Analysis of the interaction among weed, inorganic fertilizer and herbivore in paddy ecosystem in fallow season

2017 ◽  
Vol 10 (08) ◽  
pp. 1750120 ◽  
Author(s):  
Meihong Xiang ◽  
Zhaohua Wu ◽  
Tiejun Zhou
Keyword(s):  
Hopf Bifurcation ◽  
Equilibrium Point ◽  
Equation Model ◽  
Inorganic Fertilizer ◽  
Stable Region ◽  
Positive Equilibrium ◽  
Bifurcation Phenomenon ◽  
Bifurcation Phenomena ◽  
Paddy Ecosystem ◽  
Positive Equilibrium Point

Paddy growth is influenced by the amount of inorganic fertilizer in paddy ecosystem in fallow season. To discover the interaction among weed, inorganic fertilizer and herbivore in the system, we put forward a differential equation model and investigate its properties. Results show that the system has a weed and herbivore extinct equilibrium and a herbivore extinct equilibrium. The two equilibria are proven to be unstable using the center manifold method. Under certain conditions, the system also has a positive equilibrium point. We give the stable region and the unstable region of the positive equilibrium point, which are determined by some parameters. We find that the system has the Hopf bifurcation phenomenon, and give the critical value of Hopf bifurcation by taking a system parameter as the bifurcation parameter. By comparing the equilibrium states between a paddy ecosystem with herbivore and one without herbivore, we find that the content of inorganic fertilizer can be improved by putting herbivore into a paddy field. An example is given to illustrate the feasibility of the results. Numerical simulation shows that Hopf bifurcation phenomena exist in the system.

scholarly journals Dynamics Analysis of a Delayed Rumor Propagation Model in an Emergency-Affected Environment

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Chunru Li ◽  
Zujun Ma
Keyword(s):  
Hopf Bifurcation ◽  
Equilibrium Point ◽  
Propagation Model ◽  
Positive Equilibrium ◽  
Center Manifold Theorem ◽  
Rumor Propagation ◽  
Boundary Equilibrium ◽  
Normal Form Method ◽  
Positive Equilibrium Point ◽  
The Stability

Rumors influence people’s decisions in an emergency-affected environment. How to describe the spreading mechanism is significant. In this paper, we propose a delayed rumor propagation model in emergencies. By taking the delay as the bifurcation parameter, the local stability of the boundary equilibrium point and the positive equilibrium point is investigated and the conditions of Hopf bifurcation are obtained. Furthermore, formulas for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by applying the normal form method and center manifold theorem. Finally, some numerical simulations are also given to illustrate our theoretical results.

scholarly journals Evaluation of the Number of Visits to Chinese Medical Institutions Using a Logistic Differential Equation Model

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Xiaoxia Zhao ◽  
Wei Li ◽  
Yanyang Wang ◽  
Lihong Jiang
Keyword(s):  
Differential Equation ◽  
Equilibrium Point ◽  
Equilibrium Points ◽  
Equation Model ◽  
Positive Equilibrium ◽  
Model Parameters ◽  
Unknown Parameters ◽  
Differential Equation Model ◽  
Number Of Visits ◽  
Positive Equilibrium Point

In this study, we established a two-dimensional logistic differential equation model to study the number of visits in Chinese PHCIs and hospitals based on the behavior of patients. We determine the model's equilibrium points and analyze their stability and then use China medical services data to fit the unknown parameters of the model. Finally, the sensitivity of model parameters is evaluated to determine the parameters that are susceptible to influence the system. The results indicate that the system corresponds to the zero-equilibrium point, the boundary equilibrium point, and the positive equilibrium point under different parameter conditions. We found that, in order to substantially increase visits to PHCIs, efforts should be made to improve PHCI comprehensive capacity and maximum service capacity.

