Ratio-dependent Predator Prey Model Research Articles

Diffusion-induced regular and chaotic patterns in a ratio-dependent predator-prey model with fear factor and prey refuge.

Population distribution of interacting species in a large scale natural system is heterogeneous and subject to change for various reasons. Here, we explore how behavioral modification in prey species due to fear of predator and mutual interference between predators can create different spatiotemporal patterns in population distribution. We show that the fear factor and diffusion in a ratio-dependent predator-prey model may show more complex dynamics than observed earlier. It is shown that when prey diffusivity is low, prey remains concentrated at different patches throughout the domain. However, prey density becomes low at the patches as they disperse at a higher rate. Mixed and stripe patterns are observed during the transition from a hot spot pattern at the lower prey diffusivity to a cold spot pattern at its higher value. Pattern transition is, however, completely opposite if the antipredator behavior is gradually increased. Our simulation results reveal that the spatiotemporal chaotic pattern may also be observed in the Hopf-Turing region of instability provided prey shows a higher level of antipredator behavior. The chaotic pattern of the Hopf-Turing region may be shifted to a spot type pattern of the Turing region depending on the refuge level of the habitat.

A free boundary problem with a Stefan condition for a ratio-dependent predator-prey model

<abstract><p>In this paper we study a ratio-dependent predator-prey model with a free boundary caused by predator-prey interaction over a one dimensional habitat. We study the long time behaviors of the two species and prove a spreading-vanishing dichotomy; namely, as $ t $ goes to infinity, both prey and predator successfully spread to the whole space and survive in the new environment, or they spread within a bounded area and eventually die out. The criteria governing spreading and vanishing are obtained. Finally, when spreading occurs we provide some estimates to the asymptotic spreading speed of the moving boundary $ h(t) $.</p></abstract>

Stationary distribution of a stochastic ratio-dependent predator-prey system with regime-switching

Abstract This article explores a stochastic ratio-dependent predator-prey model with regime-switching. We testify that the model admits a unique stationary distribution, and demonstrate that the transition probability of the solution of the model converges to the stationary distribution in exponent rate. We also discuss the biological implications of the results by aid of some numerical simulations.

Invasion Traveling Waves for a Discrete Diffusive Ratio-Dependent Predator-Prey Model

This article is concerned with the existence of traveling wave solutions for a discrete diffusive ratio-dependent predator-prey model. By applying Schauder’s fixed point theorem with the help of suitable upper and lower solutions, we prove that there exists a positive constant c* such that when c > c*, the discrete diffusive predator-prey system admits an invasion traveling wave. The existence of an invasion traveling wave with c = c* is also established by a limiting argument and a delicate analysis of the boundary conditions. Finally, by the asymptotic spreading theory and the comparison principle, the non-existence of invasion traveling waves with speed c < c* is also proved.

Analysis for a Delayed Three-Species Predator-Prey Model with Feedback Controls and Prey Diffusion

In this paper, we study a class of 3-species multidelay Lotka–Volterra ratio-dependent predator-prey model with feedback controls and prey diffusion. By using the theory of delay differential inequalities and developing some new analysis methods as well as constructing a suitable Lyapunov function, some sufficient conditions are obtained to guarantee the permanence of the system and the global attractivity of the positive solution for the predator-prey system. Furthermore, the corresponding periodic system is discussed, and some conditions are established about the existence, uniqueness, and stability of the positive periodic solution for the periodic system by using the fixed-point theory and theoretical analysis. In additional, some numerical solutions of the equations describing the system are given to verify the obtained criteria. Finally, the corresponding stochastic time-delay predator-prey model with multiplicative noise sources is solved numerically, and some interesting new dynamics behavior is obtained.

