30 research outputs found
On some Free Boundary Problems of the Prey-predator Model
In this paper we investigate some free boundary problems for the
Lotka-Volterra type prey-predator model in one space dimension. The main
objective is to understand the asymptotic behavior of the two species (prey and
predator) spreading via a free boundary. We prove a spreading-vanishing
dichotomy, namely the two species either successfully spread to the entire
space as time goes to infinity and survive in the new environment, or they
fail to establish and die out in the long run. The long time behavior of
solution and criteria for spreading and vanishing are also obtained. Finally,
when spreading successfully, we provide an estimate to show that the spreading
speed (if exists) cannot be faster than the minimal speed of traveling
wavefront solutions for the prey-predator model on the whole real line without
a free boundary.Comment: 28 page
A sharp-front moving boundary model for malignant invasion
We analyse a novel mathematical model of malignant invasion which takes the
form of a two-phase moving boundary problem describing the invasion of a
population of malignant cells into a population of background tissue, such as
skin. Cells in both populations undergo diffusive migration and logistic
proliferation. The interface between the two populations moves according to a
two-phase Stefan condition. Unlike many reaction-diffusion models of malignant
invasion, the moving boundary model explicitly describes the motion of the
sharp front between the cancer and surrounding tissues without needing to
introduce degenerate nonlinear diffusion. Numerical simulations suggest the
model gives rise to very interesting travelling wave solutions that move with
speed , and the model supports both malignant invasion and malignant
retreat, where the travelling wave can move in either the positive or negative
-directions. Unlike the well-studied Fisher-Kolmogorov and Porous-Fisher
models where travelling waves move with a minimum wave speed ,
the moving boundary model leads to travelling wave solutions with . We interpret these travelling wave solutions in the phase plane and
show that they are associated with several features of the classical
Fisher-Kolmogorov phase plane that are often disregarded as being nonphysical.
We show, numerically, that the phase plane analysis compares well with long
time solutions from the full partial differential equation model as well as
providing accurate perturbation approximations for the shape of the travelling
waves.Comment: 48 pages, 21 figure