# A predator-prey model with combined death and competition terms

• Volume: 60, Issue: 1, page 283-295
• ISSN: 0011-4642

top

## Abstract

top
The existence of a positive solution for the generalized predator-prey model for two species $\begin{array}{c}\Delta u+u\left(a+g\left(u,v\right)\right)=0\phantom{\rule{1.0em}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}\Omega ,\\ \Delta v+v\left(d+h\left(u,v\right)\right)=0\phantom{\rule{1.0em}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}\Omega ,\\ u=v=0\phantom{\rule{1.0em}{0ex}}\text{on}\phantom{\rule{4pt}{0ex}}\partial \Omega ,\end{array}$ are investigated. The techniques used in the paper are the elliptic theory, upper-lower solutions, maximum principles and spectrum estimates. The arguments also rely on some detailed properties of the solution of logistic equations.

## How to cite

top

Kang, Joon Hyuk, and Lee, Jungho. "A predator-prey model with combined death and competition terms." Czechoslovak Mathematical Journal 60.1 (2010): 283-295. <http://eudml.org/doc/38007>.

@article{Kang2010,
abstract = {The existence of a positive solution for the generalized predator-prey model for two species $\{\begin\{array\}\{c\}\Delta u + u(a + g(u,v)) = 0\quad \mbox\{in\}\ \Omega ,\\ \Delta v + v(d + h(u,v)) = 0\quad \mbox\{in\} \ \Omega ,\\ u = v = 0\quad \mbox\{on\}\ \partial \Omega , \end\{array\}\}$ are investigated. The techniques used in the paper are the elliptic theory, upper-lower solutions, maximum principles and spectrum estimates. The arguments also rely on some detailed properties of the solution of logistic equations.},
author = {Kang, Joon Hyuk, Lee, Jungho},
journal = {Czechoslovak Mathematical Journal},
keywords = {predator-prey model; coexistence state; predator-prey model; coexistence state},
language = {eng},
number = {1},
pages = {283-295},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A predator-prey model with combined death and competition terms},
url = {http://eudml.org/doc/38007},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Kang, Joon Hyuk
AU - Lee, Jungho
TI - A predator-prey model with combined death and competition terms
JO - Czechoslovak Mathematical Journal
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 1
SP - 283
EP - 295
AB - The existence of a positive solution for the generalized predator-prey model for two species ${\begin{array}{c}\Delta u + u(a + g(u,v)) = 0\quad \mbox{in}\ \Omega ,\\ \Delta v + v(d + h(u,v)) = 0\quad \mbox{in} \ \Omega ,\\ u = v = 0\quad \mbox{on}\ \partial \Omega , \end{array}}$ are investigated. The techniques used in the paper are the elliptic theory, upper-lower solutions, maximum principles and spectrum estimates. The arguments also rely on some detailed properties of the solution of logistic equations.
LA - eng
KW - predator-prey model; coexistence state; predator-prey model; coexistence state
UR - http://eudml.org/doc/38007
ER -

## References

top
1. Cantrell, R., Cosner, C., On the uniqueness and stability of positive solutions in the Lotka-Volterra competition model with diffusion, Houston J. Math. 15 (1989), 341-361. (1989) Zbl0721.92025MR1032394
2. Cosner, C., Lazer, C., 10.1137/0144080, Siam J. Appl. Math. 44 (1984), 1112-1132. (1984) Zbl0562.92012MR0766192DOI10.1137/0144080
3. Conway, E., Gardner, R., Smoller, J., 10.1016/S0196-8858(82)80009-2, Adv. in Appl. Math. 3 (1982), 288-344. (1982) Zbl0505.35047MR0673245DOI10.1016/S0196-8858(82)80009-2
4. Dancer, E. N., 10.1090/S0002-9947-1984-0743741-4, Trans. Am. Math. Soc. 284 (1984), 729-743. (1984) MR0743741DOI10.1090/S0002-9947-1984-0743741-4
5. Dancer, E. N., 10.1016/0022-0396(85)90115-9, J. Differ. Equations. 60 (1985), 236-258. (1985) MR0810554DOI10.1016/0022-0396(85)90115-9
6. Kang, J. H., A cooperative biological model with combined self-limitation and cooperation terms, J. Comput. Math. Optimization 4 (2008), 113-126. (2008) MR2433652
7. Kang, J. H., Lee, J. H., 10.1007/s10587-006-0086-5, Czech. Math. J. 56 (2006), 1165-1183. (2006) Zbl1164.35351MR2280801DOI10.1007/s10587-006-0086-5
8. Lou, Y., Necessary and sufficient condition for the existence of positive solutions of certain cooperative system, Nonlinear Analysis, Theory, Methods and Applications 26 (1996), 1079-1095. (1996) Zbl0856.35038MR1375651
9. Dunninger, Dennis, Lecture Note of Applied Analysis, Department of Mathematics, Michigan State University.
10. Li, L., Logan, R., Positive solutions to general elliptic competition models, Differ. Integral Equations 4 (1991), 817-834. (1991) Zbl0751.35014MR1108062
11. Mingxin, Wang, Zhengyuan, Li, Qixiao, Ye, The existence of positive solutions for semilinear elliptic systems, Acta Sci. Nat. Univ. Pekin. 28 36-49 (1992). (1992)
12. Li, Zhengyuan, Mottoni, P. De, Bifurcation for some systems of cooperative and predator-prey type, J. Partial Differ. Equations 25-36 (1992). (1992) Zbl0769.35026MR1177527

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.