9 Frequency-dependent selection and game theory

9.1 Frequency-dependent selection in a haploid population

In many biological situations an individual’s fitness is not constant but depends on the genetic structure of its ownpopulation or populations of other species the first population interacts with. In such situations, we say that fitnessesare frequency-dependent or that the population is subject to frequency-dependent selection. For example, in the caseof warning coloration (e.g. in butterflies) individuals that have a rare coloration pattern may suffer from predationto a larger degree than “common” individuals. Another example is mating behavior of Drosophila where males withrare phenotypes often have mating advantage. Yet another example would be a population with within-populationcompetition for two renewable resources where rare individuals specializing to one resource would have advantage.

Let us consider a single haploid locus with two alleles: A1 and A2. Let p and q be the frequencies of alleles A1

and A2, respectively (p + q = 1), and w1 and w2 be the corresponding fitnesses. The dynamics of the frequency ofallele A1 are described by a difference equation

∆p =pq(w1 − w2)

w. (81)

Note that a polymorphic equilibrium w1 = w2.Let us assume that fitnesses depend linearly (the simplest case!) on frequencies

w1 = ap + bq,

w2 = cp + dp,

where a, b, c and d are coefficients (chosen in a way that guarantees that fitnesses are positive). One should be ableto see that the resulting equation is mahmetically identical to that describing constant viability selection in a diploidone-locus two-allele population. [Notice that in the diploid case, the induced fitnesses of alleles are

w1 = w11p + w12q,

w2 = w21p + w22q.

While in the diploid model we always assume w12 = w21, there are no a priori reason to set b = c!——————————————————–Homework Problem: rare allele disadvantage.3. Symmetric frequency-dependent selection against rare alleles can be modeled by assuming that

wA = 1 + sp,

wa = 1 + tq,

where s and t are positive coefficients. Show that in this case depending on the initial allele frequency p the systemevolves towards fixation of allele A or allele a. Find a threshold value p∗ separating the domains of attraction of thetwo fixation states.

———————————————————-

9.1.1 Major points

• frequency-dependent selection can easily maintain genetic variation;

• at a polymorphic equilibrium the fitnesses of all genotypes present are identical.

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9.2 Coevolution of two species with applications to mimicry

Mimicry is a number of phenomena by which a species imitates another one. There exist numerous examples ofmimicry in butterflies, birds, fish, reptiles, amphibians, mammals, moths, beetles, bugs, flies, snails and fungi.There are two general types of mimicry. In Batesian mimicry a palatable species mimics a species (the model)unpalatable to predators. In Mullerian mimicry two (or more) unpalatable species resemble each other. Thestandard explanation for the resemblance both between Batesian model and mimic and between Mullerian mimicsis natural selection by predation. The Batesian mimic increases its fitness by evolving to a phenotypic pattern thatthe predator tends to avoid. In the Mullerian situation both species improve their protection by evolving to a similarphenotypic pattern that the predator learns to avoid more effectively.

We consider two haploid populations of different species with non-overlapping generations. We will concentrateon a single locus with alleles A and a in species 1 and a single locus with alleles B and b in species 2. Let wA, wa, wB

and wb be fitnesses (viabilities) of the corresponding morphs, and p1 and p2 be the frequencies of morph A withinspecies 1 and morph B within species 2, respectively. Let qi = 1− pi, i = 1, 2. The changes in allele frequencies aredescribed by the standard equations

∆p1 = p1q1(wA − wa)/w1, (82a)

∆p2 = p2q2(wB − wb)/w2, (82b)

where wi is the mean fitness of the i-th species (for example, w1 = wAp1 + waq1). We will assume that thefitness of an individual depends on the genetic composition of its own species as well on the genetic compositionof the other species, i.e., wA = wA(p1, p2), wa = wa(p1, p2) etc. We will make the symmetry assumption thatwA(p1, p2) = wa(q1, q2), wB(p1, p2) = wb(q1, q2) and use linear (i.e., the simplest) fitness functions. Our symmetryassumption implies that there are no differences in fitness between two morphs of a species not related to theirfrequencies. These assumptions result in the fitnesses

wA = C1 + ap1 + bp2, wa = C1 + aq1 + bq2, (83a)

wB = C2 + cp1 + dp2, wb = C2 + cq1 + dq2. (83b)

Here C1 and C2 are positive constants equal to the fitnesses of morphs A and B (or morphs a and b) when fixed in thepopulations. Parameters a and d characterize the strength of (indirect) within-species interactions. Positive valuesof a (d) imply that individuals of a given morph benefit when they are common. Negative values of a (d) imply thatindividuals of a given morph benefit when they are rare. Parameters b and c characterize the strength of (indirect)between-species interactions. Positive values of b (c) imply that a given morph benefits when its counterpart inanother species is common. Negative values of b (c) imply that a given morph benefits when its counterpart inanother species is rare.

