Holling type-II Functional Response Research Articles

Spatiotemporal dynamics in a reaction–diffusion toxic-phytoplankton–zooplankton model

In this paper, we investigate the complex dynamics of a spatial toxic-phytoplankton–zooplankton model with Holling type-II functional response. Through a detailed analytical study of the reaction–diffusion model, we obtain some conditions for local and global stability and for diffusive instability with zero-flux boundary conditions of a positive equilibrium. On the basis of these results, we present the evolutionary processes of pattern formation that involve organism distribution and the interaction of a spatially distributed population with local diffusion. Then, novel numerical evidence of the time evolution of patterns controlled by diffusion and environmental carrying capability in the model are presented, and it is found that the model dynamics exhibits complex pattern replication. Additional studies reveal that pattern formation in the spatially extended model depends on the choice of the initial conditions. These results indicate that the interaction between toxin-producing phytoplankton and zooplankton in marine environments may be partly driven by the forces of diffusion or the environmental carrying capability.

Global existence of solutions to a cross-diffusion predator–prey system with Holling type-II functional response

This paper is concerned with a cross-diffusion system arising in a predator–prey population model with Holling type-II functional response in a bounded domain with no flux boundary condition. Employing the energy estimates and Gagliardo–Nirenberg type inequalities to establish W21-bounds uniform in time, the global existence and uniform boundedness of solutions for a cross-diffusion system are proved. The global asymptotic stability of the constant positive steady state is also investigated.

Modelling the Effects of Competing Toxin Producing Phytoplankton on a Zooplankton Population: Role of Holling Type-II Functional Response with Time-Lag

In this paper, a system consisting of two competing harmfu l phytoplankton and a zooplankton with Holling type-II functional response and discrete time lag is considered. A stable co-existence of all the species has been observed for the system without delay and the Hopf-b ifurcation phenomenon is observed for the interior equilibriu m po int. The Hopf-bifurcating solution is obtained for the critical values of parameters like predation rates and half saturation constants. Further, using the normal form theory, we have determined the direction and the stability of the Hopf-bifurcation solution. The introduction of time delay in the system also shows the Hopf-bifurcat ion as the delay parameter passes through a critical value. Finally, the numerical simu lation is carried out to support the theoretical results.

Stability and traveling waves of a stage-structured predator–prey model with Holling type-II functional response and harvesting

In this paper, we consider a reaction–diffusion predator–prey model with stage-structure, Holling type-II functional response, nonlocal spatial impact and harvesting. The stability of the equilibria is investigated. Furthermore, by the cross-iteration scheme companied with a pair of admissible upper and lower solutions and Schauder fixed point theorem, we deduce the existence of traveling wave solution which connects the zero solution and the positive constant equilibrium.

STABILITY AND BIFURCATION ANALYSIS FOR A TWO-COMPETITOR/ONE-PREY SYSTEM WITH TWO DELAYS

The present paper is concerned with a two-competitor/oneprey population system with Holling type-II functional response and two discrete delays. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the positive equilibrium and existence of local Hopf bifurcations are investigated. Particularly, by applying the normal form theory and the center manifold reduction for functional differential equations (FDEs) explicit formulae determining the direction of bifurcations and the stability of bifurcating periodic solutions are derived. Finally, to verify our theoretical predictions, some numerical simulations are also included at the end of this paper.

The study of predation of <I>Icerya aegyptiaca</I> by <I>Rodolia pumila</I>

Icerya aegyptiaca was regarded as a serious pest for ornamental plants especially in South China featured with very wide range of diet,strong reproduction ability and easy spreading.Predation of Icerya aegyptiaca by Rodolia pumila was investigated under the laboratory condition in this work.The result indicated that predation response of Rodolia pumila fitted Holling typeⅡ functional response with instant attacking rate is 0.577 and treating time is 0.019.According to Hassell model,the factor describing disturbing effect of increasing predator density was estimated to be m=0.434.The potential predating ability and the optimal prey density have been estimated to be a=24.7 and b=16.5 respectively by Wang's new model.Considering comprehensive parameters related to predation,Rodolia pumila might be unqualified to be independent agent for biological control of Icerya aegyptiaca.

Coexistence of a diffusive predator–prey model with Holling type-II functional response and density dependent mortality

In this paper, we consider a two competitor–one prey diffusive model in which both competitors exhibit Holling type-II functional response and one of the competitors exhibits density dependent mortality rate. First, we study the local and global existence of strong solution by using the C 0 analytic semigroup. Then, we consider the local and global stability of the positive constant equilibrium by using the linearization method and Laypunov functional method, respectively. Furthermore, we derive some results for the existence and non-existence of non-constant stationary solutions when the diffusion rate of a certain species is small or large. The existence of non-constant stationary solutions implies the possibility of pattern formation in this system.

