Michaelis Menten Type Harvesting Research Articles

Dynamical properties of single species stage structured model with Michaelis-Menten type harvesting on adult population and linear harvesting on juvenile population

A very common and effective way for investigating future demographics is the study of stage structured models. The focus of this article is to propose a modified model to study the impact of population harvesting on their juvenile and adult stages, and analyze the dynamical properties from both qualitative and numerical perspective. It studies single species stage structured model with linear harvesting on juvenile group and Michaelis-Menten type harvesting on adult group. We exploit general ideas in mathematical modeling process to study the dynamical properties and their biological, ecological, and economic implications. It discusses that bi-stability phenomena may exist, global asymptotic stability at boundary equilibrium points and internal equilibrium points are investigated from construction of suitable Lyapunov and Dulac functions. It has been observed that a suitable linear harvesting on juvenile population can feasibly be carry out along with Michaelis-Menten type harvesting on adult population without endangering extinction of any group of population.

Bifurcation analysis of a discrete type prey-predator model with Michaelis–Menten harvesting in predator

Abstract A discrete predator–prey model with square root functional response describing prey herd behavior and nonlinear predator harvesting has been considered in the present work. Three equilibria of the system have been found and observed that two equilibrium points always exist and are feasible, but the interior equilibrium point is feasible under a parametric condition. The local stability of the three equilibria has been analyzed. The interior equilibrium point is locally asymptotically stable under a parametric condition. It is examined that a flip and Neimark–Sacker bifurcations have occurred in the system at the axial equilibrium point. The flip and Neimark–Sacker bifurcations have been analyzed by the center manifold theorem and bifurcation theory, considering the harvesting coefficient as the bifurcation parameter. The proposed discrete model with a nonlinear Michaelis–Menten type harvesting effect on the predator population exhibits rich dynamics; for instance, bifurcations, chaos, and more complex dynamical behaviors. The discrete-time model also produced few numerical simulation results that are more accurate than the continuous model. The proposed discrete model will be performed better than the continuous model in populations with non-overlapping generations or smaller densities. The harvesting coefficient’s optimal value has finally been identified, and an optimal harvesting policy has been introduced. To verify the results, further numerical simulations have been performed.

Periodic dynamics of a single species model with seasonal Michaelis-Menten type harvesting

In this paper, we formulate a seasonally interactive model between closed seasons and open seasons with Michaelis-Menten type harvesting based on some management and capture methods of renewable resources. It is assumed that the population growth obeys the logistic equation in closed seasons and is captured following Michaelis-Menten type response function in open seasons. We define a length threshold of the closed season T¯⁎, which depends on the harvesting parameter. Under the extinct condition of the corresponding continuous harvesting model, by setting the closed season, theoretical results show that the origin is globally asymptotically stable if and only if T¯≤T¯⁎, and there exists a unique globally asymptotically stable T-periodic solution if and only if T¯>T¯⁎. In particular, under the critical conditions on special harvest parameters, it is found that the T-periodic solution still exists as long as an arbitrary positive close season is formulated. Numerical examples are carried out to confirm the obtained theoretical results. Brief conclusions and discussions on our findings are also provided.

A study of Michaelis–Menten type harvesting effects on a population in stochastic environment

In this paper, a study of Michaelis–Menten type harvesting effects on a population with outbreak characteristic in stochastic environment is done formulating stochastic model. The Ludwig model (Ludwig et al., 1978) is chosen as the base mathematical model for the population. Steady state probability distribution and stationary potential function of the population are determined. P-bifurcation analysis has been presented. The impact of the noises and cross correlations strength on outbreak and extinction of the population are measured numerically using probability. It is observed that random rate of Michaelis–Menten type harvesting may extinct the population. The choice of noise or type of harvesting process to population control cannot be uniquely defined and deeply depends on the exogenous factors (extrinsic and intrinsic noise), a contradiction to the earlier reports. Various complex ecological phenomena like noise enhance stability, phase regime shift, etc., are identified in the system.