[Add a game-theoretical interpretation of the fitness scheme.]We will assume that selection is weak (in the sense that |a|, |b| << C1, |c|, |d| << C2) which allows us to describe

the dynamics using differential equations:

p1 = p1q1 [a(p1 − q1) + b(p2 − q2)] , (84a)

p2 = p2q2 [c(p1 − q1) + d(p2 − q2)] , (84b)

where pi ≡ dpi/dt. Note that parameters a, b and c, d in (84) are different from those in (83) by the factors C1 andC2, respectively. The main reason for all the simplifying assumptions leading to (84) is that they allow us to studythe system analytically.

Before analyzing the whole system, it is illuminating to consider several special cases.

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a. Only a single species is allowed to evolve. If allele frequency in a species, say species 1, is somehow fixed(p1 = const), then

p2 = p2q2 [c(p1 − q1) + d(p2 − q2)]

This equation is similar to equation (??) analyzed above.

b. No between species interactions. If b = c = 0, the resulting equation is similar to equation (??) analyzed above.

c. Fitness depends only on the genetic composition of another species. This may be the case in many coevolvingsystems including host-parasite systems, competitive and cooperative interactions. In this case,

dp1

dt= p1q1b(p2 − q2), (85a)

dp2

dt= p2q2c(p1 − q1). (85b)

One can get some qualitative ideas about the dynamics of this system by inspecting the right-hand sides of theequations. For example, if both b and c are positive, p1 increases (dp1/dt > 0) or decreases (dp1/dt < 0) dependingon whether p2 is large or smaller than 0.5. Similar conclusions can be made about p2 and for other choices of thesigns of parameters b and c. This knowledge will be used below after we find the exact solutions.

We will use the separation of variables method. Dividing (85a) by (85b) one finds that

dp1

dp2

=bp1q1(p2 − q2)

cp2q2(p1 − q1).

Separating the variables

cp1 − q1

p1q1

dp1 = bp2 − q2

p2q2

dp2,

and integrating, we finds that all solutions of (85) satisfy to

c

∫

p1 − q1

p1q1

dp1 = b

∫

p2 − q2

p2q2

dp2. (86)

Because∫

x − (1 − x)

x(1 − x)dx =

∫

2x − 1

x(1 − x)dx = −

∫

d[x(1 − x)]

x(1 − x)= −

∫

du

u= − lnu, where u ≡ x(1 − x),

equation (86) can be integrated to−c ln(p1q1) = −b ln(p2q2) + const.

Thus, all solutions of (85) satisfy to the following equality

(p1q1)c

(p2q2)b= const. (87)

Now, we can combine our knowledge of the exact trajectories (given by equation (87)) with the qualitativeconclusions about the areas in the phase-plane (p1, p2) where allele frequencies increase or decrease (see above).

There are three cases to consider. (i) Parameters b and c are positive. In this case the fitness of each morphincreases with an increase in the frequency of its counterpart in the other species. This choice of parameters can beinterpreted as describing cooperative interactions between the species. (ii) Parameters b and c are negative. In thiscase the fitness of each morph decreases with an increase in the frequency of its counterpart in the other species. This

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choice of parameters can be interpreted as describing competitive interactions between the species. (iii) Parametersb and c have different signs. In this case the fitness of a morph in one species increases with an increase in thefrequency of its counterpart in the other species, but the fitness of the counterpart decreases with an increase inthe frequency of the first morph. This choice of parameters can be interpreted as describing victim-exploiter typeinteractions between the species.

