence of chaos is discussed in the sense of Marotto, and a novel chaos control method is proposed for discrete Brusselator model using an extended pole–placement approach. This modified approach is more suitable for codimension-two bifurcation situations. Numerical simulations are used to illustrate the theoretical discussion.

Neimark–Sacker bifurcation of two second-order rational difference equations

We investigate the Neimark–Sacker bifurcation of the equilibrium of two special cases of the difference Equation x n + 1 = β x n x n − 1 + γ x n − 1 2 + δ x n B x n x n − 1 + C x n − 1 2 + D x n where the parameters β, γ, δ, B, C, D are non-negative numbers which satisfy B + C + D>0 and the initial conditions x − 1 and x 0 are arbitrary nonnegative numbers such that B x n x n − 1 + C x n − 1 2 + D x n > 0 for all n ≥ 0 . More precisely, we consider special cases where either γ = D = 0 or β = D = 0 . As we will show both equations exhibit Neimark–Sacker bifurcation, where one of equations ( γ = D = 0 ) probably exhibits Chenciner bifurcation, with two invariant curves, while another Equation ( β = D = 0 ) exhibits simple Neimark–Sacker bifurcation with one invariant curve. We will also obtain some global asymptotic stability result for each equation in the subset of the parametric region of local stability.

Complex Dynamics and Bifurcations Analysis of Discrete-Time Modified Leslie–Gower System

This work introduces a discrete modified Leslie–Gower prey–predator system with Holling type-II functional response. The persistence of the discrete model under certain conditions is discussed. The conditions assuring the existence of fixed points are derived and nonlinear dynamics of system are explored at these fixed points. It has been shown that the system exhibits transcritical bifurcation and flip bifurcation at semi-trivial fixed point under certain bifurcation values. In addition, the center manifold and bifurcation theories are employed to attain the conditions for existence of flip and Neimark–Sacker bifurcations at coexistence fixed point. The system is found to exhibit periodic solutions along with bifurcations leading to wide range of chaotic dynamics. The numerical simulations are performed to confirm the analytical analysis.

Dynamical behavior of a discrete-time predator–prey system incorporating prey refuge and fear effect

This paper explores the effects of fear and refuge in a discrete-time two species predator–prey model. The fraction of prey that seeks refuge is assumed to be proportional to the amount of predators. The local stability of the corresponding fixed points is analyzed. The conditions of flip bifurcation and Neimark–Sacker bifurcation at the positive fixed point are described. The sufficient conditions for the positive fixed point to be globally asymptotically stable are derived in this article. Utilization of hybrid control strategy stabilizes the system’s chaotic nature. Finally, the paper supports the theoretical analysis with a numerical simulation that demonstrates the intricate nature of the system. It is observed that both fear effect and prey refuge can destabilize or stabilize the system depending on the value of handling time. Also, fear effect can restore stability in the system that exhibits chaos due to a high prey growth rate.

Stability and bifurcations of a host–parasitoid model with general host escape function and general stocking upon parasitoids

This paper analyzes the generalization of a model presented in J. Bektešević, V. Hadžiabdić, S. Kalabušić, M. Mehuljić and E. Pilav [Dynamics of a class of host–parasitoid models with external stocking upon parasitoids, Adv. Differ. Equ. 2021(31) (2021)]. The study explores the behavior of the solution near equilibrium points when the system has different outcomes, such as extinction, infinitely many exclusion points or unique exclusion and coexistence. We prove global stability for the extinction and host-exclusion equilibrium. We also investigate the non-hyperbolic case of parasitoid-exclusion equilibrium and delve deeper into the 1:1 resonance. The transcritical bifurcation occurs at the host-exclusion equilibrium, indicating a threshold for host population invasion through transcritical bifurcation. Moreover, the local dynamics around the coexisting equilibrium can be highly complex due to the appearance of the Neimark–Sacker and period-doubling bifurcations. We provide the explicit form of the first Lyapunov exponent for the Neimark–Sacker bifurcation. Through numerical examples, we illustrate the theoretical findings.

