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Nonlinear Dynamics, Psychology, and Life Sciences [ndpl] ph104-ndpl-368840 September 27, 2002 13:46 Style file version Oct 23, 2000

Nonlinear Dynamics, Psychology, and Life Sciences, Vol. 7, No. 1, January 2003

Clines: A Reductionist Model

S. Harris1

A piecewise linear model is used to provide a caricature of the nonlinear equa-tion describing genetic variation due to migration and local natural selectionin an inhomogeneous bounded habitat. The conditions for which nontrivialspatially-dependent steady state solutions exist are analytically determined to-gether with these solutions for three distinct scenarios. For the usual case ofno flux (Neumann) boundary conditions, explicit solutions require additionalinformation to fix the genetic frequency within the habitat. This difficulty canbe bypassed in two of the scenarios considered (this is an artifact of the model),but in the remaining scenario it is necessary to take into account the connectionwith the initial data to explicitly determine the steady state frequency. Somenumerical examples are considered for each of the scenarios to illustrate theanalytical results that are the primary focus of the work presented here.

KEY WORDS: clines; genetic variation; model solutions.

INTRODUCTION

Gene frequencies in many populations often vary in space, formingpatterns that are referred to as clines; see, e.g. Roughgarden (1996). If n(x, t)is the frequency of the A allele at location x and time t (so that for a singlelocus with two alleles 1− n(x, t) is the frequency of the a allele), then thedispersion of n is described by a reaction-diffusion equation

∂n/∂t = D∂2n/∂x2 + sg(x) n(1− n)(1+ h[1− 2n]) (1)

originally formulated by Fisher (1937). Here s and h are, respectively, mea-sures of the strength of selection and dominance and g(x) describes theselection pressure. The latter quantity changes sign at least once over the

1Correspondence should be directed to S. Harris, College of Engineering and Applied Sciences,SUNY, Stony Brook, New York 11794; e-mail: [email protected].

1

1090-0578/03/0100-0001/0 C© 2003 Human Sciences Press, Inc.

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spatial domain, with the simplest case being that of a step-like behavior froma negative to positive constant, i.e. |g(x)| = g for all x. The diffusion co-efficient D is twice the genetic variance per generation due to migration. Forh = 0, g(x) = g for all values of x Eq. 1 reduces to the much-studied Fisherequation (Fisher, 1937; Kolmogorov, Petrovsky, & Piscounov, 1937; Murray,1989). Here we will consider the case where overdominance does not occurso that 0 < |h| < 1 and g(x) = g sign(x).

Not surprisingly, in a finite spatial domain where Neumann boundaryconditions are most commonly considered so that ∂n/∂x = 0 at the bound-aries, analytical solutions of Eq. 1 have not been found. Instead, a consider-able amount of effort has been focused on characterizing the stability prop-erties of steady state solutions (Flemming, 1975; Nagylaki, 1975) for whichthe linearized version of Eq. 1 provides the starting point. Related problemsinvolving pattern formation in spatially heterogeneous environments havealso been extensively studied in the ecology literature (e.g., Nisbet & Gurney,1982; Ranta, Kaitala, & Lundberg, 1997; Medvinsky, Petrovskii, Tikonova,Venturino, & Malchow, 2001).

The purpose of the work reported here is to determine the solutions of acaricature of Eq. 1 for some typical scenarios imposed by the system param-eters. The motivation for this is the expectation that, at least qualitatively,the true solutions of Eq. 1 are insensitive to the details of the nonlinear“reaction” term and that this can be replaced by a piecewise linear func-tion that retains the essential mathematical feature of the latter which isits single zero for |h| < 1 as well as continuing to describe the phenomeno-logical conditions described by that equation. This idea is not new; McKean(1970) has successfully used such a caricature of the Nagumo-Fitzhugh model(Fitzhugh, 1961; Nagumo, Aximoto, & Yoshizawa, 1962), itself a model forthe Hodgkin-Huxley (1939) theory of neural communication, to study thequalitative properties of that equation which closely resembles Eq. 1. Animportant difference is that in that the latter context the space variable ex-tends over an infinite range (Haldane, 1948), unlike the situation consideredhere where the range is finite.

