Bifurcations Of Double Homoclinic Loops In Four Dimensional Systems And Problems Of Periodic Solutions In Population Dynamics

This thesis mainly investigates bifurcations of double homoclinic loops in four dimensional systems and problems of periodic solutions in population dynamics. The thesis can be divided into two main parts including six chapters.In Chapter 1, the background and research status of bifurcation theory and problems of periodic solutions in population dynamics are briefly reviewed, and the main work of this paper is described briefly.In Chapter 2, firstly the local coordinates near the double homoclinic loops are set up in four dimensional vector fields, and then the Poincaré map are constructed, and the bifurcation equations are induced. Furthermore, codimension 2 nonresonant double-homoclinic-loop bifurcations are investigated in four dimensional vector fields. And near double homoclinic loops, i.e., large loop Γ = Γ₁ ∪ Γ₂, the existence, uniqueness and incoexistence of large 1-hom loop and large 1-per orbit are proved. Their corresponding existence regions are located and bifurcation diagrams are given. Meantime, the inexistence of the large 2-hom loop and large 2-per orbit are also demonstrated.In chapter 3, codimension 3 bifurcations of nontwisted double homoclinic loops with resonant eigenvalues in four dimensional systems are studied. The coexistence of large 1-hom orbit and large 1-per orbit near Γ = Γ₁ ∪ Γ₂ is given. The conditions of existence or inexistence of saddle-node bifurcation surfaces are obtained. Finally, we give completely the bifurcation diagrams under different conditions in details.In chapter 4, sufficient conditions of at least one, two and multiple positive periodic solutions to a scalar functional differential equation with muti-finite delays are obtained by using a fixed point theorem introduced in [27] and [85] which is different from the one of a cone compression and expansion of norm type. It generalizes and improves some related results in the literature. And with the help of matlab, our results are simulated and verified.In chapter 5, sufficient conditions for the existence of multiple positive periodic solutions of a delayed discrete predator-prey system with Holling IV functional responses are presented based on coincidence theory and prior estimates.In chapter 6, the periodicity in a ratio-dependent predator-prey system with stage-structured predator and predator-prey systems with Holling type functional responses on time scales is investigated. This main approach is also coincidence degree theory. Moreover, when the time scale T is chosen as the sets of real or integer numbers, the existence of periodic solutions for its differential or difference equation are obtained respectively. This helps to avoid proving the existence results twice, once for differential equations and once again for difference equations in these ecological systems.