In case (i), the system evolves to a state with both species monomorphic (there are two such states: p1 = p2 = 0and p1 = p2 = 1). In case (ii), the system evolves to a state with both species monomorphic (there are two suchstates: p1 = 0, p2 = 1 and p1 = 1, p2 = 0). In case (iii), the solutions are neutrally stable periodic orbits encircling thepolymorphic equilibrium (1/2, 1/2); any perturbation moves the system to a different periodic orbit. The dynamicbehavior of (84) is similar to the behavior of the classical Lotka-Volterra predator-prey system (e.g., Hofbauer &Sigmund, 1988).

9.2.1 Major points

With linear frequency dependent fitnesses and no within-species interactions

• cooperative and competitive systems do not maintain genetic variation;

• victim-exploiter type systems maintain genetic variation; these systems exhibit cycling.

0.80.60.40.2

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ii. iii.i.

p1 p1 p1

p2

Figure 8: Dynamic regimes with no within-species interactions. (i) Cooperative between-species interactions. (ii) Competitive

between-species interactions. (iii). Victim-exploiter type between-species interactions.

The two species whose coevolutionary dynamics are described by (84) can be considered as Batesian mimic andmodel or as two species belonging to the same Mullerian ring. Morph A is “similar” to morph B and morph a is“similar” to morph b. Biologically the symmetry assumption means that we include only the effects of mimicry inour model and no costs or other aspects of fitness. Hence morphs within species are alike in the sense that thereis no difference between A and a (B and b) except that they look similar to B and b (A and a), respectively.Positive values of a and/or d imply that individuals of a given morph benefit when they are common. This is thecase when species 1 and/or 2 is unpalatable. Negative values of a and/or d imply that individuals of a given morphbenefit when they are rare. This is the case when species 1 and/or 2 is palatable. Parameters b and c characterizethe strength of indirect between-species interactions. These interactions are mediated through the “predator” andthereby indirect. Positive values of b and/or c imply that a given morph benefits when its counterpart in anotherspecies is common. This is the case when the other species is unpalatable. Negative values of b and/or c imply thata given morph benefits when its counterpart in another species is rare. This is the case when the other species ispalatable or it is unpalatable but has a smaller degree of unpalatability (a weaker Mullerian mimic).

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Our results on dynamic behavior of (84) can be applied to understand the evolutionary dynamics of differentmimicry systems. We will consider three different parameter configurations:

Parameterconfiguration

Interpretation

“Classical” Mullerianmimicry

a, b, c, d > 0 Within each species a morph benefits whenits own frequency or the frequency of its“counterpart” in the other species increases

“Classical” Batesianmimicry

a, c < 0; b, d > 0 Both species 1 (palatable mimic) andspecies 2 (unpalatable model) benefit fromincreasing the model frequency and sufferfrom increasing the mimic frequency

Two unpalatable specieshave different abun-dances and/or differentdegrees of unpalatability

a, b, d > 0, c < 0 Within each species a morph benefits whenits own frequency increases. Species 1 (the“weaker” mimic) also benefits when the fre-quency of its “stronger” counterpart (fromspecies 2) increases, but the “stronger”mimic suffers when its weaker counterpartincreases in frequency

——————————————————-Homework Problem: stability of equilibria of the coevolutionary model (84).

Find all equilibria of the coevolutionary model (84). Summarize the conditions for existence and stability of theseequilibria in a table:

Equilibrium (p1, p2) Conditions for existence Conditions for stability

Use Maple (or your favorite software) to plot phase portraits corresponding to the situations when the equilibriaare stable and unstable.

Can genetic variation be maintained in the system?Are there any situations when none of the equilibria are stable?———————————————————-

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9.3 Linear frequency-dependent selection in a diploid population

We consider a deterministic model of a large randomly mating diploid population with discrete generations con-centrating on a single diallelic locus with alleles A and a. Let wAA, wAa and waa be the fitnesses (viabilities) ofgenotypes AA, Aa and aa, respectively, and p be the frequency of allele A, with q = 1 − p. With random matingthe population is in Hardy-Weinberg proportions so that the frequencies of the three genotypes are p2, 2pq and q2.The change in p in one generation is described by the standard equation

∆p =p(wA − w)

w. (88)

Here wA = pwAA + qwAa is the average fitness of allele A and w = p2wAA + 2pqwAa + q2waa is the mean fitness ofthe population. If the fitnesses are constant, the population gradually evolves to a polymorphic equilibrium (with0 < p < 1) provided there is overdominance (i.e., if wAa > wAA, waa) or to a fixation state (with p = 0 or p = 1),otherwise.

