Global dynamics of delayed intraguild predation model with intraspecific competition

2018 ◽  
Vol 11 (08) ◽  
pp. 1850116
Author(s):  
Zhenzhen Li ◽  
Binxiang Dai
Keyword(s):  
Periodic Solution ◽  
Global Existence ◽  
Intraspecific Competition ◽  
Intraguild Predation ◽  
Critical Value ◽  
Existence Results ◽  
Positive Equilibrium ◽  
Global Dynamics ◽  
The Stability ◽  
Theoretical Results

A delayed intraguild predation (IGP) model with intraspecific competition is considered. It is shown that the delay has a destabilizing effect and induces oscillations. The global existence results of periodic solutions bifurcating from the positive equilibrium are established. It is shown that there exists at least one nontrival periodic solution when the delay passes through a certain critical value. Numerical simulations are performed to illustrate our theoretical results and show that intraspecific competition can also affect the stability of the positive equilibrium of the system.

scholarly journals Complex Dynamics of an Impulsive Chemostat Model

2019 ◽  
Vol 29 (08) ◽  
pp. 1950101 ◽  
Author(s):  
Jin Yang ◽  
Yuanshun Tan ◽  
Robert A. Cheke
Keyword(s):  
Periodic Solution ◽  
Global Stability ◽  
Substrate Concentration ◽  
Complex Dynamics ◽  
Control Strategies ◽  
Phase Portraits ◽  
Global Dynamics ◽  
Chemostat Model ◽  
Existence And Stability ◽  
Theoretical Results

We propose a novel impulsive chemostat model with the substrate concentration as the basis for the implementation of control strategies, and then investigate the model’s global dynamics. The exact domains of the impulsive and phase sets are discussed in the light of phase portraits of the model, and then we define the Poincaré map and study its complex properties. Furthermore, the existence and stability of the microorganism eradication periodic solution are addressed, and the analysis of a transcritical bifurcation reveals that an order-1 periodic solution is generated. We also provide the conditions for the global stability of an order-1 periodic solution and show the existence of order-[Formula: see text] [Formula: see text] periodic solutions. Moreover, the PRCC results and bifurcation analyses not only substantiate our results, but also indicate that the proposed system exists with complex dynamics. Finally, biological implications related to the theoretical results are discussed.

scholarly journals Dynamics in a delayed diffusive cell cycle model

2018 ◽  
Vol 23 (5) ◽  
pp. 691-709
Author(s):  
Yanqin Wang ◽  
Ling Yang ◽  
Jie Yan
Keyword(s):  
Cell Cycle ◽  
Periodic Solutions ◽  
Center Manifold ◽  
Spatial Dynamics ◽  
Positive Equilibrium ◽  
Cycle Model ◽  
Center Manifold Theorem ◽  
Diffusive Model ◽  
The Stability ◽  
Theoretical Results

In this paper, we construct a delayed diffusive model to explore the spatial dynamics of cell cycle in G2/M transition. We first obtain the local stability of the unique positive equilibrium for this model, which is irrelevant to the diffusion. Then, through investigating the delay-induced Hopf bifurcation in this model, we establish the existence of spatially homogeneous and inhomogeneous bifurcating periodic solutions. Applying the normal form and center manifold theorem of functional partial differential equations, we also determine the stability and direction of these bifurcating periodic solutions. Finally, numerical simulations are presented to validate our theoretical results.

scholarly journals The Study of a Predator-Prey Model with Fear Effect Based on State-Dependent Harvesting Strategy

Complexity ◽  
2022 ◽  
Vol 2022 ◽  
pp. 1-19
Author(s):  
Y. Tian ◽  
H. M. Li
Keyword(s):  
Numerical Simulation ◽  
Periodic Solution ◽  
Prey Population ◽  
Global Dynamics ◽  
Predator Population ◽  
Positive Order ◽  
Predator Prey ◽  
Predator Prey Model ◽  
State Dependent ◽  
The Stability

