### Abstract

This article is concerned with reaction-diffusion systems with nonlinear diffusion effects, which describe competition models and prey-predator models of Lotka-Volterra type in population biology. The system consists of two nonlinear diffusion equations where two unknown functions denote population densities of two interacting species. The main purpose is to discuss the existence and nonexistence of positive steady state solutions to such systems. Here a positive solution corresponds to a coexistence state in population models. We will derive a priori estimates of positive solutions by maximum principle for elliptic equations and employ the degree theory on a positive cone to show the existence of a positive solution. The existence results can be reconsidered from the view-point of bifurcation theory. We will give some information on the direction of bifurcation of positive solutions and their stability properties in terms of some biological coefficients. Moreover, we will also study the existence of multiple positive solutions for a certain class of prey-predator systems with nonlinear diffusion by making one of cross-diffusion coefficients sufficiently large.

Original language | English |
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Title of host publication | Handbook of Differential Equations: Stationary Partial Differential Equations |

Pages | 411-501 |

Number of pages | 91 |

Volume | 6 |

DOIs | |

Publication status | Published - 2008 |

### Publication series

Name | Handbook of Differential Equations: Stationary Partial Differential Equations |
---|---|

Volume | 6 |

ISSN (Print) | 18745733 |

### Fingerprint

### Keywords

- 35B50
- 35J65
- 37C25
- 92D25
- Bifurcation theory
- Competition model
- Cross-diffusion
- Degree theory
- Maximum principle
- Prey-predator model
- primary 35K57
- Reaction-diffusion system
- secondary 35B32

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics
- Numerical Analysis

### Cite this

*Handbook of Differential Equations: Stationary Partial Differential Equations*(Vol. 6, pp. 411-501). (Handbook of Differential Equations: Stationary Partial Differential Equations; Vol. 6). https://doi.org/10.1016/S1874-5733(08)80023-X

**Chapter 6 Positive solutions for Lotka-Volterra systems with cross-diffusion.** / Yamada, Yoshio.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*Handbook of Differential Equations: Stationary Partial Differential Equations.*vol. 6, Handbook of Differential Equations: Stationary Partial Differential Equations, vol. 6, pp. 411-501. https://doi.org/10.1016/S1874-5733(08)80023-X

}

TY - CHAP

T1 - Chapter 6 Positive solutions for Lotka-Volterra systems with cross-diffusion

AU - Yamada, Yoshio

PY - 2008

Y1 - 2008

N2 - This article is concerned with reaction-diffusion systems with nonlinear diffusion effects, which describe competition models and prey-predator models of Lotka-Volterra type in population biology. The system consists of two nonlinear diffusion equations where two unknown functions denote population densities of two interacting species. The main purpose is to discuss the existence and nonexistence of positive steady state solutions to such systems. Here a positive solution corresponds to a coexistence state in population models. We will derive a priori estimates of positive solutions by maximum principle for elliptic equations and employ the degree theory on a positive cone to show the existence of a positive solution. The existence results can be reconsidered from the view-point of bifurcation theory. We will give some information on the direction of bifurcation of positive solutions and their stability properties in terms of some biological coefficients. Moreover, we will also study the existence of multiple positive solutions for a certain class of prey-predator systems with nonlinear diffusion by making one of cross-diffusion coefficients sufficiently large.

AB - This article is concerned with reaction-diffusion systems with nonlinear diffusion effects, which describe competition models and prey-predator models of Lotka-Volterra type in population biology. The system consists of two nonlinear diffusion equations where two unknown functions denote population densities of two interacting species. The main purpose is to discuss the existence and nonexistence of positive steady state solutions to such systems. Here a positive solution corresponds to a coexistence state in population models. We will derive a priori estimates of positive solutions by maximum principle for elliptic equations and employ the degree theory on a positive cone to show the existence of a positive solution. The existence results can be reconsidered from the view-point of bifurcation theory. We will give some information on the direction of bifurcation of positive solutions and their stability properties in terms of some biological coefficients. Moreover, we will also study the existence of multiple positive solutions for a certain class of prey-predator systems with nonlinear diffusion by making one of cross-diffusion coefficients sufficiently large.

KW - 35B50

KW - 35J65

KW - 37C25

KW - 92D25

KW - Bifurcation theory

KW - Competition model

KW - Cross-diffusion

KW - Degree theory

KW - Maximum principle

KW - Prey-predator model

KW - primary 35K57

KW - Reaction-diffusion system

KW - secondary 35B32

UR - http://www.scopus.com/inward/record.url?scp=67649652043&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=67649652043&partnerID=8YFLogxK

U2 - 10.1016/S1874-5733(08)80023-X

DO - 10.1016/S1874-5733(08)80023-X

M3 - Chapter

AN - SCOPUS:67649652043

SN - 9780444532411

VL - 6

T3 - Handbook of Differential Equations: Stationary Partial Differential Equations

SP - 411

EP - 501

BT - Handbook of Differential Equations: Stationary Partial Differential Equations

ER -