In the course of assessing the consequences of nonequilibrium dynamics to the theory of
life history evolution, [sb04a] found themselves compelled to
reconsider a classic evolutionary question:
When will a favorable mutation spread through a
population? The traditional answer afforded by population genetics
[ck70] is that the
probability of a beneficial mutation's being fixed is proportional to its selective
advantage (generally assumed to be small),
s = r2 -
r1
(1)
where ri = ln wi,
and w1 and w2
are respectively wild type and mutant fitness values.1 The assumptions underlying this result are notable both for what they
include and what they omit. Included is the idea of demographgic stochasticity,
which leads to the conclusion that most mutants are lost by chance (sampling error),
fitness advantage or no.
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Figure 1. Dynamically mediated coexistance in life history evolution, where X1 and X2 are the numbers of competing haploid genotypes with growth rates, Xi(t+1) = wi(Ei, X) Xi(t), Ei is reproductive effort and X, the the total population. The functional form of wi(·) is given by Equations (2) with B0 = 20, p0 = 0.4, c = 0.125 and d = 0. In this picture, E1 = 0.9 and E2 = 0.207. The three invariant sets (two 2-cycles and a fixed point) are color coded: red denotes an attractor; blue, a saddle. Click the figure to enlarge. |
Excluded is the possibility of frequency-dependent selection, i.e., that the w's in Equation (1) depend on the relative abundances of the two types.2
Also excluded is any kind of explicit consideration of complex (periodic or chaotic) population dynamics [m76]. This is important, because it turns out that non-stationary dynamics introduce a form of frequency dependence even in the case of haploids and that this, in turn, can result in the creation of new attractors.
An example is given in Figure 1. Here we study the competition between haploid genotypes with different life histories. For concreteness, we assume that
wi = Bi(Ei,X) + pi(Ei,X) (2a)
where, Bi(·) is the effective fecundity [s04] of the ith genotype, pi(·) is the probability of post reproductive survival, Ei is reproductive effort [w66] and X, the combined density of both genotypes. For concreteness, we further assume that
B(Ei,X) = B0(2Ei - Ei2)e-cX (2b)
p(Ei,X) = p0(1-Ei2)e-dX (2c)
In Figure 1, B0 = 20, p0 = 0.4, c = 0.125, d = 0, E1 = 0.9 and E2 = 0.207. For these, parameter values, there are two invariant sets on the axes - a 2-cycle and a fixed point - both of which are saddles, and an interior 2-cycle that is an attractor. The boundary sets, correspond to the long-term behavior of each genotype in the absence of the other, while the interior attractor gives the long-term behavior of the two genotypes when they co-occur. Notice that because the boundary invariant sets are saddles, they correspond to systems that are invasible by the other genotype.
To summarize, Figure 1 is an example of dynamically mediated coexistence. Put another way, the existence of an interior invariant set is incompatible with constant values of the total density, X. In detail, the condition for genotype i to outcompete genotype j,
{[(Bi - Bj)] / [(pj - pi)]} > e(c-d)X, (3)
is a "yes or no" proposition for fixed values of X.
Mechanism.
By varying E2, we can determine the dynamical origins of coexistance. This is shown in Figure 2, which is an animation. Here, one can watch the interior attractor emerge from the cycle on the X1-axis via what is called a transcritical bifurcation [gh83]. Prior to the bifurcation, the axis cycle is an attractor and the cycle that will become the interior attractor, a saddle (not shown) in the 4th quadrant of the X1-X2 plane. At this point, all initial conditions corresponding to positive numbers of both genotypes tend to the attractor on the X1 axis, essentially because the fixed point on the X2 axis is a saddle with its unstable manifold pointing down and to the right, and because there are no other invariant sets in the quadrant.
With increasing values of E2, the cycles collide and there is an exchange of stability: the axis cycle becomes a saddle (with its unstable manifold pointing up and to the left) and, what has become an interior cycle in the 1st quandrant, an attractor. At this point, all initial conditions within the quadrant tend to the interior cycle, and the genotypes coexist.
With still further increases in E2, the interior cycle moves up and to the left. Meanwhile, the saddle on the X2 axis undergoes a period-doubling bifurcation, emitting a period-2 saddle (also on the axis) and itself becoming a repeller (unstable in 2 directions). Subsequently, the interior cycle collides with the axis cycle, and there is a second exchange of stability. This stabilizes the boundary cycle, while, what was formerly the interior attractor, becomes a saddle as it passes into the biological never-never land of the second quadrant. At this point, all initial conditions corresponding to positive numbers of both genotypes tend to the cycle on the X2 axis, and the system has exited the parameter regieme of coexisting genotypes.
Click the figure to view the animation. For a non-animated figure showing the system for representative values of E2 go here.
It is worth emphasizing that the bifurcation sequence shown in Figure 2 is but one of many. For example, the interior 2-cycle can vanish before colliding with the saddle, which in this case is a fixed point, on the X2axis. Precisely at the bifurcation value of E2, there exists a line of neutrally stable equilibria connecting the fixed point on the axis and the bifurcating interior point. An animation illustrating this scenario (same parameters, but with p0 = 0.70), is given in Figure 3.
