Nonlinear Diffusion in Population Genetics P. C. FrEE & L. A. PELETIER Communicated by J. B. MCLEOD

1. Introduction Consider a population distributed in space and having a continuum of possible states, measured by a single parameter u which takes on values on the interval [0, 1]. The state u will depend on both position and time, so we can speak of the state function u(x, t). For a population near a sea shore for instance, x may denote the distance to the sea shore. It is assumed that some kind of diffusion of state characteristics occurs, and that the time rate of change of u is also influenced by the state of the population. The simplest continuum mathematical model for the time evolution of such a state function leads to the non-linear diffusion equation u t = uxx + f ( x , u).

(1.1)

Here f ( x , u) is a nonlinear source term, which can be thought of as giving the rate of change of u at a fixed point x, if the influence from neighbouring segments of the population were ignored. The effect of an inhomogeneous environment is allowed for by the x-dependence in the function f. The situation described above occurs in FISHER'S model of the propagation of genetic composition in a population [4]. In this model, each individual of the population belongs to one of three possible genotypes a a, aA, and AA. The parameter u represents the fraction of alleles of type a amongst the total number of alleles in the population at a given time and place. It is thus a certain measure of the genetic composition of the population. The function f ( x , u) is derived from a knowledge of the relative survival fitnesses of the three genotypes, and the diffusion term arises from the effect of random migration of the individuals. In the present context, however, it should be emphasized that population models of this type are highly idealized. It is clear that the genetic composition of an isolated population will not change with time if the population consists entirely of individuals of genotype aa or genotype AA. Thus f ( x , 0) = f ( x , 1)=0. (1.2) Throughout this paper we shall assume that (1.2) holds. If the environment is uniform, the function f in equation (1.1) does not depend on x and (1.1) can be written as ut = uxx + f (u).

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P.C. FIFE & L.A. PELET1ER

Recently this equation has attracted considerable attention, and the behaviour of its solutions has been investigated in some detail [1, 3, 6]. In this paper we shall be interested in the situation in which the environment is not uniform, and we shall assume that the genotype AA has a selective advantage over the other genotypes for large negative values of x, and that genotype aa has a selective advantage over the other genotypes for large positive values of x. This means that 1

ff(x, u) du>O

for tel0, 1) and large positive x

t t

~f(x, u)du
for te(0, 1] and large negative x.

0

In section 2, we shall show that if fx(x,u)>O for large ]xl there exists an equilibrium solution ~b(x) of equation (1.1) such that ~b(- or)=0 and ~b(oo)= 1. This solution is monotonically increasing if fx (x, u)> 0 for all x e IR; in that case ~b is usually referred to as a cline. This generalizes a result due to CONLEY [2], who established the existence of a cline when f is of the form

f(x, u)= s(x) u(l - u),

(1.3)

in which sgn s(x)= sgn x and s is not integrable near _ oo. Many other interesting results about clines were obtained by FLEMING 1,5] and NAGYLAKI (see, for instance, I-7] and I-8]). In section 3 we shall show that this cline is asymptotically stable. By this we mean the following: Let u(x, t; ~) be the solution of equation (1.1) which takes on the initial value if, namely

u(x,O;O)=O(x),

xelR.

Let C(N) denote the space of bounded continuous functions defined on P., endowed with the usual supremum norm II. II. Then we can define the region of attraction A(q~) of q~ to be the set of functions OeC(IR) such that Ilu(.,t;O)-4~ll--'O

as t ~ .

In section 3 we shall give a partial characterization of the set A(q~); in particular we shall show that q5 is an interior point of A (~b). In addition we shall show that q~ is exponentially stable, i.e. there exist positive numbers 6, # and K such that if ]lqJ-qSlq <6, then Ilu(., t; ~)-~bll < K e -~' for all t > 0. The authors are grateful to H.F. WEIr~BERGER for helpful discussions about the question of exponential stability.

2. Existence and Uniqueness of a Cline In this section we shall consider the problem

u"+f(x,u)=O, (I)

u(-~)=0,

-~
(2.1 a) (2.1 b)

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For the existence of a cline, we shall make the following assumptions about f. Uniqueness and stability will be established later under somewhat different assumptions. A 1. f has continuous derivatives fx and fu, uniformly bounded in R x [0, 1]. A 2. f(x, 0) = f ( x , 1)=0 for all x ~ R . A 3. There exists a positive constant N such that

fx(x,u)>=O

for ]x[>N,

ue[0,1].

