Michaelis Menten Type Functional Response Research Articles

Dynamical analysis of a two-dimensional discrete predator–prey model

In this paper, we explore the existence of periodic solutions, local dynamics at equilibrium solutions, chaos and bifurcations of a discrete prey–predator model with Michaelis–Menten type functional response. More specifically, we explore local dynamical characteristics at equilibrium solutions, and existence of periodic solutions for the under consideration model. It is also studied the existence of bifurcations at equilibrium solutions, and investigated that at semitrivial and trivial equilibrium solutions model does not undergo flip bifurcation, but at positive equilibrium solution it undergoes flip and Neimark–Sacker bifurcations when parameters goes through the certain curves. It is also investigated that fold bifurcation does not exist at positive equilibrium, and we have studied these bifurcations by center manifold theorem and bifurcation theory. We also studied chaos by feedback control method. The theoretical results are confirmed numerically. Furthermore, we use 0–1 chaos test in order to quantify whether chaos exists in the model or not.

Existence of positive periodic solutions for a periodic predator–prey model with fear effect and general functional responses

This paper investigates the existence of positive periodic solutions for a periodic predator-prey model with fear effect and general functional responses. The general functional responses can cover the Holling types II and III functional response, the Beddington–DeAngelis functional response, the Crowley–Martin functional response, the ratio-dependent type with Michaelis–Menten type functional response, etc. Some new sufficient conditions for the existence of positive periodic solutions of the model are obtained by employing the continuation theorem of coincidence degree theory and some ingenious estimation techniques for the upper and lower bounds of the a priori solutions of the corresponding operator equation. Our results considerably improve and extend some known results.

Bifurcations in a diffusive predator–prey system with linear harvesting

Complex spatiotemporal dynamical behaviors of a diffusive predator–prey system with Michaelis–Menten type functional response and linear harvesting are investigated. Firstly, the critical conditions for the occurrence of Turing instability, which are necessary and sufficient, are derived in a novel way. Then, the existence conditions of codimension-1 Turing bifurcation, Hopf bifurcation, and codimension-2 Turing–Turing bifurcation, Turing–Hopf bifurcation are established. Furthermore, the detailed bifurcation set is given by calculating the amplitude equation with the method of the multiple time scale near the Turing–Hopf bifurcation. We find that the system may exhibit nonconstant steady-state solutions, spatially homogeneous periodic solutions, and spatially inhomogeneous periodic solutions, which can be verified by a series of numerical simulations. These investigations not only explain the effect of diffusion and harvesting on the dynamic behavior of the system, but also reveal the mechanism of spatiotemporal complexity in the diffusive predator–prey system.

Complex dynamical behavior of a ratio-dependent eco-epidemic model with Holling type-II incidence rate in the presence of two delays

We propose an eco-epidemiological model with disease in the prey population and the disease is not inherited. The disease spreads over the susceptible prey through the Holling type-II incidence rate and the prey becomes infected so that the number of infected prey is expected to be saturated. Here, the infected prey is assumed to recover by their natural immune. The predator consumes both susceptible and infected preys according to the ratio-dependent Michaelis–Menten type functional response. We incorporate two-time delays: incidence delay for the conversion of susceptible prey into infected one and gestation delay for the predator growth. Mathematical requirements (positivity, boundedness, permanence of the solution, existences of equilibrium points (graphically based on nullclines concept) and local as well as global stability around equilibrium points) for our proposed model to behave well are taken care here. Numerical simulation is carried out extensively to validate the analytical restrictions, imposed additional conditions and find the approximate solution of our highly nonlinear differential equations. System dynamicity is performed with special attention to newly introduced parameters. Population variation due to some of the parameters is presented and interpreted eco-epidemiologically. When the population starts oscillating, their time-averaged values are presented to show their variation. In order to make the present model more realistic, consumption of susceptible by predator and conversion from susceptible to predator rates are varied simultaneously by considering a linear relationship among these two rates. We apply the necessary tools of nonlinear dynamics such as bifurcation diagram, phase diagram, spectrum and maximum Lyapunov exponent to affirm system’s complex dynamical behavior. Stable/unstable zones are proposed in xτ1c-plane where x is any considered parameter, and τ1c is the corresponding critical value of the incidence delay. Bifurcation diagrams of population with respect to both the delays are presented, and their relative contribution (incident or/and gestation delay) to the system is explored.

