Lecture 15 Lotka-Volterra

Outline Predator and prey. Competition between species. The n-dimensional Lotka-Volterra equation. Replicator dynamics and evolutionary stable strategies. Evolutionary stable states. Entropy.

Lecture 15Lotka-Volterra.

Shlomo Sternberg

April 14 - 21, 2009

Shlomo Sternberg

Lecture 15 Lotka-Volterra.


1 Predator and prey.

1 Competition between species.

2 The n-dimensional Lotka-Volterra equation.

3 Replicator dynamics and evolutionary stable strategies.

4 Evolutionary stable states.

5 Entropy.

Shlomo Sternberg



The Lotka - Volterra predator prey equations were discoveredindependently by Alfred Lotka and by Vito Volterra in 1925-26.These equations have given rise to a vast literature, some of whichwe will sample in this lecture.

Here is how Volterra got to these equations: The number ofpredatory fishes immediately after WWI was much larger thanbefore the war. The question as to why this was so was posed tothe mathematician Volterra by his prospective son-in-law Anconawho was a marine biologist.

Shlomo Sternberg



Much of the more recent results (in the second part of this lecture)are taken from the book Evolutionary Games and PopulationDynamics by Josef Hofbauer and Karl Sigmund.

For a discussion of some of these issues at a level requiring lessmathematics that we require in these lectures (and hence withoutsome of the proofs) see the book Evolutionary Dynamics by MartinNowak.

Shlomo Sternberg



The L-V equations.

Here is Volterra’s solution to the problem: Let x denote thedensity of prey fish and y denote the density of predator fish.Assume the equations

x = x(a− by)

(1)

y = y(−c + dx)

wherea, b, c , d > 0.

Shlomo Sternberg



x = x(a− by)

y = y(−c + dx)

The idea of the first equation is that in the absence of predators,the prey would grow at a constant rate a, but decreases linearly asa function of the density y of the predators. Similarly, in theabsence of prey, the density of predators would decrease but therate increases proportional to the density of the prey.

We are interested in solutions to these differential equations in thefirst quadrant

R2+ = {(x , y)|x ≥ 0, y ≥ 0}.

Shlomo Sternberg



The zeros of the vector field.

x = x(a− by)

y = y(−c + dx)

The null-clines (where either x = 0 or y = 0 are zero) are thecoordinate axes and the lines y = a/b and x = c/d . The firstquadrant is invariant . The origin is a saddle point.

Shlomo Sternberg



x = x(a− by)

y = y(−c + dx)

The other point where the right hand side is zero is(xy

):=

(cdab

)where the linearized equation has matrix(

0 −bc/dda/b 0

)with purely imaginary eigenvalues ±i

√ac.

Shlomo Sternberg



x = x(a− by)

y = y(−c + dx)

If we multiply the first equation by (c − dx)/x and the second by(a− by)/y and add we get(c

x− d

)x +

(a

y− b

)y = 0

ord

dt[c log x − dx + a log y − by ] = 0.

LetH(x) := x log x − x , G (y) := y log y − y ,

andV (x , y) := dH(x) + bG (y).

Then V is constant on flow lines.Shlomo Sternberg



All trajectories in the interior of the quadrant are periodic.

H(x) := x log x − x , G (y) := y log y − y ,

andV (x , y) := dH(x) + bG (y).

Then V is constant on flow lines. Since

dH

dx=

x

x− 1,

d2H

dx2= − x

x2< 0

we see that H achieves a maximum at x and similarly G assumes amaximum at y . Thus V has a unique maximum in the interior ofthe quadrant at the critical point. Thus the level curves of V ,which are solution curves, are closed curves: all trajectories areperiodic.

Shlomo Sternberg



LV vector field with a=2, b=1, c=.25,d=1.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

3

Shlomo Sternberg



Some trajectories.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

3

Shlomo Sternberg



The fixed point as an average.

