9 Frequency-dependent selection and game theory

Transcription

1 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 own population or populations of other species the first population interacts with. In such situations, we say that fitnesses are frequency-dependent or that the population is subject to frequency-dependent selection. For example, in the case of warning coloration (e.g. in butterflies) individuals that have a rare coloration pattern may suffer from predation to a larger degree than common individuals. Another example is mating behavior of Drosophila where males with rare phenotypes often have mating advantage. Yet another example would be a population with within-population competition for two renewable resources where rare individuals specializing to one resource would have advantage. Let us consider a single haploid locus with two alleles: A 1 and A 2. Let p and q be the frequencies of alleles A 1 and A 2, respectively (p + q = 1), and w 1 and w 2 be the corresponding fitnesses. The dynamics of the frequency of allele A 1 are described by a difference equation p = pq(w 1 w 2 ). (81) w Note that a polymorphic equilibrium w 1 = w 2. Let us assume that fitnesses depend linearly (the simplest case!) on frequencies w 1 = ap + bq, w 2 = cp + dp, where a, b, c and d are coefficients (chosen in a way that guarantees that fitnesses are positive). One should be able to see that the resulting equation is mahmetically identical to that describing constant viability selection in a diploid one-locus two-allele population. [Notice that in the diploid case, the induced fitnesses of alleles are w 1 = w 11 p + w 12 q, w 2 = w 21 p + w 22 q. While in the diploid model we always assume w 12 = w 21, 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 w A = 1 + sp, w a = 1 + tq, where s and t are positive coefficients. Show that in this case depending on the initial allele frequency p the system evolves towards fixation of allele A or allele a. Find a threshold value p separating the domains of attraction of the two fixation states Major points frequency-dependent selection can easily maintain genetic variation; at a polymorphic equilibrium the fitnesses of all genotypes present are identical. 71

2 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 of mimicry 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 Müllerian mimicry two (or more) unpalatable species resemble each other. The standard explanation for the resemblance both between Batesian model and mimic and between Müllerian mimics is natural selection by predation. The Batesian mimic increases its fitness by evolving to a phenotypic pattern that the predator tends to avoid. In the Müllerian situation both species improve their protection by evolving to a similar phenotypic pattern that the predator learns to avoid more effectively. We consider two haploid populations of different species with non-overlapping generations. We will concentrate on a single locus with alleles A and a in species 1 and a single locus with alleles B and b in species 2. Let w A, w a, w B and w b be fitnesses (viabilities) of the corresponding morphs, and p 1 and p 2 be the frequencies of morph A within species 1 and morph B within species 2, respectively. Let q i = 1 p i, i = 1, 2. The changes in allele frequencies are described by the standard equations p 1 = p 1 q 1 (w A w a )/w 1, p 2 = p 2 q 2 (w B w b )/w 2, (82a) (82b) where w i is the mean fitness of the i-th species (for example, w 1 = w A p 1 + w a q 1 ). We will assume that the fitness of an individual depends on the genetic composition of its own species as well on the genetic composition of the other species, i.e., w A = w A (p 1, p 2 ), w a = w a (p 1, p 2 ) etc. We will make the symmetry assumption that w A (p 1, p 2 ) = w a (q 1, q 2 ), w B (p 1, p 2 ) = w b (q 1, q 2 ) and use linear (i.e., the simplest) fitness functions. Our symmetry assumption implies that there are no differences in fitness between two morphs of a species not related to their frequencies. These assumptions result in the fitnesses w A = C 1 + ap 1 + bp 2, w a = C 1 + aq 1 + bq 2, w B = C 2 + cp 1 + dp 2, w b = C 2 + cq 1 + dq 2. (83a) (83b) Here C 1 and C 2 are positive constants equal to the fitnesses of morphs A and B (or morphs a and b) when fixed in the populations. Parameters a and d characterize the strength of (indirect) within-species interactions. Positive values of a (d) imply that individuals of a given morph benefit when they are common. Negative values of a (d) imply that individuals 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 in another species is common. Negative values of b (c) imply that a given morph benefits when its counterpart in another 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 << C 1, c, d << C 2 ) which allows us to describe the dynamics using differential equations: ṗ 1 = p 1 q 1 [a(p 1 q 1 ) + b(p 2 q 2 )], ṗ 2 = p 2 q 2 [c(p 1 q 1 ) + d(p 2 q 2 )], (84a) (84b) where ṗ i dp i /dt. Note that parameters a, b and c, d in (84) are different from those in (83) by the factors C 1 and C 2, respectively. The main reason for all the simplifying assumptions leading to (84) is that they allow us to study the system analytically. Before analyzing the whole system, it is illuminating to consider several special cases. 72

