In the paper we study a class of two-dimensional integral equations on a quarter-plane with monotone nonlinearity and substochastic kernel. With specific representations of the kernel and nonlinearity, an equation of this kind arises in various fields of natural science. In particular, such equations occur in the dynamical theory of p-adic open-closed strings for the scalar field of tachyons, in the mathematical theory of the geographical spread of a pandemic, in the kinetic theory of gases, and in the theory of radiative transfer in inhomogeneous media. We prove constructive theorems on the existence of a nontrivial nonnegative and bounded solution. For one important particular case, the existence of a one-parameter family of nonnegative and bounded solutions is also established. Moreover, the asymptotic behavior at infinity of each solution from the given family os studied. At the end of the paper, specific particular examples (of an applied nature) of the kernel and nonlinearity that satisfy all the conditions of the proven statements are given.