It should be emphasized from the outset that the simplification of Eq. 1described above is quite different from the straightforward linearization thatis the basis of the stability analysis cited above although, in some situations(scenario I below), the analysis is very similar. The essential difference isthat in the latter case the linearization is about an unknown steady state so-lution whereas here our objective is to determine these solutions and theirqualitative behavior relative to the system parameters. We do this for threedistinct scenarios. First, we consider the case (Scenario I) where the “reac-tion” term has the same functional dependence on n over the full spatialdomain. Next, (Scenario II) we consider solutions for which this functionaldependence differs according to whether x is positive or negative. Finally,

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Clines: A Reductionist Model 3

a more complicated situation (Scenario III) in which the functional depen-dence changes in the interior of the sub-domain for which x > 0 is studiedfollowed by a discussion of all the results obtained. Many other scenariosare possible, but the basic analysis of these will not differ appreciably fromthose considered here.

Although our primary interest is in determining the steady state clinesand how they qualitatively depend on the system parameters, we also showthat in certain cases we cannot avoid also considering the full initial valueproblem if we are to obtain explicit results. This is a consequence of theNeumann boundary conditions which do not allow us to “calibrate” thefrequency at some domain location but rather contain a constant that mustbe determined from the initial condition or some other information aboutthe system.

Scenario I: n(x) < 1/2 Throughout the Domain

We consider Eq. 1 in the linear habitat −1 < x < L with g(x) = −1,x < 0, g(x) = α, x > 0 and ∂n/∂x = 0 for x = 0, L. The underlying premisefor what follows is that solutions of Eq. 1 do not sensitively depend on thedetails of the “reaction” term n(1− n)(1+ h[1− 2n]). Then, for the case ofno overdominance, so that |h| < 1, we can replace this term by the piecewiselinear function n(1+ h)sg(x) when n < 1

2 and by (1− n)(1+ h)sg(x) whenn > 1

2 (McKean, 1970). For the first scenario we consider, n < 12 throughout

the habitat, Eq. 1 in the steady state is replaced by

0 = ∂2n/∂x2 + K2+n, 0 < x < L (2a)

0 = ∂2n/∂x2 − K2−n, −1 < x < 0 (2b)

with ∂n/∂x = 0 at x = −1, L, and where K2+ ≡ D−1sα(1+ h), K2

− ≡D−1s(1+ h). Two additional conditions at x = 0, n(0−) = n(0+) and∂n/∂x|0+ = ∂n/∂x|0− are required to insure continuity at x = 0. Unfortu-nately, the boundary and continuity conditions alone do not provide suf-ficient information to allow us to uniquely determine the solution to thisNeumann problem and we will have to consider the evolution from the ini-tial state to do so. Before we proceed we note that the solution to Eq. 2 isconsistent with that for Scenario I* where n > 1

2 throughout the habitat asis the case here for the a allele frequency n∗ = 1− n, as we discuss later.

Continuing now with Eqs. 2, the solutions to these equations satisfyingthe boundary and continuity conditions are

n = n(0)[tan K+L sin K+x + cos K+x], x > 0 (3a)

n = n(0)[1+ e−2K− ]−1[e−K−(2+x) + eK−x], x < 0 (3b)

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subject to the following conditions

LK+ < π/2 (4a)

K+ tan K+L= K− tanh K− (4b)

The first condition insures that n > 0 everywhere while the second is thecontinuity requirement at x = 0. It is also necessary that

n(0) < (1/2)cos K+L (4c)

since n(x) assumes its maximum at x = Land this must be less than 12 . Eq. 4b

specifies the unique value of L for which a steady state cline is establishedfor fixed values of the Ki . Some numerical values are presented in the in theconcluding section illustrating the qualitative implications of this relation-ship.

Despite the fact that the conditions for a steady state cline to exist areuniquely determined independent of the initial condition, not surprisinglyn(x) itself depends on n(x, 0) through n(0). While there are an infinite num-ber of initial conditions that can evolve to a single value of n(0), most ofthese are unlikely to occur in nature. To proceed we will consider the familyof initial conditions n(x, 0) = n+ < 1

2 , x > 0, n(x, 0) = n− < 12 , x < 0. In the

analysis that follows we must assume that n remains less than one half forall time, e.g. that n+ ∼= n− ¿ 1

2 . This might describe a situation where pre-vailing conditions suddenly change in one part of a habitat, disrupting anexisting state of equilibrium. A more detailed analysis of the initial valueproblem than follows below would be desirable, but is beyond the scope ofthe present paper.

