Interior Steady State Research Articles

Spatiotemporal Dynamics of an Intraguild Predation Model with Intraspecies Competition

Different species in the environment compete with one another for shared resources and suitable habitats. As a result, it becomes crucial to identify the essential components involved in preserving biodiversity. This paper describes the emergence of a wide range of temporal and spatiotemporal dynamics of an intraguild predation (IGP) model with intraspecies competition and Holling type-II response function between the IG-prey and IG-predator. To explore the dynamics of the temporal model, we study the local stability of all the biologically feasible steady states, global stability of the interior steady state, and Hopf bifurcation. Performing partial rank correlation coefficient (PRCC) sensitivity analysis, we picked more sensitive parameters and plotted the stability regions in two-parameter spaces. The dynamics exhibited by the system can demonstrate some important hallmarks noticed in the environments, such as the existence and nonexistence of oscillatory behavior in the IGP model. The analytical formulations for the stability and bifurcation of the coexistence of the steady-state are not easy to obtain, but we have verified numerically that the limit cycle bifurcates from the coexistence of the steady-state and there eventually emerges a chaotic behavior through period-doubling bifurcations. By plotting the largest Lyapunov exponent (LLE), we have verified that the temporal system experiences chaotic behavior. It is demonstrated that in the case of a diffusive system, diffusion may cause the coexistence of the steady state, which is otherwise stable, to become unstable. Criteria for the occurrence of Turing instability connected to the IGP model are derived. Our numerical illustrations demonstrate that diffusion-driven instability develops different stationary patterns based on the different initial conditions and diffusion parameters. The spatiotemporal patterns are observed in the Turing, Hopf–Turing, and Hopf regions.

Global stability and bifurcation analysis on the dynamics of three interacting populations in prey-predator system

A mathematical model was proposed and analysed to study the dynamics of a system of a prey, primary predator and a super predator in which the predators show Holing Type II functional response to their respective prey. The study substantially provides mechanism for improving the survivability of honeybee population and to halt the negative impact of Varroa-mite infestation on honeybee population by introducing Pseudoscorpion to serve as a super predator of Varroa-mite. Also, the study significantly shows the impact of virus on honeybee infested with virus-carrying Varroa-mite. The global stability analysis and Hopf-bifurcation of the interior steady state existence of the system are studied and explored. The analytical results matched with the numerical results of the model developed.

Fishery resource management with migratory prey harvesting in two zones- delay and stochastic approach

In this work, we looked at a two-zone aquatic habitat where both prey and predators can access the zones. The prey alternates between two zones at random. The growth of prey in the absence of a predator is believed to be logistic in each zone. The inner steady state is determined. Around the interior steady state, the deterministic model’s local and global stability is investigated. Furthermore, a stochastic stability study is performed in the neighbourhood of a positive steady state, using analytical estimates of population mean square fluctuations to investigate the system’s dynamics in the presence of Gaussian white noise.

Qualitative Analysis of a Nicholson-Bailey Model in Patchy Environment

We studied a host-parasite model qualitatively. The host-parasitoid model is obtained by modifying the Nicholson-Bailey model so that the number of hosts that parasitoids can't attack is fixed. Topological classification of equilibria is achieved with the implementation of linearization. Furthermore, Neimark-Sacker bifurcation is explored using the bifurcation theory of normal forms at interior steady-state. The bifurcation in the model is controlled by implementing two control strategies. The theoretical studies are backed up by numerical simulations, which show the conclusions and their importance.

Fear induced dynamics on Leslie-Gower predator-prey system with Holling-type IV functional response

This paper analyzes the effect of fear in a Leslie-Gower predator-prey system with Holling type IV functional response. Firstly, we show positivity and boundedness of the system. Then we discuss the structure of the positive equilibrium point, dynamical behavior of all the steady states and long term survival of all the populations in the system. It is shown that fear factor has an impact on the prey and predator equilibrium densities. We have shown the occurrence of transcritical bifurcation around the axial steady state. The presence of a Hopf bifurcation near the interior steady state has been developed by choosing the level of fear as a bifurcation parameter. Furthermore, we discuss the character of the limit cycle generated by Hopf bifurcation. A global stability criterion of the positive steady state point is derived. Numerically, we checked our analytical findings.

