Holling type-II Functional Response Research Articles

A Fractional Approach to a Computational Eco-Epidemiological Model with Holling Type-II Functional Response

Eco-epidemiological can be considered as a significant combination of two research fields of computational biology and epidemiology. These problems mainly take ecological systems into account of the impact of epidemiological factors. In this paper, we examine the chaotic nature of a computational system related to the spread of disease into a specific environment involving a novel differential operator called the Atangana–Baleanu fractional derivative. To approximate the solutions of this fractional system, an efficient numerical method is adopted. The numerical method is an implicit approximate method that can provide very suitable numerical approximations for fractional problems due to symmetry. Symmetry is one of the distinguishing features of this technique compared to other methods in the literature. Through considering different choices of parameters in the model, several meaningful numerical simulations are presented. It is clear that hiring a new derivative operator greatly increases the flexibility of the model in describing the different scenarios in the model. The results of this paper can be very useful help for decision-makers to describe the situation related to the problem, in a more efficient way, and control the epidemic.

Predator-dependent transmissible disease spreading in prey under Holling type-II functional response

Abstract This paper focusses on developing two species, where only prey species suffers by a contagious disease. We consider the logistic growth rate of the prey population. The interaction between susceptible prey and infected prey with predator is presumed to be ruled by Holling type II and I functional response, respectively. A healthy prey is infected when it comes in direct contact with infected prey, and we also assume that predator-dependent disease spreads within the system. This research reveals that the transmission of this predator-dependent disease can have critical repercussions for the shaping of prey–predator interactions. The solution of the model is examined in relation to survival, uniqueness and boundedness. The positivity, feasibility and the stability conditions of the fixed points of the system are analysed by applying the linearization method and the Jacobian matrix method.

Qualitative Analysis for a Phytoplankton-Zooplankton Model with Allee Effect and Holling Type II Response

This paper investigates a predator-prey system with Allee effect and Holling type-II functional response consisting of phytoplankton as prey and zooplankton as a predator. For the non-spatial system, we analyzed the stability, the existence of Hopf bifurcation at the coexistence equilibrium and stability of bifurcating periodic solutions are obtained. Moreover, the diffusion driven instability, Hopf bifurcation of the corresponding spatial system with zero flux boundary conditions and Turing instability region regarding parameters are established. Numerical simulations are provided to illustrate theoretical results.

Prey-predator model in drainage system with migration and harvesting

Abstract In this paper, we consider a prey-predator model with a reserve region of predator where generalist predator cannot enter. Based on the intake capacity of food and other factors, we introduce the predator population which consumes the prey population with Holling type-II functional response; and generalist predator population consumes the predator population with Beddington-DeAngelis functional response. The density-dependent mortality rate for prey and generalist predator are considered. The equilibria of proposed system are determined. Local stability for the system are discussed. The environmental carrying capacity is considered as a bifurcation parameter to evaluate Hopf bifurcation in the neighbourhood at an interior equilibrium point. Here the fishing effort is used as a control parameter to harvest the generalist predator population of the system. With the help of this control parameter, a dynamic framework is developed to investigate the optimal utilization of resources, sustainability properties of the stock and the resource rent. Finally, we present a numerical simulation to verify the analytical results, and the system is analyzed through graphical illustrations. The main findings with future research directions are described at last.

Stability of a fractional order harvested prey–predator model in the existence of infection in prey

The growing relationship between prey and their predator is one of the important aspects in the field of ecology and mathematical biology. On the other hand, the utility of fractional calculus in different types of mathematical modelling have been applied extensively. In this paper, a fractional order prey–predator model is developed with the consideration of Holling type-I and Holling type-II functional response of the predator. As infection spreads through prey, the prey population is divided into two parts. In addition, we exploit the effect of harvesting to control the excessive spread of the infection. The existence and uniqueness criteria, the boundedness of the solution of the proposed model are investigated. A number of five possible equilibrium points of the proposed model are determined along with the feasibility conditions for each equilibrium points. The local stability at these equilibrium points and global stability at interior equilibrium point are investigated. Numerical simulation is presented with the help of modified Predictor-corrector method in MATLAB software to understand the dynamics of the proposed model.

Dynamical analysis of a prey-predator model incorporating a prey refuge with variable carrying capacity

A prey-predator model incorporating prey refuge with variable carrying capacity and Holling type-II functional response is proposed and analyzed. The model includes a case of increasing carrying capacity as well as a decreasing carrying capacity case. Sufficient conditions are derived to ensure the existence and local stability of the equilibrium points of the proposed model. Moreover, the occurrence of transcritical bifurcation as well as Hopf bifurcation are investigated. The effect of some model parameter related to the prey refuge and the variable carrying capacity on the prey-predator dynamics has been examined. Numerical simulations are presented to demonstrate the theoretical results and to illustrate the effect of these parameters on the model dynamics. Moreover, a comparison with the constant carrying case has been presented.

