type-II Functional Response Research Articles

Sex-specific functional responses of dragonfly naiads Rhodothemis rufa on Culex quinquefasciatus larvae in laboratory bioassay

ABSTRACTLaboratory bioassay was conducted to establish the biocontrol potentiality of naiads (aquatic nymphal stage) of Rhodothemis rufa (Rambur, 1842) against larvae of Culex quinquefasciatus, a common vector of filariasis in Tropical countries. From the study, it was noticed that in laboratory condition, the rate of predation of males of R. rufa was higher than that of females of almost same size and same species. The results of the present study revealed that both sexes displayed a density-dependent decelerating type-II functional response as the logistic regression estimated a significant negative linear parameter (P1 value of −0.330 and −0.151 for males and females, respectively). Attack rate was almost similar for both sexes (0.082); however, handling time is less in males (0.62 min) than in females (0.852 min). The predator species usually coexist in the same aquatic habitat to that of mosquito larvae and can be effectively used in field condition to reduce the larval densities of mosquitoes in temporary or permanent aquatic water bodies.

Time delay can enhance spatio-temporal chaos in a prey–predator model

In this paper we explore how the time delay induced Hopf-bifurcation interacts with Turing instability to determine the resulting spatial patterns. For this study, we consider a delayed prey–predator model with Holling type-II functional response and intra-specific competition among the predators. Analytical criteria for the delay induced Hopf-bifurcation and for the delayed spatio-temporal model are provided with numerical example to validate the analytical results. Exhaustive numerical simulation reveals the appearance of three types of stationary patterns, cold spot, labyrinthine, mixture of stripe-spot and two non-stationary patterns, quasi-periodic and spatio-temporal chaotic patterns. The qualitative features of the patterns for the non-delayed and the delayed spatio-temporal model are the same but their occurrence is solely controlled by the temporal parameters, rate of diffusivity and magnitude of the time delay. It is evident that the magnitude of time delay parameter beyond the Hopf-bifurcation threshold mostly produces spatio-temporal chaotic patterns.

Effects of harvesting and predator interference in a model of two-predators competing for a single prey

In the present paper we formulate a mathematical model of two predators living on a single biotic prey. The two predators interfere directly with each other. The predation function for the two predators are assumed to be different - one follows the mass action kinetics while the other obeys Holling type-II functional response. We also assume that one of the predators is economically viable and undergo harvesting at a rate proportional to its density. We perform local stability analysis about the boundary states where one of the predator species is absent. It is seen that, the existence and/or extinction criteria of each predator is controlled by the predation rates of one or both the predators. The interior equilibrium point exhibits unstable oscillatory behavior in the local analysis. We also perform a global analysis using Lyapunov function and LaSalle’s principle which shows global stability around this point for limited carrying capacity of prey. Finally we model the phenomenon of random harvesting by perturbing the harvest rate using Gaussian white noise. The resulting stochastic differential equation model is analyzed for exponential mean square stability which is shown to depend on the harvesting effort. Numerical simulation study of the model equations are also carried out.

Predator-prey interactions in selected slow and fast developing females of a ladybird, Propylea dissecta.

Development rate polymorphism describes the scenario in which individuals exhibit distinct differences in their rate of development resulting in slow and fast developers even from the same clutch of eggs. Previously we showed that in ladybird, Propylea dissecta fast developers have higher foraging and predation rates than slow developers. But correlation between foraging efficacies with reproductive output of female remains unexplored. We selected slow and fast developmental rate for 15 generations in a P. dissecta and assessed female functional response and numerical response by using varying prey biomasses (A. pisum). We evaluated predatory parameters: prey consumption, attack rate, handling time, and the reproductive measures: number of eggs laid, egg, and body biomass conversion efficiencies. Overall, both group of P. dissecta showed increased prey biomasses curvilinear for consumption rate demonstrating the physiological capacity of foraging for food are mutually exclusive behaviors (i.e., Holling's Type-II functional response). Consumption rate and proportion of prey consumed was higher, and prey handling time was shorter, in experimental fast developers. However, prey attack rate was higher in experimental slow developers. The functional response of experimental fast developers got elevated whereas got depressed for control slow-fast developers. Our results suggest that slow developers may perform better at low prey biomass than fast developers due to their high attack rate whereas high density prey may favour fast developers due to their shorter prey handling time and higher consumption rates. This study is first attempt to evaluate predatory responses of experimentally selected lines of slow and fast developers. J. Exp. Zool. 323A: 745-756, 2015. © 2015 Wiley Periodicals, Inc.