1

A4. ~f(N,u)du>O

for tE[0,1),

t t

S f ( - N , u)du
for te(0, 1].

0

Theorem 1.

Let f satisfy assumptions A1-4. Then there exists a solution

of Problem I. Proof. Let v (x) be defined for x__>N by the relation

x-N=

v(x) ( 1 }-~ So 12Sf(N,r)dr ds.

In view of assumption A4, v(x) is a solution of the problem

v"+f(N,v)=O

v(N)=0,

v(~)=l.

Next, define 0 u_(x)= v(x)

for x<=N, for x> N.

Then u(x) is a subsolution of Problem I. To verify this we observe that for x>N, by virtue of Assumption A 3,

u" + f(x, u_)= v" + f(x, v) > v" + f(N, v) = O, whereas, for x
fi(x)=fw(x)

ll

for x < - N for x > - N ,

where w is defined as the solution of the problem

w"+f(-N,w)=O,

w ( - oo)=0,

w(-N)=l.

Since u
ut=uxx+f(x,u), u(x, O)=u(x),

- oo < x < oo, t > 0 -~
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Then u(x, t) is strictly increasing in t and bounded above by fi(x). Therefore u (x, t) will tend pointwise to a function u(x) as t ~ oo. It can easily be shown that u(x) is a solution of Problem I. The solution so constructed is not necessarily the only solution of Problem I; if there are more, however, then u is the "minimal" solution. That the above assumptions are not sufficient to ensure uniqueness is evident from the example f ( x , u) = u(1 - u) (u- ~) + f(x, u)

in which f is a smooth bounded function which has the following properties: (i) f(x,u)=O on ( - o% L] • [0,89 w [ - L , oo)• [89 1], where L is a positive number, (ii) f (x, O)= f (x, 1)=0 for all xeIR, and (iii) f x > 0 whenever fq=0. It is easily seen that the function f now satisfies the assumptions A 1-4. However, let u be the solution of the problem

u"+u(1-u)(u-89 u ( - oo)=0,

u(+oo)=l

such that u (0)= 89 Then it is plain from the structure of f that the function u (x + e) is a solution of Problem I whenever e e [ - L , L]. To exclude such examples we strengthen assumption A 3: A3*. fx(x, u)>O for all xeIR and u~[0, 1]. In addition one of the following properties holds: (i) there exists an interval I ~ IR such that

fx(X,u)>O

for all x e I

and

ue(0,1);

or

(ii) there exists an interval J c [0, 1] such that f~(x,u)>O

for all x~IR

and

u~J.

Theorem 2. Suppose f satisfies assumptions A 1 and A 3*. Then Problem I has at most one solution with range in [0, 1]. Before proving this theorem, we establish the monotonicity of all possible solutions by a pair of lemmas. L e m m a l . Suppose f satisfies assumption A1 and fx(x,u)>O for all xelR and u~[0, 1]. Let u be a solution of Problem I with range in [0, 1]. Suppose there exists an interval (a, b) such that b < oo, u (x)< u (b) for x ~(a, b) and u' (b)= O. Then a>

--oo.

Proof. First we suppose that u is strictly increasing on some interval terminating at b. Relabelling b = x 2 , we let (xl,x2) (with xl possibly - o o ) be the maximal such open interval. Here and below for i=0, 1, 2, 3 let ui=u(xi). Let y~(u) be the inverse function yl(u(x))=x for xe[xl,x2]. Multiplying (2.1 a) by 2 u' and integrating, we find that U2

{u'(x)}2=2 f f(Yl(S),s)ds, u(x)

xe(xl,x2),

(2.2)

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and therefore

X2 --X

2 S f(yl(s), s)ds

= u (x)

~_

dt,

x e(x:, x2).

(2.3)

t

But

f(yl(s), s) = f ( x 2 , u2) - (x 2 - yl(s))f~ 03a, ~ ) - (u2 - s) f~0)p 8) for some )3~ and g depending on s. Since fx > 0 and fu is bounded, this implies that

f(Yl (s), s) < f (x2, u2) + M(u 2 - s) = - u" (x2) + M(u 2 - s) for some positive constant M. If u" (x2)= 0 it follows that

f (Yl (s), s) <=M (u z - s). Hence, using (2.3), U2

x a - x > = M -89 ~ ( u 2 - t ) - l dt=oo

for

X(~(XI,X2).

u (x)

This is impossible, so u" (X2)::~0. Since u is increasing to x 2 we must have u" (xz)< 0. Hence there is an interval (x2, x3) on which u is strictly decreasing. Suppose it is maximal, so u ' ( x 3 ) = 0 . Then necessarily x 3 < o0, for otherwise (2.1 b) would be violated. Therefore by the above a r g u m e n t we find u" (x3)> 0. Let y2(u) be the inverse function of u(x) on [u3, u2]. Then U2

f(Y2 (s), s)ds = 0.

