Qualitative Study Of Limit Cycles For Type Ⅲ Quadratic System And A Discrete Biomathematical Model

This paper includes two parts: the first part is about the uniqueness of limit cycles for type â…¢ quadratic system; the second is about a discrete biomathematical model with age-structure.In the first part, using the results of uniqueness of limit cycles for type I and type (â…¢)_a=0 quadratic system, we analyze the variations of topological structure around O for type (â…¢)_a=0 system when b changes from zero to nonzero, and find that the difference between the two systems dues to the emergence of an integral line 1 + by = 0, on which there might appear one or two saddles. When d changes from zero to nonzero and satisfies dW₁ < 0, a unique limit cycle appears and expands monotolically with the increasing of \d\, at last becomes a bounded heteroclinic cycle passing through N (defined by Hec(N) in the following), and the two systems are topological equivalent in the region around O having limit cycle.For a general type â…¢ system, when a≠0 and very small, using the theorem of structural stability and perturbation theorem we prove that for fixed l,m,n with m(l + n) ≠ 0 (namely when d = a = 0, W₁ ≠ 0) and dW₁ < 0, there exists an a₀, positive but very small, for all |a| < a₀, the type â…¢ system has at most one limit cycle which is bifurcated from O and expands with the incresing of |d| at last disappear into a heteroclinic cycle. Thus in such a procedure the two systems are topologically equivalent.When |a| is not very small, the limit cycle can not be unique. For the reason the first is that O can be a weak focus of high order or a center, then with the Bautin method the limit cycles around O is larger than one and the second is that a separatrix cycle can be formed around O with a limit cycle inside at the same time, which has the different stability of the the limit cycle inside. Besides the two cases above there may appear a semistable limit cycle around O and splits into two limit cycles with the incresing of |d|.In order to gurantee the uniqueness of limit cycles for general type â…¢ system we must give additional conditions to avoid the above phenomena. First we need the condition 0 < n < 1, since otherwise N(0, 1/n) may be a saddle and the stability ofHec(N) is contrary with that of limit cycle bifurcated by 0 thus at least two limit cycles appear;on the other hand we must let W\ ^ 0 so that O is a focus of order one and exclude the the cases having at least two limit cycles.Then we consider the following four cases: 1) / > \, m < 0; 2) I < \, m > 0; 3) I < \, m < 0; A)m > 0, / > \ in the (l,m) plane for fixed a and get the uniqueness of limit cycle under appropriate additional conditions. The process of creation and disappearance of the limit, cy is the same as the system (III)n=o-In the second part, we consider a discrete biomathematical model with age-structure for predator and classfy individuals of predator as belonging to either immature (phase 1 — k) which are raised by their parents or mature which feed on prey.this seems reasonable for a number of mammals, using the theory of overlapping,invariability , theory of discrete semi-dynamical systems, results on monotone and continuous operators and Perron-Frobnius theory, we get the boundness and persistence permanence of this system.Furthermore, we simulate the dynamics of three dimesion case, find that the region of stability for system with age-structure is larger compared with the system without age-structre by using Maple software. We also know that bi is a sensitive coefficient in this age-structured system, because of its existence, the system appears complicated dynamics, such as double-period bifurcation and chaos, using this results especially the critical values of bifurcation, we can control the balance of nature and protect the population of biological population especially the rare ones.