Type Functional Response Research Articles

An investigation of delay induced stability transition in nutrient-plankton systems

Abstract In this paper, a nutrient-plankton interaction model is proposed to explore the characteristic of plankton system in the presence of toxic phytoplankton and discrete time delay. Anti-predator efforts of phytoplankton by toxin liberation act as a prominent role on plankton dynamics. Toxicity controls the system dynamics and reduces the grazing rate of zooplankton. The toxic substance released by phytoplankton is not an instantaneous process, it requires some time for maturity. So, a discrete time delay is incorporated in the toxin liberation by the phytoplankton. The choice of functional response is important to understand the toxin liberation and it depends on the nonlinearity of the system, which follows the Monod-Haldane type functional response. Theoretically, we have studied the boundedness condition along with all the feasible equilibria analysis and stability criteria of delay free system. We have explored the local stability conditions of delayed system. The existence criterion for stability and direction of Hopf-bifurcation are also derived by using the theory of normal form and center manifold arguments. The essential features of time delay are studied by time series, phase portrait and bifurcation diagram. We perform a global sensitivity analysis to identify the important parameters of the model having a significant impact on zooplankton. Our numerical investigation reveals that the toxin liberation delay switches the stability of the system from stable to limit cycle and after a certain interval chaotic dynamics is observed. High rate of toxic substances production shows extinction of zooplankton. Further, the negative and positive impacts of other control parameters are studied. Moreover, to support the occurrence of chaos, the Poincare map is drawn and the maximum Lyapunov exponents are also computed.

Discrete-time prey-predator model with θ-logistic growth for prey incorporating square root functional response

This article presents the dynamics of a discrete-time prey-predator system with square root functional response incorporating θ-logistic growth. This type of functional response is used to study the dynamics of the prey--predator system where the prey population exhibits herd behavior, i.e., the interaction between prey and predator occurs along the boundary of the population. The existence and stability of fixed points and Neimark-Sacker Bifurcation (NSB) are analyzed. The phase portraits, bifurcation diagrams and Lyapunov exponents are presented and analyzed for different parameters of the model. Numerical simulations show that the discrete model exhibits rich dynamics as the effect of θ-logistic growth.

Mathematical Analysis of an Eco-Epidemic Model with Different Functional Responses of Healthy and Infected Predators on Prey Species

This article describes a mathematical model for a predator-prey system with two types of functional responses and a transmissible disease in the predator species. The effect of stocking of healthy predator in healthy predator species class and harvesting of healthy predator are analyzed. The essential mathematical features of the proposed eco-epidemic system such as boundedness, positivity, local and global stability and Hopf bifurcation are discussed. Numerical simulations are also performed to validate the theoretical results obtained.

Energetic requirements of the transition from solitary to group living

Synergy is known to be vital for the group collaboration among non-kin individuals. In order to evaluate the condition of synergy that initiates group living, we build a model of food intake based on three types of functional response. We show that type III functional response is prerequisite for synergy to allow group living. The optimal number of gathering individuals can be also evaluated from Type III functional response curve. Type III functional response consists of terms depending linearly and bilinearly on the number of individuals and the bilinear term represents synergy. For a fixed value of the linear coefficient, there are upper and lower boundaries of the bilinear coefficient for synergistic collaboration. The dilution effect can be incorporated into the model through the functional response of the predator. Thus, the functional response of the predator as well as that of the prey contribute to the group living of the prey. Our model shows that group livings can be categorized into three types, namely those due to (1) synergy effect of the own group, (2) dilution effect against predators, and (3) both effects contributing together. The predator's functional response plays a decisive role in the last two types, where the predator response should be of anti-Type III (i.e., Type II).

Lethal and Sublethal Effects of Two Commercial Insecticides on Egg Parasitoids (Hymenoptera: Scelionidae) of Green Stink Bugs (Hem: Pentatomidae).