(2.4)

U3

Suppose ul 0 , for if u ' ( O = 0 the a b o v e a r g u m e n t shows that u " ( ~ ) > 0 and ~=Xx. In view of (2.2) this implies that U2

I U(Ya (s), s) ds = 89 (4)) 2 > 0. U3

However, Yl < Y2 on [u3, Ul) and fx > 0. Hence U2

I12

I f(Y2(S),S) ds>= I f(Yl(S),S) ds>O, U3

u3

which contradicts (2.4). Therefore u 1 > u 3 . It follows that x~ > - o0, for if x 1 -- - 0% then u 1 - - u 3 = 0 , and since fx__>0, we would have u"(x3)= -f(x3,0)__< - f ( - 0% 0) --0, contradicting the fact that u " ( x s ) > 0. We also have u'(x 0--0 and u " ( x j > 0 . Let Xo be the largest value of x u 2 = u (b). Hence a > x o > - o0. This completes the p r o o f in the case considered. Finally consider the case when u is not strictly increasing on any interval terminating at x = b. Then x 2 is the limit from the left of a sequence of points (~n), n = 1, 2 . . . . . in (a, b) where u attains local minima. By continuity u (~,) ~ u (b) as n ~ oo, and by a s s u m p t i o n U(~n)< u(b) for all n__>1. But this cannot occur,

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P. C. FIFE & L.A. PELETIER

because the a b o v e a r g u m e n t shows that u (4, + 1)< u (4,) for all n > 1. This completes the p r o o f of the lemma. L e m m a 2. Suppose f satisfies assumption A 1 and fx(x, u)>O for all xeF, and u e [ 0 , 1]. Let u be a solution of Problem1 with range [0, 1]. Then u ' ( x ) > 0 for all

finite x. Proof. Let u o be any n u m b e r in (0, 1), and let x o the smallest value of x for which u ( x ) = u o. In view of the b o u n d a r y condition at x = - o0, we have x 0 > - o0. If there exist values of x in [Xo, o0) for which u ' ( x ) = 0 , let Xl be the smallest of them. Then u(xl)>U(Xo). The hypotheses of L e m m a 1 are satisfied with a = - o o and b=xl. The conclusion of that l e m m a is that a > - 0 % which contradicts our hypotheses. Therefore u' ( x ) > 0 for x > x o. Since u 0 was arbitrary, we conclude that u' ( x ) > 0 for all finite x. P r o o f of T h e o r e m 2. Suppose there exist two solutions u~, u 2 with u 1 < u 2 on some interval (a, h), where a and b m a y be infinite. By L e m m a 2 it is seen that ul and u 2 are strictly increasing. Suppose the interval is maximal, so that

ul(a)=u2(a)=~, u'l(a)=u2(a),

ul(b)=u2(b)=fl ul(b)>u2(b).

(2.5)

Analogously to (2.2), we have for i = 1, 2, {u'i(b)} 2 - {u;(a)} 2 = - 2~

f(yi(u), u) du,

in which Yl and Y2 are the respective inverse functions of ul and u 2. Subtracting the equation for i = 1 from the one for i = 2, we obtain [{u~ (b)} z - {u'l (b)} 2] - [{u~ (a)} z - {u~ (a)}] z = - 2 ~ { f (Y2 (u), u ) - f(y~ (u), u)}

(2.6)

du.

ag

Since ul(x)y2(u) on (a, fl); thus the right h a n d side of (2.6) is nonnegative. The inequalities (2.5) imply that the left hand side of (2.6) is nonpositive. Therefore, both sides must be zero. This implies that the derivatives of u 1 and u 2 match at x = a and x = b. By the uniqueness property of initial value problems, this can only h a p p e n if a = - oo and b = oo. Thus (a, fl) = (0, 1). Since the right hand side of (2.6) is zero we find that

f j x , u)=-O

for ue(0, 1),

xe(yz(u), yl(u)).

H o w e v e r this contradicts a s s u m p t i o n A 3*. This completes the proof.

