Holling type-II Functional Response Research Articles

The modeling and mathematical analysis of the fractional-order of Cholera disease: Dynamical and Simulation

In this study, a cholera model with asymptomatic carriers was examined. A Holling type-II functional response function was used to describe disease transmission. For analyzing the dynamical behavior of cholera disease, a fractional-order model was developed. First, the positivity and boundedness of the system’s solutions were established. The local stability of the equilibrium points was also analyzed. Second, a Lyapunov function was used to construct the global asymptotic stability of the system for both endemic and disease-free equilibrium points. Finally, numerical simulations and sensitivity analysis were carried out using matlab software to demonstrate the accuracy and validate the obtained results.

Dynamics of a controlled discrete prey-predator model with Holling type-II functional response

Dynamics of a controlled discrete prey-predator model with Holling type-II functional response

Effect of fear and non-linear predator harvesting on a predator–prey system in presence of environmental variability

In this paper, we have proposed and analyzed a predator–prey system introducing the cost of predation fear into the prey reproduction with Holling type-II functional response in the stochastic environment with the consideration of non-linear harvesting on predators. The system experiences Transcritical, Saddle–node, Hopf, and Bogdanov-Taken (BT) bifurcation with respect to the intrinsic growth rate and competition rate of the prey populations. We have discussed the existence and uniqueness of positive global solution of the stochastic model with the help of Ito’s integral formula and the long-term behavior of the solution is derived here. The existence of stationary distribution and explicit form of the density function is established here when only prey populations survive or both populations. We have shown that due to high fluctuation, the regime changes from one stable state to another state when bistability occurs in the system. The paper ends with some conclusions.

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.

Hopf bifurcation and patterns formation in a diffusive two prey-one predator system with fear in preys and help

Recognizing the relationship between the spatial patterns in species concentrations and ecological heterogeneity is crucial for understanding demographics and species governance in a given domain, as ecological patterning processes are believed to be imitated in real ecosystems. In this present article, we have considered a two-prey-one-predator system with Holling type-II functional response where both the preys have fear of predator and compete in the absence of predator. Additionally, we have assumed that both the prey help each other when the predator attacks them. We have incorporated self-diffusion in this system under Neumann boundary conditions. The existence and stability conditions of different equilibria are discussed. We have studied the Hopf bifurcation around positive offset and derived the stability and direction of periodic solution. We have done a series of numerical simulations to ensure our theoretical finding. Diffusion-driven instability conditions are obtained. Different instability regions and fascinating patterns, such as spots, mixtures, and stripes, are depicted. Spatiotemporal dynamics also reflects that prey populations are located in isolated areas of low population concentration with increasing level of fear, whereas an increase in handling time forms lower-density stripes. But with increasing help, prey becomes more concentrated in some stripe regions. In the Hopf-Turing region, it is observed that the diffusion coefficient of predators can stabilize the system. Overall, pattern creation in predator-prey systems can help anticipate long-term system behaviour and analyze the influence of ecological factors on system dynamics.

On the stability of a Caputo fractional order predator-prey framework including Holling type-II functional response along with nonlinear harvesting in predator

In this study, we look at a fractional-order two-species predator-prey framework that uses Caputo Sense's fractional derivative to try to understand its dynamics, which include a simplified Holling type II functional response and nonlinear harvesting in the predator. Initially, we establish certain mathematical facts, such as the solutions of fractional order dynamical systems being unique, non-negative, bounded, and existing. The dissipativeness of the FDE system solution is addressed. Our investigation also includes a look at the local stability requirements for every possible equilibrium point. As a bifurcation constraint, the order of derivatives has been considered in our discussion of the existence condition of Hopf bifurcation. Our analysis has been strengthened by the incorporation of fractional Hopf bifurcation, and the dynamics of the proposed model are influenced by selective non-linear harvesting. Furthermore, the investigation focuses on the impact of the predator's pace of harvesting about the complexities of the prey-predator relationship. Through the use of certain setting values, empirical evidence indicates as the non-integer order framework displays stable to unstable, when the non-integer order is increased. The analytical findings are shown by various instances presented in the numerical part.

