Stability and bistability in a one dimensional model of coastal foredune height
On sandy coastlines, foredunes provide protection from coastal storms, potentially sheltering low areas including human habitat from elevated water level and wave erosion. In this contribution we develop and explore a one dimensional model for coastal dune height based on an impulsive differential equation. In the model, coastal foredunes continuously grow in a logistic manner as the result of a biophysical feedback and they are destroyed by recurrent storm events that are discrete in time. Modeled dunes can be in one of two states: a high "resistant dune" state or a low "overwash flat" state. The number of stable states (equilibrium dune heights) depends on the value of two parameters, the nondimensional storm frequency (the ratio of storm frequency to the intrinsic growth rate of dunes) and nondimensional storm magnitude (the ratio of total water level during storms to the maximum theoretical dune height) Three regions of phase space exist (1) when nondimensional storm frequency is small, a single high resistant dune attracting state exists; (2) when both the nondimensional storm frequency and magnitude are large, there is a single overwash flat attracting state; (3) within a defined region of phase space model dunes exhibit bistable behavior both the resistant dune and the low overwash flat states are stable. Comparisons to observational studies suggest that there is evidence for each state to exist independently, the coexistence of both states (i.e. segments of barrier islands consisting of overwash flats and segments of islands having large dunes that resist erosion by storms) as well as transitions between states.