Suppose we are on a trajectory of period T , so x(T ) = x(0). From

d

dtlog x =

x

x= a− by

it follows by integration that

0 = log x(T )− log x(0) = aT − b

∫ T

0y(t)dt

or1

T

∫ T

0y(t)dt = y

and similarly1

T

∫ T

0x(t)dt = x .

Shlomo Sternberg



Volterra’s explanation of why fishing decreases the numberof predators.

Fishing reduces the rate of increase of the prey, so a is replaced bya− k and increases the rate of decrease of the predator, so c isreplaced by c + m, but does not change b or d - the interactioncoefficients. So a/b is replaced by (a− k)/b - the average numberof predators is decreased by fishing and the average number of preyis increased. Stoppage of fishing increases the average number ofpredators and decreases average the number of prey.

Shlomo Sternberg



A moral lesson.

If the prey are “pests” and the predators are their natural enemies,applying non-specific insecticides may actually increase the pestpopulation.

Shlomo Sternberg



A more realistic model.

Suppose we make the equations more realistic by adding selfcompetition terms and so get the equations

x = x(a− ex − by)

(2)

y = y(−c + dx − fy)

wherea, b, c , d , e, f > 0.

The first quadrant is still invariant, and there is an equilibriumpoint on the x-axis at x = a/e. There is no equilibrium point onthe positive y -axis. The null-clines are now the axes and the twolines

ex + by = a, and dx − fy = c

the first with negative slope and the second with positive slope.Shlomo Sternberg



The null-clines are now the axes and the two lines

ex + by = a, and dx − fy = c

the first with negative slope and the second with positive slope.

All hinges on whether or not they intersect in the first quadrant. Ifthey don’t, the quadrant is divided into three regions: in the regionI to the right of the line y = 0 of positive slope, we have x < 0 soa trajectory starting in this region moves to the left, enteringregion II. It keeps moving to the left until it crosses the line x = 0,entering region III, where it points down and to the right andheads toward the fixed point on the x-axis. The predators becomeextinct.

Shlomo Sternberg



The predators become extinct.

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

III

III

Shlomo Sternberg



The second alternative is that the lines intersect in the firstquadrant, dividing it into four regions:

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

IV

I

II

III

Shlomo Sternberg



Some trajectories from ode45.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

IV

I

II

III

Shlomo Sternberg



It looks as if trajectories (except those on the axes) are spiraling into the zero of the vector field. Let’s prove this: Label the fixedpoint as (

xy

).

With the same H,G and V as before, namely

H(x) := x log x − x , G (y) := y log y − y ,

andV (x , y) := dH(x) + bG (y)

we have

V =∂V

∂xx +

∂V

∂yy

= d(x

x− 1)x(a− by − ex) + b(

y

y− 1)y(−c + dx − fy).

Shlomo Sternberg



V = d(x

x− 1)x(a− by − ex) + b(

y

y− 1)y(−c + dx − fy).

Write a = ex + by and c = dx − f y . We get

d(x − x)(by + ex − by − ex) + b(y − y)(−dx + f y + dx − fy)

which simplifies giving

V (x , y) = de(x − x)2 + bf (y − y)2.

This is non-negative, and strictly positive except at the equilibirumpoint. Hence V is steadily increasing along each orbit, which mustthen head to the equilibrium point. �

Shlomo Sternberg



Simple equations of competition.

If x and y denote the density of populations of two speciescompeting for the same resources, then the rates of growth x/xand y/y will be decreasing functions of both x and y . Thesimplest assumption is that these decreases be linear which leadsto the equations

x = x(a− bx − cy)

y = y(d − ex − fy)

witha, b, c, d , e, f > 0.

Shlomo Sternberg



The null clines.

Again the first quadrant R2+ is invariant. The x and y null-clines

are given by the lines

a− bx − cy = 0

d − ex − fy = 0

this time both of negative slope.

We will ignore the degenerate case where these lines are parallel.So we are left with two possibilities:

The point of intersection does not lie in R2+.