Allee effect can simplify the dynamics of a prey-predator model

In this work, we investigate a prey-predator model which includes the Allee effect phenomena in prey growth function, density dependent death rate for predators and ratio dependent functional response. we fulfill a comprehensive bifurcation analysis, constructing the two-parametric bifurcation diagrams which describes the effect of density dependent death rate parameter, and also show possible phase portraits. We have also investigated the model in the absence of Allee effect and corresponding bifurcation diagram has been presented to show the dynamical changes in the system. Then we compare the properties of the ratio dependent prey-predator model with and without the Allee effect and show that Allee effect has a significant role in the dynamics. Allee effect can preserve local extinction of populations and suppress the stability of interior equilibrium point. Finally, all the analytical results are validated with the help of numerical simulations.

Persistence of a Lotka-volterra Ratio-dependent Predator-prey Model with Delays and Feedback Controls

The aim of this article is to study the dynamical behavior of a Lotka-volterra 3-species ratio-dependent predator-prey system with delays and feedback controls. By using the comparison theorem, the differential inequalities and developing new analysis method, some sufficient conditions are obtained to ensure the permanence of the solutions for the delayed predator-prey system, and some known results are generalized.

Impact of Allee effect on an eco-epidemiological system

In this paper, we propose a general ratio-dependent prey-predator model with disease in predator subject to the strong Allee effect in prey. We obtain the complete dynamics of both models: (a) full model with Allee effect; (b) full model without Allee effect. Model (a) may have more than one interior equilibrium point, but model (b) has only one interior equilibrium point. Numerical results reveal that the coexistence of all the populations at the endemic state is possible for both the models. But for the model with Allee effect, the coexistence can be destroyed by an increased supply of alternative food for the predators. It can also be proved that for the full model with Allee effect, the disease can be suppressed under certain parametric conditions. Also by comparing models (a) and (b), we conclude that Allee effect can create or destroy the interior attractor. Finally, we have studied the disease free-submodel (prey and susceptible predator model) with and without Allee effect. The comparative study between these two submodels leads to the following conclusions: 1) In the presence of Allee effect, the number of interior equilibrium points can change from zero to two whereas the submodel without Allee effect has unique interior equilibrium point; 2) Both with and without Allee effect, initial conditions play an important role on the survival and extinction of prey as well as its corresponding predator; 3) In the presence of Allee effect, bi-stability occurs with stable or periodic coexistence of prey and susceptible predator and the extinction of prey and susceptible predator; 4) Allee effect can generate or destroy the interior equilibrium points.

Existence of Traveling Waves for Ratio-dependent Predator-prey System with Nonlocal Diffusion

In this paper, we study the traveling waves for the ratio-dependent predator-prey model with nonlocal diffusion, which is devoted to the existence and nonexistence of traveling wave solution. This model incorporates the ratio-dependent functional response into the Lotka-Volterra type system, and both species obey the logistic growth. Firstly, we construct a nice pair of upper and lower solutions when the wave speed is greater than the minimal wave speed. Then by applying Schauder's fixed point theorem with the help of suitable upper and lower solutions, we can obtain the existence of traveling waves when the wave speed is greater than the minimal wave speed. Moreover, in order to prove the limit behavior of the traveling waves at infinity, we construct a sequence that converges to the coexistence state. Finally, by using the comparison principle, we obtain the nonexistence of the traveling waves when the wave speed is greater than 0 and less than the minimal wave speed. The difficulty of this paper is to construct a suitable upper and lower solution, which is also the novelty of this paper. Under certain restricted condition, this paper concludes the existence and the nonexistence of the traveling waves for the ratio-dependent predator-prey model with nonlocal diffusion.

A Ratio-Dependent Nonlinear Predator-Prey Model With Certain Dynamical Results

In order to see the dynamics of prey-predator interaction, differential or difference equations are frequently used for modeling of such interactions. In present manuscript, we explore some qualitative aspects of two-dimensional ratio-dependent predator-prey model. Taking into account the non-overlapping generations for class of predator-prey system, a novel consistency preserving scheme is proposed. Our study reveals that the implemented discretization is bifurcation preserving. Some dynamical aspects including local behavior of equilibria, phase-plane analysis and emergence of Hopf bifurcation for continuous predator-prey model are studied. Moreover, existence of biologically feasible fixed points, their local asymptotic behavior and phase-plane classification of interior (positive) fixed point are carried out. Furthermore, bifurcation theory of normal forms is implemented to prove that proposed discrete-time model undergoes Neimark-Sacker bifurcation around its unique positive fixed point. Taking into account the bifurcating and fluctuating behaviour of discrete system, three chaos control strategies are implemented. Numerical simulations are provided to illustrate the theoretical discussion and effectiveness of introduced chaos control methods.

