Chapter 12: Using an Evolutionary Invasion Analysis to Understand the Evolution of Anisogamy

Inputs (Enter first)

SP[x_] := Simplify[PowerExpand[x]]

Off[General::"spell"]

Off[General::"spell1"]

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Model set up (Enter second)

Example: Evolution of Anisogamy

Most sexually reproducing species are anisogamous, meaning that the gametes that unite to produce a diploid are different in size (e.g. sperm and eggs, or pollen and ovules). There are exceptions to this rule, however, and some species have gametes that are equal in size (isogamy). Why has anisogamy evolved and why is it so prevalent  (Parker et al. 1972; Randerson and Hurst 2001; Bulmer and Parker 2002)? To address this question, we analyze a model of gamete size in a sexually reproducing species.  Specifically, we assume that the species exhibits an alternation of generations (see Figure) and that gametes are produced by the haploid phase of the life cycle.

Suppose that there is a single gene coding for the size of the gamete and that a gamete carrying allele s has size S.  At time t, the population consists of G(t) gametes uniting to produce G(t)/2 zygotes.  When two gametes of size S1 and S2 fuse, the resulting diploid zygotes survive with some probability, P(S1+S2) (e^(-α G(t)/2)), where P is an increasing function of the combined gamete size and (e^(-α G(t)/2)) is a density-dependent factor similar in form to the Ricker model.  The mature diploid individuals then gather resources, undergo meiosis, and produce haploid offspring. Through its haploid offspring, a diploid individual with alleles s1 and s2 produces R/(S1+c) gametes of size S1 and R/(S2+c) gametes of size S2, where R represents the reproductive resources of a haploid and c represents a fixed amount of resources that are required for gametogenesis, regardless of gamete size.

Suppose the population is monomorphic for allele s. There will be G(t)/2 zygotes of genotype ss, each of which survive to maturity with probability P(2S) (e^(-α G(t)/2))and produce a total of 2R/(S+c) gametes of size S.  The recursion equation for the number of resident gametes is:
			
				G(t+1) 	= G(t)/2	P(2S) (e^(-α G(t)/2))2R/(S+c)
					= G(t) P(2S) (e^(-α G(t)/2))R/(S+c)
					
We can write this recursion equation in a form that is easy to use in Mathematica:

resident = 
        G P[2  S] Exp[-α G/2] R/(S + c);

Questions

Question 1:  Determine the nonzero equilibrium of the resident population and specify any conditions that must be met for this equilibrium to be positive (assume that P, α, R, c, S, and G are positive)

Question 2:  Determine whether the nonzero equilibrium of the resident population is stable.

Question 3:  What is the reproductive factor of a mutant allele?

Suppose that the resident population has reached the stable equilibrium identified above and that a mutant allele, sm, arises (see Figure).  Letting M(t) represent the number of mutants at time t and SM represent the size of gametes produced by a haploid mutant, the recursion equation for the number of resident gametes is (fill in BLANK and state assumptions):
			
				M(t+1) 	= M(t) BLANK
					= M(t)  λ
					
where λ is the reproductive factor of the mutant.
					
Write this reproductive factor in a form that is easy to use in Mathematica:

mutantλ =  BLANK;

Question 4:  Determine the potential ESS under the assumption that P(x) = (x^n) where n is some positive constant.

From now on, we will use the following function for the survival of a zygote as a function of its size:

P[x_]=(x^n);

Using this function in your equation for mutantλ, identify any potential ESS.

What conditions on the parameters must be met for the above ESS to be valid (i.e., give a positive value for the gamete size).?

Question 5:  When is this potential ESS convergence stable?

Question 6:  When does this potential ESS satisfy the second derivative condition in Recipe 12.2?

Under what parameter values is there a gamete size, S*, that is convergence stable but that does not satisfy the second derivative condition for an ESS?  In this case, the population eventually branches and becomes anisogamous, with some small gametes and some large gametes.

Question 7:  Plot the mutant's reproductive factor, λ, as a function of SM for c = 1 and values of n right above and below where you expect branching to occur.

Fill in the "BLANK"s in the following command to generate the appropriate graphs:

Plot[mutantλ /. G -> BLANK /. S -> BLANK /. 
          n -> BLANK /. c -> 1, {SM, 0, 2}]

Think about what the shape of these curves tell you about the fate of mutant alleles when the resident population is at the potential ESS (see also Figure 12.5).

Why might n in the survival function, P(x) = (x^n), play such a large role in the evolution of isogamy?

Answers

Question 1:  Determine the nonzero equilibrium of the resident population and specify any conditions that must be met for this equilibrium to be positive (assume that P, α, R, c, S, and G are positive)

Solve[resident == G, G]

As all of the parameters are positive, Log[((c + S)/(R P[2 S]))] must be negative for the equilibrium to represent a positive number of gametes.  Consequently, the number of gametes produced per haplod, R/(S+c), times the probability of survival P[2 S] must be greater than one.

Question 2:  Determine whether the nonzero equilibrium of the resident population is stable.

