Stability of Positive Equilibrium Point of Nonlinear System of Type of Predator–Prey with Aftereffect and Stochastic Perturbations

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.

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

Advances in Mathematics: Scientific Journal ◽

10.37418/amsj.9.1.37 ◽

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 ◽

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.

Dynamic Analysis of Stochastic Lotka–Volterra Predator-Prey Model with Discrete Delays and Feedback Control

Complexity ◽

10.1155/2019/4873290 ◽

2019 ◽

Vol 2019 ◽

pp. 1-15 ◽

Cited By ~ 3

Author(s):

Jinlei Liu ◽

Wencai Zhao

Keyword(s):

Feedback Control ◽

Deterministic Model ◽

Sufficient Conditions ◽

Positive Equilibrium ◽

Theoretical Derivation ◽

Predator Prey ◽

Predator Prey Model ◽

Prey Model ◽

Discrete Delays ◽

In this paper, a stochastic Lotka–Volterra predator-prey model with discrete delays and feedback control is studied. Firstly, the existence and uniqueness of global positive solution are proved. Further, we investigate the asymptotic property of stochastic system at the positive equilibrium point of the corresponding deterministic model and establish sufficient conditions for the persistence and extinction of the model. Finally, the correctness of the theoretical derivation is verified by numerical simulations.

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

International Journal of Biomathematics ◽

10.1142/s1793524517501200 ◽

2017 ◽

Vol 10 (08) ◽

pp. 1750120 ◽

Cited By ~ 4

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 ◽

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.

Dynamical Models For Prices With Distributed Delays

Timisoara Journal of Economics and Business ◽

10.1515/tjeb-2015-0010 ◽

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 ◽

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.

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

IOP Conference Series Earth and Environmental Science ◽

10.1088/1755-1315/921/1/012005 ◽

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 ◽

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

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127421502345 ◽

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 ◽

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.

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

Mathematical Problems in Engineering ◽

10.1155/2014/428523 ◽

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 ◽

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.

Stability Analysis of a Prey-Predator Model by Using Homotopy Analysis Method

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.166-169.2855 ◽

2012 ◽

Vol 166-169 ◽

pp. 2855-2858

Author(s):

Hong Yan Cao

Keyword(s):

Nonlinear Dynamics ◽

Equilibrium Point ◽

Homotopy Analysis Method ◽

Zero Point ◽

Positive Equilibrium ◽

Analysis Method ◽

Homotopy Analysis ◽

Predator Model ◽

The Stability

A prey-predator model was considered. Using the methods of the modern nonlinear dynamics and homotopy analysis method (HAM), its stability was discussed. Firstly, we found the system’s positive equilibrium point and shifted it to zero point through transformation. Secondly, we analyzed the stability of the system at the equilibrium point. Lastly, we analyzed the transformed system by HAM. We support our analytical findings with numerical simulation.

On Stationary Solutions Of Delay Differential Equations: A Model Of Local Translation In Synapses

Математическая биология и биоинформатика ◽

10.17537/2019.14.554 ◽

2019 ◽

Vol 14 (2) ◽

pp. 554-569 ◽

Cited By ~ 1

Author(s):

V.A. Likhoshvai ◽

T.M. Khlebodarova

Keyword(s):

Equilibrium Point ◽

Relative Stability ◽

Equilibrium Points ◽

Stationary Solutions ◽

Translation System ◽

Positive Equilibrium ◽

Absolute Instability ◽

Local Translation ◽

Delayed Arguments ◽

The results of analytical analysis of stationary solutions of a differential equation with two delayed arguments τ1 and τ2 are presented. Such equations are used in modeling of molecular-genetic systems where the delay of arguments appear naturally. Conditions of existence of non-negative solutions are described, and dependence of stability of these solutions on the values of delayed arguments is studied. This stability theory allows to give complete characterization of these solutions for all values of the parameters of the model, and ensures instability of a positive equilibrium point for any values of the delays τ2 ≥ τ1 ≥ 0 in the case when it is unstable for τ2 = τ1 = 0 (absolute instability). If this positive equilibrium point is stable only for τ2 = τ1 = 0, then this domain τ2 ≥ τ1 ≥ 0 is the domain of absolute instability as well. For positive equilibrium points which are stable at τ2 = τ1 = 0, we find domains of absolute stability were the equilibrium points remain stable for all values of the parameters τ1 and τ2. The domains of relative stability, where these points become unstable for some values of these parameters are also described. We show that when the efficiency of translation, and non-linearity and complexity of its regulation mechanisms grow, the domains of the absolute and relative stability of the positive equilibrium point shrink, while the domains of its instability expand. So, enhanced activity of the local translation system can be a factor of its instability and that of the risk of neuro-psychical diseases related to distortions of plasticity of the synapse and memory, where importance of stability of the proteome in the synapse is postulated.