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Qualitative Analysis on the Predator-prey Systems with Undercrowding Effect

Author: AoYanYan
Tutor: LiDongMei
School: Harbin University of Science and Technology
Course: Basic mathematics
Keywords: Sparse Effect Global stability Limit cycle Persistence Periodic solution
CLC: O175.1
Type: Master's thesis
Year: 2010
Downloads: 42
Quote: 0
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Abstract


In this paper, the qualitative theory of differential equations , coincidence degree theory and Lyapunov function method, for the three types of predator system with sparse effect , discuss their qualitative behavior , the main work is as follows : The first part , taking into account the impact of sparse effects on populations studied a class of sparse effect with the Holling Ⅱ functional response predator system . Qualitative analysis using the method of differential equations given equilibrium conditions for the existence , use the form on the central focus series method were determined . Use Theorem and Dulac solution is bounded , positive equilibrium obtained sufficient conditions for global stability . Use Bendixson theorem and Zhang Zhifen annulus uniqueness theorem proved existence and uniqueness of limit cycles . By algebraic equations between roots and coefficients , and focus Calculation results are obtained sufficient conditions for existence of Hopf bifurcation . The use of Matlab software for the numerical simulation . The second part , consider the impact on the system running , a class of Sparse Effect prey predator with a constant delivery rate system . Use isocline geometric properties of the behavior of the equilibrium point , there have been positive equilibrium sufficient condition . Qualitative analysis using the method of differential equations obtained is globally stable equilibrium point and the existence and uniqueness of limit cycles for a sufficient condition . And given the system 's ecological significance. By algebraic equations between roots and coefficients , and focus Calculation results are obtained sufficient conditions for existence of Hopf bifurcation . Using Matlab software for the numerical simulation . The third part, taking into account many of the behaviors and the nature of the population changes over time are affected , a class of non-autonomous Sparse Effect predator system . The system is obtained using the comparison principle sufficient condition for persistence . For non- autonomous periodic system , using the coincidence degree theory continuation theorem to obtain periodic system is a sufficient condition for existence of periodic solutions . Finally, by constructing a suitable Lyapunov function, the system is discussed global stability of positive periodic solution . Through the study of ecosystem stability , reasonable people can better guide the sustainable use of biological resources and scientific management , which for maintaining ecosystem biological diversity and the sustainable development of the ecological environment has important theoretical and practical significance .

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CLC: > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Differential equations, integral equations > Ordinary Differential Equations
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