Taking the derivative of the recursion equation with respect to the variable and plugging in the equilibrium:

D[resident, G] /. 
      G -> -((2 Log[((c + S)/(R P[2 S]))])/α)

We know that Log[((c + S)/(R P[2 S]))] must be negative for the equilibrium to be valid (see Question 1).  Consequently, the above derivative is less than one and the equilibrium is stable (see Recipe 5.4).

Question 3:  What is the reproductive factor of a mutant allele?

Suppose that the resident population has reached the stable equilibrium identified above and that a mutant allele, sm, arises (see Figure).  Letting M(t) represent the number of mutants at time t and SM represent the size of gametes produced by a haploid mutant, the recursion equation for the number of resident gametes is:
			
				M(t+1) 	= M(t) P(SM+S) (e^(-α G(t)/2))R/(SM+c)
					= M(t)  λ
					
where λ is the reproductive factor of the mutant.  This assumes that every mutant gamete finds and unites with a resident gamete (i.e., the number of mutant zygotes produced is just M(t)), that the amount of density dependence is largely determined by the number of residents in zygotes (G(t)/2), and that only the haploids that carry the sm allele produce mutant gametes, at a size equal to SM.
					
Write this reproductive factor in a form that is easy to use in Mathematica:

mutantλ = 
        P[SM + S] Exp[-α G/2] R/(SM + c);

Question 4:  Determine the potential ESS under the assumption that P(x) = (x^n) where n is some positive constant.

P[x_]=(x^n);

Using equation (12.15c) in Recipe 12.2:

D[mutantλ, SM] /. SM -> S // Factor

Solve[% == 0, S]

For the above ESS to be valid (i.e., give a positive value for the gamete size), n must be less than 2.  In other words, the survival probability of a zygote must not rise faster than a quadratic function of its size.

Question 5:  When is this potential ESS convergence stable?

Using equation (12.15b) in Recipe 12.2:

D[D[mutantλ, SM] /. SM -> S, S] /. 
        S -> ((c n)/(2 - n)) // Factor

Assuming that n < 2 (for the potential ESS to be valid), this is always negative.  Therefore, the potential ESS is always convergence stable.

Question 6:  When does this potential ESS satisfy the second derivative condition in Recipe 12.2?

Using equation (12.15d) in Recipe 12.2:

D[mutantλ, {SM, 2}] /. SM -> S /. 
        S -> ((c n)/(2 - n)) // Factor

%25 /. 
            G -> -((2 Log[((c + S)/(R P[2 S]))])/α) /. 
          S -> ((c n)/(2 - n)) /. n -> 1.1 /. 
      c -> 1

Assuming that n < 2 (for the potential ESS to be valid), this is negative only when n < 1.  Therefore, the point  S = ((c n)/(2 - n)) is an ESS only when n < 1.  When n > 1, S = ((c n)/(2 - n))represents an evolutionary branching point.

Question 7:  Plot the mutant's reproductive factor, λ, as a function of SM for c = 1 and values of n right above and below where you expect branching to occur.

Plot[mutantλ /. 
              G -> -((2 Log[((c + S)/(R P[2 S]))])/α) /. S -> ((c n)/(2 - n)) /. n -> 1.1 /. c -> 1, {SM, 0, 
        2}]

Here the potential ESS lies at about 1.2 (where λ = 1).  Mutant alleles that cause either smaller gamete sizes or larger gamete sizes than this resident strategy can invade (λ > 1).

Plot[mutantλ /. 
              G -> -((2 Log[((c + S)/(R P[2 S]))])/α) /. S -> ((c n)/(2 - n)) /. n -> 0.9 /. c -> 1, {SM, 0, 
        2}]

Here the potential ESS lies at about 0.8 (where λ = 1).  Mutant alleles that cause either smaller gamete sizes or larger gamete sizes than this resident strategy cannot invade (λ < 1).

What can we conclude from this analysis? First, when 0 < n < 1, the survival function P is concave down, meaning that survival does not rise very rapidly with zygote size.  In this case, there exists an ESS gamete size of \(\(S\^*\) = \(c\ n\)\/\(2 - n\)\), and we expect isogamy to be evolutionarily stable. On the other hand, if the survival function is concave up (i.e., 1 < n < 2) then we expect natural selection to drive the population toward a gamete size of ((c n)/(2 - n)), which is convergence stable. Once there, however, natural selection becomes disruptive, favouring the evolutionary diversification of gamete size. Thus, we expect the evolution of anisogamy to occur when the intrinsic rate of survival of zygotes rises faster than linearly with zygote size, but not too fast (1 < n < 2). This model therefore suggests that the shape of the relationship between zygote size and survival is critical in determining whether isogamy or anisogamy is favoured by natural selection (Randerson and Hurst 2001).

References

Bulmer, M. G., and G. A. Parker. 2002. The evolution of anisogamy: a game-theoretic approach. Proceedings of the Royal Society of London Series B-Biological Sciences 269:2381-2388.

Parker, G. A., R. R. Baker, and V. G. F. Smith. 1972. The origin and evolution of gamete dimorphism and the male-female phenomenon. Journal of Theoretical Biology 36:181-198.

Randerson, J. P., and L. D. Hurst. 2001. The uncertain evolution of the sexes. Trends in Ecology & Evolution 16:571-579.