The point of intersection does lie in R2+.

Shlomo Sternberg



If point of intersection does not lie in R2 then one species tends toextinction. The other species is said to be dominant. Here are thetwo possible cases:

Shlomo Sternberg



Shlomo Sternberg



Shlomo Sternberg



If the two null-clines intersect in R2 the point of intersection is at

x =af − cd

bf − ce, y =

bd − ae

bf − ce

The Jacobian matrix at this point is

J =

(−bx −− cx−ey −f y

)with determinant

det(J) = xy(bf − ce).

We can have (case 3)- a saddle, or (case 4) a sink.

In case 3 one or the other species dominates depending on theinitial conditions:

Shlomo Sternberg



Shlomo Sternberg



In case 4 all trajectories lead to the stable fixed point of“equilibrium co-existence”.

Shlomo Sternberg



In two dimensions, because of Poincare-Bendixon, we can get moreor less complete answers to the global behavior of flows.We will now embark on the study of the higher dimensional versionof the Lotka-Volterra equations where the answers are far lesscomplete.But we can say something.

Shlomo Sternberg



A theorem of Liapounov.

We will use a theorem of Liapounov describing the ω-limit set inthe presence of a “Liapounov function”.

Theorem

Let X be a vector field on some open set O ⊂ Rn. Let V : O → Rbe a continuously differentiable function. Let t 7→ x(t) be atrajectory of X . If the derivative V of the map t 7→ V (x(t))satisfies V ≥ 0 (for all t) then ω(x) ∩ O is contained in the setwhere XV = 0.

Remark.

Along x(t) we have V (t) = X (x(t))V .

Shlomo Sternberg



Proof of Liapounov’s theorem.

If y ∈ ω(x) ∩ O, there is a sequence tk →∞ with x(tk)→ y .Hence, by continuity, (XV )(y) ≥ 0. If XV does not vanish at y ,then (XV )(y) > 0 and we must show that this can not happen.

If (XV )(y) > 0, then for small positive values of t we would haveV (y(t)) > V (y). Now by hypothesis V (x(s)) is a monotoneincreasing function of s, and since x(tk)→ y and tk →∞, for anys and sufficiently large k we have

V (x(s)) ≤ V (x(tk))→ V (y).

SoV (x(s)) ≤ V (y)

for all s.

Shlomo Sternberg



Proof of Liapounov’s theorem, completed.

V (x(s)) ≤ V (y) for all s.

Since x(tk)→ y we have (for small t > 0)

x(tk + t)→ y(t)

and henceV (x(tk + t))→ V (y(t)) > V (y),

a contradiction. �

Shlomo Sternberg



The Lotka-Volterra equations for n species.

These are

xi = xi

ri +∑

j

aijxj

, i = 1, . . . , n.

xi denotes the density of the i-th species, ri is its intrinsic growth(or decay) rate and the matrix A = (aij) is called the interactionmatrix.

Shlomo Sternberg



The positive orthant and its faces are invariant.

xi = xi

(ri +

∑j aijxj

), i = 1, . . . , n.

If xi (0) = 0, then xi (t) ≡ 0 is a solution of the i-th equation, andhence by the uniqueness theorem of differential equations the onlysolution. So each of the faces xi = 0 of the positive orthant Rn

+ isinvariant under the flow, and hence so is the (interior of) thepositive orthant itself.

The vector field given by the right hand side of the equationsvanishes when

ri +∑

j

aijxj = 0

and so zeros of the vector field in the positive orthant correspondto solutions of these equations with all xi > 0.

Shlomo Sternberg



An interior α or ω point implies a solution ofri +

∑j aijxj = 0 with all positive entries.

To prove this, it is enough to show that if there is no solution tothe above equation, then there is a “Liapounov function” V withXV > 0 everywhere, since Liapounov’s theorem tells us that at anyinterior ω point we must have XV = 0 (and similarly for α points).To construct V , let L be the map Lx = y where

yi = rixi +∑

j

aijxj .

