Title: Volume Scavenging: A Model of Fluid-Dynamical Competition
Abstract: The Florida tortoise beetle (Hemisphaerota cyanea) is a purplish-blue beetle which feeds on palm trees and is common to the Southeastern United States. It would be rather unremarkable if it were not for its ingenious defense mechanism against predatory ants: When attacked, the beetle clings to the ground with more persistence than the ants muster in their assault. The beetle achieves this strong adhesion by excreting an oily liquid through thousands of openings at its leg endings, thus forming liquid bridges with the substrate. In this way the beetle can withstand pulling forces up to 60 times and more of its body mass. Motivated by this defense mechanism, I present a dynamical system modeling surface tension-induced flows of liquids in networks of interconnected channels. These channel flows are driven by surface-tension induced pressure imbalances between fluid droplets of varying sizes (“volume scavenging”). These pressure differences push liquid from one droplet to another along the network of channels. The analysis of the nonlinear dynamics will be accompanied by animations, highlighting the similarities of microfluidic behavior and the emergence of inequality in socioeconomic competition.

Title: The diffusive Lotka-Volterra competition model in fragmented patches I: Coexistence
Abstract: We consider a lotka-volterra competition model where two species compete in a domain with the strengths of competitions b1 and b2. A parameter \lambda in the model represents the square of the patch size and two parameters \gamma1 and \gamma2 represent the matrix hostility. We analyze the positive solutions of the model as the parameters b1, b2 and \gamma1, \gamma2 vary.

Title: Modeling effects of matrix heterogeneity on population persistence at the patch-level
Abstract: We study the structure of positive solutions to steady state reaction diffusion equation of the form: \begin{equation*} \left\lbrace \begin{matrix} -u” =\lambda u(1-u);~~ (0,1) \\ -u'(0) + \sqrt{\lambda} \gamma_1 u(0) = 0 \\ ~~ u'(1) + \sqrt{\lambda} \gamma_2 u(1) = 0 \end{matrix} \right. \hspace{.05in} , \end{equation*} where $\lambda>0$ is a parameter that encompasses patch size and $\gamma_1$, $\gamma_2$ are positive parameters related to the hostility at the boundaries $0$ and $1$, respectively. Note here that the parameter $\lambda$ influences both the equation and the boundary conditions. In this paper, we establish the existence, nonexistence, and uniqueness results for this model. In particular, we establish exact bifurcation diagrams for this model, first when $\gamma_2$ is fixed and $\gamma_1$ is evolving, and then when the Dirichlet boundary condition is satisfied at $x = 0$ ($u(0)=0$) and $\gamma_2$ is evolving. In each case, our results are established by combining a quadrature method and the method of sub-super solutions. Finally, we present some numerical results that we obtained for this model. Here, we numerically simulate how the parameters $\gamma_1$ and $\gamma_2$ affect the minimum size for $\lambda$ beyond which a positive solution exists. This model arises in the study of steady states for a population satisfying a logistic growth reaction and diffusing in a region surrounded by two exterior hostile matrices, and $\lambda$ is related to the minimum patch size for the existence of a positive steady state.

Title: On the controllability of some systems modeling cell dynamics related to leukemia
Abstract: The subject I choose to approach in this talk refers to my latest research topic: two control problems for a model of cell dynamics related to leukemia. The first control problem is in connection with classical chemotherapy, which indicates that the evolution of the disease under the treatment should follow a prescribed trajectory assuming that the drug works by increasing the cell death rates of both malignant and normal cells. In the second control problem, as for targeted therapies, the drug is assumed to work by decreasing the multiplication rate of malignant (leukemic) cells only. The control objective is that the disease state reaches the desired endpoint. The solvability of these two problems is proved by using a general method of analysis, and the solution of these problems is obtained within an abstract scheme, by using operator-based techniques. Numerical simulations are included to illustrate the theoretical results and prove their applicability.