Bifurcation and Turing pattern formation in a diffusive ratio-dependent predator–prey model with predator harvesting

The ratio-dependent predator–prey model exhibits rich dynamics due to the singularity of the origin. Harvesting in a ratio-dependent predator–prey model is relatively an important research project from both ecological and mathematical points of view. In this paper, we study the temporal, spatial and spatiotemporal dynamics of a ratio-dependent predator–prey diffusive model where the predator population harvest at catch-per-unit-effort hypothesis. For the spatially homogeneous model, we derive conditions for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solution by the center manifold and the normal form theory. For the reaction–diffusion model, firstly it is shown that Turing (diffusion-driven) instability occurs, which induces spatial inhomogeneous patterns. Then it is demonstrated that the model exhibit Hopf bifurcation which produces temporal inhomogeneous patterns. Finally, the existence and non-existence of positive non-constant steady-state solutions are established. Moreover, numerical simulations are performed to visualize the complex dynamic behavior.

Turing\u2013Hopf bifurcation of a ratio-dependent predator-prey model with diffusion

In this paper, the Turing–Hopf bifurcation of a ratio-dependent predator-prey model with diffusion and Neumann boundary condition is considered. Firstly, we present a kind of double parameters selection method, which can be used to analyze the Turing–Hopf bifurcation of a general reaction-diffusion equation under Neumann boundary condition. By analyzing the distribution of eigenvalues, the stable region, the unstable region (including Turing unstable region), and Turing–Hopf bifurcation point are derived in a double parameters plane. Secondly, by applying this method, the Turing–Hopf bifurcation of a ratio-dependent predator-prey model with diffusion is investigated. Finally, we compute normal forms near Turing–Hopf singularity and verify the theoretical analysis by numerical simulations.

Canards, heteroclinic and homoclinic orbits for a slow-fast predator-prey model of generalized Holling type III

For a classical ratio-dependent predator-prey model with the generalized Holling type III functional response, it was previously investigated in [20] by Hsu and Huang for global stability of an equilibrium, and in [21] by Huang, Ruan and Song for subcritical Hopf and Bogdanov-Takens bifurcations. Here in this model when prey reproduces much faster than predator, by using geometric singular perturbation theory, we achieve much richer new dynamical phenomena than the existing ones, such as the existence of canard cycles, canard explosion and relaxation oscillations, heteroclinic and homoclinic orbits, cyclicity of slow-fast cycles, and the coexistence of the Hopf cycle and the relaxation oscillation. On global stability of the equilibrium we also provide less restricted conditions than the existing ones.

Spatiotemporal dynamics in a ratio-dependent predator-prey model with time delay near the Turing–Hopf bifurcation point

Spatiotemporal dynamics of a ratio-dependent predator-prey model with time delay and the Neumann boundary condition are considered. The existence of the codimension-two Turing–Hopf bifurcation point is proved, so that both the Turing and Hopf bifurcations will occur simultaneously. Then, through the global Hopf bifurcation results, the global Hopf bifurcation of the ratio-dependent predator-prey model is investigated when diffusion is absent. Further, to identify the spatiotemporal dynamics during the Turing–Hopf bifurcation, the method of the multiple time scale analysis is employed to derive the amplitude equations near the codimension-two bifurcation point, instead of employing the center manifold theory and the normal form reduction. By analyzing the amplitude equations, it is found that the original ratio-dependent predator-prey model may exhibit the spatial, temporal or the spatiotemporal patterns, such as the nonconstant steady state, spatially homogeneous periodic solutions as well as the spatially inhomogeneous periodic solutions. Numerical simulations are carried out to validate the theoretical results.