We assume that selection is frequency-dependent and will use linear (that is the simplest) functions to model thefrequency-dependence:

wAA = p2W11 + 2pqW12 + q2W13,

wAa = p2W21 + 2pqW22 + q2W23,

waa = p2W31 + 2pqW32 + q2W33,

where Wij are parameters (i, j = 1, 2, 3) characterizing the extent to which changes in the frequencies of threegenotypes influence their fitnesses.

Asmussen and Basnayake (1990):

a b cb d bc b a

,

where all coefficients are positive. Dynamics: multiple stable equilibria; plenty of opportunities for the maintenanceof variation.

Altenber (1991):

a (a + c)/2 cb b bc (a + c)/2 a

,

where some coefficients can be negative.Gavrilets and Hastings (1995):

δ β αγ η γα β δ

.

Under this symmetric model, the dependence of fitness of a heterozygote wAa on the allele frequency p is describedby a (quadratic) function symmetric about 1/2, while wAA and waa considered as (quadratic) functions of p arereflections of each other about 1/2. For this symmetric model to produce feasible (i.e., non-negative) fitnesses, onehas to assume that α, γ, δ > 0, β > −

√αδ, η > −γ. [Note that β and η are allowed to be negative.]

The dynamics are not changed if fitnesses are multiplied or divided by a constant. This allows one to assumewithout loss of generality that δ = 1. For the symmetric model, the dynamic equation (106) can be represented as

∆p =pq(p − q)(1 − γ − Ωpq)

w, (89)

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where the mean fitness w can be written as w = 1− 2(2− β − γ)pq + 2Ωp2q2 with Ω = 1 + α− 2β − 2γ + 2η. In theAltenberg model, Ω = 0.

The allele frequency p does not change if p = 0, q = 0, p = q or pq = (1−γ)/Ω. Thus, equation (107) always hastwo monomorphic equilibria at p = 0 and p = 1 and a polymorphic equilibrium at p = 1/2. If 0 < (1 − γ)/Ω < 1/4,it has two additional polymorphic equilibria with allele frequencies satisfying p(1− p) = (1− γ)/Ω. We shall denotethese equilibria p− and p+. An equilibrium of (107) is stable if the corresponding eigenvalue lies between -2 and 0.These eigenvalues can be found in a straightforward manner.

−6 −5 −4 −3 −2 −1 0 1 2 3 4beta

0

1

2

3

4

5

6

7

8

C

−6 −5 −4 −3 −2 −1 0 1 2 3 40

1

2

3

4

5

6

7

8

CUSU

UUU USUSU

SUSUS

SUUUS SUS

γ>1

γ<1

Figure 9: Areas in parameter space corresponding to different patterns of existence and stability of equilibria in a model of

linear frequency-dependent selection. Only areas with C ≥ 0 are shown.

Figure 14 summarizes conditions for existence and stability of different equilibria in terms of γ, β and a parameterC that combines several parameters. The parameter C is defined as C = c1 + c2 with c1 ≡ α− β2 and c2 ≡ 2(γ + η)or, alternatively, as C = Ω + 4γ− (1−β)2. Parameter c2 is the minimal possible value of the fitness of heterozygote.If β < 1, c1 determines the minimal possible value of the fitness of homozygotes, c1/(c1 +(1−β2)), while if β > 1, c1

determines the maximum possible value of the fitness of homozygotes, c1 + β2. Thus, parameter C characterizes theoverall strength of selection. For fitnesses to be feasible c2 must be non-negative and c1 must be non-negative if β < 0.These conditions imply that C must be larger or equal to 0 if β is negative, and must be larger or equal to −β2 if βis positive. Figure 1 shows areas in parameter space corresponding to different patters of existence and stability ofequilibria in a model of linear frequency-dependent selection. Each pattern is described by a string of S’s (for stable)and U’s (for unstable). The leftmost, the middle and the rightmost entries indicate the stability of the monomorphicequilibrium at p = 0, the polymorphic equilibrium at p = 1/2 and the monomorphic equilibrium at p = 1, whilethe remaining entries (if any) indicate the stability of the polymorphic equilibria p− and p+. The left parabola isdescribed by equation C = −(β + 5)(β + 1). The right parabola is described by equation C = −(β + 1)(β − 3).

