Michaelis-Menten Functional Response Research Articles

Flip bifurcation and Neimark-Sacker bifurcation in a discrete predator-prey model with Michaelis-Menten functional response

<abstract><p>In this paper, we use a semi-discretization method to explore a predator-prey model with Michaelis-Menten functional response. Firstly, we investigate the local stability of fixed points. Then, by using the center manifold theorem and bifurcation theory, we demonstrate that the system experiences a flip bifurcation and a Neimark-Sacker bifurcation at a fixed point when one of the parameters goes through its critical value. To illustrate our results, numerical simulations, which include maximum Lyapunov exponents, fractal dimensions and phase portraits, are also presented.</p></abstract>

Bifurcation analysis of a food chain chemostat model with Michaelis-Menten functional response and double delays

<abstract><p>In this paper, we study a food chain chemostat model with Michaelis-Menten function response and double delays. Applying the stability theory of functional differential equations, we discuss the conditions for the stability of three equilibria, respectively. Furthermore, we analyze the sufficient conditions for the Hopf bifurcation of the system at the positive equilibrium. Finally, we present some numerical examples to verify the correctness of the theoretical analysis and give some valuable conclusions and further discussions at the end of the paper.</p></abstract>

Periodic trajectories for an age-structured prey–predator system with Michaelis–Menten functional response including delays and asymmetric diffusion

This paper studies the periodic trajectories of a novel age-structured prey–predator system with Michaelis–Menten functional response including delays and asymmetric diffusion. To begin with, the system is turned into an abstract non-densely defined Cauchy problem, and a time-lag effect in their interaction is investigated. Next, we acquire that this system appears a periodic orbit near the positive steady state by employing the method of integrated semigroup and the Hopf bifurcation theory for semilinear equations with non-dense domain, which is also the main result of this article. Finally, in order to illustrate our theoretical analysis more vividly, we make some numerical simulations and give some discussions.

Spatiotemporal Patterns Formed by a Discrete Nutrient-Phytoplankton Model with Time Delay

In this research, a continuous nutrient-phytoplankton model with time delay and Michaelis–Menten functional response is discretized to a spatiotemporal discrete model. Around the homogeneous steady state of the discrete model, Neimark–Sacker bifurcation and Turing bifurcation analysis are investigated. Based on the bifurcation analysis, numerical simulations are carried out on the formation of spatiotemporal patterns. Simulation results show that the diffusion of phytoplankton and nutrients can induce the formation of Turing-like patterns, while time delay can also induce the formation of cloud-like pattern by Neimark–Sacker bifurcation. Compared with the results generated by the continuous model, more types of patterns are obtained and are compared with real observed patterns.

A Michaelis–Menten Predator–Prey Model with Strong Allee Effect and Disease in Prey Incorporating Prey Refuge

Here, we have proposed a predator–prey model with Michaelis–Menten functional response and divided the prey population in two subpopulations: susceptible and infected prey. Refuge has been incorporated in infected preys, i.e. not the whole but only a fraction of the infected is available to the predator for consumption. Moreover, multiplicative Allee effect has been introduced only in susceptible population to make our model more realistic to environment. Boundedness and positivity have been checked to ensure that the eco-epidemiological model is well-behaved. Stability has been analyzed for all the equilibrium points. Routh–Hurwitz criterion provides the conditions for local stability while on the other hand, Bendixson–Dulac theorem and Lyapunov LaSalle theorem guarantee the global stability of the equilibrium points. Also, the analytical results have been verified numerically by using MATLAB. We have obtained the conditions for the existence of limit cycle in the system through Hopf Bifurcation theorem making the refuge parameter as the bifurcating parameter. In addition, the existence of transcritical bifurcations and saddle-node bifurcation have also been observed by making different parameters as bifurcating parameters around the critical points.

Hopf bifurcation and Turing instability in a predator–prey model with Michaelis–Menten functional response

In this paper, the predator–prey model with Michaelis–Menten functional response subject to the Neumann boundary conditions is considered. The stability of the positive equilibrium and the direction of the Hopf bifurcations for the corresponding ordinary differential equation and partial differential equation are studied, respectively. To this end, the normal form method and the center manifold theorem are applied to determine the direction of the Hopf bifurcation and the stability of the bifurcated limit cycle when the diffusion terms are present. It is shown that the corresponding system can undergo either the supercritical or subcritical Hopf bifurcation at the equilibrium point within certain parameter range. Meanwhile, the Turing instability conditions for the equilibrium and the limit cycle bifurcated from the Hopf bifurcation are derived. As a result, some patterns occur in the model. Numerical simulations are carried out to verify the theoretical analysis.