Fuzzy Analysis of SVIRS Disease System with Holling Type-II Saturated Incidence Rate and Saturated Treatment

This article presents a fuzzy SVIRS disease system with Holling type-II saturated incidence rate and saturated treatment, in which all parameters related to population dynamics have been considered as fuzzy numbers. Then, the existence condition and permanence of the SVIRS model have been discussed and we derived disease-free and endemic equilibrium points of the proposed fuzzy system. The local stability conditions of the fuzzy system around these equilibrium points using Routh–Hurwitz criteria are discussed. We also verified global stability around the interior steady state using Lozinskii measure. Computer simulations are provided to understand the dynamics of the proposed system. Parameter analysis is carried out with the help of computer simulation. Fuzzy provides better solution for any disease modeling in many ways like disease detection and transmission, different stages of disease, risk analysis (through parameter analysis), and optimal recovery solutions. Earlier literature acts as background to startup this disease modeling and optimal solutions provided by fuzzy logic is one of the motivational key element behind this fuzzy SVIRS model with Holling type-II and its analysis by both analytical and computer simulation.

Dynamics of an HTLV-I infection model with delayed CTLs immune response

This paper deals with a four dimensional mathematical model for Human T-cell leukemia virus type-I (HTLV-I) infection that includes delayed CD8+ cytotoxic T-cells (CTLs) immune response. The proposed system has three biologically feasible steady states, namely disease-free steady state, CTL-inactive steady state and an interior steady state. Our theoretical analysis demonstrates that local and global stability analysis are established by the two critical parameters R0 and R1, basic reproduction numbers due to viral infection and due to CTLs immune response, respectively. The disease-free steady state E0 is globally stable if R0≤1, and the HTLV-I infections are eliminated. The asymptotic-carrier steady state E1 is globally stable if R1≤1<R0, which indicates HTLV-I infection is chronic without persistence of CTLs immune response. The interior steady state E2 is globally asymptotic stable if R1>1, which implies that the HTLV-I infection is choric in persistence of CTLs immune response. Due to immune response delay, our proposed model undergoes a destabilization of the interior steady state leading to Hopf bifurcation and periodic oscillations. We estimate the length of time delay that preserve the stability of period-1 limit cycle. We also derived the direction and stability of Hopf bifurcation around the interior steady state by center manifold theory and normal form method. To determine the robustness of the model, we performed normalized forward sensitivity analysis with reference to R0 and R1. Our proposed model undergoes Hopf bifurcation with respect to the production rate of uninfected CD4+T cells h, removal rate of virus-specific CTLs d4, spontaneous infected CD4+T cell activation d2 and transmissibility coefficient β. Implications of our numerical illustrations to the pathogenesis of HTLV-I infection and the development of HTLV-I related HAM/TSP are explored.

Dynamic analysis of the e-SITR model for remote wireless sensor networks with noise and Sokol-Howell functional response

Wireless Sensor network (WSN) is used for a wide range of uses, including defense, healthcare, environmental tracking etc. In this work, we have formulated a Susceptible-infected-terminally infected-recovered (SITR) model to investigate the offensive worm dynamics in remote sensor networks utilizing Sokol-Howell functional response. The positive invariance and boundedness of the model have been investigated in the proposed work. In addition to that, the existence of possible equilibrium points with feasible conditions has been evaluated. The local stability of all equilibrium points and the global stability of the interior steady state of nodes are measured with the Rougth-Hurwitz criteria, Lasalle’s invariance principle, and Bendixson-Dulac criteria. The impact of noise on the proposed model is being investigated. The numerical imitations are carried out, which strengthens the diagnostic discoveries. Our numerical simulation reveals that growth rate, internode interference coefficients, and crashing rates are critical to the model's dynamics.

ROLE OF ALLEE EFFECT AND HARVESTING OF A FOOD-WEB SYSTEM IN THE PRESENCE OF SCAVENGERS

The role of scavengers, which consume the carcasses of predators along with predation of the prey, has been ignored in comparisons to herbivores and predators. It has now become a topic of high interest among researchers working with food-web systems of prey–predator interactions. The food-web considered in these works contains prey, predators, and scavengers as the third species. In this work, we attempt to study a food-web model of these species in the presence of the multiplicative Allee effect and harvesting. It is observed that this makes the model more complex in the form of multiple co-existing steady states. The conditions for the existence and local stability of all possible steady states of the proposed system are analyzed. The global stability of the steady state lying on the x-axis and the interior steady state have been discussed by choosing suitable Lyapunov functions. The existence conditions for saddle-node and Hopf bifurcations are derived analytically. The stability of Hopf bifurcating periodic solutions with respect to both Allee and harvesting constants is examined. It is also observed that multiple Hopf bifurcation thresholds occur for harvesting parameters in the case of two co-existing steady states, which indicates that the system may regain its stability. The proposed model is also studied beyond Hopf bifurcation thresholds, where we have observed that the model is capable of exhibiting period-doubling routes to chaos, which can be controlled by a suitable choice of Allee and harvesting parameters. The largest Lyapunov exponents and sensitivity to initial conditions are examined to ensure the chaotic nature of the system.