———————————-Homework: Verify the conditions for existence and stability of different equilibria presented in Figure 2.———————————-

Figure 14 shows that the system can have up to three different stable equilibria simultaneously, that a polymorphic

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equilibrium can be stable simultaneously with two monomorphic equilibria, and that two different polymorphicequilibria can be stable simultaneously. Simultaneous stability of different equilibria implies that the outcome ofevolution strongly depends on the initial conditions and population history. If parameters change in such a way thatthe system moves from one area to another, the dynamic system undergoes a bifurcation. For example, a changefrom USU to USUSU corresponds to a pitchfork bifurcation. Figure 14 also shows that there are two areas withnon-standard patterns of stability of equilibria. In the first area (marked UUU), none of the three equilibria (twomonomorphic and one polymorphic at p = 1/2) are stable. If parameters change in such a way that the systemmoves from from USU to UUU, the dynamic system undergoes a period-doubling bifurcation. In the second area(marked SUUUS), the two monomorphic equilibria are stable, while none of the three polymorphic equilibria arestable. Numerical iterations of (107) with parameter values corresponding to these areas reveal a variety of complexdynamic behaviors (e.g., cycles and chaos that arises via period-doubling route) similar to those observed in classicalecological models (e.g., May, 1974, 1976; May and Oster, 1976; Hastings et al., 1993). Figure 14 shows that sufficientconditions for the complex dynamic behavior to occur is sufficiently strong overall selection (i.e., small C) andsufficiently strong deleterious effect of heterozygote on homozygotes (i.e., β < −1).

0 1000 2000 3000 4000 5000generation number

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a

b

Figure 10: Dynamics of allele frequency with parameter values from the area marked SUUUS in Figure 1. Both monomorphic

equilibria (p = 0 and p = 1) are stable to small perturbations. a) β = −1.002, γ = .999, C = 0. Three trajectories are shown

with initial allele frequencies p(0) = .09, p(0) = .55 and p(0) = .92. b) β = −1.001, γ = .9985, C = 0, p(0) = .55.

There are also two unusual types of behavior described in Figure 15. In Figure 15a depending on the initialconditions the population evolves to a fixation state or remains polymorphic indefinitely. In the latter case, thegradual changes in the allele frequency towards p = 1/2 are interrupted by apparently chaotic fluctuations that movep away from 1/2. Such alterations between an apparently deterministic behavior and apparently chaotic fluctuations,repeated at apparently random intervals, are called intermittency (Pomeau and Manneville, 1979; Olsen and Degn,1985). In Figure 15b, these alterations end at some time point with the population settling down to a monomorphicstate. In this case the system exhibits transient chaos (Grebogi et al., 1983; Tel, 1990). In the examples presentedin Figure 15, the deterministic phase of the dynamics can last for hundreds of generations, the chaotic phase isextremely short and transient chaos (in Figure 15b) persists for a very long time. In general, the durations of all

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these stages depend both on the parameter values and initial allele frequency (see below the discussion of Figure16). Both intermittency and transient chaos are known to occur in various dynamical models including the simplestnonlinear model, the single logistic map. Our model allows a very simple explanation of the mechanisms underlyingintermittency and transient chaos using the graphical “cobwebbing” method (May and Oster, 1976).

0.0 0.2 0.4 0.6 0.8 1.0p

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p’

0.0 0.2 0.4 0.6 0.8 1.0p

a

p−

p+

p−

b

min

max

min

maxp+

Figure 11: Graphs of the allele frequency in the next generation, p′ = p + ∆p, as function of the allele frequency at this

generation, p. Also shown are the diagonal and the lines corresponding to the unstable polymorphic equilibria p− and p+.

Parameter values are γ = .9, C = 0, β = −1.15 (in Figure 15a) and β = −1.05 (in Figure 15b).