Dynamics of a Delayed Fractional-Order Predator–Prey Model with Cannibalism and Disease in Prey

In this study, a class of delayed fractional-order predation models with disease and cannibalism in the prey was studied. In addition, we considered the prey stage structure and the refuge effect. A Holling type-II functional response function was used to describe predator–prey interactions. First, the existence and uniform boundedness of the solutions of the systems without delay were proven. The local stability of the equilibrium point was also analyzed. Second, we used the digestion delay of predators as a bifurcation parameter to determine the conditions under which Hopf bifurcation occurs. Finally, a numerical simulation was performed to validate the obtained results. Numerical simulations have shown that cannibalism contributes to the elimination of disease in diseased prey populations. In addition, the size of the bifurcation point τ0 decreased with an increase in the fractional order, and this had a significant effect on the stability of the system.

Stability and Hopf Bifurcation of a Delayed Predator–Prey Model with a Stage Structure for Generalist Predators and a Holling Type-II Functional Response

In this paper, we carry out some research on a predator–prey system with maturation delay, a stage structure for generalist predators and a Holling type-II functional response, which has already been proposed. First, for the delayed model, we obtain the conditions for the occurrence of stability switches of the positive equilibrium and possible Hopf bifurcation values owing to the growth of the value of the delay by applying the geometric criterion. It should be pointed out that when we suppose that the characteristic equation has a pair of imaginary roots λ=±iω(ω>0), we just need to consider iω(ω>0) due to the symmetry, which alleviates the computation requirements. Next, we investigate the nature of Hopf bifurcation. Finally, we conduct numerical simulations to verify the correctness of our findings.

Dynamic Complexity of a Nicholson–Bailey Bioeconomic Model with Holling Type-II Functional Response

In this paper, we propose a host–parasitoid model with a Holling type-II functional response and incorporate harvest effort. The Holling type-II response leads to saturation in parasitized hosts, creating a potential economic harvesting opportunity. To address overexploitation risks, we integrate a harvest effort function, determining an optimal threshold to prevent depletion. We explore model dynamics and bifurcations, including co-dimension one behaviors such as flip and Neimark–Sacker bifurcations, we provide numerical examples for validation. Our suggested difference-algebraic model, compared to continuous-time models, exhibits rich dynamics within the Nicholson–Bailey host–parasitoid framework.

Role of Allee and Fear for Controlling Chaos in a Predator–Prey System with Circulation of Disease in Predator

This paper explores a predator–prey system featuring fear and disease within the predator population,utilizing the Rosenzweig–MacArthur model with Holling type-II functional response. The primary focus lies in investigating the impact of a fear factor, wherein the prey’s growth rate is hindered due to predator-induced fear. Additionally, the model accounts for the spread of disease among predators,leading to a division between susceptible and infected predator subpopulations. The inclusion of an Allee effect in the susceptible predator further enriches the model. The study involves a thorough examination, encompassing local and global stability analysis as well as Hopf bifurcation analysis around the interior equilibrium point. Numerical simulations underscore a noteworthy observation: an escalation in interaction force propels the system into chaotic dynamics,marked by stable focus, limit cycles and period-doubling phenomena. A noteworthy finding pertains to the influence of the Allee parameter ([Formula: see text]) on chaotic dynamics. As the Allee parameter values increase, the system tends to stable focus through a sequence of chaotic states, period-doubling and limit cycles. Subsequently, the paper introduces the role of another pivotal parameter, the fear factor, into the chaotic dynamics. Intriguingly, chaos transforms into stable focus through diverse nonlinear phenomena, including period-doubling and limit cycles. This nuanced exploration of parameters sheds light on the intricate dynamics governing the predator–prey system, offering a comprehensive understanding of the interplay between fear, disease and ecological factors. So our observation throughout this paper that how chaos behaves here after one by one injection of our new features: fear factor and Allee parameter?

Codimension-one bifurcation analysis and chaos of a discrete prey–predator system

In this paper, we explore local dynamics at equilibrium points, the existence of bifurcation sets and codimension-one bifurcation analysis of a discrete prey–predator system with Holling type-II functional response. Further, OGY and Hybrid control strategies are utilized to control chaos in the under study discrete model due to the occurrence of Neimark–Sacker and flip bifurcations. Finally, numerical simulations are given to verify theoretical results.