Analysis and simulation of numerical schemes for nonlinear hyperbolic predator–prey models with spatial diffusion

Departing from a two-dimensional parabolic system that describes the spatial dynamics in a predator–prey system with Michaelis–Menten-type functional response, we investigate a general form of that model using a finite-difference approach. The model under investigation is a hyperbolic nonlinear system consisting of two coupled partial differential equations with generalized reaction terms. We impose initial conditions on a closed and bounded rectangle, and a fully discrete finite-difference methodology is proposed. Among the most important results of this work, we establish analytically the existence and uniqueness of discrete solutions, along with the second-order consistency of our scheme. Moreover, a discrete form of the energy method is employed to prove the stability and the quadratic convergence of the technique. Some numerical simulations obtained through our method show the appearance of Turing patterns in the parabolic case, in agreement with some reports found in the literature. Moreover, our simulations also show that Turing patterns are present in the hyperbolic scenario.

Simple efficient simulation of the complex dynamics of some nonlinear hyperbolic predator–prey models with spatial diffusion

In this manuscript, we will consider a parabolic partial differential equation that models the spatial-temporal dynamics in predator–prey systems with Michaelis–Menten-type functional response. We study numerically an extension of those systems by employing finite differences. The extension includes the presence of inertial times with extended reaction components in a diffusive system consisting of two populations. The spatial domain is a closed and bounded rectangle, and initial-boundary conditions are imposed on the model. An easy-to-implement fully discrete finite-difference methodology is designed here. We establish analytically the existence and uniqueness of discrete solutions as a consequence of the explicit character of the scheme, along with the second-order consistency. To prove the stability and the second-order convergence of the technique, the energy method will be applied conveniently in a discrete manner. From the point of view of the applications, we provide various numerical simulations that show the appearance of complex patterns in the parabolic case. The results will resemble qualitatively those of some papers already published in the specialized literature. Our simulations also show that different patterns are present in the hyperbolic scenario. We also perform a numerical analysis of the convergence of the method.

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.

Maturation delay for the predators can enhance stable coexistence for a class of prey–predator models

Maturation time delay for the predators is introduced in prey–predator models to implicitly model the stage-structure of predators. Most of the prey–predator models with maturation delay are known to exhibit destabilization of coexistence steady-state. Discrete time delay induced destabilization is a common finding, however, this is due to the introduction of time delay with lack of ecological justification. The main objective of the present work is to show the stabilizing role of maturation delay for a class of delayed prey–predator model. To be specific, we consider prey–predator models with strong and weak Allee effects in prey growth and Michaelis–Menten type functional response. We provide ecological justification for the introduction of maturation delay parameter in predator's growth equation. We obtain the conditions for stable and oscillatory coexistence of prey and their specialist predator in case of strong as well as weak Allee effect for non-delayed and delayed models. Apart from the analytical results for the models under consideration, we perform extensive numerical simulations to construct the relevant bifurcation diagrams. Our analytical and supportive numerical findings reveal that delay is not always a destabilizing factor rather the stable coexistence in the presence of time delay depends upon the formulation of the delayed model. The biological implications of the current investigation are provided in the conclusion section. We also explain the validity of obtained results for other types of prey–predator models with a specialist predator.

