The influence of autotoxicity on the dynamics of vegetation spots

Plant autotoxicity has proved to play an essential role in the behaviour of local vegetation. We analyse a reaction-diffusion-ODE model describing the interactions between vegetation, water, and autotoxicity. The presence of autotoxicity is seen to induce movement and deformation of spot patterns in some parameter regimes, a phe-nomenon which does not occur in classical biomass-water models. We aim to analytically quantify this novel feature, by studying travelling wave solutions in one spatial dimension. We use geometric singular perturbation theory to prove the existence of symmetric, stationary and non-symmetric, travelling pulse solutions, by constructing appropriate homoclinic orbits in the associated 5-dimensional dynamical system. In the singularly perturbed context, we perform an extensive scaling analysis of the dynamical system, identifying multiple asymptotic scaling regimes where (travelling) pulses may or may not be constructed. We show that, while the analytically constructed stationary pulse corresponds to its numerical counterpart, there is a discrepancy between the numerically observed travelling pulse and its analytical counterpart. Our findings indicate how the inclusion of an additional ODE may significantly influence the properties of classical biomass-water models of Klausmeier/Gray-Scott type.