Uniform ultimate boundedness of solutions of predator-prey system with Michaelis-Menten functional response on time scales

In this paper, a predator-prey system with Michaelis-Menten functional response on time scales is investigated. First of all, we generalize the semi-cycle concept to time scales. Second, we obtain the uniformly ultimate boundedness of solutions of this system. Our results demonstrate that when the death rate of the predator is rather small without prey, whereas the intrinsic growth rate of the prey is relatively large, the species could coexist in the long run. In particular, if $\mathbb{T}=\mathbb{R}$ or $\mathbb{T}=\mathbb{Z}$ , some well-known results have been generalized. In addition, for the continuous case, we provide a new idea to prove its permanence. Finally, a numerical simulation is given to support our main results.

Stability and bifurcation of a predator–prey model with disease in the prey and temporal–spatial nonlocal effect

In this paper, we consider the dynamics of a predator–prey model with disease in the prey and ratio-dependent Michaelis–Menten functional response. The model is a reaction–diffusion system with a nonlocal term representing the temporal–spatial weighted average for the prey density. The limiting case of the system reduces to the Lotka–Volterra diffusive system with logistic growth of the prey. We study the linear stability of the two non-trivial steady states either with or without nonlocal term. The bifurcations to three types of periodic solutions occurring from the coexistence steady state are investigated for two particular kernels, which reveal the important significance of temporal–spatial nonlocal effects.

Positive Solutions of a Diffusive Predator-Prey System including Disease for Prey and Equipped with Dirichlet Boundary Condition

We study a three-dimensional system of a diffusive predator-prey model including disease spread for prey and with Dirichlet boundary condition and Michaelis-Menten functional response. By semigroup method, we are able to achieve existence of a global solution of this system. Extinction of this system is established by spectral method. By using bifurcation theory and fixed point index theory, we obtain existence and nonexistence of inhomogeneous positive solutions of this system in steady state.

Permanence and stability of a diffusive predator–prey model with disease in the prey

We investigate a 3-dimensional system of the predator–prey dynamics with disease for the prey. This system is a reaction–diffusion model with ratio-dependent Michaelis–Menten functional response and diffusion in a bounded domain. By applying the comparison principle, Liapunov function and linearized method, we are able to achieve the sufficient conditions of the permanence, the global stability of the boundary equilibria and the positive equilibrium, and the local stability of the positive equilibrium. Numerical simulations are also given to illustrate the applications.

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.

Dynamics of an impulsively controlled Michaelis–Menten type predator–prey system

We study a predator–prey system with a Michaelis–Menten functional response and impulsive perturbations which contain chemical and biological control terms. By applying the Floquet theory, we establish conditions for the existence and stability of prey-free solutions of the system. We also show the existence of a positive periodic solution of the system by using the bifurcation theorem and find a sufficient condition that makes the system permanent. Moreover, numerical results on impulsive perturbations show that the system we consider can give birth to various kinds of dynamical behaviors.

Permanence and global attractivity of delay diffusive prey-predator systems with the michaelis-menten functional response

In this paper, the author investigates two-species nonautonomous delay diffusive preypredator models with Michaelis-Menten functional response. Sufficient conditions are derived for permanence of the species. Further, the author also establishs the sufficient conditions that guarantee the arbitrary positive periodic solution of the systems is global attractive.

Stability in a diffusive food-chain model with Michaelis–Menten functional response

This paper deals with the behavior of positive solutions to a reaction–diffusion system with homogeneous Neumann boundary conditions describing a three species food chain. A sufficient condition for the local asymptotical stability is given by linearization and also a sufficient condition for the global asymptotical stability is given by a Lyapunov function. Our result shows that the equilibrium solution is globally asymptotically stable if the net birth rate of the first species is big enough and the net death rate of the third species is neither too big nor too small.

Exploitative competition in the chemostat for two perfectly substitutable resources

After formulating a general model involving two populations of microorganisms competing for two nonreproducing, growth-limiting resources in a chemostat, we focus on perfectly substitutable resources. León and Tumpson considered a model of perfectly substitutable resources in which the amount of each resource consumed is assumed to be independent of the concentration of the other resource. We extend their analysis and then consider a new model involving a class of response functions that takes into consideration the effects that the concentration of each resource has on the amount of the other resource consumed. This new model includes, as a special case, the model studied by Waltman, Hubbell, and Hsu in which Michaelis-Menten functional response for a single resource is generalized to two perfectly substitutable resources. Analytical methods are used to obtain information about the qualitative behavior of the models. The range of possible dynamics of model I of León and Tumpson and our new model is then compared. One surprising difference is that our model predicts that for certain parameter ranges it is possible that one of the species is unable to survive in the absence of a competitor even though there is a locally asymptotically stable coexistence equilibrium when a competitor is present. The dynamics of these models for perfectly substitutable resources are also compared with the dynamics of the classical growth and two-species competition models as well as models involving two perfectly complementary resources.