Pistacia vera L. is one of the most important horticultural crops in Iran. The stink bugs Acrosternum arabicum and Brachynema germari are two of the key pests that cause significant direct and indirect damage on Pistacia vera. Egg parasitoids have been considered as potential biological control agents of pistachio green stink bugs. Among them, Trissolcus semistriatus and Psix saccharicola are the most abundant and efficient parasitoid for A. arabicum in pistachio orchards. In this study, we assessed lethal and sublethal effects of two commonly used insecticide products (fenitrothion and a binary mixture of lambda-cyhalothrin + thiamethoxam) on these two parasitoid wasps under laboratory conditions. The median lethal concentration (LC50) values for fenitrothion and thiamethoxam + lambda-cyhalothrin in P. saccharicola and T. semistriatus were estimated as 1.03, 0.48 and 0.87, 0.98 mg a.i./liter, respectively. In terms of sublethal effects, insecticide treatments altered the type of functional response from type III to type II in T. semistriatus. However, P. saccharicola exhibited a type II functional to density of A. arabicum for all treatments, although attack rates were lower for insecticide-exposed wasps while handling times increased. Our results show that sublethal effects of insecticides further reduce the efficacy of biological control agents. Effective integrated pest management programs should avoid antagonistic interactions between chemical and biological control methods. The results of this study provide useful information to develop comprehensive integrated pest management programs for stink bugs in pistachio orchards.

Global Stability of a Predator-Prey Model with Ivlev-Type Functional Response

A predator-prey model with Ivlev-Type functional response is studied. The main purpose is to investigate the global stability of a positive (co-existence) equilibrium, whenever it exists. A recently developed approach shows that for certain classes of models, there is an implicitly defined function which plays an important rule in determining the global stability of the positive equilibrium. By performing a detailed analytic analysis we demonstrate that a crucial property of this implicitly defined function is governed by the local stability of the positive equilibrium, which enable us to show that the global and local stability of the positive equilibrium, whenever it exists, is equivalent. We believe that our approach can be extended to study the global stability of the positive equilibrium for predator-prey models with some other types of functional responses.

Incorporating Prey Refuge in a Prey-Predator Model with Beddington-DeAngelis Type Functional Response: A Comparative Study on Intra-Specific Competition

Incorporating Prey Refuge in a Prey-Predator Model with Beddington-DeAngelis Type Functional Response: A Comparative Study on Intra-Specific Competition

Global dynamics of a fractional-order SIR epidemic model with memory

In this paper, an investigation and analysis of a nonlinear fractional-order SIR epidemic model with Crowley–Martin type functional response and Holling type-II treatment rate are established along the memory. The existence and stability of the equilibrium points are investigated. The sufficient conditions for the persistence of the disease are provided. First, a threshold value, [Formula: see text], is obtained which determines the stability of equilibria, then model equilibria are determined and their stability analysis is considered by using fractional Routh-Hurwitz stability criterion and fractional La-Salle invariant principle. The fractional derivative is taken in Caputo sense and the numerical solution of the model is obtained by L1 scheme which involves the memory trace that can capture and integrate all past activity. Meanwhile, by using Lyapunov functional approach, the global dynamics of the endemic equilibrium point is discussed. Further, some numerical simulations are performed to illustrate the effectiveness of the theoretical results obtained. The outcome of the study reveals that the applied L1 scheme is computationally very strong and effective to analyze fractional-order differential equations arising in disease dynamics. The results show that order of the fractional derivative has a significant effect on the dynamic process. Also, from the results, it is obvious that the memory effect is zero for [Formula: see text]. When the fractional-order [Formula: see text] is decreased from [Formula: see text] the memory trace nonlinearly increases from [Formula: see text], and its dynamics strongly depends on time. The memory effect points out the difference between the derivatives of the fractional-order and integer order.

Comparative study about the sublethal effects of chemical and botanical insecticides on the functional response of Habrobracon hebetor Say (Hym.: Braconidae) to larvae of Ephestia kuehniella Zeller (Lep.: Pyralidae)

In this research, we evaluated sublethal effects of five commercial insecticides of fenvalerate, propargite, buprofezin, dayabon and palizin which are used in chemical control of target pests of Habrobracon hebetor Say (an idiobiont and gregarious ectoparasitoid), on the functional response of this important biocontrol agent. First, 80 mated females of H. hebetor were treated by LC30 of selected insecticides for 24 h. Then, among them six wasps were selected and introduced to different densities of Ephestia kuehniella Zeller larvae (flour moth, an economical stored pest of cereals) for 24 h. The regression analyses based on Holling model indicated the functional response type II in the control, palizin and buprofezin and type III in fenvalerate, propargite and dayabon. Moreover, the highest and lowest attack rates were obtained in palizin and fenvalerate treatments, respectively. In addition, the treated wasps by dayabon and fenvalerate showed the shortest and longest handling times, respectively. Accordingly, palizin and dayabon due to the lowest negative effects on H. hebetor are compatible insecticides for combination with this biocontrol agent in integrated pest management.