In presence of predator population, the prey population may significantly change their behavior. Fear for predator population enhances the survival probability of prey population, and it can greatly reduce the reproduction of prey population. In this study, we propose a predator-prey fishery model introducing the cost of fear into prey reproduction with Holling type-II functional response and prey-dependent harvesting and investigate the global dynamics of the proposed model. For the system without harvest, it is shown that the level of fear may alter the stability of the positive equilibrium, and an expression of fear critical level is characterized. For the harvest system, the existence of the semitrivial order-1 periodic solution and positive order- q ( q ≥ 1 ) periodic solution is discussed by the construction of a Poincaré map on the phase set, and the threshold conditions are given, which can not only transform state-dependent harvesting into a cycle one but also provide a possibility to determine the harvest frequency. In addition, to ensure a certain robustness of the adopted harvest policy, the threshold condition for the stability of the order- q periodic solution is given. Meanwhile, to achieve a good economic profit, an optimization problem is formulated and the optimum harvest level is obtained. Mathematical findings have been validated in numerical simulation by MATLAB. Different effects of different harvest levels and different fear levels have been demonstrated by depicting figures in numerical simulation using MATLAB.

scholarly journals Preventing extinction in Rastrelliger brachysoma using an impulsive mathematical model

2021 ◽  
Vol 7 (1) ◽  
pp. 1-24
Author(s):  
Din Prathumwan ◽  
◽  
Kamonchat Trachoo ◽  
Wasan Maiaugree ◽  
Inthira Chaiya ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Periodic Solution ◽  
Population Density ◽  
Upper Bound ◽  
Existence And Uniqueness ◽  
Periodic Behavior ◽  
Toxic Environment ◽  
Pacific Mackerel ◽  
The Stability ◽  
Theoretical Results

<abstract><p>In this paper, we proposed a mathematical model of the population density of Indo-Pacific mackerel (<italic>Rastrelliger brachysoma</italic>) and the population density of small fishes based on the impulsive fishery. The model also considers the effects of the toxic environment that is the major problem in the water. The developed impulsive mathematical model was analyzed theoretically in terms of existence and uniqueness, positivity, and upper bound of the solution. The obtained solution has a periodic behavior that is suitable for the fishery. Moreover, the stability, permanence, and positive of the periodic solution are investigated. Then, we obtain the parameter conditions of the model for which Indo-Pacific mackerel conservation might be expected. Numerical results were also investigated to confirm our theoretical results. The results represent the periodic behavior of the population density of the Indo-Pacific mackerel and small fishes. The outcomes showed that the duration and quantity of fisheries were the keys to prevent the extinction of Indo-Pacific mackerel.</p></abstract>

Dynamic Analysis in a Plankton Ecosystem Involving Two Delays

2021 ◽  
Vol 31 (15) ◽  
Author(s):  
Zhichao Jiang ◽  
Weicong Zhang ◽  
Xueli Bai ◽  
Maoyan Jie
Keyword(s):  
Dynamic Analysis ◽  
Numerical Simulations ◽  
Periodic Solutions ◽  
Positive Equilibrium ◽  
Switching Phenomenon ◽  
Wide Range ◽  
Two Delays ◽  
Existence Of Periodic Solutions ◽  
The Stability ◽  
Theoretical Results

In this paper, a phytoplankton and zooplankton relationship system with two delays is investigated whose coefficients are related to one of the two delays. Firstly, the dynamic behaviors of the system with one delay are given and the stability of positive equilibrium and the existence of periodic solutions are obtained. Using the fact that the system may occur, the stable switching phenomenon is verified. Under certain conditions, the periodic solutions will exist in a wide range as the delay gets away from critical values. Fixing the delay [Formula: see text] in the stable interval, it is revealed that the effect of [Formula: see text] can also cause the vibration of system. This explains that two delays play an important role in the oscillation behavior of the system. Furthermore, using the crossing curve methods, the stable changes of the positive equilibrium in two-delays plane are given, which generalizes the results of systems for which the coefficients do not depend on delay. Some numerical simulations are provided to verify the theoretical results.

scholarly journals Modelling and Analysis of a Pest-Control Pollution Model with Integrated Control Tactics

2010 ◽  
Vol 2010 ◽  
pp. 1-21 ◽  
Author(s):  
Yiping Chen ◽  
Zhijun Liu ◽  
Wenjie Qin
Keyword(s):  
Periodic Solution ◽  
Pest Control ◽  
Natural Enemies ◽  
Integrated Control ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Critical Value ◽  
Globally Asymptotically Stable ◽  
Holling Ii Functional Response ◽  
Theoretical Results

