Understanding hermaphrodite species through game theory

Abstract

We investigate the existence and stability of sexual strategies (sequential hermaphrodite, successive hermaphrodite or gonochore) at a proximate level. To accomplish this, we constructed and analyzed a general dynamical game model structured by size and sex. Our main objective is to study how costs of changing sex and of sexual competition should shape the sexual behavior of a hermaphrodite. We prove that, at the proximate level, size alone is insufficient to explain the tendency for a pair of prospective copulants to elect the male sexual role by virtue of the disparity in the energetic costs of eggs and sperm. In fact, we show that the stability of sequential vs. simultaneous hermaphrodite depends on sex change costs, while the stability of protandrous vs. protogynous strategies depends on competition cost.

Appendix: Proof of Theorem 2.1

Let $X^*=(x_1^*,\ldots ,x_4^*)$ an equilibrium point of (1). As for all $s\in S, \sum _i x^*_i(s)=1$, then at equilibrium and for all $s\in S$ the system (1) is equivalent to:

$$\begin{aligned} \left\{ \!\begin{array}{ll} x_1^*(s )((1\!-\!x_1^*(s ))f_{1,4}(s,X^*(s))\!-x_2^*(s )f_{2,4}(s,X^*(s))-x_3^*(s )f_{3,4}(s,X^*(s)))&{}=0 \\ x_2^*(s )((1\!-\!x_2^*(s ))f_{2,4}(s,X^*(s))\!-x_1^*(s )f_{1,4}(s,X^*(s))-x_3^*(s )f_{3,4}(s,X^*(s)))&{}=0 \\ x_3^*(s )((1\!-\!x_3^*(s ))f_{3,4}(s,X^*(s))\!-x_1^*(s )f_{1,4}(s,X^*(s))-x_2^*(s )f_{2,4}(s,X^*(s)))&{}=0 \end{array} \right. \end{aligned}$$

(15)

where $f_{i,4}=(f_i-f_4), \forall i\in \left\{ 1\ldots 3\right\}$.

Now, let $\{A_i\}_{i=1\ldots 4}$ a partition of $S$ and let $x_i^*(s)=1_{A_i}(s), \forall i\in \{1, \ldots , 4\}$, we have then $\forall s\in S$:

$$\begin{aligned}&x_1^*(s)x_2^*(s)=x_1^*(s)x_3^*(s)=x_2^*(s)x_3^*(s)=0,\nonumber \\&x_1^*(s )(1-x_1^*(s )) =x_2^*(s )(1-x_2^*(s ))=x_3^*(s )(1-x_3^*(s ))=0 \text{ and } \nonumber \\&x_1^*(s )+x_2^*(s )+x_3^*(s )+x_4^*(s )=1 \end{aligned}$$

(16)

Using latter Eq. (16), we can easily prove that for all $s \in S$, $X^*(s)$ satisfies (15) and that the Jacobian matrix, $J$, of the system (1) at the equilibrium point $X^*$ is:

$$\begin{aligned} {J}(X^*)= \left( \begin{array}{cccc} j(X^*(s_0))&{}0&{}\cdots &{}0\\ 0&{}\ddots &{}\ddots &{}\vdots \\ \vdots &{}\ddots &{}\ddots &{}0\\ 0&{}\cdots &{}0&{}j(X^*(s_{max}))\\ \end{array} \right) \end{aligned}$$

(17)

Where $\forall s \in S$,

$$\begin{aligned} j(X^*(s))= \left( \begin{array}{cll} \lambda _1(s)&{} -x_1^*(s)f_{2,4}(s,X^*(s))&{}-x_1^*(s)f_{3,4}(s,X^*(s)) \\ -x_2^*(s)f_{1,4}(s,X^*(s))&{}\lambda _2(s) &{} -x_2^*(s)f_{3,4}(s,X^*(s))\\ -x_3^*(s)f_{1,4}(s,X^*(s)) &{} -x_3^*(s)f_{2,4}(s,X^*(s))&{} \lambda _3(s) \end{array} \right) \end{aligned}$$

(18)

and

$$\begin{aligned} \lambda _1(s)= & {} (1-2x_1^*(s))f_{1,4}(s,X^*(s))-x_2^*(s)f_{2,4} (s,X^*(s))-x_3^*(s)f_{3,4}(s,X^*(s))\\ \lambda _2(s)= & {} (1-2x_2^*(s))f_{2,4}(s,X^*(s))-x_1^*(s)f_{1,4} (s,X^*(s))-x_3^*(s)f_{3,4}(s,X^*(s))\\ \lambda _3(s)= & {} (1-2x_3^*(s))f_{3,4}(s,X^*(s))-x_1^*(s)f_{1,4} (s,X^*(s))-x_2^*(s)f_{2,4}(s,X^*(s)) \end{aligned}$$

Therefore, the characteristic polynomial of (18) is,

$$\begin{aligned} \det (J(X^*(s))-\lambda I)=\displaystyle \prod _{s\in S}\det (j(X^*(s))-\lambda (s) I_3)=0. \end{aligned}$$

and then $\forall s \in S$, $\lambda _1(s), \lambda _2(s) \text{ and } \lambda _3(s)$ are the eigenvalues of the matrix (17).

Finally, by following we replace the value of $X^*$ and we show that $\forall i\in \{1,\ldots , 4\} \text{ and } \forall s \in A_i$:

$$\begin{aligned} \{\lambda _j(s), j\in \{1,2,3\}\}=\{(f_j-f_i)(s, X^*(s)),j\in \{1,2,3\}{\setminus } i\} \end{aligned}$$

which prove the result in Theorem 2.1.