Carrying capacity in sexual and asexual populations
I assume discrete, non-overlapping generations in the host population. I also assume that resources limit population size, rather than space (e.g. access to nest sites). I first show the standard solution for carrying capacity in an asexual population. I then show how carrying capacity is reduced in a sexual population, and how this stems from the cost of producing males. The implication here is that invasion by an asexual lineage will increase the population size, which may, in turn, increase virulence.
Using standard terminology, let b be the number of offspring produced by females in the absence of any competition. Let d be the death rate following reproduction, also in the absence of competition. Now let a represent the sensitivity of the birth rate to density, so that the actual birth rate, B, is equal to: b – aN, where N is in the number of hosts competing for resources in the relevant patch. Similarly, let c represent the sensitivity of the death rate to density, so that the actual death rate, D, is equal to: d + cN. Note that D is constrained to be less than or equal to one.
The recursion equation for the number of individuals at time t + 1, as a function of Nt is:
The change in population size is calculated as:
The equilibrium density, Kasex, is thus attained when B = D. Hence as reviewed by Gotelli (1995), the carrying capacity is:
Using this value for Kasex, eqn 1 can be converted into the more familiar form of the discrete-time logistic equation:
where r = b – d. This formulation assumes that all individuals are reproducing (e.g. asexual). For a dioecious (or gonochoric) population, however, males are not directly producing offspring. Hence the calculation for equilibrium in a sexual population is attained when (1–s)B = D, where s is the frequency of males in the population. It is easy to show that the carrying capacity for a sexual population, Ksex, is equal to:
Clearly, an asexual population would have a higher carrying capacity (see also Doncaster et al., 2000). Assuming the death rate is density independent (i.e. c = 0), the ratio of the carrying capacity for an asexual population divided by the carrying capacity for a sexual population reduces to:
Note that for d = 0 (an immortal population), Kasex = Ksex (see also Doncaster et al., 2000). For d > 0 and s = ½, we get
Therefore, the asexual population would be expected to be larger than the sexual population, and invasion of a sexual population by an asexual lineage would be expected to increase the population size. For example, for b = 4 and d = 1, the asexual population would be expected to be 1.5 times larger than the sexual population (Kasex/Ksex =(4 – 1)/(4 – 2) = 1.5).
This result seems counterintuitive at first. Why would an asexual population be more numerous on the same resource base than a sexual population? The result, however, makes more sense when one realizes that sexual females must, on average, produce two offspring at carrying capacity (Ksex, assuming a 1 : 1 sex ratio), whereas asexuals females need to make only 1 offspring at carrying capacity (Kasex). Hence, asexual populations have a lower R* (Tilman, 1982). And this is precisely why asexuals can invade a sexual population at its carrying capacity. Specifically, at Ksex, asexuals produce more offspring per capita than sexuals, which is what allows the clone to spread in the sexual population (Fig. 1).
Figure 1. Graphical representation of the cost of sex. The red line gives the per capita birth rate for the sexual population; the blue line gives the per capita birth rate for the asexual population. The horizontal black line gives the death rate (here assumed to be one for an annual). The open circle gives the carrying capacity for the sexual population (n = 20 000), and the open triangle gives the carrying capacity for the asexual population (n = 30 000). The carrying capacities were calculated from the equations for Ksex and Kasex in the text. Parameter values as follows: b = 4; a = 0.0001; s = 0.5. Note that the per capita birth rate for the asexual population is equal to two at the carrying capacity for the sexual population.
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I constructed a computer-simulation model to determine the outcome of competition between sexual and asexual individuals in the presence of potentially coevolving parasites. The goal was to determine whether: (1) invasion of sexual populations by asexual clones increased population size, (2) whether such an increase in population size lead to increased parasite virulence and (3) whether parasites could prevent the fixation of clonal mutants.
