Ratio-dependent Functional Response Research Articles

Patterns dynamics in a diffusive prey–predator system with ratio-dependent functional response in a planar domain

This paper is concerned with a classical two-species prey–predator reaction–diffusion system with ratio-dependent functional response and subject to homogeneous Neumann boundary condition in a two-dimensional rectangle domain. By analyzing the associated eigenvalue problem, the spatially homogeneous Hopf bifurcation curve and Turing bifurcation curve of system at the constant coexistence equilibrium are established. Then when the bifurcation parameter is in the interior of range for Turing instability and near Turing bifurcation curve, the amplitude equations of the original system near the constant coexistence equilibrium are obtained by multiple-scale time perturbation analysis. On the basis of the obtained amplitude equations, the stability and classifications of spatiotemporal patterns of the original system at the constant coexistence equilibrium are discussed. Finally, to verify the validity of the obtained theoretical results, numerical simulations are also carried out.

Influence of dispersal on the dynamics of an eco-epidemiological two-patch predator-prey model in deterministic and stochastic environment with ratio-dependent functional responses

Influence of dispersal on the dynamics of an eco-epidemiological two-patch predator-prey model in deterministic and stochastic environment with ratio-dependent functional responses

Dynamic Analysis of a Ratio-Dependent Food Chain Model with Prey-Taxis

In this paper, we consider a food chain model with ratio-dependent functional response and prey-taxis. We first investigate the global existence and boundedness of the unique positive classical solutions of the system in a bounded domain with smooth boundary and Neumann boundary conditions. Then, we analyze the local stability of the system and the existence of Hopf bifurcation. In addition, we prove the global asymptotic stability of steady states under some conditions by constructing a Lyapunov functional, and investigate convergence rates. Finally, we present several numerical simulations to illustrate the results.

Bifurcation Patterns in a Discrete Predator–Prey Model Incorporating Ratio-Dependent Functional Response and Prey Harvesting

Bifurcation Patterns in a Discrete Predator–Prey Model Incorporating Ratio-Dependent Functional Response and Prey Harvesting

Laplace transform-homotopy perturbation method for fractional time diffusive predator–prey models in ecology

In this paper, we consider some reaction–diffusion systems arising from two-component predator–prey models with various kinetics ranging from prey-dependent, ratio-dependent functional responses and subdiffusion. The goal of the present work is to simulate the time-dependent predator–prey model of Lotka–Volterra. Analytical solutions of this model are performed using the Laplace transform-homotopy perturbation method. The proposed scheme finds the solution without any discretization or restrictive assumptions and avoids the round-off errors. The fact that the proposed technique solves nonlinear problems without using Adomian’s polynomials can be considered as a clear advantage of this algorithm over the decomposition method. We show all the possible equilibria and examine their stability in line with the ecological parameters. Numerical simulations of the diffusive predator–prey model in one-, two- and three-dimensions are given to compare and demonstrate the asymptotic behaviour of the time-dependent reaction–diffusion systems. The results show that proposed technique is a powerful tool to solve systems of nonlinear ordinary and partial differential equations of predator–prey models in ecology.

Complicate dynamical properties of a discrete slow-fast predator-prey model with ratio-dependent functional response

Using a semidiscretization method, we derive in this paper a discrete slow-fast predator-prey system with ratio-dependent functional response. First of all, a detailed study for the local stability of fixed points of the system is obtained by invoking an important lemma. In addition, by utilizing the center manifold theorem and the bifurcation theory some sufficient conditions are obtained for the transcritical bifurcation and Neimark-Sacker bifurcation of this system to occur. Finally, with the use of Matlab software, numerical simulations are carried out to illustrate the corresponding theoretical results and reveal some new dynamics of the system. Our results clearly demonstrate that the system is very sensitive to its fast time scale parameter variable.

Analytical detection of stationary and dynamic patterns in a prey-predator model with reproductive Allee effect in prey growth.

Allee effect in population dynamics has a major impact in suppressing the paradox of enrichment through global bifurcation, and it can generate highly complex dynamics. The influence of the reproductive Allee effect, incorporated in the prey's growth rate of a prey-predator model with Beddington-DeAngelis functional response, is investigated here. Preliminary local and global bifurcations are identified of the temporal model. Existence and non-existence of heterogeneous steady-state solutions of the spatio-temporal system are established for suitable ranges of parameter values. The spatio-temporal model satisfies Turing instability conditions, but numerical investigation reveals that the heterogeneous patterns corresponding to unstable Turing eigenmodes act as a transitory pattern. Inclusion of the reproductive Allee effect in the prey population has a destabilising effect on the coexistence equilibrium. For a range of parameter values, various branches of stationary solutions including mode-dependent Turing solutions and localized pattern solutions are identified using numerical bifurcation technique. The model is also capable to produce some complex dynamic patterns such as travelling wave, moving pulse solution, and spatio-temporal chaos for certain range of parameters and diffusivity along with appropriate choice of initial conditions. Judicious choices of parametrization for the Beddington-DeAngelis functional response help us to infer about the resulting patterns for similar prey-predator models with Holling type-II functional response and ratio-dependent functional response.