Spatial Pattern of Ratio-Dependent Predator–Prey Model with Prey Harvesting and Cross-Diffusion

This study focuses on the spatial-temporal dynamics of predator–prey model with cross-diffusion where the intake rate of prey is per capita predator according to ratio-dependent functional response and the prey is harvested through nonlinear harvesting strategy. The permanence analysis and local stability analysis of the proposed model without cross-diffusion are analyzed. We derive the conditions for the appearance of diffusion-driven instability and global stability of the considered model. Also the parameter space for Turing region is specified by keeping the cross-diffusion coefficient as one of the crucial parameters. Numerical simulations are given to justify the proposed theoretical results and to show that the cross-diffusion term plays a significant role in the pattern formation.

Fixed Point Theory and Positive Solutions for a Ratio-Dependent Elliptic System

We consider a ratio-dependent predator-prey model under zero Dirichlet boundary condition. By using topological degree theory and fixed index theory, we study the necessary and sufficient conditions for the existence of positive solutions. Then we present the asymptotic behavior analysis of positive solutions, by bifurcation theory and energy estimates.

Spatiotemporal Patterns and Bifurcation Analysis of a Diffusive Predator-prey Model with Prey Refuge

In this paper, we examine a diffusive ratio dependent predator-prey model with prey refuge subject to Neumann boundary condition. We discuss the stability of the positive equilibrium and the existence of spatially homogeneous and inhomogeneous periodic solutions through distribution of the eigenvalues. The direction and stability of Hopf bifurcation are determined by the normal form theory and the center manifold theory. Finally numerical simulations are given to validate our analytical results.

BOGDANOV-TAKENS BIFURCATION IN A DELAYED MICHAELIS-MENTEN TYPE RATIO-DEPENDENT PREDATOR-PREY SYSTEM WITH PREY HARVESTING

In this paper, we study a delayed Michaelis-Menten Type ratio-dependent predator-prey model with prey harvesting. By considering the characteristic equation associated with the nonhyperbolic equilibrium, the critical value of the parameters for the Bogdanov-Takens bifurcation is obtained. The conditions for the characteristic equation having negative real parts are discussed. Using the normal form theory of Bogdanov-Takens bifurcation for retarded functional differential equations, the corresponding normal form restricted to the associated two-dimensional center manifold is calculated and the versal unfolding is considered. The parameter conditions for saddle-node bifurcation, Hopf bifurcation and homoclinic bifurcation are obtained. Numerical simulations are given to support the analytical results.

Periodic oscillations in age-structured ratio-dependent predator–prey model with Michaelis–Menten type functional response

A novel Michaelis–Menten type ratio-dependent predator–prey model with predator-age structure is investigated. The predator fertility function is considered to be the piecewise function related to predator maturation period τ. Our methods rely upon integrated semigroup theory and Hopf bifurcation theory for semilinear equations with non-dense domain. By local analysis and bifurcation analysis of the model, a non-trivial periodic oscillation phenomenon through Hopf bifurcation appears when parameter τ passes through some critical values. Numerical simulations and sensitivity analysis on several important parameters are also performed to illustrate the conclusions.

On a delay ratio-dependent predator–prey system with feedback controls and shelter for the prey

In this paper, a class of three-species multi-delay Lotka–Volterra ratio-dependent predator–prey model with feedback controls and shelter for the prey is considered. A set of easily verifiable sufficient conditions which guarantees the permanence of the system and the global attractivity of positive solution for the predator–prey system are established by developing some new analysis methods and using the theory of differential inequalities as well as constructing a suitable Lyapunov function. Furthermore, some conditions for the existence, uniqueness and stability of positive periodic solution for the corresponding periodic system are obtained by using the fixed point theory and some new analysis techniques. In addition, some numerical solutions of the equations describing the system are given to show that the obtained criteria are new, general, and easily verifiable. Finally, we still solve numerically the corresponding stochastic predator–prey models with multiplicative noise sources, and obtain some new interesting dynamical behaviors of the system. At the same time, the influence of the delays and shelters on the dynamics behavior of the system is also considered by solving numerically the predator–prey models.