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Two-parameter bifurcation in a predator-prey system of Ivlev type. (English) Zbl 0894.34025
This paper considers a predator-prey system of the form $$\dot x= rx(1- x)-(1- e^{-ax})y,\quad \dot y= y[(1- e^{-ax})- D],$$ where $D< 1-e^{-a}$, give a necessary and sufficient condition for the uniqueness of the limit cycle, which is $$a>-{2D+ (1-D)\log(1- D)\over D+(1- D)\log(1- D)} \log(1- D).$${}.

##### MSC:
 34C05 Location of integral curves, singular points, limit cycles (ODE) 92D25 Population dynamics (general)
##### Keywords:
predator-prey systems; limit cycles
Full Text:
##### References:
 [1] Gasull, A.; Guillamon, A.: Non-existence of limit cycles for some predator--prey systems. (1993) · Zbl 0938.34515 [2] Ivlev, V. S.: Experimental ecology of the feeding of fishes. (1961) [3] Kooij, R. E.; Zegeling, A.: A predator--prey model with ivlev’s functional response. J. math. Anal. appl. 198, 473-489 (1996) · Zbl 0851.34030 [4] Kuang, Y.; Freedman, H. I.: Uniqueness of limit cycles in gause-type models of predator--prey systems. Math. biosci. 88, 67-84 (1988) · Zbl 0642.92016 [5] May, R. M.: Stability and complexity in model ecosystems. (1974) [6] Rosenzweig, M. L.: Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science 171, 385-387 (1971) [7] Sugie, J.; Hara, T.: Non-existence of periodic solutions of the Liénard system. J. math. Anal. appl. 159, 224-236 (1991) · Zbl 0731.34042 [8] Sugie, J.; Kohno, R.; Miyazaki, R.: On a predator--prey system of Holling type. Proc. amer. Math. soc. 125, 2041-2050 (1997) · Zbl 0868.34023