Multiple Bifurcations in a Discrete Bazykin Predator–Prey Model with Predator Intraspecific Interactions and Ratio-Dependent Functional Response

Multiple Bifurcations in a Discrete Bazykin Predator–Prey Model with Predator Intraspecific Interactions and Ratio-Dependent Functional Response

The Ratio Predator-Prey Model with Random Initial Conditions

In this work, the predator-prey model with the ratio-dependent functional response is considered, where the randomness enters into the equations only through their initial conditions. It is done by assuming normal distribution as the initial states of the model to treat the randomness. The passage from the deterministic situation to the random one for these equations is also the most transparent. In addition, a numerical simulation will be offered using the modified approach founded on the fifth-order improved Runge-Kutta method. Furthermore, the stability of the equilibrium points, and certain statistical properties related to the random behaviour of predators and their prey, will be analyzed and discussed.

Dynamical Behaviour Changes in Response to Various Functional Responses: Temporal and Spatial Plankton System

In this paper, we focus on the comparison between ratio-dependent and Holling type-II functional response in the phytoplankton dynamics with the susceptible and infected populations with or without diffusion. The dynamics are investigated in terms of local stability using both the response function for the systems and the definition of the Hopf-bifurcation, which has been demonstrated to occur in the Holling type-II functional response using the half-saturation time (i.e. $$\alpha$$ ) as the bifurcation parameter. We observed that the system’s dynamical behaviour is affected by the response functions and the system is more stable in the case of the ratio-dependent functional response. Furthermore, the study of higher-order perturbation terms reveals the heterogeneity of in-homogeneous spatiotemporal perturbation around temporal steady state outside the linear regime. The system behaviours to both types of response functions are studied numerically in the temporal and spatiotemporal domain (i.e. pattern formation in the diffusive population). Numerical simulation is used to examine the results.

Analysis of Delay-Aware Worm Propagation Model in Wireless IoT Systems With Ratio-Dependent Functional Response

This article presents a delayed susceptibility, exposure, infectivity, recovery, and vaccination (SEIRV) model with nonlinear incidence and ratio-dependent functional responses. Model limitations and local stability analyses were examined with strict consideration of delay awareness. In addition, the presence of Hopf bifurcations with delay as a bifurcation parameter was investigated along with feature distributions with appropriate constraints. Numerical simulations are presented to verify the proposed theoretical results. In particular, if the latency exceeds the threshold, worm propagation in the system may become out of control. We demonstrated that the propagation characteristics of worms can easily be predicted and eliminated if the delay values are below a suitable threshold. Finally, we conclude that worm propagation is controllable by shifting the presence of Hopf bifurcations.

Stage structured prey-predator model incorporating mortal peril consequential to inefficiency and habitat complexity in juvenile hunting

Dynamic exploration for a predator-prey biosystem of two species with ratio-dependent functional response together with the capability for predation by both juvenile and adult stage of the predator is carried out; coupled with incentive from all the natural documentation acquainting habitat complexity as an armour for the prey and also as a factor effectuating mortality of only the juvenile predators, attributable to their inefficiency. So as to avoid extinction of either of the species and to preserve the food chain of the ecological system, the results pertaining to the existence and stability of all the equilibrium points of the biosystem along with permanence, transcritical and Hopf bifurcation have been thoroughly studied. Excessive predation by juveniles exhibits destabilization of the biosystem through supercritical Hopf bifurcation; likewise, minimal inefficiency of juveniles and habitat complexity spawns extinction and fluctuation of each species, contrarily an abundance of them decimates the predator species. The obtained results and the dependence of the biosystem on some vital parameters are corroborated from a biological viewpoint, through numerical simulation, for implementation of the system in real-life scenario. Interestingly, an illustration reveals an intriguing limit cycle between the proposed system's trivial and axial equilibriums of the proposed system along with the co-existing periodic point for specific parametric values of inefficiency of juvenile predators.

Transcritical Bifurcation and Flip Bifurcation of a New Discrete Ratio-Dependent Predator-Prey System

After a discrete two-species predator-prey system with ratio-dependent functional response is topologically and equivalently reduced, some new dynamical properties for the new discrete system are formulated. The one is for the existence and local stability for all equilibria of this new system. Although the corresponding results for the equilibrium \(E_3\) have been presented in a known literature, our results are more complete. The other is, what’s more important and difficult, to derive some sufficient conditions for the transcritical bifurcation and period-doubling bifurcation of this system at the equilibria \(E_1\), \(E_2\) and \(E_3\) to occur, which are completely new. Numerical simulations are performed to not only illustrate the theoretical results obtained but also find new dynamics—chaos occuring. Our results sufficiently display that this system is very sensitive to its parameters. Namely, the perturbations of different parameters in this system will produce different bifurcations.