3. Stability of a Cline We now investigate the stability of the cline constructed in the previous section. T h r o u g h o u t this section we shall denote it by qS(x). Thus consider the C a u c h y p r o b l e m

ut=uxx+f(x,u), u(x,O)=~,(x),

-oo
t>0

(3.1) (3.2)

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in which ~ is a continuous function which takes on values in [0, 1], and f satisfies the assumptions A 1, A 2, A 3* and A 4. The existence and uniqueness of a classical solution of problem (3.1), (3.2) was established in [6]; since its dependence on ~, will play a central role in what follows, we shall denote this solution by u(x, t; ~). We shall show that for a large class of functions ~, u(x, t; ~) - q~(x)

as t ~ oo,

uniformly with respect to x ~ ( - 0% oo). The main ingredient in the proof of this result is the construction of sub- and supersolutions of equation (3.1). One family of sub- and supersolutions can be obtained very simply, by shifting the function ~b(x) along the x-axis. Thus consider the family of functions

v(x,h)=q6(x +h),

heiR.

Let h > 0. Then, writing x + h = y, we have

..~ v=--vx~+f (x, v)-vt=dPrr + f (y-h, q~) = - f(y, q~(y)) + f ( y - h, ~o(y)) < 0 by assumption A 3*. Therefore v(x, h) is a supersolution of equation (3.1) if h=>0. Similarly v(x, h) is a subsolution of (3.1) if h<0. Finally, because By(x, h)/Sh= ~b'(x+h)>0 on IR,

v(X, hl)
v(x, hO<~
-~
(3.3)

Then u(x,t;~)~c~(x)

as t ~

uniformly with respect to x ~ ( - 0% ~). Proof. It follows from (3.3) and the maximum principle that U(X, t; t) (., hi) ) < u (x, t; ~) =0. However it can be shown by means of an argument similar to one used by ARONSON and WEINBERGER [1] that the functions u(x, t; v (.,hi)) (i = 1, 2) both tend to a solution of Problem I, the convergence being uniform on compact subsets of JR. Since, by Theorem 2, q~ is the only solution of Problem I it follows that u (x, t; ~k)--* q~(x) uniformly on compact subsets of IR.

as t ~ oo

(3.4)

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P.C. FIFE & L.A. PELETIER

Finally let e > 0. By the maximum principle

v(x, hO <=u(x,t; O) <=v(x, h2) for all xelR and t>O. Thus there exists a number A > 0 such that

lu(x,t;@)-49(x)l
for Ixl>A,

t>0.

By (3.4) there exists a number T > 0 such that for t > T

lu(x,t;@)-(o(x)l<~

for I x l < a .

It follows that for t > T Ilu (.,t; ~b)-q~l[
as

x--,

-

oo

and

l-O(x)=O(1-c~(x))

as x - . oo.

We shall now prove a result which demands much less of the function 0. However, in return we need to make two further assumptions about f. Consider the equation f (x, u) = 0 (3.5) Then u(x)=O and u(x)- 1 are solutions of (3.5). A 5. There exist positive constants ~ and N such that

f,(x,O)<-o~ f i x , 1)< -c~

if x < - N if x > + N .

Remark. The function f(x, u) = s (x) u (1 - u), which is of particular importance in population genetics [2, 5, 7], satisfies A1-5, A3* provided sECI(IR), s'>O, s > 0 for large positive x, and s < 0 for large negative x. Moreover for such s(x) the function f(x,u)=s(x)u2(1-u) satisfies A l - 4 , A3*. In the former case, the conclusions of all our theorems hold; in the latter, only those of Theorems 1-3. Define the functions a o (x) = max {u~ (0, 1] : f(x, s) < 0 for s~ (0, u)}, a l ( x ) = m i n {ue[0, 1): f(x,s)>O for se(u, 1)}. By A5, ao(x) is defined for x < - N and al(x) is defined for x>N. By A3*, a o and al are nonincreasing, so we can define the limits a + = lira a(x),

a - = lira a(x).

Theorem 4. Let u(x, t; O) be the solution of problem (3.1), (3.2) in which the function f satisfies the assumptions A1, A2, A3*, and A4-6. Suppose that lim i n f O ( x ) > a +, x ~ o O

lim sup ~k(x)
--

ct)

(3.6a, b)

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Then as t o ~ .

