The first method for determining frequencies of haploids and diploids for dioecious, isomorphic species was proposed by (Richerd et al. 1993). In this model, each individual's lifespan is constrained within a single time step; there is no survival of adults from one step to the next. This was modeled as follows:
Where is the number of haploids (both females and males), is the number of diploids, is the per capita fecundity rate, and is the survivorship rate from birth to reproduction. Richerd did not examine the conditions necessary for equilibrium of these populations and their resulting haploid-diploid ratio. However, to display the differences from this version to our models, we must explore these conditions in some detail.
From the Richerd model, we determined that a population of this form will only reach a steady equilibrium and a stable haploid-diploid ratio with two restrictive conditions. Solving for equilibrium by setting and yields
When these two conditions (Eqs. A.2, and A.3 or A.4) are met, the population will be at equilibrium with a constant haploid-diploid ratio. However, the only way this equilibrium can be reached or maintained is if the initial values of and already satisfy these conditions. When the above conditions are met, and , the ratio of haploids to diploids will be . If only (Eq. 7) is met, the population will exhibit constant oscillations around an equilibrium, which corresponds with fluctuating haploid-diploid ratios. Both have a period of two time steps. An oscillation of this type was also noted by (Scrosati and DeWreede 1999) for the same type of model with no survivorship from one time step to the next.
The drawback to this model is that a very limited range of values will yield equilibria. This is partly because it does not allow for adult survivorship from one time step to the next, which is a biologically implausible assumption for many species with haploid-diploid life cycles. As a result of this restriction, the number of haploids at a given time step cannot affect the number of haploids at the next time step; this is determined solely by the abundance of diploids. The reverse is also true, which explains why the behavior of these populations, unless started at equilibrium values, will persist as stable oscillations with a period of two timesteps. Thus, this model cannot yield general predictions about haploid-diploid ratio.
We simulated the Richerd model to
illustrate the oscillations predicted; we chose parameters so that the population
would cycle around an equilibrium
. For this simulation, we used , .
Richerd, S., D. Couvet, and M. Valero. 1993. Evolution of the alternation of haploid and diploid phases in life cycles. II. Maintenance of the haplo-diplontic cycle. Journal of Evolutionary Biology 6:263280.
Scrosati, R., and R. E. DeWreede. 1999. Demographic models to simulate the stable ratio between ecologically similar gametophytes and tetrasporophytes in populations of the Gigartinaceae (Rhodophyta). Phycological Research 47:153157.