Dynamics of a prey predator disease model with Holling type functional response and stochastic perturbation

2020 ◽  
pp. 471-488
Author(s):  
M. N. Srinivas ◽  
G. Basava Kumar ◽  
V. Madhusudanan
Keyword(s):  
Equilibrium Point ◽  
Disease Model ◽  
Stochastic Perturbation ◽  
Positive Equilibrium ◽  
Type Ii ◽  
Asymptotically Stable ◽  
Unique Positive Solution ◽  
Stochastic Nature ◽  
Research Article ◽  
Positive Equilibrium Point

The present research article constitutes Holling type II and IV diseased prey predator ecosystem and classified into two categories namely susceptible and infected predators.We show that the system has a unique positive solution. The deterministic and stochastic nature of the dynamics of the system is investigated. We check the existence of all possible steady states with local stability. By using Routh-Hurwitz criterion we showed that the positive equilibrium point $E_{7}$ is locally asymptotically stable if $x^{*} > \sqrt{m_{1}}$ .Moreover condition of the global stability of positive equilibrium point $E_{7}$ are also entrenched with help of Lyupunov theorem. Some Numerical simulations are carried out to illustrate our analytical findings.

On Generalized Hopf Bifurcations

1984 ◽  
Vol 106 (4) ◽  
pp. 327-334 ◽  
Author(s):  
K. Huseyin ◽  
A. S. Atadan
Keyword(s):  
Hopf Bifurcation ◽  
Additional Condition ◽  
System Parameter ◽  
Transversality Condition ◽  
Existence Condition ◽  
The Other ◽  
Hopf Bifurcations ◽  
Bifurcation Phenomenon ◽  
Bifurcation Phenomena ◽  
Lumped Parameter

Two distinct degenerate Hopf bifurcation phenomena associated with autonomous lumped-parameter systems are explored in great detail via the intrinsic harmonic balancing method. It is assumed that the Hopf’s transversality condition is violated and certain other conditions prevail. In one of the cases, the system exhibits a cusp shape bifurcation path which exists for either positive or negative values of the system parameter. On the other hand, the second case is concerned with a tangential bifurcation phenomenon which may not be exhibited unless an additional condition is satisfied. This existence condition is obtained in the course of analysis. The distinctive feature of the paper is that the results concerning the bifurcating paths and limit cycles are given in general, explicit forms which are expected to be very useful in a variety of applications.

scholarly journals Dynamic Behaviors of a Discrete Two Species Predator-Prey System Incorporating Harvesting

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Ting Wu
Keyword(s):  
Equilibrium Point ◽  
Hamiltonian Function ◽  
Economic Benefits ◽  
Practical Significance ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
The Maximum Principle ◽  
Prey System ◽  
Positive Equilibrium Point ◽  
Jury Criterion

A discrete two species predator-prey captured system is studied. Firstly, a sufficient condition of a positive equilibrium point for this system is obtained. Secondly, we observe that the two nonnegative equilibriums of the system are unstable through the eigenvalue discriminant method, and the positive equilibrium point is asymptotically stable by Jury criterion. Lastly, we obtain the optimal capture strategy of the system from the maximum principle by constructing a discrete Hamiltonian function. To show the feasibility of the main results, a suitable example together with its numerical simulations is illustrated in the last part of the paper. The example with certain practical significance might give an optimal scheme of the greatest economic benefits for the captors.

scholarly journals Dynamical Models For Prices With Distributed Delays

2015 ◽  
Vol 8 (1) ◽  
pp. 91-102
Author(s):  
Gabriela Mircea ◽  
Mihaela Neamţu ◽  
Laura Mariana Cismaş
Keyword(s):  
Time Delay ◽  
Equilibrium Point ◽  
Supply And Demand ◽  
Positive Equilibrium ◽  
Asymptotically Stable ◽  
Dynamical Models ◽  
Demand Functions ◽  
Distributed Time Delay ◽  
Commodity Demand ◽  
Positive Equilibrium Point