3 a. Only a single species is allowed to evolve. If allele frequency in a species, say species 1, is somehow fixed (p 1 = const), then ṗ 2 = p 2 q 2 [c(p 1 q 1 ) + d(p 2 q 2 )] 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 coevolving systems including host-parasite systems, competitive and cooperative interactions. In this case, dp 1 dt = p 1q 1 b(p 2 q 2 ), dp 2 dt = p 2q 2 c(p 1 q 1 ). One can get some qualitative ideas about the dynamics of this system by inspecting the right-hand sides of the equations. For example, if both b and c are positive, p 1 increases (dp 1 /dt > 0) or decreases (dp 1 /dt < 0) depending on whether p 2 is large or smaller than 0.5. Similar conclusions can be made about p 2 and for other choices of the signs 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 Separating the variables dp 1 = bp 1q 1 (p 2 q 2 ) dp 2 cp 2 q 2 (p 1 q 1 ). c p 1 q 1 p 1 q 1 dp 1 = b p 2 q 2 p 2 q 2 dp 2, and integrating, we finds that all solutions of (85) satisfy to (85a) (85b) p1 q 1 c p 1 q 1 dp 1 = b p2 q 2 p 2 q 2 dp 2. (86) Because x (1 x) x(1 x) dx = equation (86) can be integrated to 2x 1 d[x(1 x)] du x(1 x) dx = = = ln u, where u x(1 x), x(1 x) u c ln(p 1 q 1 ) = b ln(p 2 q 2 ) + const. Thus, all solutions of (85) satisfy to the following equality (p 1 q 1 ) c = const. (87) (p 2 q 2 ) b Now, we can combine our knowledge of the exact trajectories (given by equation (87)) with the qualitative conclusions about the areas in the phase-plane (p 1, p 2 ) 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 morph increases with an increase in the frequency of its counterpart in the other species. This choice of parameters can be interpreted as describing cooperative interactions between the species. (ii) Parameters b and c are negative. In this case the fitness of each morph decreases with an increase in the frequency of its counterpart in the other species. This 73

4 choice of parameters can be interpreted as describing competitive interactions between the species. (iii) Parameters b and c have different signs. In this case the fitness of a morph in one species increases with an increase in the frequency of its counterpart in the other species, but the fitness of the counterpart decreases with an increase in the frequency of the first morph. This choice of parameters can be interpreted as describing victim-exploiter type interactions between the species. In case (i), the system evolves to a state with both species monomorphic (there are two such states: p 1 = p 2 = 0 and p 1 = p 2 = 1). In case (ii), the system evolves to a state with both species monomorphic (there are two such states: p 1 = 0, p 2 = 1 and p 1 = 1, p 2 = 0). In case (iii), the solutions are neutrally stable periodic orbits encircling the polymorphic equilibrium (1/2, 1/2); any perturbation moves the system to a different periodic orbit. The dynamic behavior of (84) is similar to the behavior of the classical Lotka-Volterra predator-prey system (e.g., Hofbauer & Sigmund, 1988) 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. i. ii. iii. p2 Figure 8: Dynamic regimes with no p1 within-species interactions. (i) Cooperative p1 between-species interactions. p1 (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 and model or as two species belonging to the same Müllerian 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 in our model and no costs or other aspects of fitness. Hence morphs within species are alike in the sense that there is 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 the case when species 1 and/or 2 is unpalatable. Negative values of a and/or d imply that individuals of a given morph benefit when they are rare. This is the case when species 1 and/or 2 is palatable. Parameters b and c characterize the strength of indirect between-species interactions. These interactions are mediated through the predator and thereby indirect. Positive values of b and/or c imply that a given morph benefits when its counterpart in another species is common. This is the case when the other species is unpalatable. Negative values of b and/or c imply that a given morph benefits when its counterpart in another species is rare. This is the case when the other species is palatable or it is unpalatable but has a smaller degree of unpalatability (a weaker Müllerian mimic). 74