Genetic differentiation of prey predator interaction model along with an Holling type-II functional response

Prey–predator models play a vital role in mathematical biology. The competition that evolves between them is addressed in a broad scenario. According to genetics theory, various specie’s genotypes yield different outcomes in competition. This insight motivated us to introduce a genetic distinction between both predators and prey. The well-posedness of the system was determined for the proposed genetic variation prey–predator model in this research. With the use of equilibrium points, the model’s stability was examined using the center manifold theorem. Both local and global stability also proved for the system. Finally, numerical simulations are carried out with the suitable parameter values of the system.

Dynamics of a Class of Prey-Predator Interaction Model with Holling Type-II Functional Response Incorporatingthe Effect of Fear on Prey and Non-Linear Predator Harvest

Dynamics of a Class of Prey-Predator Interaction Model with Holling Type-II Functional Response Incorporatingthe Effect of Fear on Prey and Non-Linear Predator Harvest

Complex dynamics of a four-species food-web model: An analysis through Beddington-DeAngelis functional response in the presence of additional food

Abstract The four-dimensional food-web system consisting of two prey species for a generalist middle predator and a top predator is proposed and investigated. The middle predator is predating over both the prey species with a modified Holling type-II functional response. The food-web model is effectively formulated, exhibits bounded behavior, and displays dissipative dynamics. The proposed model’s essential dynamical features are studied regarding local stability. We investigated the four species’ survival and established their persistence criteria. In the proposed model, a transcritical bifurcation occurs at the axial equilibrium point. The numerical simulations reveal the persistence of a chaotic attractor or stable focus. The conclusion is that increasing the food available to the middle predator may make it possible to manage and mitigate the chaos within the food chain.

Exploring dynamics of plant-herbivore interactions: bifurcation analysis and chaos control with Holling type-II functional response.

In this study, we examine the plant-herbivore discrete model of apple twig borer and grape vine interaction, with a particular emphasis on the extended weak-predator response to Holling type-II response. We explore the dynamical and qualitative analysis of this model and investigate the conditions for stability and bifurcation. Our study demonstrates the presence of the Neimark-Sacker bifurcation at the interior equilibrium and the transcritical bifurcation at the trivial equilibrium, both of which have biological feasibility. To avoid unpredictable outcomes due to bifurcation, we employ chaos control methods. Furthermore, to support our theoretical and mathematical findings, we develop numerical simulation techniques with examples. In summary, our research enhances the comprehension of the dynamics pertaining to interactions between plants and herbivores in the context of discrete-time population models.

Invasion analysis of a reaction–diffusion-advection predator–prey model in spatially heterogeneous environment

In this paper, we study a reaction–diffusion–advection predator–prey model with Holling type-II functional response in heterogeneous environment is considered, where the prey are subject to both random and directed movements. We show the effect of changes in diffusion rates of predator and prey on whether prey can be invaded when advection rate is small. Moreover, we show that the predator with a large advection rate is driven to extinction.

Spatiotemporal dynamics of a multi-delayed prey-predator system with variable carrying capacity.

This paper presents the temporal and spatiotemporal dynamics of a delayed prey-predator system with a variable carrying capacity. Prey and predator interact via a Holling type-II functional response. A detailed dynamical analysis, including well-posedness and the possibility of coexistence equilibria, has been performed for the temporal system. Local and global stability behavior of the co-existence equilibrium is discussed. Bistability behavior between two coexistence equilibria is demonstrated. The system undergoes a Hopf bifurcation with respect to the parameter β, which affects the carrying capacity of the prey species. The delayed system exhibits chaotic behavior. A maximal Lyapunov exponent and sensitivity analysis are done to confirm the chaotic dynamics. In the spatiotemporal system, the conditions for Turing instability are derived. Furthermore, we analyzed the Turing pattern formation for different diffusivity coefficients for a two-dimensional spatial domain. Moreover, we investigated the spatiotemporal dynamics incorporating two discrete delays. The effect of the delay parameters in the transition of the Turing patterns is depicted. Various Turing patterns, such as hot-spot, coldspot, patchy, and labyrinth, are obtained in the case of a two-dimensional spatial domain. This study shows that the parameter β and the delay parameters significantly instigate the intriguing system dynamics and provide new insights into population dynamics. Furthermore, extensive numerical simulations are carried out to validate the analytical findings. The findings in this article may help evaluate the biological revelations obtained from research on interactions between the species.

