# A Single Species Model with Impulsive Diffusion

A Single Species Model with Impulsive Diffusion In most models of population dynamics, diffusion between patches is assumed to be continuous or discrete, but in practice many species diffuse only during a single period. In this paper we propose a single species model with impulsive diffusion between two patches, which provides a more natural description of population dynamics. By using the discrete dynamical system generated by a monotone, concave map for the population, we prove that the map always has a globally stable positive fixed point. This means that a single species system with impulsive diffusion always has a globally stable positive periodic solution. This result is further substantiated by numerical simulation. Under impulsive diffusion the single species survives in the two patches. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

# A Single Species Model with Impulsive Diffusion

, Volume 21 (1) – Jan 1, 2005

## A Single Species Model with Impulsive Diffusion

Acta Mathematicae Applicatae Sinica, English Series Vol. 21, No. 1 (2005) 43–48 A Single Species Model with Impulsive Diﬀusion 1 2 Jing Hui ,Lan-sun Chen Department of Mathematics, East China Normal University, Shanghai 200062, China & Department of Information & Computation Sciences, Guangxi University of Technology, Liuzhou 545006, China (E-mail:jinghui@amss.ac.cn) Department of Applied Mathematics, Dalian University of Technology, Dalian, 116024, China (E-mail: lschen@math.ac.cn) Abstract In most models of population dynamics, diﬀusion between patches is assumed to be continuous or discrete, but in practice many species diﬀuse only during a single period. In this paper we propose a single species model with impulsive diﬀusion between two patches, which provides a more natural description of population dynamics. By using the discrete dynamical system generated by a monotone, concave map for the population, we prove that the map always has a globally stable positive ﬁxed point. This means that a single species system with impulsive diﬀusion always has a globally stable positive periodic solution. This result is further substantiated by numerical simulation. Under impulsive diﬀusion the single species survives in the two patches. Keywords impulsive diﬀusion; stroboscopic map; boundedness; monotone concave map; positive ﬁxed point; global stability 2000 MR Subject Classiﬁcation 92B05; 34D05 1 Introduction The eﬀect of spatial factors in population dynamics is a topic of interest (see for instances, [13,15]). In reality, dispersal between patches often occurs in ecological environments, so more realistic models should include dispersal process. In recent years, the analysis of these models [2−7,12,19] focus on the coexistence of population and local (or global) stability of equilibria ,in particular a single population was considered in [3,5,6,8-10]. Spatial factors play a fundamental role in the persistence and stability...

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Publisher
Springer Journals
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
D.O.I.
10.1007/s10255-005-0213-3
Publisher site
See Article on Publisher Site

### Abstract

In most models of population dynamics, diffusion between patches is assumed to be continuous or discrete, but in practice many species diffuse only during a single period. In this paper we propose a single species model with impulsive diffusion between two patches, which provides a more natural description of population dynamics. By using the discrete dynamical system generated by a monotone, concave map for the population, we prove that the map always has a globally stable positive fixed point. This means that a single species system with impulsive diffusion always has a globally stable positive periodic solution. This result is further substantiated by numerical simulation. Under impulsive diffusion the single species survives in the two patches.

### Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Jan 1, 2005

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