5 Our results on dynamic behavior of (84) can be applied to understand the evolutionary dynamics of different mimicry systems. We will consider three different parameter configurations: Classical Müllerian mimicry Classical Batesian mimicry Two unpalatable species have different abundances and/or different degrees of unpalatability Parameter Interpretation configuration a, b, c, d > 0 Within each species a morph benefits when its own frequency or the frequency of its counterpart in the other species increases a, c < 0; b, d > 0 Both species 1 (palatable mimic) and species 2 (unpalatable model) benefit from increasing the model frequency and suffer from increasing the mimic frequency a, b, d > 0, c < 0 Within each species a morph benefits when its own frequency increases. Species 1 (the weaker mimic) also benefits when the frequency of its stronger counterpart (from species 2) increases, but the stronger mimic suffers when its weaker counterpart increases 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 these equilibria in a table: Equilibrium (p 1, p 2 ) Conditions for existence Conditions for stability Use Maple (or your favorite software) to plot phase portraits corresponding to the situations when the equilibria are stable and unstable. Can genetic variation be maintained in the system? Are there any situations when none of the equilibria are stable? - 75

6 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 concentrating on a single diallelic locus with alleles A and a. Let w AA, w Aa and w aa be the fitnesses (viabilities) of genotypes AA, Aa and aa, respectively, and p be the frequency of allele A, with q = 1 p. With random mating the population is in Hardy-Weinberg proportions so that the frequencies of the three genotypes are p 2, 2pq and q 2. The change in p in one generation is described by the standard equation p = p(w A w). (88) w Here w A = pw AA + qw Aa is the average fitness of allele A and w = p 2 w AA + 2pqw Aa + q 2 w aa is the mean fitness of the population. If the fitnesses are constant, the population gradually evolves to a polymorphic equilibrium (with 0 < p < 1) provided there is overdominance (i.e., if w Aa > w AA, w aa ) 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 the frequency-dependence: w AA = p 2 W pqW 12 + q 2 W 13, w Aa = p 2 W pqW 22 + q 2 W 23, w aa = p 2 W pqW 32 + q 2 W 33, where W ij are parameters (i, j = 1, 2, 3) characterizing the extent to which changes in the frequencies of three genotypes influence their fitnesses. Asmussen and Basnayake (1990): a b b d c b, c b a where all coefficients are positive. Dynamics: multiple stable equilibria; plenty of opportunities for the maintenance of variation. Altenber (1991): a (a + c)/2 c b b b, c (a + c)/2 a where some coefficients can be negative. Gavrilets and Hastings (1995): δ β α γ η γ α β δ Under this symmetric model, the dependence of fitness of a heterozygote w Aa on the allele frequency p is described by a (quadratic) function symmetric about 1/2, while w AA and w aa considered as (quadratic) functions of p are reflections of each other about 1/2. For this symmetric model to produce feasible (i.e., non-negative) fitnesses, one has 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 assume without 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), (89) w 76