Figure 16 shows graphs of the allele frequency in the next generation, p′ = p+∆p, as function of the allele frequencyat this generation, p. Also shown are the diagonal and the lines corresponding to the unstable polymorphic equilibriap− and p+. First note that for p in the neighborhood of p− or p+, the graph of p′ lies very close to the diagonal(at which p′ = p). That means that in these areas the changes in the allele frequency p are very small. For p < p−,p′ < p, and p slowly moves towards fixation of allele a. For p > p+, p′ > p, and the allele frequency slowly movestowards fixation of allele A. For p values slightly larger than p−, p′ is slightly larger than p, and, thus, p slowlymoves towards 1/2. In the neighborhood of p = 1/2, however, the dynamics are chaotic as suggested by the factthat the slope of the graph of p′ at p = 1/2 is smaller than -1. It takes many generations to leave the neighborhoodof p− or p+ and once the system has left this neighborhood, the dynamics can abruptly become fully chaotic in theneighborhood of p = 1/2, only to get caught in the neighborhood of p− or p+ again, sooner or later. Between 0and 1, the graph of p′ has a minimum, marked min, and a maximum, marked max. During the chaotic phase theallele frequency remains between these points that represent the boundaries of the chaotic attractor. In Figure 15athese boundaries lie closer to 1/2 than the unstable equilibria and the allele frequency cannot cross the values p+

and p−. In this case the intermittent chaos in the system (in the form similar to that one in Figure 15a) is presentforever. A different situation is described in Figure 15b, where the boundaries of the chaotic attractor lie furtherfrom 1/2 than the unstable equilibria p− and p+. Now the allele frequency can cross the values p+ and p− duringthe chaotic phase. Once this has happened, p slowly evolves to a fixation state (as in Figure 15b). The situationwhen the boundaries of the chaotic attractor coincide exactly with the unstable equilibria is called a crisis (Grebogiet al., 1983). In the model considered here, if C = 0, the crisis occurs when γ ≈ β + 2. Note that in general, thedependence of the length of chaotic transients on the system parameter is proportional to (a− ac)

−b, where a is theparameter value, ac is the parameter value at which the crisis occurs, and b is the “critical exponent,” which is equalto 0.5 for a broad class of one-dimensional systems (Grebogi et al., 1987). The parameter values for Figure 15 andFigure 17 below were chosen to result in long transients. The parameters for Figure 16 were chosen slightly differentfrom those for Figure 15 in order to produce a “smoother” graph of p′ as function of p.

Numerical analysis of (107) has also shown, perhaps surprisingly, that complex dynamic behavior occurs even

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0 100 200 300 400 500 600 700generation number

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Figure 12: Dynamics of allele frequency with parameter values from the area marked SUSUS in Figure 14. Both monomorphic

equilibria (p = 0 and p = 1) and the polymorphic equilibrium at p = 1/2 are stable to small perturbations. a) β = −5, γ =

.9, C = .1, p(0) = .55. b) β = −5, γ = .5, C = .1, p(0) = .55.

outside areas marked UUU and SUUUS in Figure 14. In the areas marked USU and SUSUS, cycles and chaos cannot only exist simultaneously with stable equilibria, but the former can have much larger domains of attraction thanthe latter (see Figure 17). For parameter values used in computing the dynamics in Figure 17, both monomorphicequilibria and the polymorphic equilibrium at p = 1/2 are stable to small perturbations. In Figure 17a, the growingregular oscillations in allele frequency are interrupted by apparently chaotic fluctuations that move p back to theneighborhood of 1/2, i.e., one observes intermittency. In Figure 17b, the alterations between apparently deterministicbehavior and apparent chaos end at some time point with the population settling down to a polymorphic state atp = 1/2, i.e., one observes transient chaos. The graphical “cobwebbing” method can be used to understand thesekinds of behavior as well.

A necessary condition for dramatic changes in allele frequency described in Figures 15 and 17 is strong (at leastoccasionally) selection. If selection is very weak (i.e., if the differences among coefficients α, β, γ, δ and η are verysmall) then the difference equation (107) can be approximated by the corresponding differential equation and theonly possible outcome of the dynamics is gradual evolution towards an equilibrium.

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9.4 Cycling in systems of ordinary differential equations

Let us consider a system of two ordinary differential equations

x = f(x, y), (90a)

y = g(x, y), (90b)

defined on an open set G ⊆ R2.

9.5 Dulac’s criterion

Theorem. If there exists a continuously differentiable function h(x, y) such that

∂(hf)

∂x+

∂(hg)

∂y

is continuous and non-zero on some simply connected domain D ⊆ G, then no periodic orbit can lie entirely in D.Example. Consider the Lotka-Volterra model

x = x(A − a1x + b1y),

y = y(B − a2y + b2x),

with h = 1/(xy).

