Holling type-IV Functional Response Research Articles

Bifurcation and hybrid control in a discrete predator–prey model with Holling type-IV functional response

Bifurcation and hybrid control in a discrete predator–prey model with Holling type-IV functional response

Stochastic time-optimal control and sensitivity studies for additional food provided prey-predator systems involving Holling type-IV functional response

In this study we consider an additional food provided prey-predator model exhibiting Holling type-IV functional response incorporating the combined effects of both the continuous white noise and discontinuous Lévy noise. We prove the existence and uniqueness of global positive solutions for the proposed model. We perform the stochastic sensitivity analysis for each of the parameters in a chosen range. Later we do the time optimal control studies with respect quality and quantity of additional food as control variables. Making use of the arrow condition of the sufficient stochastic maximum principle, we characterize the optimal quality of additional food and optimal quantity of additional food. We then perform the sensitivity of these control variables with respect to each of the model parameters. Numerical results are given to illustrate the theoretical findings with applications in biological conservation and pest management. At the end we briefly study the influence of the noise on the dynamics of the model.

SELF-DIFFUSIVE SPATIOTEMPORAL PATTERNS IN A FOOD CHAIN MODEL THROUGH WEAKLY NONLINEAR ANALYSIS

Density distributions of populations with self-diffusion and interaction in a spatial domain are dynamically visualized with coupled nonlinear reaction–diffusion equations. Incorporating self-diffusion terms creates a more pragmatic modeling paradigm and provides meaningful descriptions of influences on spatiotemporal pattern formation phenomena. This paper examines the effect of self-diffusion in a food chain system with a Holling type-IV functional response and the type of spatial structures forms on a geographical scale due to the random movement of species. We discussed the existence and uniqueness of a positive equilibrium solution and obtained the Turing instability conditions for the self-diffusive food chain model. Moreover, weakly nonlinear analysis close to the Turing bifurcation boundary is used to derive the amplitude equations. The stability of the amplitude equations and sufficient conditions for the emanation of spatiotemporal patterns (such as spots, stripes, and blended patterns) are investigated. The analytical results are verified with numerical simulations. The results are applicable to all environments and can be used to understand the effects of self-diffusion in other food chain models both qualitatively and quantitatively.

A non-standard finite difference scheme for a class of predator–prey systems with non-monotonic functional response

We design and analyse a non-standard finite difference (NSFD) scheme for a class of predator–prey models with non-monotonic functional response. Such functional response can be used, for instance, to model the prey's group defence ability. The numerical scheme is unconditionally dynamically consistent with respect to the positivity and boundedness of the solutions. We establish analytically the convergence of the scheme and provide numerical simulations for a Holling type-IV functional response that support the theoretical findings. The simulations further show that the NSFD scheme is more efficient than some standard numerical methods. In particular, the standard methods show dynamic inconsistencies when large step-sizes were used. On the other hand, the NSFD scheme replicated different behaviours, including the oscillatory dynamics common to predator–prey systems.

Stability switching and chaos in a multiple delayed prey–predator model with fear effect and anti-predator behavior

Recent studies demonstrate that the density of prey population is not only affected by direct killing by the predator, but the fear in prey caused by predator also reduces it by cutting down the reproduction rate of prey community, and prey shows anti-predator behavior in response to this fear. In this study, we propose a prey–predator model with fear in prey due to predator and anti-predator behavior by prey against the predator with fear response delay and gestation delay. It is assumed that the predator consumes prey via simplified Holling Type-IV functional response. We evaluate the equilibrium points and study the local and global stability behavior of the system around them. It is observed that our system undergoes Hopf-bifurcation with respect to the fear parameter. Moreover, the system shows the attribute of bi-stability involving two stable equilibriums. Further, we study the dynamics of the delayed system by incorporating fear response delay and gestation delay. We observe that the delayed system suffers Hopf-bifurcation with respect to both delays. Using the normal form method and center manifold theory, the direction and stability of Hopf-bifurcation are studied. Chaotic behavior for delayed system is observed for large values of fear response delay. All these findings are supported by numerical simulation.

Complete classification of bifurcation curves for a multiparameter diffusive logistic problem with generalized Holling type-IV functional response

We study exact multiplicity and bifurcation curves of positive solutions for the diffusive logistic problem with generalized Holling type-IV functional response $$\displaylines{ u''(x)+\lambda \big[ ru(1-\frac{u}{q})-\frac{u}{1+mu+u^2}\big] =0,\quad-1<x<1, \cr u(-1)=u(1)=0, }$$ where the quantity in brackets is the growth rate function and \(\lambda >0\) is a bifurcation parameter. On the \((\lambda ,||u||_{\infty })\)-plane, we give a complete classification of two qualitatively different bifurcation curves: a C-shaped curve and a monotone increasing curve.
 For more information see https://ejde.math.txstate.edu/Volumes/2021/10/abstr.html

A discrete-time model with non-monotonic functional response and strong Allee effect in prey

In this paper, we investigate the impact of strong Allee effect on the stability of a discrete-time predator–prey model with a non-monotonic functional response. The dynamics of discrete-time predator–prey models with strong Allee effect is studied earlier. But, the mathematical investigations of predator–prey dynamics in discrete-time set up with Holling type-IV functional response and strong Allee effect in prey are lacking. The proposed model supports the coexistence of two steady states, and the mathematical features of the model are analyzed based on local stability and bifurcation theory. By considering the Allee parameter as the bifurcation parameter, we provide sufficient conditions for the flip and the Neimark–Sacker bifurcations. We observe that Allee parameter plays a significant role in the dynamics of the system.