Proceeding, we must solve the time-dependent equivalent of Eqs. 2; tosimplify the notation we will consider units for which D= 1. Introducing theLaplace transform of n as n, for x > 0 we have

sn− n+ = ∂2n/∂x2 + K2+ n (5)

with a similar equation for x < 0. From the boundary and continuity condi-tions it then follows that, e.g. for x > 0

n = B [tan q+x + cos q+x]+ n+(s − K 2

+)−1 (6)

where qi = (K 2i − (i)s)1/2, i = +, − and

B = [n+(s − K 2+)−1 − n−

(s + K 2

−)−1]

q+tan q+L[q+(1+ exp− 2q−)

× tan q+L− q−(1− e − 2q−)]−1 (7)

Since n(0) = B+ n+(s − K 2+)−1 it then follows from the limiting properties

of the Laplace transform that n(0) can be found by expanding B for s ¿ Ki .The condition for which n(0) does not go to zero is that the O(s0) term in

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the denominator on the right hand side of Eq. 7 vanish, which is the casewhen Eq. 4b is satisfied. Then n(0) is given as

n(0) = Lims→0sB(s)

= (n−K−2− + n+K−2

+)

tan K+L[(1/2K+)(1+ exp− 2K−) tan K+L

+ (K+/K−)e−2K − tan K+L+ K+(1+ e−2K−)(L/2K+)

× (1+ tan2 K+L)+ (1/2K−)(1− e−2K−)− e−2K− ]−1

< (1/2) cos K+L (8)

Eq. 8 serves to both specify n(0) and also provide a further condition beyondEqs. 4 on the allowable values of the Ki and L for which a cline can exist.Some specific numerical results together with a discussion will be given inthe concluding section.

Scenario II: n(x > 0) > 1/2 > n(x < 0)

This scenario requires that Eq. 2a be replaced by

0 = ∂2n/∂x2 + K2+(1− n), 0 < x < L (9)

For x < 0 Eq. 2b remains valid. Although the boundary conditions againleave n(0) arbitrary, the continuity condition here requires that n(0) = 1

2and we then find

n = 1− 12

[exp K+(2L− x)+ exp K+x][exp 2K+L+ 1]−1, 0 < x < L

(10a)

n = 12

[exp−K−(2+ x)+ exp K−x][exp−K− + 1]−1, −1 < x < 0 (10b)

subject to the second continuity condition

K+ tanh K+L= K− tanh K− (11)

The symmetric cline specified by K+ = K−, L= 1 is but one of the manyallowable states that will satisfy Eq. 11.

The transparency of the results for this scenario make it worthwhileto consider the related cases where the boundary conditions are differentthan those specified in Eqs. 2. If the A allele is lethal for x < −1, thenthe boundary condition in Eq. 2b is replaced by n(−1) = 0 and Eq. 10b isreplaced by

n = 12

[exp K−x − exp− K−(2+ x)][1− exp−2K−]−1 (12)

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with the continuity condition now

K+ tanh K+L= K− coth K− (13)

In the case where the a allele is lethal for x > L Eq. 10a is replaced by

n = 12

[exp K+x − exp K+(2L− x)][1− exp 2K+L]−1 (14)

with the continuity condition

K+ coth K+L= K− tanh K− (15)

It is clear that n will have its maximum value, n(L) = 1, in the latter casehere; in the other two cases the maximum also occurs at L and when K+L isfinite this is given by 1

2 < n(L) < 1− [exp 2K+L+ 1]−1expK+L< 1. Also,the minimum value n(−1) = 0 occurs in the second case considered in thissection while in the other two cases it is given by the inequality 0 < n(−1) <2 cosh K−(1+ exp−2K−)−1 < 1

2 for K− finite. The continuity conditions im-posed by Eqs. 11 and 13 imply that for fixed K− there is a critical value ofK+ below which no clines can exist when n(0) = 1

2 regardless of the valueof L. In the first case Eq. 11 requires that K+ > K− tanh K− while for thesecond case Eq. 13 imposes the condition K+ > K− coth K−. In the last caseconsidered here, Eq. 15 requires that K+ must be smaller than the criticalvalue given by K+ < K−/ coth K−. In figure 1 we illustrate these conclusionsby comparing the allowable values of K+, L when K− = 1 for each of thethree cases discussed above.

Scenario III: n(x1) = 1/2 < n(L), x1 > 0

In this scenario n(x) is both greater and less than 12 in that part of the

habitat where g(x) = α. Then Eq. 9 applies only for x1 < x < L, and Eq. 2amust be used for 0 < x < x1; the boundary condition at Lnow applies in theformer case and continuity conditions must be satisfied at both x = 0 andx = x1. Equation 2b still describes the region −1 < x < 0.