7 where the mean fitness w can be written as w = 1 2(2 β γ)pq + 2Ωp 2 q 2 with Ω = 1 + α 2β 2γ + 2η. In the Altenberg 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 has two 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 denote these 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. 8 7 γ>1 6 5 USU C UUU USUSU γ<1 5 C 4 SUSUS 3 2 SUUUS SUS beta 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 parameter C that combines several parameters. The parameter C is defined as C = c 1 + c 2 with c 1 α β 2 and c 2 2(γ + η) or, alternatively, as C = Ω + 4γ (1 β) 2. Parameter c 2 is the minimal possible value of the fitness of heterozygote. If β < 1, c 1 determines the minimal possible value of the fitness of homozygotes, c 1 /(c 1 + (1 β 2 )), while if β > 1, c 1 determines the maximum possible value of the fitness of homozygotes, c 1 + β 2. Thus, parameter C characterizes the overall strength of selection. For fitnesses to be feasible c 2 must be non-negative and c 1 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 β 2 if β is positive. Figure 1 shows areas in parameter space corresponding to different patters of existence and stability of equilibria 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 monomorphic equilibrium at p = 0, the polymorphic equilibrium at p = 1/2 and the monomorphic equilibrium at p = 1, while the remaining entries (if any) indicate the stability of the polymorphic equilibria p and p +. The left parabola is described 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 77

8 equilibrium can be stable simultaneously with two monomorphic equilibria, and that two different polymorphic equilibria can be stable simultaneously. Simultaneous stability of different equilibria implies that the outcome of evolution strongly depends on the initial conditions and population history. If parameters change in such a way that the system moves from one area to another, the dynamic system undergoes a bifurcation. For example, a change from USU to USUSU corresponds to a pitchfork bifurcation. Figure 14 also shows that there are two areas with non-standard patterns of stability of equilibria. In the first area (marked UUU), none of the three equilibria (two monomorphic and one polymorphic at p = 1/2) are stable. If parameters change in such a way that the system moves 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 are stable. Numerical iterations of (107) with parameter values corresponding to these areas reveal a variety of complex dynamic behaviors (e.g., cycles and chaos that arises via period-doubling route) similar to those observed in classical ecological models (e.g., May, 1974, 1976; May and Oster, 1976; Hastings et al., 1993). Figure 14 shows that sufficient conditions for the complex dynamic behavior to occur is sufficiently strong overall selection (i.e., small C) and sufficiently strong deleterious effect of heterozygote on homozygotes (i.e., β < 1). 1.0 allele frequency a allele frequency b generation number 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 initial conditions the population evolves to a fixation state or remains polymorphic indefinitely. In the latter case, the gradual changes in the allele frequency towards p = 1/2 are interrupted by apparently chaotic fluctuations that move p 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 monomorphic state. In this case the system exhibits transient chaos (Grebogi et al., 1983; Tél, 1990). In the examples presented in Figure 15, the deterministic phase of the dynamics can last for hundreds of generations, the chaotic phase is extremely short and transient chaos (in Figure 15b) persists for a very long time. In general, the durations of all 78

9 these stages depend both on the parameter values and initial allele frequency (see below the discussion of Figure 16). Both intermittency and transient chaos are known to occur in various dynamical models including the simplest nonlinear model, the single logistic map. Our model allows a very simple explanation of the mechanisms underlying intermittency and transient chaos using the graphical cobwebbing method (May and Oster, 1976). 1.0 a p + max b p + max p min p p min p p 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 frequency at this generation, p. Also shown are the diagonal and the lines corresponding to the unstable polymorphic equilibria p 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 moves towards fixation of allele A. For p values slightly larger than p, p is slightly larger than p, and, thus, p slowly moves towards 1/2. In the neighborhood of p = 1/2, however, the dynamics are chaotic as suggested by the fact that the slope of the graph of p at p = 1/2 is smaller than -1. It takes many generations to leave the neighborhood of p or p + and once the system has left this neighborhood, the dynamics can abruptly become fully chaotic in the neighborhood of p = 1/2, only to get caught in the neighborhood of p or p + again, sooner or later. Between 0 and 1, the graph of p has a minimum, marked min, and a maximum, marked max. During the chaotic phase the allele frequency remains between these points that represent the boundaries of the chaotic attractor. In Figure 15a these 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 present forever. A different situation is described in Figure 15b, where the boundaries of the chaotic attractor lie further from 1/2 than the unstable equilibria p and p +. Now the allele frequency can cross the values p + and p during the chaotic phase. Once this has happened, p slowly evolves to a fixation state (as in Figure 15b). The situation when the boundaries of the chaotic attractor coincide exactly with the unstable equilibria is called a crisis (Grebogi et al., 1983). In the model considered here, if C = 0, the crisis occurs when γ β + 2. Note that in general, the dependence of the length of chaotic transients on the system parameter is proportional to (a a c ) b, where a is the parameter value, a c is the parameter value at which the crisis occurs, and b is the critical exponent, which is equal to 0.5 for a broad class of one-dimensional systems (Grebogi et al., 1987). The parameter values for Figure 15 and Figure 17 below were chosen to result in long transients. The parameters for Figure 16 were chosen slightly different from 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 79