COVID-19 and Stigma: Evolution of Self-restraint Behavior.

Social stigma can effectively prevent people from going out and possibly spreading COVID-19. Using the framework of replicator dynamics, we analyze the interaction between self-restraint behavior, infection with viruses such as COVID-19, and stigma against going out. Our model is analytically solvable with respect to an interior steady state in contrast to the previous model of COVID-19 with stigma. We show that a non-legally binding policy reduces the number of people going out in a steady state.

Global Dynamics of a Delayed Fractional-Order Viral Infection Model With Latently Infected Cells

In this paper, we propose a fractional-order viral infection model, which includes latent infection, a Holling type II response function, and a time-delay representing viral production. Based on the characteristic equations for the model, certain sufficient conditions guarantee local asymptotic stability of infection-free and interior steady states. Whenever the time-delay crosses its critical value (threshold parameter), a Hopf bifurcation occurs. Furthermore, we use LaSalle’s invariance principle and Lyapunov functions to examine global stability for infection-free and interior steady states. Our results are illustrated by numerical simulations.

Spatiotemporal dynamics of a glioma immune interaction model

We report a mathematical model which depicts the spatiotemporal dynamics of glioma cells, macrophages, cytotoxic-T-lymphocytes, immuno-suppressive cytokine TGF-β and immuno-stimulatory cytokine IFN-γ through a system of five coupled reaction-diffusion equations. We performed local stability analysis of the biologically based mathematical model for the growth of glioma cell population and their environment. The presented stability analysis of the model system demonstrates that the temporally stable positive interior steady state remains stable under the small inhomogeneous spatiotemporal perturbations. The irregular spatiotemporal dynamics of gliomas, macrophages and cytotoxic T-lymphocytes are discussed extensively and some numerical simulations are presented. Performed some numerical simulations in both one and two dimensional spaces. The occurrence of heterogeneous pattern formation of the system has both biological and mathematical implications and the concepts of glioma cell progression and invasion are considered. Simulation of the model shows that by increasing the value of time, the glioma cell population, macrophages and cytotoxic-T-lymphocytes spread throughout the domain.

Exploring the dynamics of a tumor-immune interplay with time delay

With the effect of discrete time delay in deliberation, we propose and analyze a conceptual mathematical model for the tumor-immune interaction. The proposed model is delineated by a system of three coupled non-linear ordinary differential equations (ODEs), namely tumor cells, effector cells and cytokine Interleukin-2 (IL-2). Though simple, the model can have complicated dynamical behaviors. We have discussed the qualitative properties of the mathematical model including existence and positivity of the solutions. We find out tumor-free singular point and interior steady states in our mathematical model. We have studied the local stability analysis of the biological feasible steady states in both of the delayed and non-delayed system. By using transversality condition, we have analyzed Hopf bifurcation by using time delay τ as a bifurcation parameter. We have estimated the length of time delay parameter applying Laplace transformation for preserving the stability of period-1 limit cycle that provides the idea about the mode of action in controlling oscillations in the growth of tumor cells. We performed numerical simulations and explored their biological implications to validate our theoretical analysis. We have also drawn bifurcation diagram of delayed model with reference to the intrinsic growth rate α of tumor cell, deactivation rate d1 of tumor cell, activation rate c2 of effector cell and death rate d2 of effector cell. Theoretical and numerical analysis show that in presence of IL-2 the effector cells can cause the tumor cell population to regress.

Delay differential model of one-predator two-prey system with Monod-Haldane and holling type II functional responses

In this paper, we study the dynamics of a delay differential model of predator-prey system involving teams of two-prey and one-predator, with Monod-Haldane and Holling type II functional responses, and a cooperation between the two-teams of preys against predation. We assume that the preys grow logistically and the rate of change of the predator relies on the growth, death and intra-species competition for the predators. Two discrete time-delays are incorporated to justify the reaction time of predator with each prey. The permanence of such system is proved. Local and global stabilities of interior steady states are discussed. Hopf bifurcation analysis in terms of time-delay parameters is investigated, and threshold parameters τ1* and τ2* are obtained. Sensitivity analysis that measures the impact of small changes in the model parameters into the model predictions is also investigated. Some numerical simulations are provided to show the effectiveness of the theoretical results.

