| Title:
|
A predator-prey model with combined death and competition terms (English) |
| Author:
|
Kang, Joon Hyuk |
| Author:
|
Lee, Jungho |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
60 |
| Issue:
|
1 |
| Year:
|
2010 |
| Pages:
|
283-295 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The existence of a positive solution for the generalized predator-prey model for two species $$ \begin{gathered} \Delta u + u(a + g(u,v)) = 0\quad \mbox {in}\ \Omega ,\\ \Delta v + v(d + h(u,v)) = 0\quad \mbox {in} \ \Omega ,\\ u = v = 0\quad \mbox {on}\ \partial \Omega , \end{gathered} $$ 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. (English) |
| Keyword:
|
predator-prey model |
| Keyword:
|
coexistence state |
| MSC:
|
35J47 |
| MSC:
|
35J57 |
| MSC:
|
35Q92 |
| MSC:
|
92D25 |
| idZBL:
|
Zbl 1224.35100 |
| idMR:
|
MR2595089 |
| . |
| Date available:
|
2010-07-20T16:35:29Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140568 |
| . |
| Reference:
|
[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. Zbl 0721.92025, MR 1032394 |
| Reference:
|
[2] Cosner, C., Lazer, C.: Stable coexistence states in the Volterra-Lotka competition model with diffusion.Siam J. Appl. Math. 44 (1984), 1112-1132. Zbl 0562.92012, MR 0766192, 10.1137/0144080 |
| Reference:
|
[3] Conway, E., Gardner, R., Smoller, J.: Stability and bifurcation of steady state solutions for predator-prey equations.Adv. in Appl. Math. 3 (1982), 288-344. Zbl 0505.35047, MR 0673245, 10.1016/S0196-8858(82)80009-2 |
| Reference:
|
[4] Dancer, E. N.: On positive solutions of some pair of differential equations.Trans. Am. Math. Soc. 284 (1984), 729-743. MR 0743741, 10.1090/S0002-9947-1984-0743741-4 |
| Reference:
|
[5] Dancer, E. N.: On positive solutions of some pair of differential equations II.J. Differ. Equations. 60 (1985), 236-258. MR 0810554, 10.1016/0022-0396(85)90115-9 |
| Reference:
|
[6] Kang, J. H.: A cooperative biological model with combined self-limitation and cooperation terms.J. Comput. Math. Optimization 4 (2008), 113-126. MR 2433652 |
| Reference:
|
[7] Kang, J. H., Lee, J. H.: Steady state coexistence solutions of reaction-diffusion competition models.Czech. Math. J. 56 (2006), 1165-1183. Zbl 1164.35351, MR 2280801, 10.1007/s10587-006-0086-5 |
| Reference:
|
[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. Zbl 0856.35038, MR 1375651 |
| Reference:
|
[9] Dunninger, Dennis: Lecture Note of Applied Analysis.Department of Mathematics, Michigan State University. |
| Reference:
|
[10] Li, L., Logan, R.: Positive solutions to general elliptic competition models.Differ. Integral Equations 4 (1991), 817-834. Zbl 0751.35014, MR 1108062 |
| Reference:
|
[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). |
| Reference:
|
[12] Li, Zhengyuan, Mottoni, P. De: Bifurcation for some systems of cooperative and predator-prey type.J. Partial Differ. Equations 25-36 (1992). Zbl 0769.35026, MR 1177527 |
| . |