Ilu(.,t;~')-4~ll--'0

We shall prove T h e o r e m 4 in several steps. The first and most basic step is the following. L e m m a 3 . Suppose ~k satisfies conditions (3.6). Then there exist numbers z such that

hx, h2, h l < 0 < h

lira inf u(x,t;~)>(9(x+hl)

on IR

limsupu(x,t;~,)
on IR.

t ~

oO

L e m m a 3 is a generalization of a result due to FIFE and McL~oD 1-3]. Proof. To prove the first statement of L e m m a 5, we consider the function z (x, t) = q~(x - r (t)) - q (t). We shall show that the positive functions ~ and q can be chosen so that the function

v(x,t)={;(x,t)

when z(x,t)>O when z(x, t)
is a subsolution of equation (3.1) with the properties (i) v(x,O)a + + e

if x > L 1.

We now choose q(0) such that 1 -q(O)=a + +e.

Next, let ~ be the root of the equation ~b(x)=q(0) and increase L~, if necessary, so that La > ft. Because q(0)~(0, 1) a root exists, and because ~b'> 0 there exists only one. Then z (x, 0) = ~b(x - r (0)) - q (0) =<0 on ( - ~ , ff + r (0)). Hence if we choose ~ (0) > 0 such that X+~(0)=L~, then z(x, 0 ) < 0 on ( - ~ , L1]. However, on (L 1, ~), ~b(x) > a + + e = l

-q(O)>(a(x-~(O))-q(O)=z(x,O).

Therefore, for this choice of q(0) and ~(0),

v(x,O)
o n IR.

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It is clear that v---0 is a solution of (3.1). Moreover, because zx>0 the function convex ridge. Thus v is a subsolution of (3.1) if z is. We have

v has a

5s z=zx~+ f (x, z)-z,=4)" + f (x, 4)-q)+4)' ~' +q' = - f ( x - ~, 4))+f(x, 4)-q)+ 4)' ~' +q' > - f ( x , 4))+f(x, 4)-q)+4)' ~'+q'

(3.7)

provided ~ > 0 ; here we have used assumption A 3*. Define the function

q~(x, u~

q)=S[f(x,, u - q ) - f ( x , u)]/q, [ - f , ( x , u),

q >0 q=0.

is a continuous function for q > 0. Because of A 5 and the definition of q (0), there exists a number L 2 > N such that 9 (x, 1, q)> 0 for 0 < q < q(0) and x > L z. Also, ~ ( x , l , 0 ) = - f , ( x , 1)>0. Hence there exists a constant ~1>0 such that rb(x, 1, q ) > 2 e 1 for O0 such that ~(x,u,q)>a 1 for x > L z and 1-61=
O<=q<=q(O).

We then have

f(x, u - q ) - f ( x , u) > ~1 q"

(3.8)

We now choose 61 so small that x > L 2 if 1 - 6 1 < 4 ) ( x - ~ ( t ) ) < 1. To do this we set 1 - 6* = 4)(L2 -

(0)).

Then if 61 < 6* we have 4)(x- r

> 1 - 6 1 > 1 - 6 * =4)(L 2 - ~ (0));

hence, because 4)'> 0, it follows that

x-~(t)>L2-r Consequently x > L2 + ~ ( t ) - ~ ( O ) > L 2

if we can choose ~ so that 4'> 0. Thus if 1 - 61 < 4) (x - ~ (t)) < 1 we obtain

f(x, 4 ) - q ) - f ( x , 4))>=alq,

O<=q<=q(O).

If ~' >0, (3.7) now becomes

~ z>=al q+q'. By a similar argument, there exist positive numbers N1, 62 and ~2 such that

f (x, u - q ) - f (x, u)>=otzq whenever O 0 then

5s z>=a2 q+q'.

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Thus, setting ~ = m i n {~, ~2} and defining q (t) = q (0) e- ",

(3.9)

we obtain C~z>O, if q)(x-4(t))e(O, 6 2 ) w ( 1 - 5 ~, 1)and z > 0 . Next, let 62 < ~b( x - 4 ( 0 ) < 1 - 6~. Then there exist positive constants k and fl such that f,(x, (o)> - k on IR and ck'(x-4(t))>fl. Substitution into (3.7) yields

Y z> f l 4 ' - ( a + k ) q. Hence, if we choose c~+k 4 (t) = ~ (0) + ~ - q (0) (1 - e- ~t),

(3.1 O)

we obtain ~ z > 0 when z > O. W e note that C > 0 and that a+k 4(00)= ~ (0) + ~ f i - q(0). Thus, with the functions q and 4 given by (3.9) and (3.10), z(x, t) is a subsolution of (3.1) whenever z>0. It follows that v(x, t) is a subsolution of (3.1). Since v(x, 0)< ~(x) it follows by the maximum principle that

u(x,t;~J)>v(x,t),

xelR, t > 0 ;

hence, in particular, lim inf u(x, t; ~J)__>lim v(x, t)= ~b( x - ~(oo)). t~OO

t~oO

The second estimate can be proved in an identical manner. This completes the proof of Lemma 3. We shall also need local stability in the space C (IR). Lemma 4. For each ee(O, %), where eo is small enough, there exists a constant 6 > 0 such that if It~b-~b[I <8, then [[u(.,t; ~)-~b[[ O. Proof. Let /~ be a small positive number, and let [[~J-~b[[
< o~(/~).