Stability and Bifurcation Analysis of a Three-Species Food Chain Model with Delay

In the present paper, we study the effect of gestation delay on a tri-trophic food chain model with Holling type-II functional response. The essential mathematical features of the proposed model are analyzed with the help of equilibrium analysis, stability analysis, and bifurcation theory. Considering time-delay as the bifurcation parameter, the Hopf-bifurcation analysis is carried out around the coexisting equilibrium. The direction of Hopf-bifurcation and the stability of the bifurcating periodic solutions are determined by applying the normal form theory and center manifold theorem. We observe that if the magnitude of the delay is increased, the system loses stability and shows limit cycle oscillations through Hopf-bifurcation. The system also shows the chaotic dynamics via period-doubling bifurcation for further enhancement of time-delay. Our analytical findings are illustrated through numerical simulations.

Long-time behavior and Turing instability induced by cross-diffusion in a three species food chain model with a Holling type-II functional response

In this paper, we study a strongly coupled reaction-diffusion system describing three interacting species in a food chain model, where the third species preys on the second one and simultaneously the second species preys on the first one. An intra-species competition b2 among the second predator is introduced to the food chain model. This parameter produces some very interesting result in linear stability and Turing instability. We first show that the unique positive equilibrium solution is locally asymptotically stable for the corresponding ODE system when the intra-species competition exists among the second predator. The positive equilibrium solution remains linearly stable for the reaction diffusion system without cross diffusion, hence it does not belong to the classical Turing instability scheme. But it becomes linearly unstable only when cross-diffusion also plays a role in the reaction-diffusion system, hence the instability is driven solely from the effect of cross diffusion. Our results also exhibit some interesting combining effects of cross-diffusion, intra-species competitions and inter-species interactions. Numerically, we conduct a one parameter analysis which illustrate how the interactions change the existence of stable equilibrium, limit cycle, and chaos. Some interesting dynamical phenomena occur when we perform analysis of interactions in terms of self-production of prey and intra-species competition of the middle predator. By numerical simulations, it illustrates the existence of nonuniform steady solutions and new patterns such as spot patterns, strip patterns and fluctuations due to the diffusion and cross diffusion in two-dimension.

Predatory responses of selected lines of developmental variants of ladybird, Propylea dissecta (Coleoptera: Coccinellidae) in relation to increasing prey and predator densities

Individuals of the same species, population and generation frequently exhibit sub-maximal and significant genetic and phenotypic variation in their rate of development, showing slow and fast developers. Fast developers commonly have higher foraging and predation rates than slow developers. The consequence of such differences and foraging for the efficacy of biocontrol species remains under-explored. Slow and fast developers from a population of the ladybird, Propylea dissecta were separated and selected experimentally for F15 generations, and the predatory response of fourth instar larvae of control and experimentally selected slow and fast developers was then assessed at differing levels of prey (pea aphid, Acyrthosiphon pisum) and conspecific predator abundance. All individuals, whether slow or fast developers, showed a Type-II functional response, decrease in proportion of prey consumed with increasing prey biomass and an increase in proportion of prey consumed with increasing predator density. The proportion of prey consumed was highest in experimental fast developers and lowest in experimental slow developers. Attack rate was highest and handling time longest in slow developers of control/experimental groups. Mutual interference was least while area of discovery was highest in experimental fast developers. Thus, selection of fast developers for F15 generations led to higher functional responses, slower attack rates and faster prey consumption. This lower mutual interference and high searching efficiency indicates that they can be experimentally selected and used for better control of the pea aphids. This study is the first attempt to evaluate predatory responses of selected lines of an aphidophagous ladybird.

Alternative food and external source of infection stabilize predator–prey oscillations — A conclusion drawn from an eco-epidemiological model

Das et al. [Effect of disease-selective predation on prey infected by contact and external sources, Biosystems 95(3) (2009) 188–199] proposed an eco-epidemiological model where the prey species is infected through the external source of infection and contact of the species. In this present study we have modified their model by assuming that the predator consumes both the susceptible as well as the infected prey following the modified Holling type-II functional response. Our main focusing points of this study are the role of infection rate (both internal and external), alternative food, and half-saturation constant in the predator–prey dynamics with disease in the prey population. We have shown the local stability of the boundary as well as the interior equilibrium point under certain conditions. We have also worked out the permanence of the system. Our simulation results show that the system enters into limit cycle oscillations from stable position for higher values of the contact rate. But it is also shown that the external infection rate, enrichment of the alternative food of the predator population and the half-saturation constant can prevent limit cycle oscillations and stabilize the system. Thus external disease propagation, enrichment of the alternative food resource, and the half-saturation constant are the key factors for preventing the oscillatory behavior of the species.

