The Biological Dynamics Of Population Dynamics Model

Biological mathematics theory and method, for the development of modern sciencehas made enormous contribution. As an important branch of Mathematical Biology-thebiological dynamics, are now widely used to study the law of life science. With thedynamic point of view to analyze the biological objective phenomena, research therelationship between organisms and their environment, and the study of biologicalrelations between groups, thereby utilizing the dynamics modeling method to build theecological and population between the mathematical model, and population andenvironment between the mathematical model, use this model to study some ecologicalphenomenon, so as to achieve on some ecological phenomenon and control.This paper first introduces the population dynamic model. First introduced thesingle population dynamic model of continuous models in two main types: adensity-dependent model of single population and density of single population model,and analyzes the equilibrium and stability of. It then introduces the two populationdynamical models of Kolmogorov model and Lotka-Volterra model, and analyzes thetwo models of biological significance and equilibrium, limit ring. Finally introduced inthe Lotka-Volterra model is established based on the three kind functional reactionsystem, briefly introduced its equilibrium point stability and limit cycles.In order to better research, this article then introduced the dynamic system of someof the basic terminology. Then it introduces the dynamic system equilibrium point,two-dimensional linear system equilibrium point are geometric classification: stablefocus, stable point, unstable focus, unstable point, saddle point, the center, thenintroduces the nonlinear dynamic system and constant power system equivalentconditions. Finally introduced the continuous time dynamic system equilibrium point ofthe single parameter branch, two classes of General Yu dimension is a branch: foldbranch and Hopf branch, and in detail introduced the Hopf existence condition andparameter, and using the coordinate transformation, the time scale and nonlinear timeweight parameter of Hopf branch of the Poincare canonical form of normative.The stability and Hopf bifurcation of a Volterra model with sparse effect arediscussed. It is shown that the system has up to three equilibria．By normal form theory,these equilibria can be saddle，stable node，unstable node，saddle-node and weak centerdepending on the ranges of parameters．By using the first Lyapunov number，it is proved that the system undergoes a supercritical Hopf bifurcation and hence a uniquestable limit cycle occursï¼›when the system have a unstable equilibria,By using thePoincare-Bendison theorem,it is proved that it always have limit cycle．Based on theeffect of sparse Volterra dynamic analysis, know when the prey population issparse,Namely the N is big enough, the parameters of the biological significance ofknowledge:N, said prey sparse extent more serious.When N is largeenough,Because n=N/K,n is large enough,in this time m₀=（n²+n）^1/2-nâ†'1/2.Bytheorem3.1: E₁₁=（m₀,m₀（1-m₀）/（m₀+n））â†'（1/2,0）,And in the nearbyE₁₁stableperiodic solution.That is to say, when the prey the sparsity degree is very serious,predator and prey populations in nearbyE11cycle phenomenon, predator numbersaround0, predators become extinct. So this paper used the dynamics method isillustrated by a sparse effect rule of life science.