The image of the positive orthant is some convex cone C . Theassumption is that C does not contain the origin.

Shlomo Sternberg



L is the map x 7→ y , yi = rixi +∑

j aijxj . The image of thepositive orthant is some convex cone C . The assumption is that Cdoes not contain the origin. Since C is convex, there is ahyperplane separating it from the origin. Put another way, there isa vector c such that c · y > 0 for all y ∈ C . Now define

V (x) :=∑

i

ci log xi

for all x in the positive orthant. Then

V =∑

cixi

xi=∑

ciyi > 0

at all points, proving our claim.

In particular, if there is a periodic solution in the positive orthant,there must also be a fixed point.

Shlomo Sternberg



Food chains.

A food chain is a system where the first species is prey for thesecond, the second is prey for the third , etc. up to the n-th whichis at the top of the pyramid. Taking competition within eachspecies into account, the differential equations are:

Shlomo Sternberg



The food chain equations.

x1 = x1(r1 − a11x1 − a12x2)

x2 = x2(−r2 + a21x1 − a22x2 − a23x3)...

......

xj = xj(−rj + aj ,j−1xj−1 − ajjxj − aj ,j+1xj+1)

......

...

xn = xn(−rn + an,n−1xn−1 − annxn)

with all the ri and aij positive.

Shlomo Sternberg



Theorem

If the food chain equations have an interior rest point

p =

p1...

pn

,

i.e. a point p where the right hand side of the food chainequations vanish, then p is globally stable in the sense that allorbits in the interior of the positive orthant converge to p.

The proof will consist of constructing a Liapounov function of theform

V (x) =∑

ci (xi − pi log xi )

for suitably chosen constants ci .

Shlomo Sternberg



Proof of the food chain theorem, I.

Let V (x) =∑

ci (xi − pi log xi ). Then

V (x) =∑

ci

(xi − pi

xi

xi

).

If we write the food chain equations as xi = xiwi this becomes

V (x) =∑

ci (xi − pi )wi .

By assumption, the wi vanish at p. So, for example,r1 = a11p1 + a22p2 so

w1 = r1 − a11x1 − a12x2 = a11(p1 − x1) + a12(p2 − x2).

Shlomo Sternberg



Proof of the food chain theorem, II.

V (x) =∑

ci (xi − p log xi ), xi = xiwi , V (x) =∑

ci (xi − pi )wi .

w1 = a11(p1 − x1) + a12(p2 − x2)

w2 = a21(x1 − p1)− a22(x2 − p2)− a23(x3 − p3)...

......

wn = an,n−1(xn−1 − pn−1)− ann(xn − pn).

So if we set yi := xi − pi we get

V = −c1a11y21 − y1y2c1a12 − c2a22y2

2 + c2a21y1y2c2 − a23y2y3

−c3a33y23 + y2y3c3a32 − c3a34y3y4 + · · ·

Shlomo Sternberg



Proof of the food chain theorem, III.

V (x) =∑

ci (xi − p log xi ), xi = xiwi , V (x) =∑

ci (xi − pi )wi .yi := xi − pi Then

V = −n∑

j=1

cjajjy2j +

n−1∑j=1

yjyj+1(−cjaj ,j+1 + cj+1aj+1,j).

Shlomo Sternberg



Proof of the food chain theorem, completed.

V = −n∑

j=1

cjajjy2j +

n−1∑j=1

yjyj+1(−cjaj ,j+1 + cj+1aj+1,j).

Since all the aj ,j+1 and aj+1,j are positive, we can choose cj > 0recursively such that −cjaj ,j+1 + cj+1aj+1,j = 0, i.e.cj+1/cj = aj ,j+1/aj+1,j . Then the second summand abovevanishes, and we have

V = −n∑

j=1

cjajjy2j ≤ 0

with strict inequality unless all the yi = 0.

