Holling Type Functional Response Research Articles

Coexistence of populations in a Leslie-Gower tritrophic model with Holling-type functional responses

The dynamics of a Leslie-Gower type tritrophic model are analyzed. The model considers the interaction among three populations, a resource, a predator and a superpredator. It is assumed that the predator is generalist and its interaction with the resource is according to a general Holling-type functional response. Furthermore, it is assumed that the superpredator is specialist and its interaction with the predator follows a Holling type II functional response. The goal of this work is to show conditions that guarantee the coexistence of the three species. To do this, the existence of a stable equilibrium point or a stable limit cycle is demonstrated, which appears via a bifurcation. In addition, the analytical results are exemplified through numerical simulations.

Impact of Spatial Variability in Zooplankton Grazing Rates on Carbon Export Flux

AbstractThe biological carbon pump is a key controller of how much carbon is stored within the global ocean. This pathway is influenced by food web interactions between zooplankton and their prey. In global biogeochemical models, Holling Type functional responses are frequently used to represent grazing interactions. How these responses are parameterized greatly influences biomass and subsequent carbon export estimates. The half‐saturation constant, or k value, is central to the Holling functional response. Empirical studies show k can vary over three orders of magnitude, however, this variation is poorly represented in global models. This study derives zooplankton grazing dynamics from remote sensing products of phytoplankton biomass, resulting in global distribution maps of the grazing parameter k. The impact of these spatially varying k values on model skill and carbon export flux estimates is then considered. This study finds large spatial variation in k values across the global ocean, with distinct distributions for micro‐ and mesozooplankton. High half‐saturation constants, which drive slower grazing, are generally associated with areas of high productivity. Grazing rate parameterization is found to be critical in reproducing satellite‐derived distributions of small phytoplankton biomass, highlighting the importance of top‐down drivers for this size class. Spatially varying grazing dynamics decrease mean total carbon export by >17% compared to globally homogeneous dynamics, with increases in fecal pellet export and decreases in export from algal aggregates. This study highlights the importance of grazing dynamics to both community structure and carbon export, with implications for modeling marine carbon sequestration under future climate scenarios.

Modelling effects and control of soil contaminants on the dynamics of plant–herbivore species system

In this paper, we have proposed and analysed a mathematical model to study the effects of impulsive input of contaminants in soil on the dynamics of interacting species system consisting of contaminants, soil-nutrients, plant biomass and herbivore population in order to investigate the criteria for the survival or extinction of plant biomass and herbivore population. The uptake function for herbivore population considered in the model of the paper shows saturation effect and this type of uptake function behave linearly with respect to plant biomass for low density. Also, the saturation for large plant biomass is a reflection of the limited herbivore capability or perseverance when the plant biomass is abundant. Therefore, the uptake function considered in this paper is more natural to lotka volterra and holling type functional responses. We have shown that the plant–herbivore extinction periodic solution exists and also the plant–herbivore extinction periodic solution is locally and globally stable if the impulsive depletion rate of nutrients due to contaminants is more than some critical value. Further, it is concluded that if the impulsive input period for contaminants is increased and impulsive depletion rate of nutrients due to contaminants is decreased then both the plant and herbivore population will survive otherwise they will tend to extinction.

Spatial pattern formation driven by the cross-diffusion in a predator–prey model with Holling type functional response

In this paper, we consider a cross-diffusion predator–prey system with Holling type functional response. We study the local stability, Turing instability, spatial pattern formation, Hopf and Turing–Hopf bifurcation of the equilibrium. Numerical simulation with zero-flux boundary conditions discloses that the system under consideration experiences the occurrence of cross-diffusion-driven instability. The dynamical system in Turing space emerges spots, stripe-spot mixtures and labyrinthine patterns, which reveals that the interaction of both self- and cross-diffusions play a significant role on the pattern formation of the present system in a way to enrich the pattern. We obtain the normal form of the Turing–Hopf bifurcation and observe that the system has stably spatially homogeneous periodic solutions, stable constant and nonconstant steady-state solutions, which indicates that the intrinsic growth rate coefficient and the environmental carrying capacity coefficient are two important factors for predator–prey system, and affect the stability of predator–prey system.

Complex pattern formations induced by the presence of cross-diffusion in a generalized predator–prey model incorporating the Holling type functional response and generalization of habitat complexity effect

The interaction between predator and their prey is one of the most complex processes in ecosystems due to its structure and nature. Many extrinsic and intrinsic factors can affect the population dynamics. In this manuscript, we have focused our study on the process of Turing-pattern formation in a generalized predator–prey system with cross-diffusion and habitat complexity effects. By applying the linear stability theory, sufficient conditions for the occurrence of Hopf bifurcation and Turing-driven instability are successfully established. Moreover, the amplitude equations are derived near the critical value of Turing instability according to the standard multiple-scale analysis. From the mathematical view, our investigation reveals that the whole patterns generated by the system are governed by two control parameters, which are the cross-diffusion parameter δ21 and the predator mortality rate m. Complex but interesting dynamical pattern formations are specified theoretically and numerically such as spot pattern, stripe pattern, and spot-strip pattern in terms of varying the two control parameters. To support the theoretical results, a series of numerical tests are carried out, especially, to illustrate the diverse patterns near the codimension- Turing–Hopf bifurcation point (δ21T,m(0)H) and the significant impact of δ21 and m on the distribution of the two species.

