We study positive solutions to the ecological model of the form
where
is a positive parameter proportional to the square of the habitat size, and
,
where
and
are positive parameters. This type of reaction-diffusion equation
specifically models the population density of a species living in a
habitat surrounded by a hostile exterior region. In it, our reaction term,
,
represents a weak Allee growth of the population. This means the per-capita growth
rate,
, is
increasing at smaller population densities and decreasing at larger ones. The variable
represents the population density that is a function of the spatial variable
.
In our model, the exterior region is extremely hostile, to the point
that the population on the boundary has a density of zero. In this
study, we discuss the existence of a weak Allee effect for any choices of
and
as
well as the structure of positive steady states using a quadrature method. Specifically,
we discuss a bifurcation curve modeling this scenario. Further, we present certain
numerical results that we obtain using a quadrature method and Mathematica
computations.
Keywords
boundary-value problems, ODE, population dynamics, weak
Allee growth models