By Liapounov’s theorem, the ω limit of every orbit in the interiorof the postive orthant is p. �

Shlomo Sternberg



The replicator equation.

We will let Sn denote the simplex consisting of all x =

x1...

xn

with

xi ≥ 0 and∑

i xi = 1. We want to think of a population as beingdivided into n types E1, . . . ,En and of xi as the frequency of thei-th type Ei . The “fitness” fi of Ei will be a function of thesefrequencies, i.e. of x . If the population is very large and thegenerations blend continuously into each other, we may assumethat x(t) is differentiable function of t. The rate of increase ofxi/xi is a measure of the evolutionary success of type Ei . Thebasic tenet of Darwinism says that we may express this success asthe difference between fi (x) and the average fitnessf (x) :=

∑xi fi (x). We obtain the replicator equation

xi = xi (fi (x)− f (x)).Shlomo Sternberg



Elementary properties of the equation xi = xi (fi (x)− f (x)).

If we set S(x) := x1 + · · ·+ xn, then summing the above equationsgives the equation

S = (1− S)f .

The (unique) solution of this equation with S(0) = 1 is S(t) ≡ 1.So the set Sn is preserved by the flow. Also, if xi (0) = 0 for somei , then xi (t) ≡ 0. Thus the faces of Sn are preserved, and hence sois the open simplex

Sn := {x ∈ Sn|xi > 0 ∀i}.

Shlomo Sternberg



Linear fitness.

For most of the rest of today’s lecture I will assume that thefitnesses are linear functions of x , i.e. there is a matrix A such thatfi (x) = (Ax)i - the i-th component of Ax . The replicatorequations are then cubic equations in x :

xi = xi ((Ax)i − x · Ax) .

We shall see that a change of variables will carry the orbits of thereplicator equation with linear fitness functions to the orbits of theLotka-Volterra equations (which are quadratic) in one fewervariables.

These equations are equivalent in the above sense, but somenotions are easier to formulate and understand in one setting, andsome in the other.

Shlomo Sternberg



Some preliminaries to the equivalence theorem.

Let us go back temporarily to the general replicator equation

xi = xi (fi (x)− f (x)).

If we add a function h = h(x) to all the fi , this has the effect ofreplacing f by f + h since

∑xih = h as

∑xi = 1. Thus the right

hand side of the above equation is unchanged. So adding acommon function to all the fi does not change the replicatorequation.

In case fi = (Ax)i , if we add a constant c to all the entries in thej-th column of A, this has the effect of adding the function cxi toall the fi , so does not change the replicator equation. In particular,this means that (by subtracting off the entry in the last row fromeach column) we can assume that our matrix A has all entries inthe bottom row zero, without changing the replicator equation.

Shlomo Sternberg



Here is another useful fact about the general replicator equationxi = xi (fi (x)− f (x)). We have(

xi

xj

).

=xixj − xi xj

x2j

=(fi − f )xixj − (fj − f )xixj

x2j

=(fi − fj)xixj

x2j

So (xi

xj

).

=

(xi

xj

)(fi (x)− fj(x)).

Shlomo Sternberg



Consider the map of the set {y ∈ Rn+|yn = 1} onto the set

Sn := {x ∈ Sn|xn > 0}

given by

xi =yi∑j yj

, i = 1, . . . , n.

The inverse map x 7→ y is given by

yi =yi

yn=

xi

xn.

If x satisfies the general replicator equation then

yi = yi (fi (x)− fn(x))

by the results of the previous slide.

Shlomo Sternberg



yi = yi (fi (x)− fn(x)).

Now suppose that fi (x) = (Ax)i and we have chosen the matrix Ato have its bottom row all zero (which we can do without changingthe equations). Then fn(x) ≡ 0 and the preceding equationsbecome

yi = yi

n∑j=1

aijxj = yi

n−1∑j=1

aijyj

xn.

Now the positive factor xn affects the speed with which thetrajectories are traveled, but not the shape of the trajectoriesthemselves.