A delayed synthetic drug transmission model with two stages of addiction and Holling Type-II functional response

This paper gropes the stability and Hopf bifurcation of a delayed synthetic drug transmission model with two stages of addiction and Holling Type-II functional response. The critical point at which a Hopf bifurcation occurs can be figured out by using the escalating time delay of psychologically addicts as a bifurcation parameter. Directly afterwards, properties of the Hopf bifurcation are explored with aid of the central manifold theorem and normal form theory. Specially, global stability of the model is proved by constructing a suitable Lyapunov function. To underline effectiveness of the obtained results and analyze influence of some influential parameters on dynamics of the model, some numerical simulations are ultimately addressed.

Disease control prey–predator model incorporating prey refuge under fuzzy uncertainty

Ecological system depends on prey predator interaction. Sometimes diseases are spread among prey or predator or both species. In this paper, a disease control prey–predator model with Holling type-II functional response incorporating refuge in the susceptible prey is proposed. The impact of additional food for predator species in the system is investigated. Using some assumptions, the crisp model is formulated and it is converted into fuzzy model and defuzzified models by using basic defuzzification method. In the theoretical section, the condition of boundedness, existence of equilibrium points, global stability of interior equilibrium point and Hopf bifurcation on a refuge parameter are investigated. Theoretical results are verified in our numerical simulation section. Using MATLAB package we examined the behaviour of the species in the presence and absence of additional food in both crisp and fuzzy environments. The phase trajectories for different initial conditions in both environments and variations in population of the species are presented. Using MATCONT package, we present the bifurcation scenarios when the additional food parameter and refuge parameter vary. We compute the existence of Hopf point (H) and branch point (BP) in the model for suitable supply of additional food and intensity of refuge parameter. Finally the sensitivities of the parameters are plotted graphically. This theoretical and numerical study may be useful to control the infectious diseases in real world ecological systems.

A multi-delay model for pest control with awareness induced interventions — Hopf bifurcation and optimal control analysis

Farming awareness is an important measure for pest controlling in agricultural practice. Time delay in controlling pest may affect the system. Time delay occurs in organizing awareness campaigns, also time delay may takes place in becoming aware of the control strategies or implementing suitable controlling methods informed through social media. Thus we have derived a mathematical model incorporating two time delays into the system and Holling type-II functional response. The existence and the stability criteria of the equilibria are obtained in terms of the basic reproduction number and time delays. Stability changes occur through Hopf-bifurcation when time delays cross the critical values. Optimal control theory has been applied for cost-effectiveness of the delayed system. Numerical simulations are carried out to justify the analytical results. This study shows that optimal farming awareness through radio, TV etc. can control the delay induced bifurcation in a cost-effective way.

Order and Chaos in a Prey-Predator Model Incorporating Refuge, Disease, and Harvesting

In this paper, a mathematical model consisting of a prey-predator system incorporating infectious disease in the prey has been proposed and analyzed. It is assumed that the predator preys upon the nonrefugees prey only according to the modified Holling type-II functional response. There is a harvesting process from the predator. The existence and uniqueness of the solution in addition to their bounded are discussed. The stability analysis of the model around all possible equilibrium points is investigated. The persistence conditions of the system are established. Local bifurcation analysis in view of the Sotomayor theorem is carried out. Numerical simulation has been applied to investigate the global dynamics and specify the effect of varying the parameters. It is observed that the system has a chaotic dynamics.

Effect of habitat complexity on rhinoceros and tiger population model with additional food and poaching in Kaziranga National Park, Assam

This paper studies the effect of habitat complexity on Greater one-horned rhinoceros (Rhinoceros unicornis) and tiger (Panthera tigris) population model in Kaziranga National Park (KNP), Assam, India. Based on the analysis of the data collected from PCCF, Wildlife, Assam, three mathematical models are formulated and studied. In view of ecology, the main objective of the study is to increase the size of rhino population in aforesaid park. The mathematical stability and the complex dynamical behavior of systems are analyzed here. In KNP, the immature rhinos are killed by tiger, so the first system describes a stage structured prey–predator interaction , where the rhino is the prey and is divided into immature (horn not developed) and mature (horn developed) category and the tiger is the predator. The immature rhino is killed by the tiger following Holling type-II functional response, while the mature rhino falls prey to human induced poaching activities [Source: PCCF Wildlife, Assam, India]. Though Kaziranga is famous all over the world for the Greater one-horned rhinoceros, many other herbivores are also present in really good number and the tiger preys on them too. Thus, the previous system is modified by modifying the Holling type-II functional response incorporating the effect of additional food availability. To increase the number of rhino and to maintain the ecological balance of KNP, the second system is further extended by introducing habitat complexity in the Holling type-II functional response. In each system, equilibrium points have been obtained and their stability are discussed. Finally, numerical simulations are carried out to illustrate the analytical results. Based on the simulation results, it can be stated that the size of the rhino population increases in the presence of additional food and habitat complexity in spite of the poaching activities to a certain extent. The system shows complex dynamical behavior like Hopf bifurcation with respect to Poaching activity. Sensitivity analysis with respect to four important parameters viz., poaching effect, quality and quantity of the additional food and habitat complexity is also discussed.