Since n(x1) = 12 we have

n = 1− 12

[exp K+(2L− x)+ exp K+x]

× [exp K+ + exp K+(2L− x1)]−1, x1 < x < L (16)

n = (1/sin x1 K+)[1/2 sin xK+ + n(0)(cos xK+ sin x1 K+

− cos x1 K+ sin xK+)], 0 < x < x1 (17)

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The inequality x1 K+ < π/2 follows from Eq. 18 since we require n(x) < 12

for x < x1 and the continuity condition at x1 imposes the further condition

tanh K+(L− x1) = (1− β) cot x1 K+ (18)

where n(0) ≡ (β/2) cos x1 K+, 0 < β < 1.A final condition is imposed by thecontinuity condition at x = 0,

βK− cos x1 K+ sin x1 K+ tanh K− = K+(1− β cos2 x1 K+) (19)

The situation here is different than in the first scenario where we did notrequire n(0) to find the allowable values of K+, K−, L for a cline to exist.However, because n(0) appears here in the continuity conditions it must bedetermined for given values of LK+, x1 K+ from Eq. 19. Then Eq. 20 canbe used with K− given to first determine the allowable values of K+ andthen the accompanying values of L and x1. In each case the inequalitiesx1 K+ < π/2, n(0) < 1

2 must also be satisfied. The reason we can again avoiddetermining n(0) from the initial condition is that the continuity condition atx1 fixes the values of n uniquely for the allowable values of LK+, x1 K+, K−.Some numerical results for these quantities are shown in the next section.

DISCUSSION

We begin by considering Scenario I in which the a allele is in theminority throughout the habitat. Allowable values of K+, K−, L are pre-scribed by Eqs. 4 subject to restrictions on the initial conditions (Fig. 1).For the case where the latter are constant Eq. 8 must also be satisfied forthese values of the system parameters. In Fig. 2 we show L as a functionof K+ for K− = 1

2 , 1, 2. Each of the cline states indicated is possible foran infinite, but restricted, number of initial conditions. As an example wetake n− = n+ = 0.050 so that it is reasonable to expect that n(x, t) < 1

2 for

Fig. 1. Allowable values for K+, L when K− = 1for Scenario II with different boundary conditions:∂n/∂x = 0, x = −1, L(- - - - - -); ∂n/∂x = 0, x = L,n = 0, x = −1(·········); ∂n/∂x = 0, x = −1, n = 0,x = L(——).

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Fig. 2. Allowable Scenario II values for K+, L as determinedfrom Eq. 4 when K− = 1

2 (- - - - - -), 1(·········), 2(——).

all x, t if Limt→∞n(L, t) < 12 and therefore Eq. 8 is valid. Then, e.g. for

K− = K+ = 1 we have from Eq. 4b that L= 0.650 and from Eq. 8 it thenfollows that n(0) = 0.054 < 1

2 cos K+L= 0.398 so that all the necessary con-ditions are satisfied and the steady state cline is then completely defined sub-ject to the above mild caveat. The conclusion here that analysis of the steadystate problem alone is not sufficient to explicitly determine n(x) seems not tohave been previously noted. The different outcome seen here in Scenarios IIand III is an artifact of the model we are using; in general additional infor-mation will be required to supplement Neumann boundary conditions if themore difficult analysis of the initial value problem is to be avoided.

Some results for Scenario III are shown in Fig. 3 where we show K+x1

as a function of K+L for n(0) = 0.25. The inequality constraints n(0) < 12 ,

Fig. 3. Allowable Scenario III values for K+x1, K+L whenn(0) = 0.25 as determined from Eq. 18.

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Fig. 4. Allowable Scenario III values of K+, L when n(0) =0.25 as determined from Eqs. 18 and 19 when K− =12 (- - - - - -), 1(·········), 2(——).

K+x1 < π/2 limit the allowable values of these quantities rather severelyas indicated in Fig. 3. For larger values of K+L the gradient will be muchsteeper, resulting in a correspondingly much smaller value of K+x1 requiredto attain the specified frequency at x = 0. The individual values of K+, x1

and L now follow from Eq. 20 and in Fig. 4 we show L as a function of K+ asdetermined from Eqs. 19 and 20 for K− = 1

2 , 1, 2 with n(0) = 0.25 again. Ingeneral, for the Scenario III clines higher values of K− require higher valuesof K+ with correspondingly smaller values of L. Note the difference seenhere with the results shown in Fig. 1 and 2 where allowable cline states forfixed L can exist for different values of K−, K+.