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Figure 2. Origin and death of the interior period-2 attractor. With increasing values of E2 ("P" in the figure), a stable 2-cycle emerges from the attractor on the X1-axis, which loses stability and becomes a saddle (transcritical bifurcation). With further increases in E2, the attractor moves up and to the left. As it does so, the saddle equilibrium on the X2-axis period-doubles. Eventually, there is a second transcritical bifurcation. The interior cycle passes through the boundary cycle, which is thereby stabilized. Click the figure to enlarge and again to start the animation. |
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Figure 3. As in Figure 2, but for a larger value of p0 (0.7 vs. 0.4). In this case, the interior 2-cycle dies before ploughing into the E2 axis. Click the figure to enlarge and again to start the animation. |
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Figure 4. As in Figure 3, but with E1 = 0.55. In this case, the interior 2-cycle returns to the E1 axis without undergoing a bifurcation. Click the figure to enlarge and again to start the animation. |
Alternatively (Figure 4), the interior cycle can "turn around" without bifurcating and return to the X1 axis. Nor is coexistance necessarily associated with the existance of a 2-cycle. To the contrary, it would appear that over a suitably range of parameter values, there is an infinite number of interior cycles corresponding to monogenic systems that are mutually invasible by the other genotype [sb04b].
Abundance of Behavior.
How common is such behavior? To answer this question, we study the entire E1-E2 plane and determine which genotypes can persist in isolation and whether or not positive boundary solutions are invasible by the other genotype. An example of such a calculation is shown in Figure 5, wherein the remaining parameters are as above. In this, case, there is a large region (shown in violet) of parameter space for which coexistance obtains. As one expects, the size of this region depends on the remaining parameter values (Figure 6). In particular, as one moves away from values that correspond to cyclic or chaotic behavior on the boundary, the region of coexistance disappears.
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Figure 5. The E1-E2 plane divided into regions according to long term coexistance patterns: very dark red (near the origin) - neither genotype survives; red - only genotype 1 survives; orange, only genotype 2; light blue, both can survive and genotype 1 is invasible by 2; dark blue, both can survive and genotype 2 is invasible by 1; violet, both survive and each can invade the other. |
References.
[cw75] Charlesworth, B. and J. A. Williamson. 1975. The probability of survival of a
mutant gene in an age-structured population and implications for the evolution of life
histories. Genet. Res. Camb. 26: 1-10.
[ck70] Crow, J. F. and M. Kimura. 1970. An Introduction to Population Genetics Theory.
Harper and Row. New York.
[f30] Fisher, R. A. 1930. The Genetical Theory of Natural Selection. Clarendon Press.
Oxford.
[gh83] Guckenheimer, J. and P. Holmes. 1983. Nonlinear Oscillations, Dynamical Systems
and Bifurcations of Vector Fields. Springer-Verlag. New York.
[l45] Leslie, P. H. 1945. On the use of matrices in certain population mathematics.
Biometrika. 33: 183-212.
[l48] Leslie, P. H. 1948. Some further notes on the use of matrices in certain population
mathe-matics. Biometrika. 35: 213-245.
[m62] MacArthur, R. H. 1962. Some generalized theorems of natural selection. Proc. Nat.
Acad. Sci. USA. 48: 1893-97.
[m76] May, R. M. 1976. Simple mathematical models with very complicated dynamics.
Nature. 262: 459-467.
[r92] Roff, D. A. 1992. The Evolution of Life Histories. Routledge, Chapman and
Hall. New York.
[s04] Schaffer, W. M. 2004.
Life histories, evolution and salmonids. Pp. 20-51. In,
Stearns, S. and A. Hendry (eds.) Evolution Illuminated: Salmon and their Relatives.
Oxford Univeristy Press.
[sb04a] Schaffer, W. M. and T. V. Bronnikova. 2004a.
[sb04b] Schaffer, W. M. and T. V. Bronnikova. 2004b.
[s76] Stearns, S. C. 1976. Life history tactics: a review of the ideas. Q. Rev. Biol.
51: 3-47.
[s77] Stearns, S. C. 1977. The evolution of life history traits A critique of the theory and
a review of the data. Ann. Rev. Ecol. Syst. 8: 145-171.
[s92] Stearns, S. C. 1992. The Evolution of Life Histories. Oxford University Press.
Oxford, UK.
[w66] Williams, G. C. 1966. Adaptation and Natural Selection: A Critique of Some
Current Evolutionary Thought. Princeton Univ. Press. Princeton, N. J.
1. In the haploid case, genic and genotypic fitnesses (= rates of multiplication) are the same - i.e., xi(t+1) = wi xi(t). In the diploid case, wi = Sqjwij, where the summation is over j, and wij is the fitness of the ij genotype.
2. For diploids, frequency dependence is unavoidable - i.e., if the genic fitnesses differ, the geontypic frequencies necessarily depend on gene frequency. In the case of 2 alleles, the qualitative consequences of frequency dependence, i.e., the creation of new equilibria, are avoided if heterozygote fitness lies between that of the homozygotes. That is, with w00 ≤ w01 ≤ w11 or w00 ≥ w01 ≥ w11, one gene or the other prevails, and its frequency goes to 100% in the limit of large time. .