In other words, the trajectories in y space are given by

yi = yi

ain +n−1∑j=1

aijyj

i = 1, . . . n − 1.

Shlomo Sternberg



If we consider a general matrix (and then modify it to get thebottom row zero) we have proved

Theorem

[Hofbauer.] The differentiable invertible map x 7→ y given abovemaps the orbits of the replicator equation with linear fitnessfi = (Ax)i onto the orbits of the Lotka-Volterra equation

yi = yi

ri +n−1∑j=1

bijyj

i = 1, . . . n − 1

whereri = ain − ann and bij = aij − anj .

The steps in this passage from replicator equations to LV arereversible.

Shlomo Sternberg



Back to the replicator equations with linear fitnesses.

The right hand side of the equation xi = xi ((Ax)i − x · Ax)vanishes if and only if all the (Ax)i are equal (in which case theyall equal x · Ax). So the conditions for such a rest point are theequations

(Ax)1 = · · · = (Ax)n, x1 + · · · xn = 1, xi > 0 ∀ i ,

n equations in n unknowns, which, therefore, will generically haveone or no solutions. These equations are related to certainconcepts and equations in game theory:

Shlomo Sternberg



Nash equilibria.

For a given matrix A a point p ∈ Sn is called a Nash equilibrium if

x · Ap ≤ p · Ap ∀x ∈ Sn.

If p is a Nash equilibrium, then taking x = ei (the i-th unit vector)in the above inequality gives

(Ap)i ≤ p · Ap.

Multiplying by pi and summing i gives us back p · Ap. So we cannot have strict inequality for any i for which pi > 0. We must have

(Ap)i = p · Ap ∀ i for which pi > 0.

Shlomo Sternberg



Interior Nash equilibria are rest states of the replicatorequation.

In particular, if p ∈ Sn (the interior of the simplex) - so that all thepi > 0) then we have

(Ap)i = p · Ap ∀ i ,

and so the right hand side of the replicator equation vanishes at p.More generally, if we consider the face of Sn spanned by those ei

for which pi > 0, we see that p is a rest point of the replicatorequation (restricted ot the interior of that face).

We know that for the Lotka-Volterra equations the existence of aninterior ω limit point of any orbit implies the existence of aninterior rest point, and we know that the replicator orbits have thesame structure as the LV orbits. So we have proved:

Shlomo Sternberg



Theorem

If p ∈ Sn is a Nash equilibrium of A then it is a rest point for theassociated replicator equation.

We also have:

Theorem

If x(t)→ p ∈ Sn as t→∞ for some orbit then it is a Nashequilibrium.

Proof.

Suppose that p is not a Nash equilibrium. Then for some i wehave (Ap)i − p · Sp > 2ε > 0. Since x(t)→ p, this means that forsufficiently large t we have xi/xi > ε which is clearlyimpossible.

Shlomo Sternberg



Evolutionary stable states.

A point p ∈ Sn is called an evolutionary stable state if

p · Ax > x · Ax ∀ x 6= p, x ∈ Sn.

Theorem

[Zeeman.] If p is an evolutionary stable state then every orbit ofthe associated replicator equation in the open simplex Sn

converges to p.

Shlomo Sternberg



log x ≤ x − 1.

For the proof of the theorem we will use the inequality

log x ≤ x − 1

for x > 0 with strict inequality when x 6= 1.To prove this inequality observe that both sides are equal whenx = 1. For x > 1 the derivative of the right hand side is 1 whilethe derivative of the left hand side is 1/x < 1, so the right handside is increasing faster. For x < 1 we have 1/x > 1 so the lefthand side is increasing faster, and so is strictly below x − 1. �

Shlomo Sternberg


Outline Predator and prey. Competition between species. The n-dimensional Lotka-Volterra equation. Replicator dynamics and evolutionary stable strategies. Evolutionary stable states. Entropy.∑pi log xi ≤

∑pi log pi with strict inequality unless

xi = pi for all i .