Bifurcation and Stability of an Discrete-time SIS Epidemic Model with Treatment

Mathematical models are useful in examining the effect of an infection on populations. Conditions involving the spread and control of the disease are calculated by analyzing mathematical models, so that it is possible to have information about the behavior of the infection. This article includes the dynamic behavior of a discrete-time SIS epidemic model with treatment. Existence conditions of the fixed points of the model are obtained, and stability analysis is performed for these fixed points. The stability and bifurcation conditions of the obtained endemic fixed point are investigated. Depending on the infection coefficient, the flip bifurcation condition is obtained. At the same time, it is determined in which situation Neimark Sacker bifurcation may occur depending on the step size, and bifurcation is controlled. Rich dynamic behaviors are given to support our theoretical results.

Spatiotemporal Patterns in a Space–Time Discrete SIRS Epidemic Model with Self- and Cross-Diffusion

In this work, a two-dimensional Coupled Map Lattice (CML) method is applied to a continuous Reaction–Diffusion (RD) SIRS epidemic model. The model involves a complex, nonlinear incidence rate and the spatially heterogeneous self- and cross-diffusion coefficients. The CML method applied here results in a space–time discrete counterpart of the model. The basic reproduction number is derived, and the local stability of the fixed points of the discrete model is established. The two typical bifurcations of codimension 1, such as the flip and the Neimark–Sacker (NS) bifurcations, are investigated by utilizing the center manifold theory and the normal form theorem. Furthermore, the codimension-2 fold–flip bifurcation about the endemic fixed point is discussed in detail. Turing instability criterion of the cross-diffusion epidemic model is established by utilizing the CML method under periodic boundary conditions. Finally, the numerical simulations clarify both the effectiveness of all theoretical findings and the effects of time step size and cross-diffusion on the spatiotemporal patterns. The main contribution of this work is the discovery of some interesting Turing and non-Turing patterns which may be induced by flip–Turing instability, NS–Turing instability or fold–flip–Turing instability. Concretely speaking, spiral patterns are observed near the flip bifurcation threshold value and ultimately evolve into an irregular defect pattern. Stripe pattern, labyrinth pattern and circular pattern are observed near the NS bifurcation threshold value. The disorderly interwoven ribbon pattern, mosaic pattern and some irregular oscillatory patterns are observed near the fold–flip bifurcation point. Our results greatly help in predicting the long-term behavior of the space–time discrete SIRS epidemic model under study.

Dynamical properties of a small heterogeneous chain network of neurons in discrete time

We propose a novel nonlinear bidirectionally coupled heterogeneous chain network whose dynamics evolve in discrete time. The backbone of the model is a pair of popular map-based neuron models, the Chialvo and the Rulkov maps. This model is assumed to proximate the intricate dynamical properties of neurons in the widely complex nervous system. The model is first realized via various nonlinear analysis techniques: fixed point analysis, phase portraits, Jacobian matrix, and bifurcation diagrams. We observe the coexistence of chaotic and period-4 attractors. Various codimension-1 and -2 patterns for example saddle-node, period-doubling, Neimark–Sacker, double Neimark–Sacker, flip- and fold-Neimark–Sacker, and 1 : 1 and 1 : 2 resonance are also explored. Furthermore, the study employs two synchronization measures to quantify how the oscillators in the network behave in tandem with each other over a long number of iterations. Finally, a time series analysis of the model is performed to investigate its complexity in terms of sample entropy.

Complex dynamics in prey-predator systems with cross-coupling: Exploring nonlinear interactions and population oscillations

This study investigates the problem of ecosystem dynamics in fragmented landscapes, specifically focusing on a two-patch environment with interacting prey and predators. The research examines the impact of cross-predation on these interactions. Using bifurcation analysis, we explored the structural arrangement of attractors and identified complex dynamics such as symmetric, asymmetric, and asynchronous attractors induced by varying cross-coupling levels. Notably, our study describes a novel mechanism for the formation of anti-phase synchrony in the patches. Unlike typical occurrences of a cycle following Hopf bifurcation, our model reveals that the anti-phase cycle stabilizes via Neimark-Sacker (NS) bifurcation of a two-period unstable cycle branch emanating from the synchronous cycle branch. Our findings also demonstrate that cross-feeding can lead to significant ecosystem asymmetry and branching, culminating in the dominance of a single cross-feeding chain. These results challenge traditional models and highlight the presence of multistability and the potential for ecosystem evolution towards distinct subsystem branches due to cross-predation. The study’s insights offer valuable contributions to population and evolutionary biology, enhancing our understanding of the intricate dynamics within fragmented ecosystems.