Finite Difference Schemes for Fisher´s Equation
P.Chocholatý
Reaction-diffusion equations arise as mathematical models in a series of important
applications, e.g. in models of superconducting liquids, flame propagation, chemical kinetics,
biochemical reactions, predator-prey systems in ecology and so on. Both numerical and
mathematical analysis of reaction-diffusion equations are currently very active fields of
research. We start our discussion by considering a model arising in mathematical ecology. In
order to understand the foundation of this model, we first recapture the model of population
growth. This model states that the growth of a population facing limited resources is governed
by ODE:
(1)
v´(t )   v(t ) ( A  v(t )), v(0)  f 0 .
Here v (t ) is the population density,  is the growth rate, and A  0 is the so-called carrying
capacity of the environment. In deriving the model (1), it is assumed that spatial variation in
the density of the population is of little importance for the growth of the population. Thus, one
simply assumes that the population is evenly distributed over some area for all time. For real
population, this assumption is often quite dubious. So it is common to take into account the
tendency of a population to spread out over the area where it is possible to live. This effect is
incorporated by adding a Fickian diffusion term to the model and we get the following PDE:
ut  d u xx   u( A  u)
(2)
Here d is a diffusion coefficient and u ( x, t ) is the population density. In mathematical
ecology, this model of population growth is called Fisher´s equation. Obviously, the
introduction of a diffusion term leads to a PDE which in contrast to the ODE (1) cannot in
general be solved analytically.
Usually Fisher´s equation (2) is studied in conjunction with Neumann-type boundary
condition, i.e.
u x (0, t )  u x ( L, t )  0 ,
(3)
where L denotes the length of the domain. The reason for this condition is that we assume the
area to be closed, so there is no migration from the domain. We may consider a valley
surrounded by mountains, or we can simply think of an island.
So, we study the problem (2),(3) with the initial condition
u( x,0)  f ( x) , x  0, L ,
(4)
where f ( x) , 0  f ( x)  A , denotes the initial distribution of the population.
Our aim is to present Fisher´s equation and to explore some of their properties using finite
difference schemes. Let u j , n denote an approximation of u ( x j , t n ) , then finite difference
schemes are presented for some u t generated by the schemes,
u ń ´  (u n 1  u n ) / l ,
u n ´  (3u n 1  4u n  u n 1 ) /( 2l ) ,
and so on. Here l  t n 1  t n , n  0,1,2,...
.
(5)