Analysis of a Single Species Model with Dissymmetric Bidirectional Impulsive Diffusion and Dispersal Delay

In most models of population dynamics, diffusion between two patches is assumed to be either continuous or discrete, but in the real natural ecosystem, impulsive diffusion provides a more suitable manner to model the actual dispersal (or migration) behavior for many ecological species. In addition, the species not only requires some time to disperse or migrate among the patches but also has some possibility of loss during dispersal. In view of these facts, a single species model with dissymmetric bidirectional impulsive diffusion and dispersal delay is formulated. Criteria on the permanence and extinction of species are established. Furthermore, the realistic conditions for the existence, uniqueness, and the global stability of the positive periodic solution are obtained. Finally, numerical simulations and discussion are presented to illustrate our theoretical results.

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IMPULSIVE DISPERSAL OF PLANT SEEDS IN SINGLE SPECIES MODEL

Journal of Biological System ◽

10.1142/s021833900700226x ◽

2007 ◽

Vol 15 (03) ◽

pp. 385-396

Author(s):

LIMIN WANG ◽

YUANSHUN TAN ◽

LANSUN CHEN

Keyword(s):

Discrete Dynamical System ◽

Sufficient Conditions ◽

Positive Periodic Solution ◽

Single Species ◽

Plant Seeds ◽

Impulsive Diffusion ◽

Globally Stable ◽

Natural Description ◽

Negative Growth ◽

Single Species Model

In this paper, a single species model with impulsive diffusion between two patches is proposed, which provides a more natural description of plant seeds dynamics compared with the continuous or discrete ones. By using the discrete dynamical system generated by a monotone, concave map for the dispersal model and a ∊1 - ∊2 variation, it is proved that the Poincare map has a globally stable positive fixed point. This implies that the system considered here has a globally stable positive periodic solution under some sufficient conditions. Further numerical simulations show that the diffusion can save the extinction though the species has a negative growth rate in one patch.

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TWO PATCHES IMPULSIVE DIFFUSION PERIODIC SINGLE-SPECIES LOGISTIC MODEL

International Journal of Biomathematics ◽

10.1142/s1793524510000842 ◽

2010 ◽

Vol 03 (01) ◽

pp. 127-141 ◽

Cited By ~ 12

Author(s):

ZIJIAN LIU ◽

ZHIDONG TENG ◽

LONG ZHANG

Keyword(s):

Periodic Solution ◽

Iterative Method ◽

Numerical Simulations ◽

Logistic Model ◽

Global Attractivity ◽

Necessary Conditions ◽

Positive Periodic Solution ◽

Single Species ◽

Impulsive Diffusion ◽

Sufficient And Necessary Conditions

In this paper, we study a periodic single-species logistic system with impulsive diffusion in two patches. By using the iterative method, sufficient and necessary conditions on the existence, uniqueness and global attractivity of positive periodic solution and the extinction of species for this system are established. Two examples and numerical simulations are presented to illustrate the feasibility of our results.

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Complex Dynamics of an Impulsive Chemostat Model

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127419501013 ◽

2019 ◽

Vol 29 (08) ◽

pp. 1950101 ◽

Cited By ~ 1

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.

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A single species model with impulsive diffusion and pulsed harvesting

Applied Mathematical Sciences ◽

10.12988/ams.2014.48651 ◽

2014 ◽

Vol 8 ◽

pp. 6141-6149 ◽

Cited By ~ 1

Author(s):

Xiaohong Wang ◽

Jianwen Jia

Keyword(s):

Single Species ◽

Impulsive Diffusion ◽

Single Species Model

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An Impulsive Periodic Single-Species Logistic System with Diffusion

Journal of Applied Mathematics ◽

10.1155/2013/101238 ◽

2013 ◽

Vol 2013 ◽

pp. 1-9

Author(s):

Chenxue Yang ◽

Mao Ye ◽

Zijian Liu

Keyword(s):

Periodic Solution ◽

Lyapunov Function ◽

Sufficient Conditions ◽

Positive Periodic Solution ◽

Single Species ◽

Patchy Environment ◽

Estimation Technique ◽

Numerical Examples ◽

Integrable Form

We study a single-species periodic logistic type dispersal system in a patchy environment with impulses. On the basis of inequality estimation technique, sufficient conditions of integrable form for the permanence and extinction of the system are obtained. By constructing an appropriate Lyapunov function, conditions for the existence of a unique globally attractively positive periodic solution are also established. Numerical examples are shown to verify the validity of our results and to further discuss the model.

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