The complex dynamics of a diffusive prey–predator model with an Allee effect in prey

This paper investigates complex dynamics of a predator–prey interaction model that incorporates: (a) an Allee effect in prey; (b) the Michaelis–Menten type functional response between prey and predator; and (c) diffusion in both prey and predator. We provide rigorous mathematical results of the proposed model including: (1) the stability of non-negative constant steady states; (2) sufficient conditions that lead to Hopf/Turing bifurcations; (3) a prior estimates of positive steady states; (4) the non-existence and existence of non-constant positive steady states when the model is under zero-flux boundary condition. We also perform completed analysis of the corresponding ODE model to obtain a better understanding on effects of diffusion on the stability. Our analytical results show that the small values of the ratio of the prey's diffusion rate to the predator's diffusion rate are more likely to destabilize the system, thus generate Hopf-bifurcation and Turing instability that can lead to different spatial patterns. Through numerical simulations, we observe that our model, with or without Allee effect, can exhibit extremely rich pattern formations that include but not limit to strips, spotted patterns, symmetric patterns. In addition, the strength of Allee effects also plays an important role in generating distinct spatial patterns.

Hopf Bifurcation and Global Periodic Solutions in a Predator-Prey System with Michaelis-Menten Type Functional Response and Two Delays

We consider a predator-prey system with Michaelis-Menten type functional response and two delays. We focus on the case with two unequal and non-zero delays present in the model, study the local stability of the equilibria and the existence of Hopf bifurcation, and then obtain explicit formulas to determine the properties of Hopf bifurcation by using the normal form method and center manifold theorem. Special attention is paid to the global continuation of local Hopf bifurcation when the delaysτ1≠τ2.

Stability in a Simple Food Chain System with Michaelis-Menten Functional Response and Nonlocal Delays

This paper is concerned with the asymptotical behavior of solutions to the reaction-diffusion system under homogeneous Neumann boundary condition. By taking food ingestion and species' moving into account, the model is further coupled with Michaelis-Menten type functional response and nonlocal delay. Sufficient conditions are derived for the global stability of the positive steady state and the semitrivial steady state of the proposed problem by using the Lyapunov functional. Our results show that intraspecific competition benefits the coexistence of prey and predator. Furthermore, the introduction of Michaelis-Menten type functional response positively affects the coexistence of prey and predator, and the nonlocal delay is harmless for stabilities of all nonnegative steady states of the system. Numerical simulations are carried out to illustrate the main results.

Influence of prey refuge on predator–prey dynamics

A diffusive predator–prey system with Michaelis–Menten type functional response subject to prey refuge is considered. Bifurcation analysis of Hopf and Turing are carried out in detail. In particular, Turing domain is given in the two parameters space. The obtained results show that the refuges used by prey have great influence on the pattern formation of the populations. More specifically, as prey refuge being increased, spotted pattern and coexistence of spotted and stripe-like pattern emerge. It is also proved that the pattern is not dependent on the initial conditions, which means the pattern is controlled by the intrinsic mechanism.

Pattern Formation in Tri-Trophic Ratio-Dependent Food Chain Model

In this paper, a spatial tri-trophic food chain model with ratio-dependent Michaelis-Menten type functional response under homogeneous Neumann boundary conditions is studied. Conditions for Hopf and Turing bifurcation are derived. Sufficient conditions for the emergence of spatial patterns are obtained. The results of numerical simulations reveal the formation of labyrinth patterns and the coexistence of spotted and stripe-like patterns.

STABILITY AND BIFURCATION ANALYSIS IN A DELAYED PREDATOR-PREY SYSTEM

A delayed ratio-dependent one-predator and two-prey system with Michaelis–Menten type functional response is investigated. We show the existence of nonnegative equilibria under some appropriated conditions. Criteria for local stability, instability of nonnegative equilibria are obtained. The existence of Hopf bifurcations at the endemic equilibrium is established by analyzing the distribution of the characteristic values. An explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by using the normal form and the center manifold theory. At last, some numerical simulations to support the analytical conclusions are carried out.

An Analytical Foundation of the Ratio-Dependent Predator-Prey Model

In this paper we present a micro ecosystem model whose basic entities are representative prey and predator organisms which behave as if they maximize their net offspring under constraints. In the short-run periods, prices (scarcity indicators) coordinate and determine all biomass transactions and net offspring which directly translates into population growth and thus constitutes the dynamics of the predator-prey relations. For a specific parametric version, our model offers a theoretical foundation of the ratio-dependent predator-prey model with Michaelis-Menten type functional response.