Bifurcation and bio-economic analysis of a prey-generalist predator model with Holling type IV functional response and nonlinear age-selective prey harvesting

Abstract In this paper, we have studied the dynamical behaviour of a Leslie-Gower model with Holling type-IV functional response and nonlinear prey harvesting where target prey for harvesting are above a particular age which is called maturation delay. The model system has been studied using qualitative analysis, bifurcation theory, bionomic equilibrium, MSY and optimal harvesting policy. We explain that the interior equilibrium is delay dependent and locally asymptotically stable. The age-selective harvesting system under goes a Hopf-bifurcation with respect to the maturation delay. Conditions for the MSY and existence of bionomic equilibrium are analyzed and Pontryagin's maximum principle has been used to find the path of a singular control. Difficult analytical results are featured by numerical examples to better understand the system in a simple way.

Permanency in predator–prey models of Leslie type with ratio-dependent simplified Holling type-IV functional response

We consider a predator–prey model of Leslie type with ratio-dependent simplified Holling type-IV functional response. The novelty of the model is that the functional response simulates group defense of the prey kind, which in turn, affects permanency of the system and existence of limit cycles. We show that permanency of the system holds automatically for some values of parameters and provides sufficient conditions for global stability of interior equilibrium by constructing a Lyapunov function. We prove that for some values of parameters the system exhibits a Hopf cycle and provides conditions by which the corresponding stable Hopf cycle is the only cycle that model may have. Numerical simulations show that if the conditions are broken, the model may have more than one limit cycle, which is phenomenal among the predator–prey models with one interior equilibrium.

Chaos control in a discrete-time predator–prey model with weak Allee effect

The stability of the predator–prey model subject to the Allee effect is an interesting topic in recent times. In this paper, we investigate the impact of weak Allee effect on the stability of a discrete-time predator–prey model with Holling type-IV functional response. The mathematical features of the proposed model are analyzed with the help of equilibrium analysis, stability analysis, and bifurcation theory. We provide sufficient conditions for the flip bifurcation by considering Allee parameter as the bifurcation parameter. We observe that the model becomes stable from chaotic dynamics as the Allee parameter increases. Further, we observe bi-stability behavior of the model between only prey existence equilibrium and the coexistence equilibrium. Our analytical findings are illustrated through numerical simulations.

Bifurcation analysis of a modified Leslie\u2013Gower model with Holling type-IV functional response and nonlinear prey harvesting

In this work, an attempt is made to understand the dynamics of a modified Leslie–Gower model with nonlinear harvesting and Holling type-IV functional response. We study the model system using qualitative analysis, bifurcation theory and singular optimal control. We show that the interior equilibrium point is locally asymptotically stable and the system under goes a Hopf bifurcation with respect to the ratio of intrinsic growth of the predator and prey population as bifurcation parameter. The existence of bionomic equilibria is analyzed and the singular optimal control strategy is characterized using Pontryagin’s maximum principle. The existence of limit cycles appearing through local Hopf bifurcation and its stability is also examined and validated numerically by computing the first Lyapunov number. Optimal singular equilibrium points are obtained numerically for various discount rates.

On the stability and Hopf bifurcation of a predator-prey model

We consider a time delay predator-prey model with Holling type-IV functional response and stage-structured for the prey. Our aim is to observe the dynamics of this model under the influence of gestation delay of the predator. We obtain sufficient conditions for the local stability of each of feasible equilibria of the system and the existence of a Hopf bifurcation at the coexistence equilibrium. By using the normal form theory and center manifold theory we also derive some explicit formulae determining the bifurcation direction and the stability of the bifurcated periodic solutions. Finally, numerical simulations are given to explain the theoretical results.

Self-organized spatial patterns due to diffusion in a Holling–Tanner predator–prey model

In this paper, we study a two-dimensional spatio-temporal Holling–Tanner type predator–prey model incorporating ratio-dependent Holling type-IV functional response. From the analytical conditions of Hopf and diffusion-driven instability with zero flux boundary conditions, Turing region is indicated. The emergence of spatial and spatio-temporal patterns controlled by diffusion process is considered numerically. Our numerical experience suggests that the system can produce different types of patterns with carefully chosen parameters and initial conditions.