ON A PREY–PREDATOR–PATHOGEN MODEL WITH PREY PREFERENCE: A THEORETICAL STUDY

In the present paper, we study a prey–predator–pathogen model where the prey population suffer from an SI epidemic. The predator is assumed to exercise preferential predation. The prey preference mechanism ranges from predation only on the susceptible or only on the infective to proportional predation on both the prey types. The formulation maintains the native Holling type-II functional response in one hand and invokes prey switching on the other. We first describe stability and persistence results for the model without preference, which highlights the significance of some system parameters namely, predator mortality rate and predation pressure on the susceptible in shaping the system dynamics. Then we investigate the model with the preference mechanism, which shows the importance of the preference parameter (α) in controlling the stability and existence criteria of component populations. Our study reveals the existence of preference parameter ranges that (i) guarantees stable species coexistence (ii) exhibits rich dynamics in the form of oscillatory phenomena, point attractors and limit cycle attractors and (iii) causes possible extinction of the predators together with the susceptible prey. The ecological meaning of the corresponding ranges is also elaborated.

Analysis of a Stage-Structured Predator-Prey System Concerning Impulsive Control Strategy

Based on integrated pest management, a stage-structured predator-prey system with Holling type-II functional response concerning impulsive control strategy is investigated. By the Floquet theory and small amplitude perturbation skills, it is proved that there exists a globally stable pest-eradication periodic solution when the impulsive period is less than some critical values. Further, sufficient conditions for the permanence of the system is established. Numerical simulations are carried out to illustrate the effect of impulses on the dynamics of the system.

Global dynamics of a predator–prey model with time delay and stage structure for the prey

A stage-structured predator–prey system with Holling type-II functional response and time delay due to the gestation of predator is investigated. By analyzing the characteristic equations, the local stability of each of feasible equilibria of the system is discussed and the existence of a Hopf bifurcation at the coexistence equilibrium is established. By means of the persistence theory on infinite dimensional systems, it is proven that the system is permanent if the coexistence equilibrium exists. By using Lyapunov functionals and LaSalle invariant principle, it is shown that the trivial equilibrium is globally stable when both the predator-extinction equilibrium and the coexistence equilibrium are not feasible, and that the predator-extinction equilibrium is globally asymptotically stable if the coexistence equilibrium does not exist, and sufficient conditions are derived for the global stability of the coexistence equilibrium. Numerical simulations are carried out to illustrate the main theoretical results.

Coexistence states of a Holling type-II predator–prey system

This paper characterize the existence of coexistence states to a reaction–diffusion predator–prey model with Holling type-II functional response subject to Dirichlet boundary conditions. We find the necessary and sufficient conditions for existence of coexistence states by fixed point index theory and bifurcation theory.

STABILITY AND HOPF BIFURCATION OF A PREDATOR–PREY SYSTEM WITH TIME DELAY AND HOLLING TYPE-II FUNCTIONAL RESPONSE

A predator–prey model with time delay and Holling type-II functional response is investigated. By choosing time delay as the bifurcation parameter and analyzing the associated characteristic equation of the linearized system, the local stability of the system is investigated and Hopf bifurcations are established. The formulae determining the direction of bifurcations and the stability of bifurcating periodic solutions are given by using the normal form theory and center manifold theorem. Numerical simulations are carried out to illustrate the theoretical results.

Non-existence of non-constant positive steady states of two Holling type-II predator–prey systems: Strong interaction case

We prove the non-existence of non-constant positive steady state solutions of two reaction–diffusion predator–prey models with Holling type-II functional response when the interaction between the predator and the prey is strong. The result implies that the global bifurcating branches of steady state solutions are bounded loops.

