Math and Stats - Analysis Applied Math Physics Seminar - "Spike dynamics in the presence of noise and competition between two plant species"

Speaker :  Chunyi Gai, Dalhousie University

Chunyi is a graduate student finishing her PhD under the supervision of Prof. Theo Kolokolnikov and works in pattern formation.  In particular, her thesis work has focused on spikes and S-I-R models.  She will be moving onto a postdoctoral position this Fall 2021.

Abstract:  In this talk, based on the joint work with Theodore Kolokolnikov, David Iron and John Rumsey, I discuss spike dynamics of two dynamical systems. In the classical Gierer--Meinhardt model, we study the effect of noise on dynamics of a single spike on a finite interval. When spatio-temporal noise is introduced in the equation for the activator, we derive a stochastic ODE that describes the motion of a single spike on a slow time scale. For sufficiently small noise level, the spike performs random fluctuations near the center of the domain. As noise level is increased, the spike can deviate from the domain center but remains effectively "trapped" within a certain subinterval that includes the center. For even larger noise levels, the spike starts to undergo large excursions that eventually collide with the domain boundary and temporarily trap the spike there. By reformulating this problem in terms of mean first passage time, we derive the expected time for the spike to collide with the boundary.

The other dynamical system is an extension of the well-known Klausmeier model of vegetation to two plant species that consume water at different rates. In semi-arid regions, the Klausmeier model produces vegetation spot patterns. We are interested in how the competition for water affects co-existence and stability of patches of different plant species. We consider two plant types: a “thirsty” species and a “frugal” species, that only differ by the amount of water they consume, while being identical in all other aspects. We find that there is a finite range of precipitation rate for which two species can co-exist. Outside of that range, (when the rate is either sufficiently low or high), the frugal species outcompetes the thirsty species. As the precipitation rate is decreased, there is a sequence of stability thresholds such that thirsty plant patches are the first to die off, while the frugal spots remain resilient for longer. We also examine numerically what happens for very large precipitation rate.​


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Ellen Lynch