| Abstract: |
| Steady states of nonlinear reaction-diffusion-advection (RDA) models can be viewed as solutions of a system of two first order ODEs (subject to appropriate boundary conditions). Geometrically, they are represented by orbits in the phase plane, generated by the corresponding flow operator. In this talk, I will discuss applications of the phase plane technique in two extensions of a logistic reaction-diffusion-advection model. In one setting, we increase the complexity of the habitat by considering a binary river network. Here a steady state is represented by a configuration of orbits in the corresponding phase plane satisfying geometric constraints induced by junction and boundary conditions. Proving a certain concavity preservation property of the flow of the corresponding system of ODEs allows us to establish uniqueness of the positive steady state solution, as well as explicit existence conditions. In the second setting (joint work with Abby Anderson), we increase the complexity of the reaction term. Namely, we study an RDA version of the classical Ludwig-Aronson-Weinberger spatial spruce budworm model (where reaction term combines logistic growth and predation by generalist), with advection term describing biased movement of larvae due to prevailing winds. We use phase plane analysis to determine conditions for the existence of the outbreak solutions. In particular, we observe that increasing advection can prevent outbreaks while allowing persistence in the form of an endemic state. We obtain upper and lower bounds for the critical advection for outbreak steady state solutions. |
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