Bifurcation analysis of a predator–prey model involving age structure, intraspecific competition, Michaelis–Menten type harvesting, and memory effect

The complexity of the dynamical behaviors of interaction between prey and its predator is studied. The prey and predator relationship involves the age structure and intraspecific competition on predators and the nonlinear harvesting of prey following the Michaelis–Menten type term. Some biological validities are shown for the constructed model such as the existence and uniqueness as well as the non-negativity and boundedness of solutions. Three equilibrium points, namely the origin, axial, and interior points, are found including their global dynamics by employing the Lyapunov function along with the generalized Lassale invariant principle. The changes in dynamical behaviors driven by the harvesting and the memory effect are exhibited, including transcritical, saddle-node, backward, and Hopf bifurcations. The appearance of these interesting phenomena is strengthened by giving numerical simulations consisting of bifurcation diagrams, phase portraits, and their time series.

Dynamics study of nonlinear discrete predator-prey system with Michaelis-Menten type harvesting.

In this paper, we study a discrete predator-prey system with Michaelis-Menten type harvesting. First, the equilibrium points number, local stability and boundedness of the system are discussed. Second, using the bifurcation theory and the center manifold theorem, the bifurcation conditions for the system to go through flip bifurcation and Neimark-Sacker bifurcation at the interior equilibrium point are obtained. A feedback control strategy is used to control chaos in the system, and an optimal harvesting strategy is introduced to obtain the optimal value of the harvesting coefficient. Finally, the numerical simulation not only shows the complex dynamic behavior, but also verifies the correctness of our theoretical analysis. In addition, the results show that the system causes nonlinear behaviors such as periodic orbits, invariant rings, chaotic attractors, and periodic windows by bifurcation.

Global dynamics of an amensalism system with Michaelis-Menten type harvesting

<abstract><p>In this work, a new amensalism system with the nonlinear Michaelis-Menten type harvesting for the second species is studied. Firstly, we clarify topological types for all possible equilibria of the system. Then, the behaviors near infinity and the existence of closed orbits as well as saddle connections of the system are discussed via bifurcation analysis, and the global phase portraits of the model are also illustrated. Finally, for the sake of comparison, we further offer a new complete global dynamics of the model without harvesting. Numerical simulations show that the system with harvesting has far richer dynamics, like preserving the extinction of the first species or approaching the steady-state more slowly. Our research will provide useful information which may help us have a better understanding to the dynamic complexity of amensalism systems with harvesting effects.</p></abstract>

Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting

Abstract In this paper, a discrete Leslie-Gower predator-prey system with Michaelis-Menten type harvesting is studied. Conditions on the existence and stability of fixed points are obtained. It is shown that the system can undergo fold bifurcation, flip bifurcation, and Neimark-Sacker bifurcation by using the center manifold theorem and bifurcation theory. Numerical simulations are presented to illustrate the main theoretical results. Compared to the continuous analog, the discrete system here possesses much richer dynamical behaviors including orbits of period-16, 21, 35, 49, 54, invariant cycles, cascades of period-doubling bifurcation in orbits of period-2, 4, 8, and chaotic sets.

Global Attractivity of Symbiotic Model of Commensalism in Four Populations with Michaelis–Menten Type Harvesting in the First Commensal Populations

This article revisits the stability property of a symbiotic model of commensalism with Michaelis–Menten type harvesting in the first commensal populations. By constructing some suitable Lyapunov functions, we provide a thorough analysis of the dynamic behaviors of the subsystem composed of the second and third species. After that, by applying the stability results of this subsystem and combining with the differential inequality theory, sufficient conditions which ensure the global attractivity of the equilibria are obtained. The results obtained here essentially improve and generalize some known results.

Positive Periodic Solution of a Discrete Lotka-volterra Commensal Symbiosis Model with Michaelis-menten Type Harvesting

A non-autonomous discrete Lotka-volterra commensal symbiosis model with Michaelis-Menten type harvesting is proposed and studied in this paper. Under some very simple and easily verified condition, we show that the system admits at least one positive periodic solution.

