Holling type-II Functional Response Research Articles

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

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

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.

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.

ALLEE EFFECT IN A RICKER TYPE DISCRETE-TIME PREDATOR–PREY MODEL WITH HOLLING TYPE-II FUNCTIONAL RESPONSE

In recent years, the stability of the predator–prey model subject to the Allee effect has become an interesting issue. This study investigates the effect of Allee effect on the stability of a discrete-time predator–prey model with Holling type-II functional response. Using equilibrium analysis, stability analysis and bifurcation theory, the mathematical characteristics of the proposed model are examined. Model experiences flip bifurcation and Neimark–Sacker bifurcation based on the center manifold theorem and bifurcation theory. Our analytical results are demonstrated by numerical simulations.

Coexistence of triple, quadruple attractors and Wada basin boundaries in a predator–prey model with additional food for predators

In biological control programs, the provision of additional food is regarded as one of the most useful tools for species conservation and pest management. In a system, providing additional food for predators can distract the predators from the overconsumption of prey (short term) or enhance the predation rate (long term). In the present article, we consider a predator–prey model with Holling type-II functional response by incorporating additional food for the predator species in a discrete-time setup. Here, we intend to explore the impact of additional food on the growth of prey as well as the overall dynamics of the underlying system. We analyze the system dynamics by varying two control parameters, viz, the biomass of additional food and the intrinsic growth rate of prey species simultaneously with the help of Lyapunov exponent and isoperiodic diagrams. Through the simultaneous variation of two parameters, very rich and complex dynamical behaviors of a simple deterministic system can come into view with the presence of structurally stable periodic patterns in the transitional and chaotic zones, which is not looked on only by the variability of one parameter. In this study, we notice the presence of infinite families of different organized periodic structures, like Arnold tongues, shrimp-shaped domains, a new kind of ‘double fishhook’ structures, etc. in the quasiperiodic and chaotic zones of the parameter planes with various kinds of period-adding phenomena. The study also discloses the transition to chaos via shrimp-induced period-bubbling process. We observe the coexistence of double and triple attractors in several different sets of attractors. The most fascinating finding in this study is the coexistence of quadruple attractors, one of the rarest phenomena in ecological systems. The basin boundaries of all these coexisting double, triple, and quadruple attractors are of a very complex nature, they are either fractal basin boundaries or Wada basin boundaries.

Chaos Control of a Delayed Tri-Trophic Food Chain Model with Fear and Its Carry Over Effects

One of the main objectives of theoretical ecologists involves finding mechanisms to control the chaos in ecological models to maintain positive densities of the species. Numerous researchers have suggested that, apart from the direct killing in the prey–predator relationship, there are some indirect effects, such as fear of predation. Induced fear can lead to slowing down the growth rate of the prey species, and this non-chemical strategy can be carried over to successive seasons or upcoming generations. In this work, we explore the impact of fear due to predation and its carry-over effect (COE) in a delayed tri-trophic food chain model, whereas the Holling type-II functional response is used to determine the interference among the species. The proposed model is an asymmetric interaction food chain model since the species in this model only kills other species. The growth rate of prey and middle predators is affected due to the respective fear of predation by middle and special predators. The non-delayed model considered in this paper generalizes the models developed by Hastings–Powell and Panday et al. The gestation delay in the special predator’s growth term is incorporated into the proposed model. We determined the essential conditions for the existence of ecologically feasible equilibrium points and their local and global stability. Furthermore, we developed the conditions for the occurrence of the Hopf bifurcation around an interior equilibrium to seek periodic behaviors of delayed and non-delayed models. Numerical examples were performed to justify the proposed theoretical findings and to show the impacts of fear and its COE parameters on the system dynamics through phase portraits, the time series of solutions, and bifurcation diagrams. We discovered that the chaotic behavior of the food chain model can be controlled by using the fear effect and its COE parameters. The dynamics of the delayed food chain model with the fear effect and its COEs are further explored in our findings. Our theoretical findings clearly provide a mechanism to protect and control species populations in ecological systems. It is also essential for developing optimized harvesting strategies in fisheries and pest management in agriculture.