Figure 5 shows three separate clines for (K+, K−, L) = (1, 1, 0.65),(1, 1, 0.816), (1, 0.93, 0.65) so that only K− or L is different in each case.Only x > 0 is shown since the solutions are identical except for a scale factorfor x < 0. The first of these is a scenario I cline (see Fig. 2) and we havearbitrarily chosen n(0) = 0.25 < 1/2 cos K+L. In the latter two Scenario IIIcases the value of n(0) is determined from the choice of environmental pa-rameters. We cannot specify the initial conditions that evolve to these states.Therefore it is not correct to conclude that the initial conditions that evolvedto the (1, 1, 0.65) cline would have evolved to one of the Scenario III clinesshown if K− or Lwere the values shown for the latter. The solution of the ini-tial value problem for a Scenario III cline is a Stefan problem (Meirmanov,1992) since x1 must be treated as a function of time and its solution wouldpresent a formidable problem. Our results here for both Scenario II and IIItherefore are limited to a description of the equilibrium clines and theirqualitative dependence on the environmental parameters, consistent withthe current state of interest (Roughgarden, 1996).

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Fig. 5. Clines for Scenario I with K+ = K− = 1, L=0.65, n(0) = 0.25, and for Scenario III with K+ = K− = 1,L= 0.816, n(0) = 0.466 and also K+ = 1, K− = 0.93, L=0.65 n(0) = 0.448.

The results found here can be reinterpreted in terms of the a allelefrequency n∗ = 1− n, in which case K+ → −K−, K− → −K+ so that, e.g.for n∗ > 1/2 throughout the habitat

0 = ∂2n∗/∂x2 + K+(1− n∗), −1 < x < 0 (20)

0 = ∂2n∗/∂x2 − K−(1− n∗), 0 < x < L (21)

It is easily verified that the solutions of these equations are consistent withour earlier solutions for n.

In light of the considerable interest in the stability properties of solutionsto Eq. 1 (Flemming, 1975; Nagylaki, 1975; Roughgarden, 1996) it seems bothappropriate and necessary to include a brief comment on this in the contextof the work presented here. The linear model that we have based our resultson may be expected to mimic the behavior of these solutions with regard tomost of their qualitative features. However, it cannot be used as a basis foran analysis of their stability. In this sense the results found here complimentthe earlier work cited above. In our Scenario I, if the steady state solution isslightly perturbed so that, e.g. at t = 0 n→ n+ δn0 with δn0 constant, thenat long times the perturbation δn will satisfy Eqs. 2 and the steady state cline,which depends on the initial conditions, will be slightly displaced, i.e. thatcline is metastable. In Scenario II (and III) δn(t) initially satisfies a differentequation than n in all (part) of the habitat where x > 0. Then, following theargument given previously, since Eq. 4b is not satisfied in this scenario it isreasonable to expect that δn must vanish in the steady state. In this case thesteady state is then stable. However, these conclusions regarding stabilitycannot be expected to carry over to the nonlinear Eq. 1.

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REFERENCES

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croissance de la quantitie de materie et son application a un problem biologique, MoscowUniv. Bull. Math. 1, 1–25. English translation in Pelce (Ed) (1988) Dynamics of CurvedFronts (pp. 105–130). San Diego, Academic Press.

McKean, H (1970). Nagumo’s equation. Adv. in Math. 4, 209–223.Medvinsky, A., Petrovskii, S., Tikonova, I., Venturino, E. and Malchow, H. (2001). Chaos and

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Meirmanov, A (1992). The Stefan problem. Berlin:Walter de Gruyter.Murray, J. (1989). Mathematical biology. New York: Springer.Nagumo, J., Aximoto, S., & Yoshizawa, S. (1962). An active pulse transmission simulating nerve

axon. Proc. IRE 50, 2061–2071.Nagylaki, T (1975). Conditions for the existence of clines. Genetics 80, 595–615.Nisbet, R., & Gurney W. (1982). Modelling fluctuating population. New York: Wiley.Ranta, E., Kaitala, V., & Lundberg, P. (1997). The spatial dimension in population fluctuations.

Science 278, 1621–1623.Roughgarden, J. (1996). The theory of population genetics and evolutionary ecology: An intro-

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