Complex Dynamics on a Discrete Tritrophic Model of Leslie Type with General Functional Responses

In this paper, by averaging the growth rate on each state, we analyze the dynamics of a discrete dynamical system coming from a system of ODEs. This differential system corresponds to a tritrophic Leslie type model which is formed by three populations (prey (P), mesopredator (MP) and superpredator (SP)), where the last two populations are generalist predators. We give sufficient conditions where the discrete model undergoes a Neimark–Sacker bifurcation at a coexistence point. This analysis is independent of the functional responses that govern the interactions. To illustrate our results, several applications are given, under the assumptions that the population P has logistic growth and that the relations MP–P and SP–MP are carried out through Holling type functional responses. From these applications, we conclude that there are sufficient conditions to guarantee that the three species coexist by means of a supercritical Neimark–Sacker bifurcation. Moreover, numerically we can detect that the discrete system exhibits a chaotic behavior.

Dynamics of autotroph–mixotroph interactions with the intraguild predation structure

A mathematical model with the intraguild predation structure is proposed to describe the interactions of autotrophs and mixotrophs containing light and nutrients in a well-mixed aquatic ecosystem. The dissipation, existence and stability of equilibria of the model are proved, and the ecological reproductive indexes for the extinction, survival and coexistence of autotrophs and mixotrophs are established. We also consider the influence of Holling type functional responses and abiotic factors on the coexistence and biomass of autotrophs and mixotrophs. It is shown that the intraguild predation structure is beneficial to phytoplankton biodiversity and provides an explanation for the phytoplankton paradox.

Periodic solutions for an impulsive predator-prey model with Holling type functional response and time delays

In this paper we propose an impulsive predator-prey model with time delays. By applying the continuation theorem of coincidence degree theory, we establish a better estimation on the difference between the supremum and infimum of a differentiable piecewise continuous periodic function.

Time-delay effect on a diffusive predator\u2013prey model with habitat complexity

Based on the predator–prey system with a Holling type functional response function, a diffusive predator–prey system with digest delay and habitat complexity is proposed. Firstly, the stability of the equilibrium of diffusion system without delay is studied. Secondly, under the Neumann boundary conditions, taking time delay as the bifurcation parameter, by analyzing the eigenvalues of linearized operator of the system and using the normal form theory and center manifold method of partial functional differential equations, the effect of time delay on the stability of the system is studied and the conditions under which Hopf bifurcation occurs are given. In addition, the calculation formulas of the bifurcation direction and the stability of bifurcating periodic solutions are derived. Finally, the accuracy of theoretical analysis results is verified by numerical simulations and the biological explanation is given for the analysis results.

Coexistence and harvesting optimal policy in three species food chain model with general Holling type functional response

In this paper, we have discussed harvesting of prey and intermediate predator species. Both are subjected to Holling type I–V functional response. Conditions for local and global stability of the nonnegative equilibria are verified. The permanent coexistence criterion of the model system and existence of optimal equilibrium solution of the control problem are demonstrated. Maximum sustainable yield and maximal net present revenue are determined. To confirm analytical results, numerical solution has been carried out using the Matlab™ ODE solver ODE45 and the simulations show the model system reveals complex behavior (such as oscillations), which reflects the real situation. Recommendations for Resource Managers From our investigation of this study, we recommend to the management the following points. 1. Coexistence of the three species with harvesting, or persistence of the model system is possible provided that good management(treatment) of some factors (such as harvesting rate, growth rate of species, etc.) are performed. 2. The dynamics reveals complex behavior (such as oscillations), which reflects the real situation and it is sensitive to the above factors, especially the growth rate of the intermediate predator. 3. The policy makers should recommend the optimal effort to be applied and the optimal stock to harvest. This indicates that maximum profit will attain while securing sustainability of the three species in the ecosystem.

Periodic solutions for an impulsive predator-prey model with Holling type functional response and time delays

In this paper we propose an impulsive predator-prey model with time delays. By applying the continuation theorem of coincidence degree theory, we establish a better estimation on the difference between the supremum and infimum of a differentiable piecewise continuous periodic function.

A predator-prey system with Holling-type functional response

We consider the Gause-type predator-prey system with a large class of growth and response functions, in the case where the response function is not smooth at the origin. We discuss the conditions under which this system has exactly one stable limit cycle or has a positive stable equilibrium point and we describe the basin of attraction of the stable limit cycle and the stable equilibrium point, respectively. Our results correct previous results of the existing literature obtained for the Holling response function x p /(a + x p), in the case where 0 < p < 1.

