*Eulimnadia texana* is an extremely unusual species in having three sexes (at least genetically).
Sex in Eulimnadia is determined by a gene or gene region as follows:

Female Female Male

Females reproduce by one of two means: self-fertilization and/or sex with males. Males cannot self and females cannot fertilize one another.

Consequently, the following matings can occur:

Interestingly, the sex ratio varies from pool to pool, with males ranging from 0-35%.

A number of evolutionary questions arise in this system:

- Why are males maintained?
- What sex ratio is expected within the population?
- What are the costs of inbreeding? outbreeding?

- What information do you really want to know?
- Boil the question down!

- Start with the simplest, biological reasonable description
- Make sure that results from the model can address the question

- Define the variables in the model
- Consider how the variables relate to one another
- Define the parameters in the model
- Consider whether any of the parameters are redundant

- Write down the equations (a diagram or table may help)
- Perform checks (eg. do the relationships among the variables still hold?)

- Make sure analysis addresses the question
- Determine how general the results are

- Consider alternatives to the simplest model
- Expand model, if appropriate, and repeat steps 3-5

- Do the results answer the biological question?
- Are the results counter-intuitive? Why?
- Relate the results to data or to potential experiments

- u: frequency of ss males
- v: frequency of Ss females
- w: frequency of SS females

**Parameter:** "An arbitrary constant whose value describes a characteristic of a system"

- 1-: fitness of males relative to fitness of females
- 1-: fitness of individuals produced by selfing
- : proportion of eggs that will be mated with male sperm when a male is encountered
- : proportion of eggs not fertilized by males that are selfed

is the average fitness and equals the sum of the numerators. We must divide by the average fitness so that u', v' and w' remain frequency measures that sum to one.

At equilibrium u'=u, v'=v, and w'=w. This gives us three equations in three unknowns (u, v, w).

Solving these three equations indicates that there are only two biologically interesting equilibrium states.

- w=1 and only SS females exist No males, No outcrossing
- u= Males present

**Analysis Step 2:** Determine which equilibria are stable

The w=1 equilibrium is the only stable equilibrium when . When this condition fails to hold, the population will converge towards the second equilibrium with males present.

Therefore males will be maintained in the population if they are fairly fecund ( high) and fairly viable (1- high), if females are poorly able to fertilize their own eggs ( low), and if there is substantial inbreeding depression (1- low).

Notice that if there are no viability differences ( = = 0), males will only be maintained if > 2 . This reflects a phenomenon known as the "two-fold cost of sex".

Using these proportions, we estimate that males must have a lower fitness, being approximately 58% as viable as females. Furthermore, must be about an order of magnitude higher than (1-) in order to be consistent with the composition of this population.

When males are encountered, most eggs must be fertilized with male sperm or most eggs must die when they are self-fertilized.

This analysis is being used by Stephen Weeks to gain further insight into the factors that allow the maintenance of sexual reproduction in this shrimp.