To prove this:∑pi log xi −

∑pi log pi =

∑pi log

xi

pi≤∑

pi

(xi

pi− 1

)=∑

xi −∑

pi = 1− 1 = 0.

The inequality becomes strict if any xi 6= pi . �

So the function V (x) =∑

pi log xi achieves its maximum at p.We shall show that if p is an evolutionary stable state then V is aLiapounov function for the associated replicator equation.

Shlomo Sternberg



Indeed,

V =∑

pixi

xi=∑

pi ((Ax)i − x · Ax) = p · Ax − x · Ax > 0

if x 6= p. �

The function Ent(x) := −∑

xi log xi (known as the entropy) playsa key role in thermodynamics, statistical mechanics, andinformation theory. As a diversion, I will spend the last few slidesof today’s lecture trying to explain why and how this enters intocommunication theory.

Shlomo Sternberg



Codes.

We use the following notations. W will be a set of “words”,W = {a, b, c , d , . . . }. The number of elements in W will bedenoted by N. A message is just a concatenation of words, i.e. astring of elements of W .Σ will denote an alphabet of D symbols (usually D = 2 and thesymbols are 0 and 1). A code or an encoding is a map from W tostrings on Σ. It then extends by concatenation to messages.

Example.

W = {a, b, c , d} φ : a 7→ 0, b 7→ 111, c 7→ 110, d 7→ 101.

Thenφ(aba) = 01110.

Shlomo Sternberg



Uniquely decipherable codes and instantaneous codes.

A code, φ, is called uniquely decipherable (UD) if any string Son Σ has at most one preimage under φ. A code is calledinstantaneous (INS) if no φ(w) occurs as a prefix of the code forsome other word. Clearly every instantaneous code can be uniquelydeciphered, each word as it arrives. Hence

UD ⊃ INS.

Example. W = {x , y},Σ = {0, 1}, φ : x 7→ 0, y 7→ 01. Then

00010101

deciphers from the end as xxyyy . But we needed to wait until theend of the message to decode. If the last digit had been a 0 insteadof a 1, it would have decoded as xxyyxx . So the inclusion is strict.

Shlomo Sternberg



Frequencies.

We let |S | denote the length (number of elements in) a string S .Suppose that the messages sent are all such that each word woccurs with a relative frequency f (w), so we think of f as aprobability measure on W . So now f (w) denotes frequency ratherthan fitness.

Then the expectation

E (|φ(w)|) = Ef (|φ(w)|) =∑

f (w)|φ(w)|

is the “average length of the encoding φ”. We would like to makethis as small as possible.

We define

Ent(f ) = E (− log f ) = −∑w

f (w) log f (w)

as before.Shlomo Sternberg



Shannon’s “first theorem”.

Theorem

For any UD code, φ we have

E (|φ(w)|) ≥ Ent(f )

log D. (3)

There exists an INS code φ such that

E (|φ(w)|) ≤ Ent(f )

log D+ 1. (4)

Shlomo Sternberg



McMillan’s inequality.

I will prove (3) by first proving McMillan’s inequality: Let`1, . . . , `N be the code word lengths of a UD code. ThenMcMillan’s inequality says that

N∑1

D−ì ≤ 1. (5)

The proof of McMillan’s inequality will be by the method ofgenerating functions:

Shlomo Sternberg



For any integer r we have

(D−`1 + D−`2 + · · ·+ D−`N

)r=

r∑1

biD−i

where ` = max `j and where bi denotes the number of ways that astring of length i can be constructed by concatenating r codewords. Now there are D i strings of length i in all. If the code isuniquely decipherable then there can’t be more than D i messageswhose code is a string of length i . Hence

bi ≤ D i

and plugging into the preceding equality gives(D−`1 + D−`2 + · · ·+ D−`N

)r≤ r`.

Shlomo Sternberg



We have shown that(D−`1 + D−`2 + · · ·+ D−`N

)r≤ r`.

