## Doctoral Dissertations

#### Date of Award

12-2000

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy

#### Major

Mathematics

#### Major Professor

Mark Kot

#### Committee Members

Suzanne Lenhart, Thomas Hallam, Yueh-er Kuo

#### Abstract

Models that describe the spread of invading organisms often assume no Allee effect. In contrast, abundant observational data provide evidence for Allee effects. In chapter 1, I study an invasion model based on an integrodifference equation with an Allee effect. I derive a general result for the sign of the speed of invasion. I then examine a special, linear-constant, Allee growth function and introduce a numerical scheme that allows me to estimate the speed of traveling wave solutions. In chapter 2, I study an invasion model based on a reaction-diffusion equation with an Allee effect. I use a special, piecewise-linear, Allee population growth rate. This function allows me to obtain traveling wave solutions and to compute wave speeds for a full range of Allee effects, including weak Allee effects. Some investigators claim that linearization fails to give the correct speed of invasion if there is an Allee effectI show that the minimum speed for a sufficiently weak Allee may be the same as that derived by means of linearization. In chapters 3 and 4, I extend a discrete-time analog of the Lotka-Volterra competition equations to an integrodifference-competition model and analyze this model by investigating traveling wave solutions. The speed of wave is calculated as a function of the model parameters by linearization. I also show that the linearization may fail to give the correct speed for the competition model with strongly interacting competitors because of the introduction of a "weak Allee effect". A linear-constant approximation to the resulting Allee growth function is introduced to estimate the speed under this weak Allee effect. I also analyze the back of the wave for the competition model. Some sufficient conditions that guarantee no oscillation behind the wave are given.

#### Recommended Citation

Wang, Mei-Hui, "Speeds of invasion for models with Allee effects. " PhD diss., University of Tennessee, 2000.

https://trace.tennessee.edu/utk_graddiss/8445