Abstract In the present paper we study some models for the price dynamics of a single commodity market. The quantities of supplied and demanded are regarded as a function of time. Nonlinearities in both supply and demand functions are considered. The inventory and the level of inventory are taken into consideration. Due to the fact that the consumer behavior affects commodity demand, and the behavior is influenced not only by the instantaneous price, but also by the weighted past prices, the distributed time delay is introduced. The following kernels are taken into consideration: demand price weak kernel and demand price Dirac kernel. Only one positive equilibrium point is found and its stability analysis is presented. When the demand price kernel is weak, under some conditions of the parameters, the equilibrium point is locally asymptotically stable. When the demand price kernel is Dirac, the existence of the local oscillations is investigated. A change in local stability of the equilibrium point, from stable to unstable, implies a Hopf bifurcation. A family of periodic orbits bifurcates from the positive equilibrium point when the time delay passes through a critical value. The last part contains some numerical simulations to illustrate the effectiveness of our results and conclusions.

scholarly journals Stability analysis of two predators and one prey population model with harvesting in fisheries management

2021 ◽  
Vol 921 (1) ◽  
pp. 012005
Author(s):  
D Didiharyono ◽  
S Toaha ◽  
J Kusuma ◽  
Kasbawati
Keyword(s):  
Equilibrium Point ◽  
Population Model ◽  
Fishery Management ◽  
Stable Equilibrium ◽  
Prey Population ◽  
Positive Equilibrium ◽  
Stable Equilibrium Point ◽  
Maximum Profit ◽  
Positive Equilibrium Point ◽  
Prey Populations

Abstract The discussion is focussed in the interaction between two predators and one prey population model in fishery management. Mathematically model is built by involving harvesting with constant efforts in the two predators and one prey populations. The positive equilibrium point of the model is analyzed via linearization and Routh-Hurwitz stability criteria. From the analysis, there exists a certain condition that makes the positive equilibrium point is asymptotically stable. The stable equilibrium point is then related to the maximum profit problem. With suitable value of harvesting efforts, the maximum profit is reached and the predator and prey populations remain stable. Finally, a numerical simulation is carried out to find out how much the maximum profit is obtained and to visualize how the trajectories of predator and prey tend to the stable equilibrium point.

Multiple Time Delays Induced Dynamics of p53 Gene Regulatory Network

2021 ◽  
Vol 31 (15) ◽  
Author(s):  
Yuanhong Bi ◽  
Yanan Li ◽  
Jianmin Hou ◽  
Quansheng Liu
Keyword(s):  
Time Delay ◽  
Equilibrium Point ◽  
Regulatory Network ◽  
Time Delays ◽  
P53 Gene ◽  
Positive Equilibrium ◽  
Multiple Time ◽  
Multiple Time Delays ◽  
Gene Regulatory ◽  
Positive Equilibrium Point

p53 dynamics plays an important role in determining cell arrest or apoptosis upon DNA damage response. In this paper, based on a p53 gene regulatory network composed of its core regulator ATM, Mdm2 and Wip1, the effect of multiple time delays in transcription and translation of Mdm2 and Wip1 gene expression on p53 dynamics are investigated through theoretical and numerical analyses. The stability of the positive equilibrium point and the existence of Hopf bifurcation are demonstrated through analyzing the associated characteristic equation of the corresponding linearized system in five cases. Detailed numerical simulations and bifurcation analyses are performed to support the theoretical results. The results show that with the increase of a time delay, the positive equilibrium point becomes unstable, and the p53 dynamics presents an oscillating state. These results reveal that time delay has a significant impact on p53 dynamics and may provide a useful insight into developing anti-cancer therapy.

scholarly journals Stability and Hopf Bifurcation of a Predator-Prey Model with Distributed Delays and Competition Term

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Lv-Zhou Zheng
Keyword(s):  
Hopf Bifurcation ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Distributed Delays ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Positive Equilibrium Point ◽  
The Stability

A class of predator-prey system with distributed delays and competition term is considered. By considering the time delay as bifurcation parameter, we analyze the stability and the Hopf bifurcation of the predator-prey system. According to the theorem of Hopf bifurcation, some sufficient conditions are obtained for the local stability of the positive equilibrium point.

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