The role of harvesting in population control in the presence of correlated noise sources

In this paper, we compare the role of constant and Michaelis-Menten type harvesting in single species population control in presence of correlated noise sources. Our aim is to determine a statistical correlation between the noises in the model parameters and population size. The characteristic time scale over which the population size of prey and predator varies are very much different, this fact justifies to consider a single species model as the predator population can be chosen to be constant within the generation time of prey. Steady state probability distributions and stationary potentials of the population are determined by Fokker-Planck equations. Stochastic bifurcation analysis and mean first passage times have been computed. Noise induced critical transitions are observed depending on the strength of the noises. The extinction possibility of population in stochastic control with Michaelis-Menten type harvesting is higher than the constant harvesting. One of our key findings is the transition from bistable to tristable state for weak noises. Another valuable finding is the noise enhanced stability phenomenon for negatively correlated noises. Constant rate of harvesting is better in deterministic case whereas Michaelis-Menten type harvesting is better in stochastic case to reduce population size. The obtained results may be helpful to understand toxin producing phytoplankton blooms and its control in marine ecosystem. Moreover, the study can be useful to explain wild prey population outbreak or extinction in forest.

Influence of the Fear Effect on a Holling Type II Prey–Predator System with a Michaelis–Menten Type Harvesting

In this work, a prey–predator system with Holling type II response function including a Michaelis–Menten type capture and fear effect is put forward to be studied. Firstly, the existence and stability of equilibria of the system are discussed. Then, by considering the harvesting coefficient as bifurcation parameter, the occurrence of Hopf bifurcation at the positive equilibrium point and the existence of limit cycle emerging through Hopf bifurcation are proved. Furthermore, through the analysis of fear effect and capture item, we find that: (i) the fear effect can either stabilize the system by excluding periodic solutions or destroy the stability of the system and produce periodic oscillation behavior; (ii) increasing the level of fear can reduce the final number of predators, but not lead to extinction; (iii) the harvesting coefficient also has significant influence on the persistence of the predator. Finally, numerical simulations are presented to illustrate the results.

Qualitative Analysis of a Single-Species Model with Distributed Delay and Nonlinear Harvest

In this paper, a single-species population model with distributed delay and Michaelis-Menten type harvesting is established. Through an appropriate transformation, the mathematical model is converted into a two-dimensional system. Applying qualitative theory of ordinary differential equations, we obtain sufficient conditions for the stability of the equilibria of this system under three cases. The equilibrium A1 of system is globally asymptotically stable when br−c>0 and η<0. Using Poincare-Bendixson theorem, we determine the existence and stability of limit cycle when br−c>0 and η>0. By computing Lyapunov number, we obtain that a supercritical Hopf bifurcation occurs when η passes through 0. High order singularity of the system, such as saddle node, degenerate critical point, unstable node, saddle point, etc, is studied by the theory of ordinary differential equations. Numerical simulations are provided to verify our main results in this paper.

Dynamical Analysis of the Symbiotic Model of Commensalism in Four Populations with Michaelis-Menten type Harvesting in the First Commensal Population

Dynamical Analysis of the Symbiotic Model of Commensalism in Four Populations with Michaelis-Menten type Harvesting in the First Commensal Population

Delay induced dynamical behaviors in a stochastic insect outbreak model in presence of Michaelis-Menten type harvesting

In this paper, we have considered stochastic insect outbreak model in presence of Michaelis-Menten type of harvesting. The growth of the insect species is taken as delayed logistic type together with a multiplicative noise term. The impact of internal environmental disturbances on the insect population is taken into account by adding an additive noise term in the model. The effects of the noises, cross correlation strength of the noises and time delay on the insect population are investigated and observed very rich dynamical behaviors. It is ascertained that multiplicative noise reduces population size greatly than additive noise. As usual, increase of harvesting of insect species reduces the population size at faster rate. Regime shift is possible depending on multiplicative noise only in contrast it is not possible via only additive noise. One of the key finding is the noise-delayed switching phenomenon for negatively correlated noises.