10 allele frequency allele frequency generation number 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 can not only exist simultaneously with stable equilibria, but the former can have much larger domains of attraction than the latter (see Figure 17). For parameter values used in computing the dynamics in Figure 17, both monomorphic equilibria and the polymorphic equilibrium at p = 1/2 are stable to small perturbations. In Figure 17a, the growing regular oscillations in allele frequency are interrupted by apparently chaotic fluctuations that move p back to the neighborhood of 1/2, i.e., one observes intermittency. In Figure 17b, the alterations between apparently deterministic behavior and apparent chaos end at some time point with the population settling down to a polymorphic state at p = 1/2, i.e., one observes transient chaos. The graphical cobwebbing method can be used to understand these kinds of behavior as well. A necessary condition for dramatic changes in allele frequency described in Figures 15 and 17 is strong (at least occasionally) selection. If selection is very weak (i.e., if the differences among coefficients α, β, γ, δ and η are very small) then the difference equation (107) can be approximated by the corresponding differential equation and the only possible outcome of the dynamics is gradual evolution towards an equilibrium. 80

11 9.4 Cycling in systems of ordinary differential equations Let us consider a system of two ordinary differential equations ẋ = f(x, y), ẏ = g(x, y), (90a) (90b) defined on an open set G R 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 with h = 1/(xy) Poincaré-Bendixon theorem ẋ = x(a a 1 x + b 1 y), ẏ = y(b a 2 y + b 2 x), Theorem. Assume that a trajectory γ(x 0, y 0 ) enters and does not leave some closed domain D and that there are no equilibrium points in D. Then there is at least one periodic orbit in D, and this orbit is in the ω-limit set of (x 0, y 0 ). To apply the Poincaré-Bendixon theorem we need to find a region D which contains no equilibrium points and which trajectories enter but do not leave. Both the Poincaré-Bendixon theorem and Dulac s criterion do not work in higher dimensions (n > 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, cycling is 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), ṗ 1 / p 2 = 2bp 1 q 1, ṗ 2 / p 1 = 2cp 2 q 2. Thus, if b and c have the same sign so do ṗ 1 / p 2 and ṗ 2 / p 1, and stable cycling is impossible Poincaré-Andronov-Hopf bifurcation (birth of cycles from an equilibrium) Example. Let us consider a system of non-linear ODE: ẋ = αx βy (x 2 + y 2 )x, ẏ = βx + αy (x 2 + y 2 )y. (91a) (91b) 81

