M M[3 || 4]A21 Mathematical Biology:    Example sheet 1 A.J. Mestel



Population Dynamics of a single species; Fisher's equation.


1. An organism, with population density U(X, T), has a logistic growth and diffuses in space, so that

UT = R U(1- U
Q
 )+DUXX Î a < X < b ,
where R, Q and D are known positive constants.

Defining x, t and u(x, t) appropriately, obtain a non-dimensional version of this equation

ut = u(1-u)+uxx Î 0 < x < L ,     (1)
and give a formula for L.



2. The population density u(x,t) obeys the Fisher equation (1) together with the boundary conditions ,

u(0, t) = 0 = u(L, t) u(x, 0) = u0(x) ³ 0 .
You may assume that u(x, t) ³ 0 for all x and t.

Defining a sort of ``energy," E(t) = 1/2ò0L u2 dx, show that

dE
dt
 = ó
õ 		L

0 
 u2(1-u) dx- ó
õ 		L

0 
 ux2 dx .
Assuming the Poincaré inequality for smooth functions f(x) satisfying f(0) = f(L) = 0,
ó
õ 		L

0 
 fx2 dx ³ p2
L2
 ó
õ 		L

0 
 f2 dx ,     (Poincaré)
deduce that the population will die out if L < p. Write down the corresponding result in terms of the dimensional quantities a, b, R, D and Q of question 1.



3. Suppose, instead, that u(x,t) is subject to the `no flux' boundary conditions ,

ux(0, t) = 0 = ux(L, t) u(x,0) = u0(x) .
This time consider F(t) = 1/2ò0L ux2 dx. Show that if L < p, then F®0 as t®¥, and hence u must tend to a constant for any initial state u0(x).

[Hint: First differentiate (1) with respect to x, and continue as in the last question.]



4. A swimming pool occupies 0 < x < a, 0 < y < b, 0 < z < c. The density of algae within it, u(x, y, z, t), is assumed to obey the equation and boundary conditions

ut = u(1-u)+Ñ2u      with   u = 0      on   x = 0,a,    y = 0,b,    z = 0,c .
Discuss whether or not the `clean' state u º 0 is stable.

Is this a good model, do you think?

File translated from TEX by TTH, version 2.22.
On 14 Oct 1999, 16:07.