Study of Dynamical Behavior of a Delayed Stage-Structured Predator-Prey Model with Disease in Prey

The present paper deals with the dynamics of a stage-structured predator-prey model, with a ratio-dependent functional response including gestational delay in the predator. The prey is carrying an infection which affects the predator adversely. The boundedness of solutions and the stability of equilibrium points have been investigated. There is a Hopf-bifurcation arising out of the variation in the time-delay parameter. Numerical simulations of phase-plane diagrams, and bifurcation diagrams illustrate the dependence of the system on the delay -time. The effect of the disease transmission from prey to predator has also been illustrated through simulations.

Predator invasion in predator–prey model with prey-taxis in spatially heterogeneous environment

We consider a predator–prey model with prey-taxis and Holling-type II functional responses in a spatially heterogeneous environment to analyze the effects of prey-taxis and the heterogeneity of an environment on predator invasion. To achieve our goal, we investigate the stability of semi-trivial solution in which the predator is absent. It is known that both the predator diffusion and the death rate contribute to the predator invasion in a heterogeneous habitat when there is no prey-taxis. In this paper, we show that predator invasion is affected by the prey-taxis and diffusions of the prey-taxis model for a certain range of predator death rates in a heterogeneous environment. Furthermore, in cases where predator invasion by predator diffusion does not occur in a particular death rate range of the predator, predator invasion can occur by prey-taxis in a spatially heterogeneous habitat. In addition, we compare this phenomenon to the corresponding predator–prey model with ratio-dependent functional responses. It is observed that none of the predator’s diffusion and prey-taxis affect the predator’s invasion, and that only the predator’s death rate contributes to predator invasion for the model with ratio-dependent functional responses.

Analysis on Dynamics of Delayed Intraguild Predation Model with Ratio-Dependent Functional Response

Analysis on Dynamics of Delayed Intraguild Predation Model with Ratio-Dependent Functional Response

Ratio-dependent functional response of two common Cladocera present in farmland ponds to Batrachochytrium dendrobatidis

Batrachochytrium dendrobatidis is responsible for amphibian declines worldwide. Decreasing the aquatic density of this chytrid through consumption of its infectious zoospores by Cladocera (water fleas) may mitigate the impact of chytridiomycosis. Understanding this predator-prey relationship requires insights in the zoospore ingestion rate of an average water flea, but such data are almost non-existent. We investigated the functional response of Simocephalus vetulus and Chydorus sphaericus feeding on B. dendrobatidis zoospores. These Cladocera commonly occur in farmland ponds, which may represent a major habitat for disease control. Both water fleas actively ingested zoospores and their per capita ingestion rate was best modelled in function of zoospore-to-Cladocera ratio, implying mutual interference among water fleas during zoospore feeding. The larger S. vetulus substantially consumed more zoospores, characterised by a maximum ingestion rate of 2.5 × 10 5 zoospores.Cladocera −1 .h −1 .mL −1 , which is about 12 times higher than for C. sphaericus . These findings are useful to support model-based management of chytridiomycosis. • Simocephalus vetulus and Chydorus sphaericus are common Cladocera in farmland ponds. • S. vetulus and C. sphaericus consume zoospores of B. dendrobatidis . • Zoospore ingestion rate for both Cladocera depends on zoospore-to-Cladocera ratio. • The larger S. vetulus has a significant larger per capita ingestion rate than C. sphaericus .

The Spatial Pattern Dynamics of Reaction-Diffusion Instability in Tumor Cell–Immune Cell System

In this paper, spatial patterns of a diffusive Tumor cell and immune cell model with sigmoid ratio-dependent functional response are investigated. The asymptotic stability behavior of the corresponding nonspatial model around the unique positive interior equilibrium point in homogeneous steady state is obtained. We have obtained the optimal condition under which the system loses stability and a Turing pattern occurs. Numerical simulations have been carried out in order to show the significant role of reaction-diffusion coefficients and other important parameters of the system. Various contour figures of spatial patterns through Turing instability are portrayed and analyzed in order to substantiate the applicability of the present model. The paper ends with an extended discussion of biological implications of the immune system.

Dynamics in a ratio-dependent Lotka\u2013Volterra competitive-competitive-cooperative system with feedback controls and delays

This study investigates the dynamical behavior of a ratio-dependent Lotka–Volterra competitive-competitive-cooperative system with feedback controls and delays. Compared with previous studies, both ratio-dependent functional responses and time delays are considered. By employing the comparison method, the Lyapunov function method, and useful inequality techniques, some sufficient conditions on the permanence, periodic solution, and global attractivity for the considered system are derived. Finally, a numerical example is also presented to validate the practicability and feasibility of our proposed results.

Bifurcation analysis of a predator-prey model with predator intraspecific interactions and ratio-dependent functional response

We study the Bazykin predator-prey model with predator intraspecific interactions and ratio-dependent functional response and show the existence and stability of two interior equilibrium points. We prove that the model displays a wide range of different bifurcations, such as saddle-node bifurcations, Hopf bifurcations, homoclinic bifurcations and Bogdanov-Takens bifurcations. We use numerical simulations to further illustrate the impact changing the predator per capita consumption rate has on the basin of attraction of the stable equilibrium points, as well as the impact of changing the efficiency with which predators convert consumed prey into new predators.