Limit cycles in a generalized Gause-type predator-prey system

Abstract

The qualitative behavior of solutions for a generalized Gause-type predator-prey system was studied. A large number of biological and bioeconomic models are special cases of this system. The system was investigated in the region D={(x, y)|x>0, y>0} because of the biological meaning of the system. The authors derived some sufficient conditions for the boundedness of the solutions and the existence of limit cycles of the system, which ensure that the system has at least one limit cycle. The theory of limit sets of autonomous plane systems and the theorem of cycle field of Poincare-Bendixson are efficiently employed in the research. The main results and their consequences presented not only generalize some known results, but also improve some corresponding results of other authors.

Keywords

generalized Gause-type predator-prey system / boundedness / periodic solution / existence

References

[1] HuangX C. Limit cycles in a Kolmogorov-type model[J]. Intern J Math and Math Sci, 1990, 13(3): 555-566

[2] KuangY. Nonuniqueness of limit cycles of Gause-type predator-prey systems[J]. Appl Anal Math, 1988, 29: 269-287

[3] ChenLan-sunMathematical ecology models and research methods (in Chinese) [M], 1988, Beijing, Science Press: 11-33

[4] KuangY, FreedmanH I. Uniqueness of limit cycles in Gause-type models of predator-prey systems[J]. Math Biosci, 1988, 88: 67-84

[5] MayR M. Limit cycles in predator-prey systems[J]. Comm Sci, 1972, 177: 900-902

[6] ChengK S. Uniqueness of limit cycles for a predator-prey system[J]. SIAM J Math Anal, 1981, 12: 541-548

[7] HuangX C, MerrillS. Condition for uniqueness of limit cycles in a general predator-prey system [J]. Math Biosci, 1989, 96: 47-60

[8] DingS H. On a kind of predator-prey system[J]. SIAM J Math Anal, 1989, 20: 1426-1435

[9] HwangT W. Uniqueness of the limit cycle for Gause-type predator-prey systems[J]. J M A A, 1999, 238: 179-195

[10] ZhangZhi-fang, DingTong-ren, DongZhen-xi, et al.Qualitative theory of ordinary differential equation (in Chinese)[M], 1985, Beijing, Science Press: 25-36