Feeding Threshold for Predators Stabilizes Predator-Prey Systems

Since Rosenzweig showed the destabilisation of exploited ecosystems, the so called Paradox of enrichment, several mechanisms have been proposed to resolve this paradox. In this paper we will show that a feeding threshold in the functional response for predators feeding on a prey population stabilizes the system and that there exists a minimum threshold value above which the predator-prey system is unconditionally stable with respect to enrichment. Two models are analysed, the first being the classical Rosenzweig-MacArthur (RM) model with an adapted Holling type-II functional response to include a feeding threshold. This mathematical model can be studied using analytical tools, which gives insight into the mathematical properties of the two dimensional ODE system and reveals underlying stabilisation mechanisms. The second model is a mass-balance (MB) model for a predator-prey-nutrient system with complete recycling of the nutrient in a closed environment. In this model a feeding threshold is also taken into account for the predator-prey trophic interaction. Numerical bifurcation analysis is performed on both models. Analysis results are compared between models and are discussed in relation to the analytical analysis of the classical RM model. Experimental data from the literature of a closed system with ciliates, algae and a limiting nutrient are used to estimate parameters for the MB model. This microbial system forms the bottom trophic levels of aquatic ecosystems and therefore a complete overview of its dynamics is essential for understanding aquatic ecosystem dynamics.

Species extinction and permanence in a prey–predator model with two-type functional responses and impulsive biological control

By introducing impulsive biological control strategy, the dynamic behaviors of the two-prey one-predator model with defensive ability and Holling type-II functional response are investigated. By using Floquet’s Theorem and the small amplitude perturbation method, we prove that there exists an asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical minimum value, and permanence conditions (that is, the impulsive period is greater than some critical maximum value) are established via the method of comparison involving multiple Liapunov functions. It is shown that our impulsive control strategy is more effective than the classical one. Furthermore, the effect of impulsive perturbations on the unforced continuous system is studied. From simulations, we find that the system has more complex dynamic behaviors and is dominated by periodic, quasi-periodic, and chaotic solutions.

Chaos and Hopf bifurcation of a hybrid ratio-dependent three species food chain

In this paper, we propose and study a model of a hybrid ratio-dependent three species food chain, which is constituted by a hybrid type subsystem of prey and middle-predator and a middle-top predators’ subsystem with Holling type-II functional response. We investigate the persistence and Hopf bifurcation of the system. Computer simulations are carried out to explain the mathematical conclusions. The chaotic attractor is obtained for suitable choice of parametric values.

Persistence and Stability of a Food Chain Model with Mixed Selection of Functional Responses

One approach to the study of an ecological community begins with an important object: its food web. Theoretical studies of food web must contend with the question of how to couple the large number of interacting species. One line of investigation assumes that the “building blocks” are species interacting in a pairwise fashion. The model we analyze in this paper describes a tritrophic food chain composed of logistic prey, a classical Lotka-Volterra functional response for prey and predator, and a Holling type-II functional response for predator and superpredator. Dynamical behaviours such as boundedness, stability, persistence, bifurcation et cetera of the model are studied critically. Computer simulations are carried out to explain the analytical findings. Finally it is discussed how these ideas illuminate some of the observed properties of real populations in the field, and explores practical implications.

Dynamical behaviors of a partial-dependent predator–prey system☆

Abstract In this paper, the dynamical behaviors are investigated for a partial-dependent predator–prey system with Holling type-II functional response. The change of equilibria number and their stability are discussed, and several sufficient conditions are given for checking existence of two positive steady states and the existence of limit cycle for the predator–prey system. In addition, their biological explanations are presented, and some numerical simulations are given to illustrate the effectiveness of the obtained results.

Order and chaos in a food web consisting of a predator and two independent preys

A food web consisting of two independent preys and a predator is modeled incorporating modified Holling type-II functional response. The mathematical model has a unique and bounded solution. The necessary and sufficient conditions for persistence of the food web are obtained. Bifurcation diagram has been obtained for selected range of different parameters. The system exhibits chaos for a range of parametric values when long time behavior is studied. The computation of Lyapunov exponents and the existence of strange attractor also indicate the chaotic behavior of the system.

Wavefronts and global stability in a time-delayed population model with stage structure

We formulate and study a one–dimensional single–species diffusive–delay population model. The time delay is the time taken from birth to maturity. Without diffusion, the delay differential model extends the well–known logistic differential equation by allowing delayed constant birth processes and instantaneous quadratically regulated death processes. This delayed model is known to have simple global dynamics similar to that of the logistic equation. Through the use of a sub/supersolution pair method, we show that the diffusive delay model continues to generate simple global dynamics. This has the important biological implication that quadratically regulated death processes dramatically simplify the growth dynamics. We also consider the possibility of travelling wavefront solutions of the scalar equation for the mature population, connecting the zero solution of that equation with the positive steady state. Our main finding here is that our fronts appear to be all monotone, regardless of the size of the delay. This is in sharp contrast to the frequently reported findings that delay causes a loss of monotonicity, with the front developing a prominent hump in some other delay models.