A hybrid impulsive pest control model with stage structure for pest and Holling II functional response is proposed and investigated, in which the effects of impulsive pesticide input in the environment and in the organism are considered. Sufficient conditions for global attractiveness of the pest-extinction periodic solution and permanence of the system are obtained, which show that there exists a globally asymptotically stable pest-extinction periodic solution when the number of natural enemies released is more than some critical value, whereas the system can be permanent when the number of natural enemies released is less than another critical value. Furthermore, numerical simulations are carried out to illustrate our theoretical results and facilitate their interpretation.

scholarly journals Impact of the fear and Allee effect on a Holling type II prey–predator model

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Binfeng Xie
Keyword(s):  
Hopf Bifurcation ◽  
Allee Effect ◽  
Jacobian Matrix ◽  
Positive Equilibrium ◽  
Type Ii ◽  
Bifurcation Parameter ◽  
Fear Effect ◽  
Predator Model ◽  
The Stability ◽  
Theoretical Results

AbstractIn this paper, we propose and investigate a prey–predator model with Holling type II response function incorporating Allee and fear effect in the prey. First of all, we obtain all possible equilibria of the model and discuss their stability by analyzing the eigenvalues of Jacobian matrix around the equilibria. Secondly, it can be observed that the model undergoes Hopf bifurcation at the positive equilibrium by taking the level of fear as bifurcation parameter. Moreover, through the analysis of Allee and fear effect, we find that: (i) the fear effect can enhance the stability of the positive equilibrium of the system by excluding periodic solutions; (ii) increasing the level of fear and Allee can reduce the final number of predators; (iii) the Allee effect also has important influence on the permanence of the predator. Finally, numerical simulations are provided to check the validity of the theoretical results.

Persistence, Turing Instability and Hopf Bifurcation in a Diffusive Plankton System with Delay and Quadratic Closure

2016 ◽  
Vol 26 (03) ◽  
pp. 1650047 ◽  
Author(s):  
Jiantao Zhao ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
System With Delay ◽  
The Stability ◽  
Theoretical Results

A reaction–diffusion plankton system with delay and quadratic closure term is investigated to study the interactions between phytoplankton and zooplankton. Sufficient conditions independent of diffusion and delay are obtained for the persistence of the system. Our conclusions show that diffusion can induce Turing instability, delay can influence the stability of the positive equilibrium and induce Hopf bifurcations to occur. The computational formulas which determine the properties of bifurcating periodic solutions are given by calculating the normal form on the center manifold, and some numerical simulations are carried out for illustrating the theoretical results.

The global dynamics of a discrete nonlinear hierarchical population system

2019 ◽  
Vol 12 (02) ◽  
pp. 1950022
Author(s):  
Ze-Rong He ◽  
Huai Chen ◽  
Shu-Ping Wang
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Function ◽  
Population Model ◽  
Numerical Experiments ◽  
Discrete Dynamical Systems ◽  
Reproductive Number ◽  
Global Dynamics ◽  
Population System ◽  
The Stability ◽  
Theoretical Results

This paper is concerned with the global dynamics of a hierarchical population model, in which the fertility of an individual depends on the total number of higher-ranking members. We investigate the stability of equilibria, nonexistence of periodic orbits and the persistence of the population by means of eigenvalues, Lyapunov function, and several results in discrete dynamical systems. Our work demonstrates that the reproductive number governs the evolution of the population. Besides the theoretical results, some numerical experiments are also presented.

scholarly journals Global Dynamics for a Novel Differential Infectivity Epidemic Model with Stage Structure

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Yunguo Jin
Keyword(s):  
Epidemic Model ◽  
Lyapunov Functions ◽  
Reproduction Number ◽  
Stage Structure ◽  
Threshold Dynamics ◽  
Uniform Persistence ◽  
Global Dynamics ◽  
Differential Infectivity ◽  
The Stability ◽  
Theoretical Results

A novel differential infectivity epidemic model with stage structure is formulated and studied. Under biological motivation, the stability of equilibria is investigated by the global Lyapunov functions. Some novel techniques are applied to the global dynamics analysis for the differential infectivity epidemic model. Uniform persistence and the sharp threshold dynamics are established; that is, the reproduction number determines the global dynamics of the system. Finally, numerical simulations are given to illustrate the main theoretical results.

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