Temperature-dependent functional response of Euborellia annulipes (Dermaptera: Anisolabididae) preying on Plutella xylostella (Lepidoptera: Plutellidae) larvae.

Temperature mediates trophic interactions, including relationships between insect pests and predators, and functional response studies are often used to determine the suitability of predators as biocontrol agents. We investigated the effects of temperature on the functional response of Euborellia annulipes (Lucas) (Dermaptera: Anisolabididae) preying on Plutella xylostella (L.) (Lepidoptera: Plutellidae) larvae. Predation rate, type of functional response, attack rate (a'), handling time (Th), and maximum predation rate (T/Th) of the predator were estimated using seven prey densities and three thermal conditions. The functional response of E. annulipes to P. xylostella was temperature-dependent, type III under the lower temperatur (18°C and 25 °C) , and type II at 32°C. We observed increasing values of a' in 25°C and 32°C, decreasing values of Th and highest T/Th as the thermal condition increased. Our findings suggest that E. annulipes could be effective to control P. xylostella under different thermal conditions, however its predation behavior changes according to temperature variation.

Stability and Double-Hopf Bifurcations of a Gause–Kolmogorov-Type Predator–Prey System with Indirect Prey-Taxis

In this paper, we deal with the Gause–Kolmogorov-type predator–prey system with indirect prey-taxis, which means that directional movement of predators is stimulated by some chemicals emitted by preys. The existence of the positive equilibrium, the effect of the indirect prey-taxis on the stability and the related bifurcations are investigated. The critical values for the occurrence of the Hopf bifurcation, Turing bifurcation, Turing–Hopf bifurcation and double-Hopf bifurcation are explicitly determined. An algorithm for calculating the normal form of the double-Hopf bifurcation for the non-resonance and weak resonance is derived. Moreover, we apply the theoretical results to the system with Holling-II type functional response, the stable region and the bifurcation curves are completely determined in the plane of the indirect prey-taxis and self saturation coefficient. The dynamical classification near the double-Hopf bifurcation point is explicitly determined. In the neighborhood of the double-Hopf bifurcation, there are stable spatially homogeneous/inhomogeneous periodic solutions, stable spatially inhomogeneous quadi-periodic solutions and the pattern transitions from one spatial–temporal patterns to another one with the changes of the indirect taxis and semi saturation coefficients. The results show that spatially inhomogeneous Hopf bifurcations are induced by an indirect prey-taxis parameter $$\chi >0$$ , which is impossible for the reaction–diffusion predator–prey model with a direct prey-taxis.

The pest-specific effects of glyphosate on functional response of a wolf spider

Although glyphosate is widely used for weed pest control, it might have negative side effects on natural enemies. Wolf spiders are one of the most representative predators found on soybean crops in Uruguay, preying on a wide variety of potential pests. However, the sublethal effects that pesticides might have on this group have been poorly explored for South American species. Herein, we explored the sublethal effects of glyphosate on the functional response of the wolf spider Hogna cf. bivittata against three potential pest insects, namely ant (Acromyrmex sp.), caterpillar (Anticarsia gemmatalis), and cricket (Miogryllus sp.). We contaminated residually adult females of the species Hogna cf. bivittata with glyphosate (Roundup®) and compared their functional response against non-contaminated spiders. We did not observe any mortality during the study. We found that overall Hogna cf. bivittata showed a functional response type II against crickets and caterpillars but no functional response to ants. Contaminated spiders killed less ants and caterpillars in comparison to the control group, probably as a consequence of the irritating effects of glyphosate. We did not observe differences in functional response to crickets at the evaluated densities, probably as a consequence of the low capture rate against this prey. Although glyphosate does not specifically target spiders, it might have negative sublethal effects on native predators such as Hogna cf. bivittata. Further studies should explore effect of glyphosate on other native predators from South American crops.