Dynamical Behaviour Changes in Response to Various Functional Responses: Temporal and Spatial Plankton System

In this paper, we focus on the comparison between ratio-dependent and Holling type-II functional response in the phytoplankton dynamics with the susceptible and infected populations with or without diffusion. The dynamics are investigated in terms of local stability using both the response function for the systems and the definition of the Hopf-bifurcation, which has been demonstrated to occur in the Holling type-II functional response using the half-saturation time (i.e. $$\alpha$$ ) as the bifurcation parameter. We observed that the system’s dynamical behaviour is affected by the response functions and the system is more stable in the case of the ratio-dependent functional response. Furthermore, the study of higher-order perturbation terms reveals the heterogeneity of in-homogeneous spatiotemporal perturbation around temporal steady state outside the linear regime. The system behaviours to both types of response functions are studied numerically in the temporal and spatiotemporal domain (i.e. pattern formation in the diffusive population). Numerical simulation is used to examine the results.

MODELING THE EFFECTS OF INSECTICIDES ON CROP PRODUCTION IN THE PRESENCE OF INSECT POPULATION

In this research work, a nonlinear mathematical model is proposed and analyzed to study the adverse effects of insects on agricultural productivity by controlling the insect population using insecticides. In the model formulation, it is assumed that agricultural crops grow logistically and the growth rate of insects wholly depends on agricultural crops with Holling type-II functional response. It is further assumed that insects uptake insecticides; thus, the amount of insecticides decreases at a rate proportional to its amount and the density of insect population, and the growth rate of insect population decrease in the same proportion. The feasibility of all non-negative equilibria and their stability properties are discussed. Stability analysis specifies that agricultural crop consumption rate destabilizes the system; however, the spraying rate of insecticides stabilizes the system. The conditions for the existence of pitchfork and Hopf-bifurcation are derived. Considering the spraying rate of insecticides as time-dependent, we have also discussed the optimal control strategy to minimize both insect density and the associated cost. The numerical simulation validates the analytical findings.

Dynamics of a fractional plankton–fish model under the influence of toxicity, refuge, and combine-harvesting efforts

In this work, we analyze plankton–fish dynamics in the presence of toxicity, refuge, and combine-harvesting efforts by a considering Holling type-II functional response. We have considered phytoplankton, zooplankton, and fish populations, and the interdependent evolution is presented with the help of the Caputo fractional derivative. Since toxicity in phytoplankton spreads to zooplankton and hence to fish, we have introduced toxic terms in all the populations. On the other hand, to save the population from extinction harvesting is an essential tool. Theoretical aspects are studied in terms of nonnegativity, boundedness, existence, and uniqueness of the solution. Sufficient conditions are derived for the stability of various points of equilibrium. The composite behavior is studied by varying the values of different parameters and fractional derivatives. Numerical simulations are carried out to strengthen the theoretical findings.

Bifurcation Analysis and Spatio-Temporal Patterns of a Prey–Predator Model with Hunting Cooperation

Understanding the effects of cooperative hunting among the predators is an important topic in ecology. We explore a prey–predator model with cooperative hunting among predators modeled by a Holling type-II functional response having saturated encounter rate. Bifurcation analysis is carried out on the temporal model to detect local bifurcations such as transcritical, saddle-node, Hopf bifurcation and global homoclinic bifurcation. A comprehensive numerical bifurcation analysis is performed on the Turing solutions of the corresponding diffusive model. Stable and unstable branches of mode specific Turing solutions and their dependence on parameters are identified. For suitable values of parameters, spatio-temporal chaos via period doubling cascade is observed when a given parameter traverses the Turing–Hopf region.

Bifurcations and chaos control in a discrete-time prey–predator model with Holling type-II functional response and prey refuge

In this paper, we study the complex dynamical behavior in a discrete prey–predator model with Holling type II functional response with prey refuge. It is exhibited that the system shows different types of bifurcations viz. flip bifurcation, transcritical bifurcation and Neimark–Sacker bifurcation(NSB) by using center manifold theorem and bifurcation theory. Moreover, the chaos occurred in the system has been controlled by deploying control strategies viz. state feedback, pole placement and hybrid control techniques. The certain conditions have been determined under which chaos and bifurcation can be stabilized. The extensive numerical simulation is performed to demonstrate the analytical findings.