# 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)
Existence of periodic solutions in predator-prey and competition dynamic systems. (English) Zbl 1104.92057
Summary: We systematically explore the periodicity of some dynamic equations on time scales, which incorporate as special cases many population models (e.g., predator-prey systems and competition systems) in mathematical biology governed by differential equations and difference equations. Easily verifiable sufficient criteria are established for the existence of periodic solutions of such dynamic equations, which generalize many known results for continuous and discrete population models when the time scale $\Bbb T$ is chosen as $\Bbb R$ or $\Bbb Z$, respectively. The main approach is based on a continuation theorem in coincidence degree theory, which has been extensively applied in studying existence problems in differential equations and difference equations but rarely applied in dynamic equations on time scales. This study shows that it is unnecessary to explore the existence of periodic solutions of continuous and discrete population models in separate ways. One can unify such studies in the sense of dynamic equations on general time scales.

##### MSC:
 92D40 Ecology 46N60 Applications of functional analysis in biology and other sciences 37N25 Dynamical systems in biology 39A12 Discrete version of topics in analysis
Full Text:
##### References:
 [1] Berryman, A. A.: The origins and evolution of predator -- prey theory. Ecology 73, 1530-1535 (1999) [2] Bohner, M.; Peterson, A.: Dynamic equations on time scales: an introduction with applications. (2001) · Zbl 0978.39001 [3] Bohner, M.; Peterson, A.: Advances in dynamic equations on time scales. (2003) · Zbl 1025.34001 [4] Chesson, P.: Understanding the role of environmental variation in population and community dynamics. Theor. popul. Biol. 64, 253-254 (2003) [5] Fan, M.; Agarwal, S.: Periodic solutions for a class of discrete time competition systems. Nonlinear stud. 9, No. 3, 249-261 (2002) · Zbl 1032.39002 [6] Fan, M.; Kuang, Y.: Dynamics of a nonautonomous predator -- prey system with the beddington -- deangelis functional response. J. math. Anal. appl. 295, No. 1, 15-39 (2004) · Zbl 1051.34033 [7] Fan, M.; Wang, K.: Global periodic solutions of a generalized n-species gilpin -- ayala competition model. Comput. math. Appl. 40, No. 10 -- 11, 1141-1151 (2000) · Zbl 0954.92027 [8] Fan, M.; Wang, K.: Global existence of a positive periodic solution to a predator -- prey system with Holling type II functional response. Acta math. Sci. ser. A, chin. Ed. 21, No. 4, 492-497 (2001) · Zbl 0997.34063 [9] Fan, M.; Wang, K.: Periodicity in a delayed ratio-dependent predator -- prey system. J. math. Anal. appl. 262, No. 1, 179-190 (2001) · Zbl 0994.34058 [10] Fan, M.; Wang, K.: Periodic solutions of a discrete time nonautonomous ratio-dependent predator -- prey system. Math. comput. Modelling 35, No. 9 -- 10, 951-961 (2002) · Zbl 1050.39022 [11] Fan, M.; Wang, Q.: Periodic solutions of a class of nonautonomous discrete time semi-ratio-dependent predator -- prey systems. Discrete contin. Dynam. syst. Ser. B 4, No. 3, 563-574 (2004) · Zbl 1100.92064 [12] Gaines, R. E.; Mawhin, J. L.: Coincidence degree and nonlinear differential equations. Lecture notes in mathematics 568 (1977) · Zbl 0339.47031 [13] Hilger, S.: Analysis on measure chains --- a unified approach to continuous and discrete calculus. Results math. 18, 18-56 (1990) · Zbl 0722.39001 [14] Huo, H. F.: Periodic solutions for a semi-ratio-dependent predator -- prey system with functional responses. Appl. math. Lett. 18, 313-320 (2005) · Zbl 1079.34515 [15] Li, Y. K.: Periodic solutions of a periodic delay predator -- prey system. Proc. amer. Math. soc. 127, No. 5, 1331-1335 (1999) · Zbl 0917.34057 [16] Wang, Q.; Fan, M.; Wang, K.: Dynamics of a class of nonautonomous semi-ratio-dependent predator -- prey systems with functional responses. J. math. Anal. appl. 278, No. 2, 443-471 (2003) · Zbl 1029.34042 [17] 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, No. 2, 637-654 (2005) · Zbl 1064.34053 [18] Yuan, S. L.; Jin, Z.; Ma, Z.: Global existence of a positive periodic solution to a predator -- prey system. J. Xi’an jiaotong univ. 34, No. 10, 80-83 (2000) · Zbl 0978.34037