A predator-prey model with combined death and competition terms

Joon Hyuk Kang; Jungho Lee

Czechoslovak Mathematical Journal (2010)

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

Abstract

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The existence of a positive solution for the generalized predator-prey model for two species Δ u + u ( a + g ( u , v ) ) = 0 in Ω , Δ v + v ( d + h ( u , v ) ) = 0 in Ω , u = v = 0 on ∂ Ω , 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

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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

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  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

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