  In this paper the spatial-temporal dynamics of the members of interacting
populations is described by nonlinear partial differential equations. We
consider the migration as a diffusion process influenced by the changing values
of the birth rates and coefficients of interaction between the populations. For
the particular case of one population and one spatial dimension the general
model is reduced to analytically tractable PDE with polynomial nonlinearity up
to third order. By applying the modified method of simplest equation to the
described model we obtain an analytical solution which describes nonlinear kink
and solitary waves in the population dynamics.