Impact of fear on an eco-epidemiological model

Abstract In the present paper, we investigate the impact of fear in a predator-prey model with disease in the prey species. The logistically growing prey population is divided into two groups: susceptible and infected. We take the fear of predators among prey population into consideration, which costs lowering of prey’s growth rate and slims down the interactions among prey individuals. In our model, it is reflected by two decreasing factors of the fear parameter and the predator population: one in the growth term of the susceptibles and the other in the disease transmission term. We choose a general disease transmission function for which mass action, standard incidence, and saturation laws are particular cases. The predator-prey interactions are described by generalized Holling type-II functional response. We explore the effect of fear for three subcases of our model, all of which happen to be identical to three published works (without fear effect). Apart from the preliminary mathematical analysis of our model (e.g. positivity, boundedness, etc.), we find the conditions for existence and local stability of the equilibrium points and study Hopf-bifurcation around the endemic equilibrium point w.r.t. the fear parameters. We observe that fear can eliminate the chaotic oscillations of the system, produced in the absence of fear, by either making the endemic equilibrium regular (stable limit cycle or stable equilibrium point) or moving it towards the disease-free state. We also observe the presence of multiple attractors in the phase-space and different types of bistabilities. We perform extensive numerical simulations to explore the rich dynamics of our model.

Impact of fear effect on the growth of prey in a predator-prey interaction model

Several field data and experiments on a terrestrial vertebrates exhibited that the fear of predators would cause a substantial variability of prey demography. Fear for predator population enhances the survival probability of prey population, and it can greatly reduce the reproduction of prey population. Based on the experimental evidence, we proposed and analyzed a prey-predator system introducing the cost of fear into prey reproduction with Holling type-II functional response. We investigate all the biologically feasible equilibrium points, and their stability is analyzed in terms of the model parameters. Our mathematical analysis exhibits that for strong anti-predator responses can stabilize the prey-predator interactions by ignoring the existence of periodic behaviors. Our model system undergoes Hopf bifurcation by considering the birth rate r0 as a bifurcation parameter. For larger prey birth rate, we investigate the transition to a stable coexisting equilibrium state, with oscillatory approach to this equilibrium state, indicating that the greatest characteristic eigenvalues are actually a pair of imaginary eigenvalues with real part negative, which is increasing for r0. We obtained the conditions for the occurrence of Hopf bifurcation and conditions governing the direction of Hopf bifurcation, which imply that the prey birth rate will not only influence the occurrence of Hopf bifurcation but also alter the direction of Hopf bifurcation. We identify the parameter regions associated with the extinct equilibria, predator-free equilibria and coexisting equilibria with respect to prey birth rate, predator mortality rates. Fear can stabilize the predator-prey system at an interior steady state, where all the species can exists together, or it can create the oscillatory coexistence of all the populations. We performed some numerical simulations to investigate the relationship between the effects of fear and other biologically related parameters (including growth/decay rate of prey/predator), which exhibit the impact that fear can have in prey-predator system. Our numerical illustrations also demonstrate that the prey become less sensitive to perceive the risk of predation with increasing prey growth rate or increasing predators decay rate.

The Dynamics of an Eco-Epidemiological Model with Allee Effect and Harvesting in the Predator

The aim of this study was to propose and evaluate an eco-epidemiological model with Allee effect and nonlinear harvesting in predators. It was assumed that there is an SI-type of disease in prey, and only portion of the prey would be attacked by the predator due to the fleeing of the remainder of the prey to a safe area. It was also assumed that the predator consumed the prey according to modified Holling type-II functional response. All possible equilibrium points were determined, and the local and global stabilities were investigated. The possibility of occurrence of local bifurcation was also studied. Numerical simulation was used to further evaluate the global dynamics and the effects of varying parameters on the asymptotic behavior of the system.