HenceD−`1 + D−`2 + · · ·+ D−`N ≤ `1/r r1/r → 1

as r →∞. This proves McMillan’s inequality (5).

Now to the proof of Shannon’s inequality (3):

Shlomo Sternberg



Set

qi =D−ì

D−`1 + D−`2 + · · ·+ D−`N

so that ∑qi =

∑fi = 1

where fi = f (wi ) is the frequency of the i−th word. We haveproved that

Ent(f ) = −∑

fi log fi ≤ −∑

fi log qi = log D∑

fiì+log(N∑1

D−ì ).

But (∑N

1 D−ì ) ≤ 1 and hence its logarithm is negative. Weconclude that

Ent(f ) ≤ log D × E (|φ(w)|)

which is just (3). �

Shlomo Sternberg



Krafts lemma.

We now show that there exists an instantaneous code satisfying(4). For this we need Kraft’s lemma

Lemma

For any ì satisfying

N∑1

D−ì ≤ 1, (5)

there exists an instantaneous code whose word lengths are ì .

Shlomo Sternberg



Proof of Kraft’s lemma, I.

Write (5) as ∑1

njD−j ≤ 1

where ni is the number of ì which are equal to j . Multiply throughby D` and move terms to the other side so the inequality becomes

n` ≤ D` − n1D`−1 − · · · − n`−1D.

Now the n` which occurs on the left of the inequality is anon-negative (actually positive) integer. So we certainly have theinequality

0 ≤ D` − n1D`−1 − · · · − n`−1D.

Shlomo Sternberg



Proof of Kraft’s lemma, II.

0 ≤ D` − n1D`−1 − · · · − n`−1D.

Dividing by D and bringing n`−1 over to the other side gives

n`−1 ≤ D`−1 − n1D`−2 − · · · − n`−2D.

So proceding in this way we get the string of inequalities:

Shlomo Sternberg



Proof of Kraft’s lemma, III.

n` ≤ D` − n1D`−1 − · · · − n`−1D

n`−1 ≤ D`−1 − n1D`−2 − · · · − n`−2D

n`−2 ≤ D`−2 − n1D`−3 − · · · − n`−3D...

n3 ≤ D3 − n1D2 − n2D

n2 ≤ D2 − n1D

n1 ≤ D.

Shlomo Sternberg



Proof of Kraft’s lemma, IV.

Let us read these inequalities in reverse order. The last inequalitysays that we can encode n1 words each by a single letter from thealphabet Σ, with D − n1 letters left over to serve as prefixes ofcode words. The next to last inequality says that we can encode n2

words as two letter code words using the D − n1 letters as firstletters and choosing from the D letters as second letters,D2 − n1D = (D − n1)D possibilities in all. This leaveD2 − n1D − n2 possible prefixes for three letter words, and thethird from last inequality says that we have enough room toencode n3 words as three letter code words. Proceeding in this wayback up to the top proves Kraft’s lemma. �

Shlomo Sternberg



Proof of the second assertion in Shannon’s theorem usingKraft’s lemma.

Choose word lengths ì to be the smallest integers satisfying

f −1i ≤ Dì .

This is equivalent to

ì log D ≥ − log fi

and

ì ≤ 1− log filog D

since we have chosen ì as small as possible.

Shlomo Sternberg



But ∑D−ì ≤

∑fi = 1

so (5) is satisfied, and we can find an instantaneous code with theword lengths ì . For this code we have∑

fiì ≤∑

fi

(1− log fi

log D

)= 1 +

Ent(f )

log D. �

Shlomo Sternberg



Here are photographs of Lotka and Volterra:

Shlomo Sternberg



Alfred James Lotka (1880 – 1949)

Shlomo Sternberg



Vito Volterra

1860 - 1940

Shlomo Sternberg



Here is a photograph of Shannon:

Claude E Shannon

1916 - 2001

Shlomo Sternberg


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Lecture 15 Lotka-Volterra

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