This proves the Lemma. Lemma 5. Let ~ be a positive number. Then the set 7~-- {u(., t; r

C(IR): r < t < oo1

is compact in C(IR). Proof. Because O<=u(x,t;~,)<=l for all xeIR and t_>_0, the first derivative of u with respect to x, which we denote by ux(x, t; ~J), is uniformly bounded on x [z, oo), see [9]. Hence the set y~ is bounded and equicontinuous in C(N). Let Uk(X, t; ~) be the restriction of u(x, t; ~J) to the interval [ - k , k ] . Then by the

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P.C. FIFE& L.A. PELETIER

Arz61a-Ascoli theorem there exists, for each k > l , a subsequence llk(X , tk.; I[I) which converges uniformly on [ - k , k]. By a diagonalization process we can extract a sequence u (x, t.; qJ) which converges uniformly on each interval [ - k , k]. We denote the limit of this sequence by v(x). Let e > 0. Then by Lemma 3 there exist positive numbers L and T~ such that if t . > T~ lu(x,t#,ql)-v(x)l<~ for Ixl>Z. But by the construction of the sequence {t,} there exists a number T2 > 0 such that if t, > T2 then

lu(x,t.;ql)-v(x)l
for I x l < L .

Hence if t. > max { T1, Tz} we have

]u(x,t.,~k)-v(x)]<~

for x ~ .

Thus 7, is compact. We are now able to prove Theorem 4. Proof of Theorem 4. For convenience we define a family of bounded operators T(t): C(~)--* C(F,) by

T(t) ~k = u (., t; qJ). It has been shown in [6] that {T(t): t>0} is a semigroup on C(IR), that is (i) T(0) = identity, and (ii) T(s) T(t) = T(s + t) for all s, t > 0. Moreover it was shown in [6] that for any finite z > 0 there exists a constant C(z) such that [1T(z) ~1 - T(z) ~b2 l[ < C (z)l] ~1 - qJ211

(3.11)

for any ~Ol,~b2 satisfying 0<~O~(x)l. By Lemma5 there exists a sequence {t.} and a function t/eC(IR)such that

!imoI1Z(t,) qJ - tl II= 0.

(3.12)

We shall show that t/= 4. Suppose to the contrary that r/4=q5 and that II~t-q~ll--p>0. Choose e = m i n {eo, 89p}. By Lemma4 it is possible to find a value 6(e)>0 such that if I1~'-q~ll<& then IlT(t)O-c~ll 0 . (3.13) By Lemma3 there exist numbers h1< 0 and h z > 0 such that

dp(x+hz)
xelR,

and hence by Theorem 3 there exists a number s > 0 such that

]lr(t)rl-qbl[< 89

if t > s .

In view of (3.12) there exists an integer k > 1 such that

]lZ(t.)O-rlll< 89

if n > k ,

in which C(s) is the constant appearing in (3.11). Invoking (3.11) we find

IIT(tk + s) O -- r(s) rIII <896.

Population Genetics

105

Thus IIT(tk + s) ~b- 4)11< I1T(tk+ s) ~ -- T(s) t11]+ t[T(s) tl - 4)II < ~, and therefore in view of (3.13)

[]T(tk+s+t)~,--4)ll
for t_-->0.

(3.14)

Now choose E___1 such that te>=tk+S. Then by (3.14) IIT(te) ~ - 4)II < and P = 114)-t/ll --< 114)- T(t,) ~Ol[+ IIT(te) ~ -t/ll

<__e+89189 where we have used the facts that C(s)> 1 and 6
p
t~O0

=0.

This completes the proof of the theorem. We conclude this section with an estimate of the rate of convergence ofu (.,t; r towards 4). For convenience we make an additional smoothness assumption on f. A l*. f has continuous derivatives fx, fu and f,u, bounded uniformly in

~ x [0, 13. Theorem 5. Let u be the solution of problem (3.1), (3.2) in which f satisfies assumptions A 1", A 2, A 3*, A4-6, and let 4) be the solution of problem (2.1 a, b). Then there exist positive constants 6, p and K such that if 11~-4)ll _-<& then

Ilu(.,t;~)-4)ll
t>O.