Permanence for the Michaelis–Menten type discrete three-species ratio-dependent food chain model with delay

A discrete three trophic level food chain model with ratio-dependent Michaelis–Menten type functional response is investigated. It is shown that under some appropriate conditions the system is permanent. The results indicate that, to make the species coexist in the long run, it is a surefire strategy to keep the death rate of the predator and top predator rather small and the intrinsic growth rate of the prey relatively large.

A Micro-Level 'Consumer Approach' to Species Population Dynamics

In this paper we develop a micro ecosystem model whose basic entities are representative organisms which behave as if maximizing their net offspring under constraints. Net offspring is increasing in prey biomass intake, declining in the loss of own biomass to predators and Allee's Law applies. The organism's constraint reflects its perception of how scarce its own biomass and the biomass of its prey is. In the short-run periods prices (scarcity indicators) coordinate and determine all biomass transactions and net offspring which directly translates into population growth functions. We are able to explicitly determine these growth functions for a simple food web when specific parametric net offspring functions are chosen in the micro-level ecosystem model. For the case of a single species our model is shown to yield the well-known Verhulst-Pearl logistic growth function. With two species in predator-prey relationship, we derive differential equations whose dynamics are completely characterized and turn out to be similar to the predator-prey model with Michaelis-Menten type functional response. With two species competing for a single resource we find that coexistence is a knife-edge feature confirming Tschirhart's (2002) result in a different but related model.

Periodic solution of a delayed predator–prey system with Michaelis–Menten type functional response

By using the continuation theorem base on Gaines and Mawhin's coincidence degree, sufficient and realistic conditions are obtained for the global existence of positive periodic solution for a delayed predator–prey system with Michaelis–Menten type functional response. Indeed, our results are applicable to state-dependent delays.

Persistence and global stability in a delayed predator–prey system with Michaelis–Menten type functional response

A delayed three-species predator–prey food-chain model with Michaelis–Menten type functional response is investigated. It is proved that the system is uniformly persistent under some appropriate conditions. By means of constructing suitable Lyapunov functional, sufficient conditions are derived for the global asymptotic stability of the positive equilibrium of the system.

The biological control paradox

We propose an eco-epidemiological model with disease in the prey population and the disease is not inherited. The disease spreads over the susceptible prey through the Holling type-II incidence rate and the prey becomes infected so that the number of infected prey is expected to be saturated. Here, the infected prey is assumed to recover by their natural immune. The predator consumes both susceptible and infected preys according to the ratio-dependent Michaelis–Menten type functional response. We incorporate two-time delays: incidence delay for the conversion of susceptible prey into infected one and gestation delay for the predator growth. Mathematical requirements (positivity, boundedness, permanence of the solution, existences of equilibrium points (graphically based on nullclines concept) and local as well as global stability around equilibrium points) for our proposed model to behave well are taken care here. Numerical simulation is carried out extensively to validate the analytical restrictions, imposed additional conditions and find the approximate solution of our highly nonlinear differential equations. System dynamicity is performed with special attention to newly introduced parameters. Population variation due to some of the parameters is presented and interpreted eco-epidemiologically. When the population starts oscillating, their time-averaged values are presented to show their variation. In order to make the present model more realistic, consumption of susceptible by predator and conversion from susceptible to predator rates are varied simultaneously by considering a linear relationship among these two rates. We apply the necessary tools of nonlinear dynamics such as bifurcation diagram, phase diagram, spectrum and maximum Lyapunov exponent to affirm system’s complex dynamical behavior. Stable/unstable zones are proposed in xτ1c-plane where x is any considered parameter, and τ1c is the corresponding critical value of the incidence delay. Bifurcation diagrams of population with respect to both the delays are presented, and their relative contribution (incident or/and gestation delay) to the system is explored.