Rich Global Dynamics in a Prey–Predator Model with Allee Effect and Density Dependent Death Rate of Predator

In this work we have considered a prey–predator model with strong Allee effect in the prey growth function, Holling type-II functional response and density dependent death rate for predators. It presents a comprehensive study of the complete global dynamics for the considered system. Especially to see the effect of the density dependent death rate of predator on the system behavior, we have presented the two parametric bifurcation diagrams taking it as one of the bifurcation parameters. In course of that we have explored all possible local and global bifurcations that the system could undergo, namely the existence of transcritical bifurcation, saddle node bifurcation, cusp bifurcation, Hopf-bifurcation, Bogdanov–Takens bifurcation and Bautin bifurcation respectively.

Fractional-order delayed predator–prey systems with Holling type-II functional response

In this paper, a fractional dynamical system of predator–prey with Holling type-II functional response and time delay is studied. Local and global stability of existence steady states and Hopf bifurcation with respect to the delay is investigated, with fractional-order \(0< \alpha \le 1\). It is found that Hopf bifurcation occurs when the delay passes through a sequence of critical values. Unconditionally, stable implicit scheme for the numerical simulations of the fractional-order delay differential model is introduced. The numerical simulations show the effectiveness of the numerical method and confirm the theoretical results. The presence of fractional order in the delayed differential model improves the stability of the solutions and enrich the dynamics of the model.

Cross-diffusion induced instability and pattern formation for a Holling type-II predator–prey model

This paper is devoted to studying a predator–prey model with Holling type-II functional response and cross-diffusion subject to Neumann boundary condition. Our main interest lies in the effects of cross-diffusion on stability and stationary patterns. More precisely, the presented results show that cross-diffusion can not only destabilize a uniform equilibrium which is stable for the kinetic and random diffusion reaction systems, but also create spatial patterns even when the random diffusion fails to do so. Furthermore, our results also reveal that, in this kind of ecological system, instability and stationary patterns can appear only when the predators rapidly move away from a large group of preys, regardless of the speed that the preys keep away from the predators.

A predator-prey model with generic birth and death rates for the predator and Beddington-DeAngelis functional response

Since the Lotka-Volterra model was published, a lot of work has been devoted to studying the interactions between predator and preypopulations. In 2014, A. J. Terry proposed a new model [1], which is a generalization of the well-known Rosenzweig-MacArthur model [2]. In his model Terry removes the hypothesis of a constant death rate anda linear (with respect to the functional response) growth rate for the predator,that underline the Rosenzweig-MacArthur model (and other classical models). This certainly makes the model more realistic. It should be mentioned, however, that a Holling Type-II functional response is used in the Terry's model.In [3] it is discussed that in many cases the Beddington-DeAngelis functional responce is to be preferred, since it gives a better fit to experimental data.Thus, we further modify the Terry's model by using theBeddington-DeAngelis functional response and study the behaviour of the solutions of the obtainedsystem.

Spatiotemporal complexity in a predator--prey model with weak Allee effects.

In this article, we study the rich dynamics of a diffusive predator-prey system with Allee effects in the prey growth. Our model assumes a prey-dependent Holling type-II functional response and a density dependent death rate for predator. We investigate the dissipation and persistence property, the stability of nonnegative and positive constant steady state of the model, as well as the existence of Hopf bifurcation at the positive constant solution. In addition, we provide results on the existence and non-existence of positive non-constant solutions of the model. We also demonstrate the Turing instability under some conditions, and find that our model exhibits a diffusion-controlled formation growth of spots, stripes, and holes pattern replication via numerical simulations. One of the most interesting findings is that Turing instability in the model is induced by the density dependent death rate in predator.

On the Dynamical Behavior of a Two-Prey One-Predator System with Two-Type Functional Responses

In the paper, a two-prey one-predator system with defensive ability and Holling type-II functional responses is investigated. First, the stability of equilibrium points of the system is discussed and then conditions for the persistence of the system are established according to the existence of limit cycles. Numerical examples are illustrated to attest to our mathematical results. Finally, via bifurcation diagrams, various dynamic behaviors including chaotic phenomena are demonstrated.

Bifurcation behaviours in a delayed three-species food-chain model with Holling type-II functional response

This article is concerned with the local stability of a positive equilibrium and the Hopf bifurcation of a delayed three-species food-chain system with the Holling type-II functional response. Some new sufficient conditions ensuring the local stability of a positive equilibrium and the existence of Hopf bifurcation for the system are established. Some explicit formulae are derived to determine the direction of bifurcations and the stability of bifurcating periodic solutions using the normal form theory and the centre manifold theory. Numerical simulations for justifying the theoretical analysis are also provided. Finally, biological explanations and main conclusions are included.

