Moving spike patterns in a chemotaxis model and oscillatory traveling waves in a traffic flow model

We study a branch of mathematical problems in partial differential equations (PDEs) called non-linear hyperbolic conservation laws. PDEs are an important branch of mathematics because we can model and analyze complex phenomena in areas such as physics, biology, and finance. In this thesis, we study traveling wave solutions to PDE models arising from chemotaxis and traffic flow. Traveling waves are a type of solution to PDEs that maintain their shape while moving at a constant speed.

Chemotaxis describes the movement of cells or organisms due to chemical response. Some examples of chemotaxis include the movement of bacteria towards a food source or the migration of slime molds due to chemicals in the environment. In chemotaxis, traveling waves represent wave-like patterns that cells or organisms form as they follow the chemical attractant. The model we study is derived from the Keller-Segel model, the first mathematical model describing chemotaxis. We prove the existence and instability of nonmonotone traveling wave solutions which represent moving spike patterns.

Traffic flow models represent interactions between roads, vehicles, drivers’ reaction times, and human behavior. In traffic flow, traveling waves represent wave-like disturbances through a stream of vehicles causing localized areas of congestion that propagate backwards in the direction of traffic. In our model, the first equation relates the number of cars passing a specific point on the road to the velocity of the cars and the second equation describes the drivers’ acceleration behavior. We investigate the existence and instability of oscillatory traveling wave solutions representing stop-and-go traffic.