9.5.1 Poincare-Bendixon theorem

Theorem. Assume that a trajectory γ(x0, y0) enters and does not leave some closed domain D and that there areno equilibrium points in D. Then there is at least one periodic orbit in D, and this orbit is in the ω-limit set of (x0, y0).

To apply the Poincare-Bendixon theorem we need to find a region D which contains no equilibrium points andwhich trajectories enter but do not leave.

Both the Poincare-Bendixon theorem and Dulac’s criterion do not work in higher dimensions (n > 2).

9.5.2 Non-existence of periodic orbits

Theorem (Hofbauer&Sigmund). If the partial derivatives ∂f/∂x ≥ 0 and ∂g/∂y ≥ 0 for all (x, y) ⊆ G, cyclingis impossible. The same result holds if ∂f/∂x ≤ 0 and ∂g/∂y ≤ 0 for all (x, y) ⊆ G.

No cycling means the trajectories converge either to an equilibrium or to infinity.

Example. For the coevolutionary model (85), ∂p1/∂p2 = 2bp1q1, ∂p2/∂p1 = 2cp2q2. Thus, if b and c have thesame sign so do ∂p1/∂p2 and ∂p2/∂p1, and stable cycling is impossible.

9.5.3 Poincare-Andronov-Hopf bifurcation (birth of cycles from an equilibrium)

Example. Let us consider a system of non-linear ODE:

x = αx − βy − (x2 + y2)x, (91a)

y = βx + αy − (x2 + y2)y. (91b)

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The origin (0, 0) is an equilibrium; the corresponding stability matrix is

S =

(

α −ββ α

)

with eigenvalues λ = α ± iβ. Thus, (0, 0) is a stable focus for α < 0 and an unstable focus for α > 0. Using polarcoordinates x = r cos θ, y = r sin θ the system can be rewritten as

r = αr − r3,

θ = β.

Thus, if β 6= 0, θ changes with constant speed β. r = 0 if r = 0 or r =√

α. If α > 0, r > 0 for 0 < r <√

α and r < 0for r >

√α. In this case, there is a periodic solution which is stable. If α < 0, r < 0 always and the only (stable)

equilibrium is r = 0.

Feb.2.General case.Theorem. Let us consider a system of n ordinary differential equations

z = Fµ(z), (92)

depending on some parameter µ and defined on an open subset of Rn. Let z∗ be an equilibrium of (92). Assumethat all eigenvalues of the Jacobian have negative real parts, with the exception of one pair of complex conjugateeigenvalues α(µ) ± iβ(µ). Let for some µ0, these two eigenvalues be pure imaginary: α(µ0) = 0, β(µ0) 6= 0, and letd

dµα(µ0) 6= 0. Then for µ sufficiently close to µ0, a periodic solution bifurcates from equilibrium z∗.

If the first Lyapunov value L1 (defined below) is negative, the bifurcation is supercritical (i.e. the periodic orbitexists for α(µ) > 0 and is stable). If L1 is positive, the bifurcation is subcritical (i.e. the periodic orbit existsfor α(µ) < 0 and is unstable). For small values of α(µ), the period and the amplitude of the periodic orbit areapproximately 2π/|β(µ)| and

√

|α(µ)|, respectively.

First Lyapunov value L1. Let us consider a system of two ordinary differential equations

dx

dt=ax + by + P (x, y), (93a)

dy

dt=cx + dy + Q(x, y). (93b)

Let TrS = a + d and det S = ad − bc be the trace and the determinant of the stability matrix for the equilibrium(0,0). Let P (x) and Q(x) be represented as power series

P (x, y) =P2(x, y) + P3(x, y) + . . . , (94a)

Q(x, y) =Q2(x, y) + Q3(x, y) + . . . , (94b)

where

P2(x, y) =a20x2 + a11xy + a02y

2, (95a)

P3(x, y) =a30x3 + a21x

2y + a12xy2 + a03y3, (95b)

Q2(x, y) =b20x2 + b11xy + b02y

2, (95c)

Q3(x, y) =b30x3 + b21x

2y + b12xy2 + b03y3. (95d)

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The first Lyapunov value L1 is defined as

L1 = − π

4(detS)3/2

1

b

[

ac(a211 + a11b02 + a02b11)

+ ab(b211 + a20b11 + a11b20) + c2(a11a02 + 2a02b02)

− 2ac(b202 − a20a02) − 2ab(a2

20 − b20b02)

− b2(2a20b20 + b11b20) + (bc − 2a2)(b11b02 − a11a20)

−(a2 + bc) (3(cb03 − ba30) + 2a(a21 + b12) + ca12 − bb21)]

(Bautin 1938).