Optimal Positive Capital Taxes at Interior Steady States

We generalize recent results of Bassetto and Benhabib (2006) and Straub and Werning (2019) in a neoclassical model with endogenous labor-leisure choice where all agents are allowed to save and accumulate capital. We provide a sufficient condition under which optimal redistributive capital taxes remain at their allowed upper bound forever, even if the resulting equilibrium trajectory converges to a unique steady state with positive and finite consumption, capital, and labor. We then provide an interpretation of our sufficient condition. Using recent evidence on wealth distribution in the United States, we argue that our sufficient condition is empirically plausible. (JEL D31, E21, H21, H23, H25, J22)

Period-Doubling and Neimark–Sacker Bifurcations of a Beddington Host-Parasitoid Model with a Host Refuge Effect

In this paper, we explore the dynamics of a certain class of Beddington host-parasitoid models, where in each generation a constant portion of hosts is safe from attack by parasitoids, and the Ricker equation governs the host population. Using the intrinsic growth rate of the host population that is not safe from parasitoids as a bifurcation parameter, we prove that the system can either undergo a period-doubling or a Neimark–Sacker bifurcation when the unique interior steady state loses its stability. Then, we apply the new theory to the following well-known cases: May’s model, [Formula: see text]-model, Hassel and Varley (HV)-model, parasitoid-parasitoid (PP) model and [Formula: see text] model. We use numerical simulations to confirm our theoretical results.

Holling-Tanner prey-predator model with Beddington-DeAngelis functional response including delay

ABSTRACT This paper is designed based on the combined bioeconomic harvesting of Holling-Tanner prey-predator competition model with Beddington-DeAngelis functional response with two different delays. The situation and existence of the steady state points for proposed model are discussed. The conditions for local stability of the interior steady state points together with the existence of Hopf bifurcation at the interior steady state point are addressed. The situations for stability and the direction of bifurcating periodic solutions are obtained by using center manifold theory. Also total revenue is calculated using bionomic equilibrium, and it is maximized by using Pontryagin’s maximum principle. Some numerical simulations together with pictorial representations are placed in the paper for showing the effects of various parameters in the considered model. Conclusions with the future study are described at last.

Modeling of Insect-Pathogen Dynamics with Biological Control

In this work, a model has been proposed to analyze the effect of wild plant species on biologically-based technologies for pest control. It is assumed that the pest species have a second food source (wild host plants) except crops. Analytical results prove that the model is well-posed as the system variables are positive and uniformly bounded. The permanence of the system has been verified. Equilibrium points and corresponding stability analysis have also been performed. Numerical figures have supported the fact that the interior steady state if it exists, remains stable for any transmission rate. Henceforth biological control has a stabilizing effect. Furthermore, the results prove that biological control is beneficial not only for wild plants but for crops too.

Residence- and source-based capital taxation in open economies with infinitely-lived consumers

In this paper we investigate tax competition in a neoclassical growth model where each country may use both residence- and source-based capital taxes. We show that both types of capital taxes are zero at any interior steady state, just as in a closed economy. For symmetric countries, and even for countries that differ only with respect to size and productivity, we prove analytically and verify numerically that the open-economy policies coincide exactly with the closed-economy policies in all time periods. For countries that are asymmetric in other dimensions, we find that source-based taxes are used to manipulate the intertemporal terms of trade in the short run. Either way, the fiscal externalities of source-based taxes vanish once residence-based taxes are allowed.

Cooperative hunting in a discrete predator–prey system

We propose and investigate a discrete-time predator–prey system with cooperative hunting in the predator population. The model is constructed from the classical Nicholson–Bailey host-parasitoid system with density dependent growth rate. A sufficient condition based on the model parameters for which both populations can coexist is derived, namely that the predator’s maximal reproductive number exceeds one. We study existence of interior steady states and their stability in certain parameter regimes. It is shown that the system behaves asymptotically similar to the model with no cooperative hunting if the degree of cooperation is small. Large cooperative hunting, however, may promote persistence of the predator for which the predator would otherwise go extinct if there were no cooperation.