Optimal Harvesting Policy of Discrete-Time Predator-Prey Dynamic System with Holling Type-IV Functional Response and Its Simulation

This paper deals with a discrete-time prey-predator system with Holling type-IV function response in the presence of some alternative food to predator and harvesting of prey species. By theoretical analysis and numerical simulation, comparing with the system without harvesting, ecological equilibrium point of the system is removed if harvesting effort is changed, and the appropriate harvesting effort can increase the stability of the system. Moreover, optimal harvesting policy is obtained using Pontryagin’s maximum principle. Meanwhile, some numerical simulations verify our analytical results. This study also gains the maximum economic profit which is based on the ecological equilibrium. The suitable price of resources can control the excessive harvest to promote the sustainable development of species.

Classification of bifurcation diagrams for a multiparameter diffusive logistic problem with Holling type-IV functional response

We study exact multiplicity and bifurcation diagrams of positive solutions for a multiparameter diffusive logistic problem with Holling type-IV functional response{u″(x)+λf(u)=0,−1<x<1,u(−1)=u(1)=0, where the growth rate function f(u)=ru(1−uq)−u1+u2, q, r are positive dimensionless parameters, and λ>0 is a bifurcation parameter. Assume that either r⩽η1q and (q,r) lies above the curve Γ1={(q,r):q(a)=1+3a22a,r(a)=1+3a2(1+a2)2,0<a<1/3} or r⩽η2q for some constants η1≈0.618 and η2≈0.601. Then on the (λ,‖u‖∞)-plane, we give a classification of four qualitatively different bifurcation diagrams: an S-shaped curve, a broken S-shaped curve, a ⊂-shaped curve and a monotone increasing curve.

Traveling Wave Solution in a Diffusive Predator-Prey System with Holling Type-IV Functional Response

We establish the existence of traveling wave solution for a reaction-diffusion predator-prey system with Holling type-IV functional response. For simplicity, only one space dimension will be involved, the traveling solution equivalent to the heteroclinic orbits inR3. The methods used to prove the result are the shooting argument and the invariant manifold theory.

Dispersal permanence of a periodic predator–prey system with Holling type-IV functional response

Abstract In this paper, we study the permanence of a class of periodic predator–prey system with Holling type-IV functional response where the prey disperses in patchy environment with two patches, and provide a sufficient and necessary condition to guarantee permanence of the system. Finally, two examples are presented to illustrate the application of our main results.

The Dynamic Complexity of a Holling Type-IV Predator-Prey System with Stage Structure and Double Delays

We invest a predator-prey model of Holling type-IV functional response with stage structure and double delays due to maturation time for both prey and predator. The dynamical behavior of the system is investigated from the point of view of stability switches aspects. We assume that the immature and mature individuals of each species are divided by a fixed age, and the mature predator only attacks the mature prey. Based on some comparison arguments, sharp threshold conditions which are both necessary and sufficient for the global stability of the equilibrium point of predator extinction are obtained. The most important outcome of this paper is that the variation of predator stage structure can affect the existence of the interior equilibrium point and drive the predator into extinction by changing the maturation (through-stage) time delay. Our linear stability work and numerical results show that if the resource is dynamic, as in nature, there is a window in maturation time delay parameters that generate sustainable oscillatory dynamics.

Stability for a Holling Type IV Food Chain System With Impulsive Perturbations

Abstract. We investigate a three species food chain system with a Holling type IVfunctional response and impulsive perturbations. We find conditions for local and globalstabilities of prey(or predator) free periodic solutions by applying the Floquet theory andthe comparison theorems. 1. IntroductionIt is currently very much in vogue to study population models with impulsiveperturbations containing biological and chemical controls. Especially, simple multi-species systems consisting of a three species food chain with impulsive perturbationshave been discussed by a number of researchers [13], [17], [18], [19], [20] and thereare also many literatures on impulsive prey-predator population models [10], [11],[12].A well-known model of such systems is a food chain system with Holling typeIV functional response [7], [14], [20], which can be described the following equation:(1.1)x 0 (t) = x(t)(a−bx(t))−c 1 x(t)y(t)1+e 1 x 2 (t),y 0 (t) = −d 1 y(t)+c 2 x(t)y(t)1+e 1 x 2 (t)−c 3 y(t)z(t)1+e

Bifurcations of a predator–prey system of Holling and Leslie types

A predator–prey model with simplified Holling type-IV functional response and Leslie type predator’s numerical response is considered. It is shown that the model has two non-hyperbolic positive equilibria for some values of parameters, one is a cusp of co-dimension 2 and the other is a multiple focus of multiplicity one. When parameters vary in a small neighborhood of the values of parameters, the model undergoes the Bogdanov–Takens bifurcation and the subcritical Hopf bifurcation in two small neighborhoods of these two equilibria, respectively. And it is further shown that by choosing different values of parameters the model can have a stable limit cycle enclosing two equilibria, or a unstable limit cycle enclosing a hyperbolic equilibrium, or two limit cycles enclosing a hyperbolic equilibrium. However, the model never has two limit cycles enclosing a hyperbolic equilibrium each for all values of parameters. Some computer simulation are presented to illustrate the conclusions.