I assumed an annual host population, so the death rate, D, was equal to one (d = 1, and c = 0). I also assumed two loci in a haploid host population, with three alleles at both loci (giving nine possible genotypes). Infection by parasites required that parasites match the host at both loci, which is the standard assumption for the matching alleles model for infection (Frank, 1993; Otto & Michalakis, 1998; Agrawal & Lively, 2002). I did not simulate the effects of infection as determined by gene-for-gene interactions; but elsewhere, we have found that the coevolutionary system behaves dynamically like a matching-alleles system over most of the continuum between matching alleles and gene-for-gene genetics (Agrawal & Lively, 2002). In general, the matching-alleles model is a good approximation for the genetic basis of infection when different parasite genotypes infect different host genotypes (for experimental evidence, see Carius et al., 2001; Dybdahl et al., 2008).
Under these assumptions, the recursion for the number of asexual individuals with genotype ij at time t + 1 is:
where gives the total number of asexual individuals with the ijth genotype at time t + 1; gives the number of infected (I) asexual individuals with the ijth genotype at time t; and Aij(U) gives the number of uninfected (U) asexual individuals for the ijth genotype at time t. In addition, b(I) gives the intrinsic birth rate of infected individuals, whereas b(U) gives the intrinsic birth rate of uninfected individuals. Finally, a(I) reflects the sensitivity to total density for infected individuals, and a(U) gives the sensitivity of uninfected individuals to total host density, where Nt is the total number of sexual and asexual individuals in the population. Similarly, the recursion for the number of sexual individuals with genotype ij at t + 1, assuming no recombination between resistance loci, is:
where is the total number of sexual individuals with the ijth genotype at time t+1; s is the frequency of males in the sexual population; Sij(I) is the number of infected individuals of the ijth genotype in the sexual population at time t; and Sij(U) is the number of uninfected individuals of the ijth genotype in the sexual population at time t. If there is recombination between loci, the number of sexual individuals of each genotype, after accounting for recombination is:
where Si is the number of individuals with allele i at the first locus; Sj is the number of individuals with allele j at the second locus; and ρ is the frequency of recombination (see Hartl & Clark, 1989). Finally, I used the standard assumption that the probability of infection for each host genotype depends on the frequency of the matching genotype in the parasite population. Hence, we get:
where Pij gives the frequency of parasite genotype ij at time t. Similarly, the number of infected and uninfected asexual individuals at time t is:
The simulation began with a sexual population at its carrying capacity (eqn 5). There were two haploid loci, each with three alleles that determined ‘self’ for the host; parasites that matched at both loci infected the host; otherwise the parasite was killed by the host’s self-nonself recognition system. Host-genotype frequencies began in linkage equilibrium, calculated from allele frequencies that were randomly assigned at the beginning of each run. Parasite-genotype frequencies were set as being equally common at the start of the simulation.
Migration into the sexual host population was allowed. In the results presented below, the probability that a single, uninfected individual entered the population was set to 0.10 for each sexual genotype. Similarly, the probability that a single, infected individual entered the population was set to 0.02 for each sexual genotype; hence, the simulation also allowed for a low level of parasite immigration. A single clonal mutant entered the simulation at generation 1000. Clones did not immigrate into the population, and the clone was considered extinct if the number of clones dropped below one in any generation. The recombination rate, ρ, in the sexual population was set to 0.2, as intermediate values of recombination were favoured in similar models (Peters & Lively, 2007).
To test the generality of the results, I also ran the simulation for a haploid, two-locus, two-allele system, thereby reducing the number of possible genotypes from nine to four. I also ran the simulation for a diploid system, assuming two alleles at each of two loci, giving nine possible genotypes, as in the original haploid model. In the diploid model, I assumed that the clone had the same genotype as one of the double homozygotes in the sexual population. All other assumptions were retained as for the haploid, two-locus, three-allele model described above.
In all simulations, the effect of infection (virulence) emerged as the difference in per capita growth rates between infected and uninfected individuals within each reproductive mode. For the sake of calculating the effect of infection over time, virulence at time t (Vt) was calculated (assuming D = 1) for both sexual and asexual populations as,
Thus, virulence depended on total host density (Nt), as well as on the birth rates of infected individuals relative to uninfected individuals (see also Lively, 2006).