(3.16)

Proof. We write

u(x, t; r

4)(x)+ v(x, t).

Then v satisfies the equation

v t = L v+ h(x, v), where

L v =- Vxx +f.(x, 4)(x)) v and

h (x, v) - f (x, 4) (x) + v ) - f (x, 4) (x))- f,(x, 4) (x)) v. Note that sup Ih(x, v)l < mlPvll 2, where M = 1 sup {f~.(x, u): xelR, ue[0, 1]}.

(3.17)

106

P.C. FIFE& L.A. PELETIER

To begin with, we consider the linear eigenvalue problem (L - 2) y = 0 in the space of functions C(~.). For convenience we write

q (x) =f, (X, 4)(x)). In view of the monotonicity of f(x, u) the limits q+ =limq(x),

q- =xfimooq(x)

exist, and by Assumption A 6 q* = m a x {q+, q-} < - ~ < 0 . By the general theory of Sturm-Liouville eigenvalue problems [11], the spectrum a(L) of L consists of a continuum in the interval ( - or, q*], and possibly a discrete part in the interval (q*,C/), where ~t=sup{q(x):xeN}. We shall show that the entire spectrum is on the negative half-line and bounded away from zero. Since q* <0, clearly the continuum part is so bounded. If there is also a discrete part, let 2 o be the largest number in it. We assert that 2 o <0. Let vo be the eigenfunction corresponding to 2 o. Because 2 0 is a discrete eigenvalue it is clear that voeL2(IR), and because 2 0 is the largest eigenvalue of L, vo (x) does not vanish anywhere on IR. We shall take vo > 0 and [Ivo II = 1. Thus we have L vo - 2 0 vo = 0 .

(3.18)

If we differentiate the equation for 4) we obtain

L 4)' +fx(x, 4)(x))=0.

(3.19)

Hence, if we multiply (3.18) by 4)' and (3.19) by Vo, then subtract and integrate over IR, we obtain

4),(x)vo(x)ax=--oO

fx(x,

vo( ) dx

~oo

Since 4)'>0, % > 0 , and fx>=0 (but fx~0), it follows that ~.o<0. We are now in position to prove (3.16). We shall do this by means of the maximum principle, using an appropriate comparison function. Let 2 0 be the largest point in the spectrum. Choose a number 2e(2o, 0), and a function g~L2(IR) such that g __>0 on IR, and g ~ 0. Then the equation (L-,~) w = - g

has a unique solution weHl(lR) such that w(x)>0 for x e ( - o v , oo), (see the Appendix). We adjust the magnitude of g so that Ilwll =1. Now consider the function

z(x, t)=fl{w(x)+ v} e -u'

Population Genetics

107

in which/~, 7 and p are positive constants which we shall determine later. By a simple computation using (3.17) we find Jlz=-Lz+h(x, z)-z t <=fie-ut [(J. +/A) w+ 7 {q(x)+ p} + M fl(1 +7)2].

(i) Let q * < q o < 0 . Then there exists a number ~ ( 0 , oo) such that q(x)<=qo for Ixl> ~. Hence ./g z<=~e -~'t [(2 +p) w+ 7(qo + l~)+ M ~(1 +7)23

for Ixl>r We now choose 0 < p < m i n { - 2 , - % } . Assuming that 7 is known (it will be specified below), we select/~ so that y(qo +#)+M/~(1 + 7 ) 5 = 0 and hence ~#z=<0,

Ixl__>r

(ii) Let Ixl__<~. Since w(x)>O on IR, and weC(IR), we have

m = m i n {w(x): Ixl_-0. Therefore Jgz<=fle -u' [(2 + p) m+y(E/+ #)+ M//(1 +7) 2]

=/~ e - " ' E(2 + p) m + 7(g/- qo)] ; hence, if we choose 2+p y= ---m cl-qo we have J/[z_<0

for [x[__<~ and

t__>0.

Thus for the above choice of fl, ?2 and 12 the function z is a supersolution of the equation ~ / v -- 0. Let sup v(x, 0)<6, where 6 = ~ y = - (qo + P) 7Z/M( 1 + 7)2. Then

v(x,O)
xeIR, t>_O.