12 The origin (0, 0) is an equilibrium; the corresponding stability matrix is ( ) α β S = β α with eigenvalues λ = α ± iβ. Thus, (0, 0) is a stable focus for α < 0 and an unstable focus for α > 0. Using polar coordinates x = r cos θ, y = r sin θ the system can be rewritten as ṙ = αr r 3, θ = β. Thus, if β 0, θ changes with constant speed β. ṙ = 0 if r = 0 or r = α. If α > 0, ṙ > 0 for 0 < r < α and ṙ < 0 for r > α. In this case, there is a periodic solution which is stable. If α < 0, ṙ < 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 ż = F µ (z), (92) depending on some parameter µ and defined on an open subset of R n. Let z be an equilibrium of (92). Assume that all eigenvalues of the Jacobian have negative real parts, with the exception of one pair of complex conjugate eigenvalues α(µ) ± iβ(µ). Let for some µ 0, these two eigenvalues be pure imaginary: α(µ 0 ) = 0, β(µ 0 ) 0, and let d dµ α(µ 0) 0. Then for µ sufficiently close to µ 0, a periodic solution bifurcates from equilibrium z. If the first Lyapunov value L 1 (defined below) is negative, the bifurcation is supercritical (i.e. the periodic orbit exists for α(µ) > 0 and is stable). If L 1 is positive, the bifurcation is subcritical (i.e. the periodic orbit exists for α(µ) < 0 and is unstable). For small values of α(µ), the period and the amplitude of the periodic orbit are approximately 2π/ β(µ) and α(µ), respectively. First Lyapunov value L 1. Let us consider a system of two ordinary differential equations dx =ax + by + P (x, y), dt (93a) dy =cx + dy + Q(x, y). dt (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) =P 2 (x, y) + P 3 (x, y) +..., Q(x, y) =Q 2 (x, y) + Q 3 (x, y) +..., (94a) (94b) where P 2 (x, y) =a 20 x 2 + a 11 xy + a 02 y 2, P 3 (x, y) =a 30 x 3 + a 21 x 2 y + a 12 xy 2 + a 03 y 3, Q 2 (x, y) =b 20 x 2 + b 11 xy + b 02 y 2, Q 3 (x, y) =b 30 x 3 + b 21 x 2 y + b 12 xy 2 + b 03 y 3. (95a) (95b) (95c) (95d) 82

13 The first Lyapunov value L 1 is defined as (Bautin 1938). L 1 = π 1 4(det S) 3/2 b [ ac(a a 11 b 02 + a 02 b 11 ) + ab(b a 20b 11 + a 11 b 20 ) + c 2 (a 11 a a 02 b 02 ) 2ac(b 2 02 a 20 a 02 ) 2ab(a 2 20 b 20 b 02 ) b 2 (2a 20 b 20 + b 11 b 20 ) + (bc 2a 2 )(b 11 b 02 a 11 a 20 ) (a 2 + bc) (3(cb 03 ba 30 ) + 2a(a 21 + b 12 ) + ca 12 bb 21 ) ] The expression for L 1 looks horrible and almost impossible to use practically. However, using modern symbolic manipulation 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 L 1 = 8. Thus, the bifurcation is supercritical (i.e. the periodic orbit exists for α(µ) > 0 and is stable) which is what we already found using exact analysis. Example 2. Let us consider a system of two ordinary differential equations x ẋ = rx(1 x/k) β x + C y, x ẏ = (kβ x + C m)y, where all coefficients are positive. One can interpret x as the density of a prey population which grows logistically in the absence of predator (if y = 0, then ẋ = rx(1 x/k)), and y as the density of predator which experiences densityindependent mortality (with rate m). Terms β x x+c y and kβ x x+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 = Kkβ K+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 the corresponding stability matrix are mr(kkβ Km Ckβ mc) T rs =, 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 Poincaré- Andronov-Hopf bifurcation. For this bifurcation, the first Lyapunov value is 4Cβk(K C)r2 L 1 = K 2 (K + C) 2, 83

14 and is positive (because at m = m, K/C = (kβ + m)/(kβ m) > 1). Thus, the bifurcation is supercritical (the periodic orbit exists for m < m and is stable). Summarizing, if m > m, the predator population cannot maintain itself whereas if m < m both species coexist. If m < m < m, the coexistence is non-oscillatory whereas if m < m, there are stable oscillations of the densities of both populations. Example 3. For the coevolutionary model (84), the first Lyapunov value corresponding to Poincaré-Andronov- Hopf bifurcation from the doubly polymorphic equilibrium (1/2, 1/2) is L 1 = 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 conditions guarantee that (i) monomorphic equilibria are unstable, (ii) singly polymorphic equilibria do not exist, and (iii) the determinant of the stability matrix evaluated at the doubly polymorphic equilibrium (1/2, 1/2) is positive. Thus, if a + d < 0, the equilibrium (1/2, 1/2) is a stable focus. If a + d > 0, this equilibrium is an unstable focus. The sign of L 1 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 = Unstable cycle: a = 1.01, b = 1.3, c = 1.2, d = 1. Illuminating initial conditions are p 1 = 5, p 2 = 0.75 and p 1 = 0.5, p 2 =. 84