The extended single-species Klausmeier model under a stochastic environment is investigated. The underlying model consists of two species, one is the tree species and the other is the amount of water. Such models have numerous applications in arid and semi-arid environments. The explicit solitary wave solutions are obtained with the \(\phi ^6\) -model expansion. This is a basic analytical tool that will give us the solutions in the form of Jacobi elliptic functions. Solitons and solitary wave solutions are obtained as continuous limits. The different forms of dark, bright, dark-bright mixed and periodic solutions are explored under the effect of noise. The computational study is carried out with the help of a finite-difference scheme. The analysis of the scheme is derived in the mean-square sense. Some analytical solutions will be physically interpreted and the effect of the noise will be discussed. The numerical solutions are obtained for different choices of the parameters, and two equilibria are successfully derived. The equilibia also have some physical connotation. Some solitary waves are plotted in 3D and 2D forms under different values of noise strength. The classical solitons are drawn by choosing noise zero and their different noise effect as well. For comparison purposes, some analytical solutions are selected. Initial-boundary-value problems are fixed and their physical behavior is examined under the stochastic environment. For the smaller strength of the noise, our solutions resemble the classical solutions. Moreover, two- and three-dimensional plots show that our theoretical results agree with the numerical solutions.
