Traveling waves and turing patterns in predator-prey equations

Abstract

In this thesis, we address traveling waves and patterns in predator-prey equations.

In Chapter 3, we investigate the existence of traveling wave pulses in Lotka-Volterra type predator-prey equations with strong Allee effect in prey population. For all positive speed $c \geq c^*$, there exist traveling wave solutions in which the population density of prey is strictly monotone and the one of predator is pulse shaped. The solutions describe extinction phenomena of prey by predator. We prove the existence of the traveling waves by means of a topological shooting argument.

In Chapter 4, we propose Lotka-Volterra type predator-prey equations which include constant harvesting and planting effects. To see the effect of the constant effects clearly we drop all other functional responses except the ones in the original Lotka-Volterra equations. We add a negative constant term for constant harvesting or the Allee effect for ecological phenomenon. A positive constant term is added to model consistent planting or supply. We find that the constant effect gives most of dynamic and static patterns observed by other models using various functional responses.