Study on a fractional order delayed predator-prey model including prey refuge and type II functional response

The fractional-order delayed predator-prey dynamical scheme studied here includes a prey refuge with Holling type-II functional response. The existence and uniqueness of a solution for our scheme are explored. Hopf bifurcation with respect to delay is reconnoitered, besides local and global stability of existing steady states of fractional order0 < q ≤ 1. It is discovered that the delay passes through a series of crucial values when Hopf bifurcation takes place. Additionally, both theoretically and statistically, the impacts of prey refuge effects and fractional order on system stability are investigated. The delayed differential scheme's dynamics and results durability are improved by fractional order. Fractional order enhances the stability of the solutions and enriches the dynamics of the delayed differential model. Additionally, both theoretically and by the use of numerical simulations, the effects of fractional order and the prey refuge effects on the stability of the system are investigated.

COMBINED IMPACT OF CANNIBALISM AND ALLEE EFFECT ON THE DYNAMICS OF A PREY–PREDATOR MODEL

In ecological environment, Allee effect is one of the important factors which cause significant changes to the system dynamics. In this paper, using the theory of dynamical systems, we analyze a variation of a standard cannibalistic two-dimensional prey–predator model with Holling type-II functional response in the presence of both weak and strong Allee effects. We have analyzed the impact of strong and weak Allee effects on the dynamics of a cannibalistic system, knowing the dynamics of the cannibalistic model without Allee effect. We have deduced that in the presence of cannibalism, both strong and weak Allee effects generate bistability between equilibrium points. For strong Allee effect, bistability occurs between trivial equilibrium point and predator-free equilibrium point as well as between trivial and coexistence equilibrium points. But for weak Allee effect, bistability occurs only between coexistence equilibrium points. We also pointed out that the cannibalistic system without Allee effect exhibits tristability among the trivial equilibrium point, coexistence equilibrium point having low prey concentration and coexistence equilibrium point having comparatively high prey concentration. But in the presence of strong Allee effect, cannibalistic system experiences tristability among trivial and two other stable coexistence equilibrium points. By a comprehensive bifurcation analysis, we have observed that Allee effect enriches both the local and global dynamics of the system. Here, we have reported all possible codimension-one and codimension-two bifurcations extensively by choosing cannibalism, Allee effect and predator natural death rate as the bifurcation parameters. In the analysis of bifurcations, we have explored the existence of transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation and Bautin bifurcation. Our analytical findings are validated through exhaustive numerical simulations. Finally, we have reported a comparative study between the impacts of strong and weak Allee effects on the dynamics of the cannibalistic system.

Analytical detection of stationary and dynamic patterns in a prey-predator model with reproductive Allee effect in prey growth.

Allee effect in population dynamics has a major impact in suppressing the paradox of enrichment through global bifurcation, and it can generate highly complex dynamics. The influence of the reproductive Allee effect, incorporated in the prey's growth rate of a prey-predator model with Beddington-DeAngelis functional response, is investigated here. Preliminary local and global bifurcations are identified of the temporal model. Existence and non-existence of heterogeneous steady-state solutions of the spatio-temporal system are established for suitable ranges of parameter values. The spatio-temporal model satisfies Turing instability conditions, but numerical investigation reveals that the heterogeneous patterns corresponding to unstable Turing eigenmodes act as a transitory pattern. Inclusion of the reproductive Allee effect in the prey population has a destabilising effect on the coexistence equilibrium. For a range of parameter values, various branches of stationary solutions including mode-dependent Turing solutions and localized pattern solutions are identified using numerical bifurcation technique. The model is also capable to produce some complex dynamic patterns such as travelling wave, moving pulse solution, and spatio-temporal chaos for certain range of parameters and diffusivity along with appropriate choice of initial conditions. Judicious choices of parametrization for the Beddington-DeAngelis functional response help us to infer about the resulting patterns for similar prey-predator models with Holling type-II functional response and ratio-dependent functional response.