The prey-dependent consumption two-prey one-predator models with stage structure for the predator and impulsive effects.

In this paper, we consider the prey-dependent consumption two-prey one-predator models with stage structure for the predator and impulsive effects. By applying the Floquet theory of linear periodic impulsive equation, we show that there exists a globally asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value, that is, the pest population can be eradicated totally. But from the point of ecological balance and saving resources, we only need to control the pest population under the economic threshold level instead of eradicating it totally, and thus, we further prove that the system is uniformly permanent if the impulsive period is larger than some critical value, and meanwhile we also give the conditions for the extinction of one of the two preys and permanence of the remaining species. Thus, we can use the stability of the positive periodic solution and its period to control insect pests at acceptably low levels. Considering population communities always are imbedded in periodically varying environments, and the parameters in ecosystem models may oscillate simultaneously with the periodically varying environments, we add a forcing term into the prey population's intrinsic growth rate. The resulting bifurcation diagrams show that with the varying of parameters, the system experiences process of cycles, periodic windows, periodic-doubling cascade, symmetry breaking bifurcation as well as chaos.

[1]  Bing Liu,et al.  Extinction and permanence of a two-prey one-predator system with impulsive effect. , 2003, Mathematical medicine and biology : a journal of the IMA.

[2]  Y. Takeuchi,et al.  Existence and bifurcation of stable equilibrium in two-prey, one-predator communities , 1983 .

[3]  W. Gurney,et al.  Period to delay ratio near stability boundaries for systems with delayed feedback , 1988 .

[4]  William Gurney,et al.  Fluctuation periodicity, generation separation, and the expression of larval competition , 1985 .

[5]  H. I. Freedman,et al.  A time-delay model of single-species growth with stage structure. , 1990, Mathematical biosciences.

[6]  R M Nisbet,et al.  Stage-structure models of populations with distinct growth and development processes. , 1985, IMA journal of mathematics applied in medicine and biology.

[7]  S B Saila,et al.  Interspecific competition, predation and species diversity. , 1970, Journal of theoretical biology.

[8]  Xinzhi Liu,et al.  Permanence of population growth models with impulsive effects , 1997 .

[9]  William Gurney,et al.  Stage Structure Models Applied in Evolutionary Ecology , 1989 .

[10]  V. Lakshmikantham,et al.  Theory of Impulsive Differential Equations , 1989, Series in Modern Applied Mathematics.

[11]  Lansun Chen,et al.  Chaos in three species food chain system with impulsive perturbations , 2005 .

[12]  R. Anderson,et al.  Pulse mass measles vaccination across age cohorts. , 1993, Proceedings of the National Academy of Sciences of the United States of America.

[13]  J. C. Lenteren,et al.  Biological and Integrated Pest control in Greenhouses , 1988 .

[14]  Lansun Chen,et al.  Density-dependent birth rate, birth pulses and their population dynamic consequences , 2002, Journal of mathematical biology.

[15]  K. Fujii,et al.  Complexity-stability relationship of two-prey-one-predator species system model: local and global stability. , 1977, Journal of theoretical biology.

[16]  Mary Louise Flint.  Integrated Pest Management for Walnuts , 1987 .

[17]  Leon G. Higley,et al.  The Economic Injury Level Concept and Environmental Quality: A New Perspective , 1992 .

[18]  Larry P. Pedigo,et al.  Entomology and pest management , 1989 .

[19]  R. van den Bosch,et al.  The Pesticide Conspiracy , 2023 .

[20]  Stephen D. Wratten,et al.  Measures of Success in Biological Control , 2000 .

[21]  William W. Murdoch,et al.  Consumer-resource dynamics , 2003 .

[22]  S. Hsu.  Predator-mediated coexistence and extinction☆ , 1981 .

[23]  J. C. van Lenteren,et al.  Success in Biological Control of Arthropods by Augmentation of Natural Enemies , 2000 .

[24]  P. Cull,et al.  Local and global stability for population models , 1986, Biological Cybernetics.

[25]  Zhilong Xiu,et al.  Dynamic complexity of a two-prey one-predator system with impulsive effect , 2005 .

[26]  Xianning Liu,et al.  Complex dynamics of Holling type II Lotka–Volterra predator–prey system with impulsive perturbations on the predator ☆ , 2003 .