Dynamic of interactive model for information propagation across social networks media

Controlling information diffusion or propagation through social networks can be challenging when dealing with information related to a subject of highest interest for the public. The complexity level of control depends on subject importance, users’ dynamic, and network structure. When two published messages or pieces of information share the same interest for targeted readers, analyzing their propagation dynamic for control and prediction is of great interest. This article proposes to model, based on a modified interactive system with Holling type functional response, the dynamic of underlying relationship between two broadcasted messages traveling through social networks media. We showed in the qualitative analysis of the proposed model that system could be stable at certain conditions, and the model-system exhibited very rich dynamical behavior. Numerical simulation results validated theoretical analyses and suggested adapting resources harvesting and assimilation efficiency for an authoritative message to stabilize the system and control the dissemination of information in a closed environment.

Dynamics of a predator–prey model with generalized Holling type functional response and mutual interference

Mutual interference and prey refuge are important drivers of predator–prey dynamics. The “exponent” or degree of mutual interference has been under much debate in theoretical ecology. In the present work, we investigate the interplay of the mutual interference exponent, and prey refuge, on the behavior of a predator–prey model with a generalized Holling type functional response — considering in particular the “non-smooth” case. This model can also be used to model an infectious disease where a susceptible population, moves to an infected class, after being infected by the disease. We investigate dynamical properties of the system and derive conditions for the occurrence of saddle–node, transcritical and Hopf-bifurcations. A sufficient condition for finite time extinction of the prey species has also been derived. In addition, we investigate the effect of a prey refuge on the population dynamics of the model and derive conditions such that the prey refuge would yield persistence of the population. We provide additional verification of our analytical results via numerical simulations. Our findings are in accordance with classical experimental results in ecology (Gause, 1934), that show that extinction of predator and prey populations is possible in a finite time period — but that bringing in refuge can effectively yield persistence.

Traveling Waves in a Stage-Structured Predator–Prey Model with Holling Type Functional Response

This article is mainly concerned with a predator–prey model with Holling type functional response, where the stage structure and harvesting are considered. Firstly, we discuss the local stability of various equilibria by using the characteristic equation technique. Then, by constructing upper and lower solutions, we prove the existence of traveling waves connecting two steady states.

Dynamics of a prey predator disease model with Holling type functional response and stochastic perturbation

The present research article constitutes Holling type II and IV diseased prey predator ecosystem and classified into two categories namely susceptible and infected predators.We show that the system has a unique positive solution. The deterministic and stochastic nature of the dynamics of the system is investigated. We check the existence of all possible steady states with local stability. By using Routh-Hurwitz criterion we showed that the positive equilibrium point $E_{7}$ is locally asymptotically stable if $x^{*} &gt; \sqrt{m_{1}}$ .Moreover condition of the global stability of positive equilibrium point $E_{7}$ are also entrenched with help of Lyupunov theorem. Some Numerical simulations are carried out to illustrate our analytical findings.

Global Dynamics of a Predator–Prey Model with General Holling Type Functional Responses

A predator–prey model with general Holling type functional response is considered. The aim of this paper is to investigate the global stability of the interior equilibrium of the model when it exists. By transforming the model to an equivalent system and with the detailed analysis of the crucial behavior of a corresponding function we are able to show that the local stability and the global stability of the interior equilibrium are equivalent.

Analysing a prey predator model for stability with prey refuge-stage structure on predator

This paper analyses a model of prey predator with Michaelis Menten- Holling type functional response with inclusion of a constant proportion of prey refuge and predator stage structure. The analysed model consists of three nonlinear ordinary differential equations which describe the interaction between the prey and juvenile & adult predators. The system is studied for its local stability by determining the equilibrium points and obtaining the conditions for its local stability. Suitable Lyapunov function has been constructed to obtain the global asymptotic stability of the system.

Dynamics of predator–prey system with fading memory

A predator–prey model with fading memory for general Holling type functional response is proposed. The fading memory term is used with the hypothesis that the predator’s growth rate at present depends on the recent past quantities of prey. The effects of predator harvesting is also considered in the model. The model is analysed theoretically as well as numerically. Two parameter bifurcation analysis are done and the existence of Hopf point bifurcation is identified. Both supercritical and subcritical Hopf bifurcations are obtained with the variation of system parameters. Maximum sustainable yield with respect to harvesting effort is also determined. It is to be observed that the system dynamics is very rich in presence of fading memory. The obtained results may be useful in the field of agriculture and fishery.

Global dynamics of stochastic predator–prey model with mutual interference and prey defense

Predator–prey interactions with stochastic forcing have been extensively investigated in the literature. However there are not many investigations of such models, that include prey defense. The goal of the current manuscript is to investigate a stochastic predator–prey model with mutual interference, and various Holling type functional responses, where the prey is able to release toxins as defense against a predator. This can also be generalized to include group or herd defense, toxin production and mimicry. We establish local and global existence for the stochastic model, and perform various numerical simulations to support our theoretical results. Our key result is that we have globally existing solutions independent of the magnitude of the toxin release parameter, or the predation rates. We also show that large enough noise intensity in solely the prey, can lead to extinction in the noisy model, for both species, whilst there is persistence in the deterministic model.