Daya Predasi dan Tanggap Fungsional Kumbang Unta (Ophionea nigrofasciata) pada Beberapa Kepadatan Wereng Batang Coklat (Nilaparvata lugens)

Camel groundbeetle (Ophionea nigrofasciata Schmidt-Goble 1846) or CGB is reported as one of predators of brown planthopper (Nilaparvata lugens Stal 1854) or BPH. The difference in BPH densities is assumed to affect the predation rate of the CGB. This study aimed to determine the predation rate of CGB on BPH at different densities. This experiment used a completely randomized design (CRD) consisting of six treatments and five replications. The treatment consisted of predation of one CGB at several BPH densities (5, 10, 20, 30, 40, and 50 individuals). BPH used was second-third instar of nymphs, and the CGB used was female. The results showed that the predation rate of CGB increased with increasing BPH density with functional response types classified as type I (Linear). The highest predation occurred at 50 density (23.6 individuals or 47.2% of BPH provided). The difference in density of BPH did not affect the bodyweight gain of CGB and tended to decrease due to different prey-seeking behavior. Keywords: Nillaparvata lugens, Ophionea nigrofasciata, predation, predator

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.

An application of the Atangana-Baleanu fractional derivative in mathematical biology: A three-species predator-prey model

In recent decades, studying the behavior of biological species has become one of the most fascinating areas of applied mathematics. The high importance of conservation of rare species in nature has prompted researchers in various fields to pay particular attention to this issue. Therefore, it is essential to develop mathematical models that examine the dynamics of their behavior. On the other hand, the development of new concepts in numerical analysis has enabled us to preserve more information on the evolutionary behavior history of a dynamic system and to use it in predicting the new features of the system. Fractional derivatives have provided such a valuable tool. This paper studies a dynamic system that models the interactions between two densities of immature and mature prey and predator populations. In the model, prey population is divided into two populations, including mature prey and immature prey. Another feature of the model is that predator depends on mature prey only and it followed by Crowley-Martin type functional response. Moreover, the fractional operator used in this model as derivative is of the Atangana-Baleanu AB type. Using this kind of fractional derivative causes the results to depend on the fractional order of the derivative. The addition of the concept of memory to the model is another highlight of using this type of derivative for the biological model. This helps the model to apply all the essential information of the phenomenon from the beginning to the desired time in the calculations. Existence and uniqueness of solutions to the fractional model are also investigated in this manuscript. The numerical method used in the article is also one of the most efficient patterns in solving problems with fractional derivatives. Using this effective method makes the results very consistent with what we actually expect to happen. Many simulations have been carried out to investigate the effect of parameters in the model on its overall behavior. Numerical results show the impressive performance of the fractional operator on the dynamic behavior of the considered predator-prey model. This efficient fractional operator can also be tested in the structure of other existing biological models.

Characterizing the existence of hydra effect in spatial predator-prey models and the influence of functional response types and species dispersal

The hydra effect in ecology corresponds to the increase in the equilibrium or time-averaged density of a population when its mortality rate is augmented. The literature presents examples of the hydra effect in continuous as well as in discrete time population models described by nonlinear systems of ordinary differential and difference equations, respectively. The novelty of this work is that we show by means of numerical simulation the occurrence of the hydra effect in some stationary dynamics of one dimensional reaction-diffusion predator-prey models described by nonlinear systems of partial differential equations with distinct nonlinear functional responses. Reaction-diffusion models are a set of mathematical models that take into account population dynamics and dispersal (diffusivity) within a predefined spatial domain. We show that for a type 2 functional response an increasing diffusivity shortens the range of the hydra effect. Another interesting result concerns the fact that, together with the hydra effect, the prey density also increases with the increase of the per capita density-independent mortality rate of the predator. This result indicates that this top-down disturbance does not necessarily create a trophic cascade in a spatial predator-prey interaction at the steady-state regime.

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 $x^{*} > \sqrt{m_{1}}$ .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.

Dynamical analysis for a fractional-order prey–predator model with Holling III type functional response and discontinuous harvest

In this letter, a fractional-order prey–predator model with Holling III type functional response and discontinuous harvest is considered. The non-negative and boundedness of the model are proved. In addition, some conditions for the existence and stability of the positive equilibrium point are proposed. Finally, numerical simulations are given to confirm the correctness of theorem.