Spatiotemporal Dynamics Induced by Michaelis–Menten Type Prey Harvesting in a Diffusive Leslie–Gower Predator–Prey Model

This paper is devoted to study the spatiotemporal dynamics of a diffusive Leslie–Gower predator–prey model with Michaelis-Menten type harvesting in the prey population. The existence and stability of possible non-negative constant equilibria are investigated. By regarding [Formula: see text] as a bifurcation parameter, the Hopf bifurcation from the positive constant equilibrium solution is investigated. The necessary and sufficient conditions of Turing instability are explicitly obtained. We show that at the critical value of the bifurcation parameter [Formula: see text] a Turing bifurcation occurs (i.e. a pattern arises). The conditions for the stability of the pattern are also derived in detail. Moreover, the global steady state bifurcation from the positive constant equilibrium solution is investigated. In particular, the local steady state bifurcation from double zero eigenvalues is also obtained by the techniques of space decomposition and the implicit function theorem. Our results show that Michaelis–Menten type harvesting in our model plays a crucial role in the formation of spatiotemporal dynamics, which is a strong contrast to the case without harvesting.

Almost periodic solutions of a commensalism system with Michaelis-Menten type harvesting on time scales

Abstract In this paper, we consider an almost periodic commensal symbiosis model with nonlinear harvesting on time scales. We establish a criterion for the existence and uniformly asymptotic stability of unique positive almost periodic solution of the system. Our results show that the continuous system and discrete system can be unify well. Examples and their numerical simulations are carried out to illustrate the feasibility of our main results.

On the existence and stability of positive periodic solution of a nonautonomous commensal symbiosis model with Michaelis-Menten type harvesting

A non-autonomous commensal symbiosis model of two populations with Michaelis-Menten type harvesting is proposed and studied in this paper. By using a continuation theorem based on Gaines and Mawhin's coincidence degree, we study the global existence of positive periodic solutions of the system. By constructing a suitable Lyapuonov function, sufficient conditions which ensure the global attractivity of the positive periodic solution are obtained. Numeric simulations are carried out to show the feasibility of the main results.

Patterns induced by super cross-diffusion in a predator-prey system with Michaelis–Menten type harvesting

Turing instability and pattern formation in a super cross-diffusion predator-prey system with Michaelis–Menten type predator harvesting are investigated. Stability of equilibrium points is first explored with or without super cross-diffusion. It is found that cross-diffusion could induce instability of equilibria. To further derive the conditions of Turing instability, the linear stability analysis is carried out. From theoretical analysis, note that cross-diffusion is the key mechanism for the formation of spatial patterns. By taking cross-diffusion rate as bifurcation parameter, we derive amplitude equations near the Turing bifurcation point for the excited modes by means of weakly nonlinear theory. Dynamical analysis of the amplitude equations interprets the structural transitions and stability of various forms of Turing patterns. Furthermore, the theoretical results are illustrated via numerical simulations.

Bifurcation of a delayed Gause predator-prey model with Michaelis-Menten type harvesting

In this paper, a Gause predator-prey model with gestation delay and Michaelis–Menten type harvesting of prey is proposed and analyzed by considering Holling type III functional response. We first consider the local stability of the interior equilibrium by investigating the corresponding characteristic equation. In succession, we derive some sufficient conditions on the occurrence of the stability switches of the positive steady state by taking the gestation delay as a bifurcation parameter. It is shown that the delay can induce instability and small amplitude oscillations of population densities via Hopf bifurcations. Furthermore, the stability and direction of the Hopf bifurcations are determined by employing the center manifold argument. Finally, computer simulations are performed to illustrate our analytical findings, and the biological implications of our analytical findings are also discussed.