2009
DOI: 10.1007/s11071-009-9594-9
|View full text |Cite
|
Sign up to set email alerts
|

Self-organized wave pattern in a predator-prey model

Abstract: In this paper, pattern formation of a predatorprey model with spatial effect is investigated. We obtain the conditions for Hopf bifurcation and Turing bifurcation by mathematical analysis. When the values of the parameters can ensure a stable limit cycle of the no-spatial model, our study shows that the spatially extended models have spiral waves dynamics. Moreover, the stability of the spiral wave is given by the theory of essential spectrum. Furthermore, although the environment is heterogeneous, the system … Show more

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...
1
1
1
1

Citation Types

0
15
0
1

Year Published

2011
2011
2016
2016

Publication Types

Select...
7

Relationship

1
6

Authors

Journals

citations
Cited by 57 publications
(16 citation statements)
references
References 42 publications
0
15
0
1
Order By: Relevance
“…In [15], the model without noise is considered, where the pattern transition was not obtained. Our results may indicate that noise plays an important role on pattern formation in predator-prey models.…”
Section: Discussionmentioning
confidence: 99%
See 2 more Smart Citations
“…In [15], the model without noise is considered, where the pattern transition was not obtained. Our results may indicate that noise plays an important role on pattern formation in predator-prey models.…”
Section: Discussionmentioning
confidence: 99%
“…The onset of Hopf bifurcation corresponds to the case where a pair of imaginary eigenvalues cross the real axis from the negative to the positive side. And this situation occurs only when the diffusion vanishes [15]. By direct calculations, we can get the critical value of the Hopf bifurcation parameter a, equal to …”
Section: Stability Analysismentioning
confidence: 98%
See 1 more Smart Citation
“…1. The spatio-temporal dynamics of interacting biological, economic or social components can generate numerous local and spatially distributed effects far from equilibrium point, such as excitability, steady-state multiplicity, regular and irregular oscillations, and pulses as well as stationary spatial patterns [7][8][9][10]. In real life, the phytoplankton and zooplankton are always moving, that is to say, the population densities have become space and time dependent [11].…”
Section: Introductionmentioning
confidence: 99%
“…The diffusion-driven instability of the equilibrium leads to a spatially inhomogeneous distribution of species concentration, which is the so-called Turing instability. Although Turing instability was first investigated in a morphogenesis, it has quickly spread to ecological systems [3,4,6,[13][14][15][16][17][18][19][20][21][22][23][24], chemical reaction system [25][26][27][28][29][30] and other reaction-diffusion system [31][32][33][34][35][36][37][38]. From [39], we know that the phenomenon of spatial pattern formation in (1) with diffusion can not occur under all possible diffusion rates.…”
mentioning
confidence: 99%