Differential equations with discontinuities and population dynamics

Aruğaslan Çinçin, Duygu
In this thesis, both theoretical and application oriented results are obtained for differential equations with discontinuities of different types: impulsive differential equations, differential equations with piecewise constant argument of generalized type and differential equations with discontinuous right-hand sides. Several qualitative problems such as stability, Hopf bifurcation, center manifold reduction, permanence and persistence are addressed for these equations and also for Lotka-Volterra predator-prey models with variable time of impulses, ratio-dependent predator-prey systems and logistic equation with piecewise constant argument of generalized type. For the first time, by means of Lyapunov functions coupled with the Razumikhin method, sufficient conditions are established for stability of the trivial solution of differential equations with piecewise constant argument of generalized type. Appropriate examples are worked out to illustrate the applicability of the method. Moreover, stability analysis is performed for the logistic equation, which is one of the most widely used population dynamics models. The behaviour of solutions for a 2-dimensional system of differential equations with discontinuous right-hand side, also called a Filippov system, is studied. Discontinuity sets intersect at a vertex, and are of the quasilinear nature. Through the B−equivalence of that system to an impulsive differential equation, Hopf bifurcation is investigated. Finally, the obtained results are extended to a 3-dimensional discontinuous system of Filippov type. After the existence of a center manifold is proved for the 3-dimensional system, a theorem on the bifurcation of periodic solutions is provided in the critical case. Illustrative examples and numerical simulations are presented to verify the theoretical results.


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In this thesis, we introduce two new classes of differential equations, which essentially extend, in several directions, impulsive differential equations and equations on time scales. Basics of the theory for quasilinear systems are discussed, and particular results are obtained so that further investigations of the theory are guaranteed. Applications of the newly-introduced systems are shown through a center manifold theorem, and further, Hopf bifurcation Theorem is proved for a three-dimensional discontin...
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Citation Formats
D. Aruğaslan Çinçin, “Differential equations with discontinuities and population dynamics,” Ph.D. - Doctoral Program, Middle East Technical University, 2009.