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)
Multiple positive periodic solutions of a delayed discrete predator-prey system with type IV functional responses. (English) Zbl 1142.39015
The authors consider the following discrete predator-prey system with type IV functional responses and delays: $$\align x_1(k+ 1)&= x_1(k)\exp\Biggl[b_1(k)-a_1(k)x_1(k- \tau_1(k))-{c(k)x_2(k-\sigma(k))\over(x^2_1(k- \tau_2(k))/n)+ x_1(k-\tau_2(k))+ a}\Biggr],\\ x_2(k+1)&= x_2(k)\exp\Biggl[-b_2(k)+ {a_2(k)x_1(k-\tau_2(k))\over (x^2_1(k-\tau_2(k))/n)+ x_1(k- \tau_2(k))+ a}\Biggr] \endalign$$ (where for $i=1,2$, $b_i: Z\to\Bbb R$, $c,a_i: Z\to\Bbb R^+$, $\tau_i,\sigma: Z\to Z^+$ are all $\omega$ periodic, $n$ and $a$ are positive constants) for the initial condition $$\align x_1(-m)&\ge 0,\quad m=1,2,\dots,\max\{\tau_1(k), \tau_2(k),\sigma(k)\},\quad x(0)> 0.\\ x_2(-m)&\ge 0,\quad m= 1,2,\dots,\max\{\tau_1(k), \tau_2(k), \sigma(k)\},\quad y(0)> 0. \endalign$$ In the paper a theorem for the existence of positive periodic solutions of the system is given.

MSC:
39A11Stability of difference equations (MSC2000)
92D25Population dynamics (general)
39A20Generalized difference equations
WorldCat.org
Full Text: DOI
References:
[1] Agarwal, R. P.: Difference equations and inequalities: theory, methods and applications. Monographs and textbooks in pure and applied mathematics 228 (2000)
[2] Arditi, R.; Perrin, N.; Saiah, H.: Functional response and heterogeneities: an experimental test with cladocerans. Oikos 60, 69-75 (1991)
[3] Berryman, A. A.: The origins and evolution of predator--prey theory. Ecology 75, 1530-1535 (1992)
[4] Chen, Y. M.: Multiple periodic solutions of delayed predator--prey systems with type IV functional responses. Nonlinear anal. 5, 45-53 (2004) · Zbl 1066.92050
[5] Fan, M.; Wang, K.: Periodic solutions of a discrete time non-autonomous ratio-dependent predator--prey system. Math. comput. Modelling 35, 951-961 (2002) · Zbl 1050.39022
[6] Freedman, H. I.: Deterministic mathematical models in population ecology. (1980) · Zbl 0448.92023
[7] Gaines, R. E.; Mawhin, J. L.: Coincidence degree and nonlinear differential equations. (1977) · Zbl 0339.47031
[8] Gopalsamy, K.: Stability and oscillations in delay differential equations of population dynamics. (1992) · Zbl 0752.34039
[9] Hanski, I.: The functional response of predator: worries about scale. Tree 6, 141-142 (1991)
[10] Holling, C. S.: The functional response of predators to prey density and its role in mimicry and population regulation. Mem. ent. Soc. can. 45, 1-60 (1965)
[11] Maynard, S. J.: Models in ecology. (1974)
[12] May, R. M.: Stability and complexity in model ecosystems. (1974)
[13] Murry, J. D.: Mathematical biology. (1989)
[14] Rosenzweig, M. L.; Macarthur, R. H.: Graphical representation and stability conditions of predator--prey interactions. Amer. nat. 47, 209-223 (1963)
[15] Rosenzweig, M. L.: Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science 171, 385-387 (1969)
[16] Xu, R.; Chaplain, M. A. J.; Davidson, F. A.: Periodic solutions for a predator--prey model with Holling-type functional response and time delays. Appl. math. Comput. 161, 637-654 (2005) · Zbl 1064.34053
[17] Wang, L. L.; Li, W. T.: Existence and global stability of positive periodic solutions of a predator--prey system with delays. Appl. math. Comput. 146, 167-185 (2004) · Zbl 1029.92025
[18] Wang, L. L.; Li, W. T.: Periodic solutions and permanence for a delayed nonautonomous ratio-dependent predator--prey model with Holling type functional response. J. comput. Appl. math. 162, 341-357 (2004) · Zbl 1076.34085