Stability Research Of Biological Models

Biological mathematics is a relatively independent and relatively complete discipline, Which plays an important role for the modern science. It is such a interdisciplinary subject, Which makes life science, public health,medicine, biology,agronomy and mathematics mutual penetration.when researching the complexity of the ecological relationship in natural,people often study through the establishment of mathematical model, Which is the interdisciplinary of mathematics and biologyBiological Mathematics. As an important branch of biological mathematics, biology power system mainly uses dynamic knowledge to study the biological mathematics modeling which has been established. Mathematical results can be used to explain phenomena existing in biology and to predict something which may happen in the future in the biological world.In this way, people can choose a more appropriate way in life, which makes human and nature more harmonious.The first part of this paper focuses on the research background and research status to do the research of biological dynamics and population dynamics to do the research.The second part of this paper mainly introduces several classic population dynamics model: Logistic model, Lotka-Volterra model and Leslie- Gower model.Finally this part mainly introduces the dynamic behavior of the three different kinds of functional response function in which the species population has density or the density restriction.and introduces its stability in equilibrium and the conditions of limit cycle.The third part of this paper mainly study the qualitative analysis of Volterra model which has a constant deposit rate.This kind of Volterra model has at least two equilibrium point, Using qualitative theory of planar system and the specification theory, we find that they can be stable nodes, unstable, saddle points, weak center, etc, under different parameters. Through specification theory of Hopf bifurcation and calculating the first Lyapunov coefficient, the system will occur supercritical Hopf bifurcation near the weak center and branch out the only stable limit cycle from the equilibrium point.Through analysis this kind of Volterra model of a constant deposit rate: When npna21+<<)1(0,)1(010npaa++<<, Internal equilibrium point is a stable node of the system.If you see n as a sparse rate of the system,then for any given spare rate,when the deposit rate p is enough big,such that it is greater than the ratio of the birthrate of prey and the death rate of predator-prey, the biological system can be coexistence for long-term. When)133(1)(nnpnnanp212++<<+,)1(10npaak++<<, the system can branch out the only stable limit cycle near the equilibrium point. this suggests the biological system can be coexistence by stable periodic solution for long-term.