# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Qualitative analysis of a predator--prey model with Holling type II functional response incorporating a prey refuge. (English) Zbl 05115330
Summary: We study a predator--prey model with Holling type II functional response incorporating a prey refuge under homogeneous Neumann boundary condition. We show the existence and non-existence of non-constant positive steady-state solutions depending on the constant $m\in (0,1]$, which provides a condition for protecting $(1 - m)u$ of prey $u$ from predation. Moreover, we investigate the asymptotic behavior of spacially inhomogeneous solutions and the local existence of periodic solutions.

##### MSC:
 35J60 Nonlinear elliptic equations 92D25 Population dynamics (general)
Full Text:
##### References:
 [1] Blat, J.; Brown, K. J.: Global bifurcation of positive solutions in some systems of elliptic equations. SIAM J. Math. anal. 17, No. 6, 1339-1353 (1986) · Zbl 0613.35008 [2] Casal, A.; Eilbeck, J. C.; López-Gómez, J.: Existence and uniqueness of coexistence states for a predator -- prey model with diffusion. Differential integral equations 7, No. 2, 411-439 (1994) · Zbl 0823.35050 [3] Crandall, M. G.; Rabinowitz, P. H.: The Hopf bifurcation theorem in infinite dimensions. Arch. ration. Mech. anal. 67, No. 1, 53-72 (1977) · Zbl 0385.34020 [4] Du, Y.; Lou, Y.: Some uniqueness and exact multiplicity results for a predator -- prey model. Trans. amer. Math. soc. 349, No. 6, 2443-2475 (1997) · Zbl 0965.35041 [5] Du, Y.; Lou, Y.: S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator -- prey model. J. differential equations 144, No. 2, 390-440 (1998) · Zbl 0970.35030 [6] Du, Y.; Lou, Y.: Qualitative behaviour of positive solutions of a predator -- prey model: effects of saturation. Proc. roy. Soc. Edinburgh sect. A 131, No. 2, 321-349 (2001) · Zbl 0980.35028 [7] Gilbarg, D.; Trudinger, N. S.: Elliptic partial differential equations of second order. (1983) · Zbl 0562.35001 [8] Hassel, M.: The dynamics of arthropod predator -- prey systems. (1978) [9] Hausrath, A.: Analysis of a model predator -- prey system with refuges. J. math. Anal. appl. 181, 531-545 (1994) · Zbl 0799.34047 [10] Henry, D.: Geometric theory of semilinear parabolic equations. Lecture notes in math. 840 (1993) [11] Kar, T. K.: Stability analysis of a prey -- predator model incorporating a prey refuge. Commun. nonlinear sci. Numer. simul. 10, 681-691 (2005) · Zbl 1064.92045 [12] Lin, C. S.; Ni, W. M.; Takagi, I.: Large amplitude stationary solutions to a chemotaxis system. J. differential equations 72, 1-27 (1988) · Zbl 0676.35030 [13] Mcnair, J.: The effects of refuges on predator -- prey interactions: A reconsideration. Theoret. population biol. 29, 38-63 (1986) · Zbl 0594.92017 [14] Nirenberg, L.: Topics in nonlinear functional analysis. (1973) [15] Sih, A.: Prey refuges and predator -- prey stability. Theoret. population biol. 31, 1-12 (1987) [16] Sith, J.: Models in ecology. (1974) [17] Smoller, J.: Shock waves and reaction -- diffusion equations. (1983) · Zbl 0508.35002 [18] Taylor, R.: Predation. (1984)