DYNAMICS AND BIFURCATIONS IN A TRITROPHIC FOOD-CHAIN MODEL WITH GROUP DEFENCE MECHANISM AND INTERN-PREDATOR COMPETITION

• Main Author: Owen, Livia
• Format: Dissertations
• Language: Indonesia
• Subjects: Ilmu alam dan matematika
• Online Access: https://digilib.itb.ac.id/gdl/view/52955
• Institution: Institut Teknologi Bandung

This research has two main topics, i.e. a new method for computing fold and cusp bifurcation in a system of two ordinary differential equations, and a mathematical model for tritrophic food chains with group defense mechanism and intern-predator competition. In the first topic, we prove that the fold bifurcation point in that system corresponds to a local maximum or minimum of a constrained optimization problem, which can be computed using the classical Lagrange multiplier method. Conversely, we provide a sufficient condition so that the solution of a particular constrained optimization problem using the Lagrange multiplier method, corresponds to a fold bifurcation of two ordinary differential equations. Similarly, for a system with two parameters, we also provide sufficient conditions for cusp bifurcation point. Then we apply the method into the subsystem of our tritrophic food-chain model. In tritrophic food chains model, we first concern on condition when the ratio of maximum per capita growth rate of predator and top predator goes to zero. This condition implies the top predator becomes constant. By choosing the top predator constant as a bifurcation parameter and applying Lagrange multiplier method, we identify all of fold bifurcation points in this subsystem. By using AUTO software, we get another codimension-one bifurcation points. Then we find more complex dynamical behaviors as an impact of intern-predator competition, i.e. cusp, Bautin, and Bogdanov-Takens bifurcations. Furthermore, we indicate a codimension-three bifurcation, i.e. Swallow-tail bifurcation and cusp Bogdanov-Takens bifurcation. The latter can be considered a new bifurcation because no research has been studied. Finally, we identify and analyze all the codimension-one and codimension-two bifurcations of the tritrophic food chains model. We also provide some interesting attractors and indicate a chaotic dynamics by varying a parameter of group mechanism defense.