In collaboration with various researchers, my work in the study of steady state solutions for reaction diffusion equations has involved use of/adaptation of various methods from nonlinear analysis such as the method of sub-super solutions, principle of linearized stability, time map analysis (quadrature methods), and degree theory to accommodate nonlinear boundary conditions present in the equations. In particular, I have studied the structure and/or stability of positive steady state solutions of reaction diffusion models arising from both combustion theory and population dynamics.

With regard to population dynamics, I have explored models with various growth terms including logistic and Allee effect, density dependent predation, and constant yield predation, all in the presence of nonlinear boundary conditions modeling conditional dispersal. An Allee effect is exhibited at the patch level when dynamics of the model predict persistence for some initial conditions but extinction for others, even with the same parameter ranges.

When the reaction diffusion equations include predation that is constant yield (i.e., not density dependent), the problem has what is known in the literature as “semipositone structure.” Proving existence of positive solutions of reaction diffusion equations with **semipositone structure** poses challenging mathematical problems.