In a similar manner we can show that if inf v(x, 0)> - 6 , then v(x,t)>=-z(x,t),

xelR, t>=O.

P.C. FIFE & L.A. PELETIER

108 Hence if I1~-r

= IIv (.,0)11 5 ~ then Ilu(.,t; 0)-r


t>=0,

where K = {1 +(1/7)} 6. Appendix

We consider the equation (A1)

(2-L) u=g, in which 2 e F , , g e L 2 ( ~ ) , and

Lu=u" +q(x) u, q being given by q(x)=f,(x, ~(x)). It follows from the assumptions on f that q e C(IR) a L~176 (~,,). It was shown in section 3 that the spectrum a(L) of L as an o p e r a t o r from L2(IR) into itself is contained in the interval ( - 0 % 20] , where 20 < 0 . Thus if 2 > 2 0 , then for any geL2(IR) there exists a unique ueHa(lR) * such that equation (A 1) is satisfied. Proposition. Suppose g(x)>__O a.e. on ~, g(x)~O, and 2 > 2 o. Then u ( x ) > 0 for all x e ( - 00, 0o). Proof. Let az(., .) denote the Dirichlet form associated with the o p e r a t o r 2 - L: az(q~ , ff)=~ [q~' ~ ' + ( 2 - q ) q ~ ~ ] T h e u p p e r b o u n d on

dx,

O, ~eH'(~..).

a(L) implies the coercivity inequality

aa(v, v)>=cllvl102 (e=;~-2o>0)

(A2)

for all v~HI(IR). First, we show that u > 0. If this were not true, there would be an open interval I on which u < 0, with u = 0 at any finite endpoint of I. Let

u_(x)={;Ix), Then u - ~ H I ( ~ ) ,

x~I xr

and (A2) yields

c IIu - II0 2 < a z ( u - , u - ) = ~ [(u;)2 + (2 - q) U 2] = I U- [-(2 -- L) u - ] = ~ u - g < 0. I

I

I

This contradiction implies u > 0. It remains to show that u ( x ) > 0 for all x ~ ( - 0 o , ~ ) . Define k = m a x { q ( x ) - 2 , 0}. Then

u" +(q(x)-2-k)u=u" +(q(x)-2)u-ku = -g(x)-ku O, k > 0 and u > 0. Because q ( x ) - 2 - k < 0 the result follows from the strong m a x i m u m principle. This work was supported by the National Science Foundation under Grant MPS-74-06835-A01. 9 H~(F.) denotes the completion with respect to the norm IIr lit = (IIr hi02+ I}r r 9 C1(~.) such that ~b~ L2(R), ~b'e L2(F.). Here II. Iio denotes the norm in L2(F.).

~ of the functions

Population Genetics

109

References 1. ARONSON,D.G. & H. F. WEINBERGER,Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. Lecture Notes in Mathematics 446, 5-49, Springer, New York, 1975. 2. COSL~Y,C., An application of Wazewski's method to a nonlinear boundary value problem which arises in population genetics, Univ, of Wisconsin Math. Research Center Tech. Summary Report No. 1444, 1975. 3. FIFE, P. C. & J. B. McL~oD, The approach of solutions of nonlinear diffusion equations to travelling front solutions. To appear. 4. FISHER,R.A., The advance of advantageous genes, Ann. of Eugenics 7, 355-369 (1937). 5. FLEMING,W. H., A selection-migration model in population genetics, J. Math. Biology 2, 219-133 (1975). 6. KOLMOGOROFF,A., ][. PETROVSKY& N. PISCOUNOFF,Etude de l'6quation de la diffusion avec croissance de la quantit6 de mati&e et son application ~ un probleme biologique, Bull. Univ. Moskou, Ser. Internat., Sec. A, 1 (1937) 6, 1-25. 7. NAGYLAKI,T., Conditions for the existence of clines, Genetics 80, 595-615 (1975). g. NAGYLAKt,T., Clines with variable migration, to appear in Genetics. 9. OLEINIK,O.A. &S.N. KRU~KOV, Quasilinear second order parabolic equations with many independent variables. Russian Math. Surveys, 16, 105-146 (1961). 10. SATTINGER,D.H., Topics in stability and bifurcation theory, Lecture Notes in Mathematics 309, Springer, New York, 1973. 11. TITCHMAI~sH,E.C., Eigenfunction expansions, Part II, Chapter XVI, Oxford University Press London, 1958. Department of Mathematics University of Arizona Tucson, and Department of Mathematics Delft University of Technology Delft

(Received August 25, 1976)