The expression for L1 looks horrible and almost impossible to use practically. However, using modern symbolicmanipulation packages such as Maple and Mathematica makes it a trivial exercise.

Example 1. The first Lyapunov value corresponding to the equilibrium (0, 0) of system (91) is

L1 = −8.

Thus, the bifurcation is supercritical (i.e. the periodic orbit exists for α(µ) > 0 and is stable) which is what wealready found using exact analysis.

Example 2. Let us consider a system of two ordinary differential equations

x = rx(1 − x/K) − βx

x + Cy,

y = (kβx

x + C− m)y,

where all coefficients are positive. One can interpret x as the density of a prey population which grows logistically inthe absence of predator (if y = 0, then x = rx(1−x/K)), and y as the density of predator which experiences density-independent mortality (with rate m). Terms −β x

x+C y and kβ xx+C y stand for the rates at which predation decreases

the prey population and increases the predator population, respectively (a Holling type II functional response).The trivial equilibrium (0, 0) is always unstable. [It is a saddle point with eigenvalues r and −m.] The equilibrium

with only the prey population present (K, 0) is stable if m > m∗ where m∗ = KkβK+C , and is unstable otherwise. [The

eigenvalues are −r and m∗ − m.] The model also has a single coexistence state with population densities

x∗ =mC

kβ − m, y∗ =

rkC(K + C)(m∗ − m)

K(kβ − m)2.

This equilibrium is feasible if m < m∗ (predator’s mortality is not too large). The trace and determinant of thecorresponding stability matrix are

TrS =mr(Kkβ − Km− Ckβ − mC)

kβ(kβ − m)K,

det S =(K + C)(m∗ − m)mr

Kkβ,

that is det S > 0 always, but the TrS can change its sign. The coexistence state is stable if m > m∗∗ ≡ kβ(K −C)/(K + C) and is unstable otherwise. [Note that m∗∗ < m∗.] If m = m∗∗, the system undergoes a Poincare-Andronov-Hopf bifurcation. For this bifurcation, the first Lyapunov value is

L1 = −4Cβk(K − C)r2

K2(K + C)2,

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and is positive (because at m = m∗∗, K/C = (kβ + m)/(kβ − m) > 1). Thus, the bifurcation is supercritical (theperiodic orbit exists for m < m∗ and is stable). Summarizing, if m > m∗, the predator population cannot maintainitself whereas if m < m∗ both species coexist. If m∗∗ < m < m∗, the coexistence is non-oscillatory whereas ifm < m∗∗, there are stable oscillations of the densities of both populations.

Example 3. For the coevolutionary model (84), the first Lyapunov value corresponding to Poincare-Andronov-Hopf bifurcation from the doubly polymorphic equilibrium (1/2, 1/2) is

L1 = −2a(b + c)

b,

which can be positive or negative depending on parameter values.Let us consider the case corresponding to classical Batesian mimicry (a < 0, b > 0, c < 0, d > 0). Assume that

|a| < b, d < |c| (i.e. between species interactions are stronger than within species interactions). These conditionsguarantee that (i) monomorphic equilibria are unstable, (ii) singly polymorphic equilibria do not exist, and (iii) thedeterminant of the stability matrix evaluated at the doubly polymorphic equilibrium (1/2, 1/2) is positive. Thus, ifa + d < 0, the equilibrium (1/2, 1/2) is a stable focus. If a + d > 0, this equilibrium is an unstable focus. The sign ofL1 coincides with that of b+c. Thus, for small |a+d|, the system should have a stable limit cycle if a+d > 0, b+c < 0.If a + d < 0, b + c > 0, then there is an unstable cycle encircling the equilibrium (1/2, 1/2).

Numerical examples. Stable cycle: a = −1, b = 1.2, c = −1.3, d = 1.01. Unstable cycle: a = −1.01, b = 1.3, c =−1.2, d = 1. Illuminating initial conditions are p1 = 0.25, p2 = 0.75 and p1 = 0.5, p2 = 0.6.

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