Pattern solutions of the Klausmeier model for banded vegetation in semi-arid environments II: patterns with the largest possible propagation speeds

Abstract

Pattern formation at the ecosystem level is a rapidly growing area of spatial ecology. The best studied example is vegetation stripes running along contours in semi-arid regions. Theoretical models are a widely used tool for studying these banded vegetation patterns, and one important model is the system of advection-diffusion equations proposed by Klausmeier. The present study is part of a series of papers whose objective is a comprehensive understanding of patterned solutions of the Klausmeier model. The author focuses on the region of parameter space in which the propagation speed of the patterns is close to its maximum possible value. Exploiting the large value of one of the model parameters, a leading order approximation is obtained for the maximum propagation speed, and the author undertakes a detailed investigation of the parameter region in which there are patterns with speeds close to this maximum.

History-dependent patterns of whole ecosystems

“…The plot has been truncated at c = 10, and the parameter region giving patterns actually extends up to c % 50 (Sherratt, 2011). Moreover for c above about 14 there is a fold in the solution branch as A is varied with c fixed, implying two different patterns for given values of A and c (see Sherratt (2011) for details); however, all patterns with c > 10 are unstable as solutions of (1). There is also a fold in the solution branch as A is varied with c fixed at less than about 0.004 (Sherratt, 2013b); again this leads to two different pattern solutions, but both are unstable as solutions of (1).…”

“…To be specific, I will consider only one model, due originally to Klausmeier (1999). A detailed ecological appraisal of this model is given by Ursino (2005), and mathematical properties of the equations are discussed by Sherratt (2005Sherratt ( , 2010Sherratt ( , 2011Sherratt ( , 2013a. The model consists of conservation equations for the plant biomass u(x, t) and the water density wðx; tÞ, and when suitably non-dimensionalised it has the form @u @t …”

“…Details of the stability change are given in Appendix A. The plot has been truncated at c = 10, and the parameter region giving patterns actually extends up to c % 50 (Sherratt, 2011). Moreover for c above about 14 there is a fold in the solution branch as A is varied with c fixed, implying two different patterns for given values of A and c (see Sherratt (2011) for details); however, all patterns with c > 10 are unstable as solutions of (1).…”

Turing Patterns in Deserts

“…For most values of c, this range is bounded by a Hopf bifurcation point for (3,4) and a homoclinic solution of (4). However for some values of c there is a fold in the branch of periodic travelling wave solutions, and this then constitutes on end of the rainfall range for patterns [31,32]. A typical result is illustrated in Figure 2, which shows the loci of the Hopf bifurcation point and the homoclinic solution in the A-c parameter plane, for fixed values of B and ν. Analytical study of (1) is made more complicated by the advective term in the u-equation.…”

Impact of Parameter Variability and Environmental Noise on the Klausmeier Model of Vegetation Pattern Formation

“…However, the analytical investigation lies beyond the scope of this study. The interested reader may find further information regarding the analytical treatment of wavetrains in the Klausmeier model in Sherratt [45].…”