Stability of a stage-structure Rosenzweig-MacArthur model incorporating Holling type-II functional response

The local stability of the Rosenzweig-MacArthur predator-prey system with Holling type-II functional response and stage-structure for prey is studied in this paper. It is shown that the model has three equilibrium points. The trivial equilibrium point is always unstable while two other equilibrium points, i.e., the predator extinction point and the coexistence point, are conditionally stable. When the predation process on prey increases, the number of predator increases. If the predation rate is less than or equal to the reduction rate of the predator, then the predator will go to extinct. By using the Routh-Hurwitz criterion, the local stability of the interior equilibrium point is investigated. It is also shown that the model undergoes a Hopf-bifurcation around the coexisting equilibrium point. The dynamics of the system are confirmed by some numerical simulations.

The impact of provision of additional food to predator in predator–prey model with combined harvesting in the presence of toxicity

In this article, the global dynamics of a predator–prey system incorporating combined harvesting has been investigated. The additional food is being provided to the predator, accordingly, modified Holling type-II functional response is considered in the model. The presence of toxicants affect the quality of food for both the species, reducing their growth. The steady states of the system are obtained under some suitable conditions. The local and global dynamics are explored. The conditions for permanence and existence for bionomic equilibrium of the system have been investigated. It is also observed that the system exhibits local bifurcations i.e., transcritical, Hopf, saddle-node as well as global bifurcations i.e., Bogdanov–Takens bifurcation and generalized Hopf bifurcation with respect to the suitable set of parameters. Optimal harvesting policy is discussed with the help of Pontryagin’s maximum principle to preserve both the species from extinction and maintain a sustainable fishery. Numerical simulations are carried out for the suitable choice of parameters.

Detection of turing patterns in a three species food chain model via amplitude equation

Study of pattern formation in three species spatio-temporal food chain models with the help of amplitude equations within Turing domain is a much less explored area compared to that of two species prey-predator models. These models play a vital role in understanding the resource-consumer interactions at the successive trophic levels of respective ecosystems. In this work, we consider the spatio-temporal extension of a three species food chain model with Holling type-II functional response to describe the grazing pattern of both the intermediate and top predator populations. Conditions for Turing instability are derived. The thresholds for emergence and stability of various Turing patterns near the Turing bifurcation boundary are obtained with the help of amplitude equations using weakly nonlinear analysis. The analytical results are further verified with the help of extensive numerical simulations for the complete nonlinear model and both the results are compared for different sets of parameter values.

Global stability and Hopf-bifurcation of prey-predator system with two discrete delays including habitat complexity and prey refuge

In this paper, we consider a two-dimensional prey-predator system with two delays. One delay is for negative feedback of the prey population and another is for gestation delay of the predator population. The predator is partially dependent on the prey followed by Holling type-II functional response. Due to habitat complexity and prey refuge, the Holling type-II functional response is modified in this work. We discuss the boundedness, permanence, local and global asymptotic behavior of the non-delayed and delayed models. The existence of periodic solutions via Hopf-bifurcation with respect to both the delays is established. The stability and direction of Hopf-bifurcation is also analyzed by using Normal form theory and Centre manifold theory. Lastly, numerical simulations have been carried out to confirm the analytical findings. The main objective of this work is to balance the prey-predator relationship in the presence of habitat complexity, prey refuge and delays.

Dynamics of a Three Species Predator-Prey Delay Differential Model with Allee Effect and Holling Type-II Functional Response

In recent years, predator-prey models appearing in various fields of mathematical biology have been proposed and studied extensively. In this paper, we propose a delay differential model for predator-prey systems of one predator and two preys with cooperation/competition among the preys. The growth of both prey populations is subjected to the Allee effect in the logistic terms (the growth rate increases with the population density while it decreases at larger densities), with Holling type I and II functional responses with the predators. Discrete time delays are incorporated into the model to justify time required by predator to interact with the preys. We study the existence of positive equilibrium points and their asymptotic stabilities. Hopfbifurcations are obtained in terms of critical values of time delays. The system may have a stable periodic orbit, depending on parameter values. The presence of Allee terms and time delays in the model improves the stability of the solutions and enriches the dynamics of the model. Some numerical examples and simulations are provided to validate the derived theoretical results.

Study of an eco-epidemiological model with Z-type control

In the present paper, an eco-epidemic model with Z-type control mechanism has been proposed and analyzed. We consider a predator-prey model with Holling type-II functional response, where prey is subjected to disease infection. We observe that disease may destabilize the system by producing chaotic oscillations. To confirm the occurrence of chaos, we draw the Poincare map and also compute the Lyapunov exponents. We further observe that if the indirect Z-controller is applied in the predator population, then the chaos as well as the disease can be eliminated from the system. To explore the global dynamics of the system and the possible applications of Z-type control mechanism, we perform extensive numerical experiments.