Period doubling cascades of prey–predator model with nonlinear harvesting and control of over exploitation through taxation

The present study investigates a prey predator type model for conservation of ecological resources through taxation with nonlinear harvesting. The model uses the harvesting function as proposed by Agnew (1979) [1] which accounts for the handling time of the catch and also the competition between standard vessels being utilized for harvesting of resources. In this paper we consider a three dimensional dynamic effort prey–predator model with Holling type-II functional response. The conditions for uniform persistence of the model have been derived. The existence and stability of bifurcating periodic solution through Hopf bifurcation have been examined for a particular set of parameter value. Using numerical examples it is shown that the system admits periodic, quasi-periodic and chaotic solutions. It is observed that the system exhibits periodic doubling route to chaos with respect to tax. Many forms of complexities such as chaotic bands (including periodic windows, period-doubling bifurcations, period-halving bifurcations and attractor crisis) and chaotic attractors have been observed. Sensitivity analysis is carried out and it is observed that the solutions are highly dependent to the initial conditions. Pontryagin’s Maximum Principle has been used to obtain optimal tax policy to maximize the monetary social benefit as well as conservation of the ecosystem.

Optimal control of the dynamics of a tumor growth model with Hollings’ type-II functional response

In this paper, a simple predator–prey type model for interaction of immune system (consisting of resting and hunting cells) with tumor cells using Hollings’ type-II functional response is considered. It is found that the model system has no positive equilibrium if the half-saturation population of resting cells is greater than a critical value. In case of violation of this condition, model system mostly has unique positive equilibrium but when the half-saturation population of cancer cells is smaller than one, it may have three positive equilibriums. Local stability of the feasible equilibriums is studied. These results suggest that a positive equilibrium can be locally asymptotically stable or unstable based on the system parameter values. In case of unstable positive equilibrium (equilibriums), long-time sustenance of immune system with tumor in terms of limit cycle through Hopf bifurcation is observed with half-saturation population of resting cells as bifurcation parameter. The necessary optimal control inputs that make the unstable equilibriums of the model system asymptotically stable and minimize the required performance measure are given as nonlinear functions of the system densities.

Diffusive Holling type-II predator–prey system with harvesting of prey

We study diffusive heterogeneous predator–prey system with Holling type-II functional response in the presence of harvesting in prey equation i.e.{−Δu=λu−a(x)u2−buv1+mu−ch(x)in Ω,−Δv=μv−v2+duv1+muin Ω,∂νu=∂νv=0on ∂Ω, where Ω is a smooth bounded region in RN and a(x) is a nonnegative continuous function on Ω¯, with Ω0 a region such that Ω¯0⊂Ω and a(x)>0 on Ω¯∖Ω¯0. We show that in the case of weak prey growth rate, i.e. λ less than the first eigenvalue of −Δ in Ω0, the system has a positive solution for the values of μ in a bounded interval and c>0 sufficiently small. We also investigate the multiplicity of positive solutions in this case. When the prey growth rate is strong, we will show that the system has a positive solution for large positive value of μ, again provided that c is sufficiently small.

Biological control of a predator–prey system through provision of a super predator

In this paper, we make a systematic analysis of the dynamics of a predator–prey system with type-II functional response, in which the predator growth rate is affected by the presence of a super predator. The main aim of this research is to study the consequences of the presence of a super predator on the system dynamics. The existence and stability of the different possible equilibrium points are studied, and we conclude that the maximum consumption rate of a super predator plays a key role in determining the eventual state of the ecosystem. A detailed bifurcation analysis is carried out through numerical simulations, and we observe that theoretically it is possible to control the dynamics of the system by manipulating the consumption rate of the super predator.

Positive solutions for a Lotka-Volterra prey-predator model with cross-diffusion and Holling type-II functional response

We consider a Lotka-Volterra prey-predator model with cross-diffusion and Holling type-II functional response. The main concern is the existence of positive solutions under the combined effect of cross-diffusion and Holling type-II functional response. Here, a positive solution corresponds to a coexistence state of the model. Firstly, we study the sufficient conditions to ensure the existence of positive solutions by using degree theory and analyze the coexistence region in parameter plane. In addition, we present the uniqueness of positive solutions in one dimension case. Secondly, we study the stability of the trivial and semi-trivial solutions by analyzing the principal eigenvalue of the corresponding linearized system, and then we characterize the stable/unstable regions of semi-trivial solutions in parameter plane.