To begin a discussion of the need for generic reaction-diffusion equations, we consider a set of simplified models relating to predator-prey population dynam- ics. These models consider the interaction of two species: predators and their prey. It should be obvious that such species will have significant impact on one another. In particular, if there is an abundance of prey, then the predator popu- lation will grow due to the surplus of food. Alternatively, if the prey population is low, then the predators may die off due to starvation.
To model the interaction between these species, we begin by considering the predators and prey in the absence of any interaction. Thus the prey population (denoted by x(t)) is governed by
dx = ax (A.2.1)
dt
where a > 0 is a net growth constant. The solution to this simple differential equation is x(t) = x(0) exp(at) so that the population grows without bound.
We have assumed here that the food supply is essentially unlimited for the prey so that the unlimited growth makes sense since there is nothing to kill off the population.
Likewise, the predators can be modeled in the absence of their prey. In this case, the population (denoted by y(t)) is governed by
dy
dt = −cy (A.2.2)
where c > 0 is a net decay constant. The reason for the decay is that the population starves off since there is no food (prey) to eat.
We now try to model the interaction. Essentially, the interaction must ac- count for the fact the the predators eat the prey. Such an interaction term can
2
∂t ∇ result in the following system:
dx
dt = ax − αxy (A.2.3a)
dy
dt = −cx + αxy , (A.2.3b)
where α > 0 is the interaction constant. Note that α acts as a decay to the prey population since the predators will eat them, and as a growth term to the predators since they now have a food supply. These nonlinear and autonomous equations are known as the Lotka–Volterra equations. There are two fundamen- tal limitations of this model: the interaction is only heuristic in nature and there is no spatial dependence. Thus the validity of this simple modeling is certainly questionable.
Spatial Dependence
One way to model the dispersion of a species in a given domain is by assuming the dispersion is governed by a diffusion process. If in addition we assume that the prey population can saturate at a given level, then the governing population equations are
∂x
∂t = a x −
∂y
x2 \
k − αxy + D1∇ x (A.2.4a)
∂t = −cx + αxy + D2∇ y , 2 (A.2.4b) which are known as the modified Lotka-Volterra equations. It includes the species interaction with saturation, i.e. the reaction terms, and spatial spreading through diffusion, i.e. the diffusion term. Thus it is a simple reaction-diffusion equation.
Along with the governing equations, boundary conditions must be speci- fied. A variety of conditions may be imposed, these include periodic bound- aries, clamped boundaries such that x and y are known at the boundary, flux boundaries in which ∂x/∂n and ∂y/∂n are known at the boundaries, or some combination of flux and clamped. The boundaries are significant in determining the ultimate behavior in the system.
Spiral Waves
One of the many phenomena which can be observed in reaction-diffusion sys- tems is spiral waves. An excellent system for studying this phenomena is the Fitzhugh-Nagumo model which provides a heuristic description of an excitable nerve potential:
∂u = u(a − u)(1 − u) − v + D 2 u (A.2.5a)
j 2 )
This reduces to a system of differential equations for which the fixed points can be considered. Fixed points occur when ∂u/∂t = ∂v/∂t = 0. The three fixed points for this system are given by
(u, v) = (0, 0) (A.2.7a)
This can be written as the linear system dx = a −1
dt b −γ
\
x . (A.2.10)
Assuming a solution of the form x = v exp(λt) results in the eigenvalue problem
which has the eigenvalues
u u
+stable
unstable x
stable u
-Figure 40: Front solution which connects the two stable branch of solutions u± through the unstable solution u = 0.
Figure 41: Spatial-temporal evolution of the solution of the Fitzhugh-Nagumo model for the parameter D = 2 × 10−6, a = 0.25, b = 10−3, and γ = 3 × 10−3. The dark regions are where u = u+ whereas the white regions are where u = u−.
In the case where b − aγ > 0, the eigenvalues are purely real with one positive and one negative eigenvalue, i.e. it is a saddle node. Thus the steady-state solution in this case is unstable.
Further, if the condition b − aγ > 0 holds, then the two remaning fixed points occur for u− < 0 and u+ > 0. The stability of these points may also be found.
For the parameter restrictions considered here, these two remaining points are found to be stable upon linearization.
The question which can then naturally arise: if the two stable solutions u± are connected, how will the front between the two stable solutions evolve?
The scenario is depicted in one dimension in Fig. 40 where the two stable u± branches are connected through the unstable u = 0 solution. Figure 41 depicts the spatial-temporal evolution of the solution of the Fitzhugh-Nagumo model for the parameter D = 2×10−6, a = 0.25, b = 10−3, and γ = 3×10−3. The dark regions are where u = u+ whereas the white regions are where u = u−. Note
Figure 42: Experimental observations in which target patterns and spiral waves are both exhibited. These behaviors are generated in the Belousov-Zhabotinskii reaction by pacemaker nuclei (a) and (b), and by the slime mold Dictyostelium (c) which emit periodic signals of the chemical cyclic AMP, which is a chemoat- tractant for the cells.
the formation of spiral waves. The boundary conditions used here are no flux, i.e. ∂u/∂x = 0 or ∂u/∂y = 0 across the left/right and top/bottom boundaries respectively.
Not just an interesting mathematical phenomena, spiral waves are also ex- hibited in nature. Figure 42 illustrates a series of experimental observations in which target patterns and spiral waves are both exhibited. These behaviors are generated in the Belousov-Zhabotinskii reaction by pacemaker nuclei and by the slime mold Dictyostelium which emit periodic signals of the chemical cyclic AMP, which is a chemoattractant for the cells. Spiral waves are seen to be naturally generated in these systems and are of natural interest for analytic study.
∇ The λ − ω Reaction-Diffusion System
The general reaction-diffusion system can be written in the vector form
∂u = f (u) + D 2u (A.2.13)
∂t
where the diffusion is proportional to the parameter D and the reaction terms are given by f (u). The specific case we consider is the λ − ω system which takes possibility of supporting three steady-state solutions as in the Fitzhugh-Nagumo model which led to spiral wave behavior. The spirals to be investigated in this of arms on the spiral. The stability of the spiral evolution under perturbation and in different parameter regimes is essential in understanding the underly- ing dynamics of the system. A wide variety of boundary conditions can also be applied. Namely, periodic, clamped, no-flux, or mixed. In each case, the boundaries have significant impact on the resulting evolution as the boundary effects creep into the middle of the computational domain.