A Large Population Parental Care Game with Asynchronous Moves

Abstract

This article considers two game-theoretic models of parental care which take into account the feedback between patterns of care and the operational sex ratio. Attention is paid to fish species which care for their young by mouthbrooding, in particular to St. Peter’s Fish. It is assumed here that individuals can be in one of the two states: searching for a mate or breeding (including caring for their offspring). However, the sets of states can be adapted to the physiology of a particular species. The length of time an individual remains in the breeding state depends on the level of care he/she gives. According to one model, parents make their decision regarding the amount of care they give simultaneously. Under the second model, one individual in a pair (for convenience, the female) makes her decision before the male makes his decision. When in the searching state, individuals find partners at a rate dependent on the proportion of members of the opposite sex searching. These rates are defined to satisfy the Fisher condition that the total number of offspring of males equals the total number of offspring of females. The operational sex ratio is not defined exogenously, but can be derived from the adult sex ratio and the pattern of parental care. The results obtained go some way to explain the variety of parental care behaviour observed in fish, in particular the high frequency of male care, although further work is required to explain the exact patterns observed.

Appendices

Appendix 1: Stability Conditions in Game with Simultaneous Moves

The left-hand side of each inequality is the reproduction rate of males (which is the reproduction rate of the females divided by the ASR), the first entry on the right-hand side is the reproduction rate of a mutant male and the second entry is the reproduction rate of a mutant female divided by the ASR.

Only male parental care is an ESS if

$$\displaystyle{ k_{m}\lambda _{1}q_{1}^{CD}p_{ 1}^{CD} >\max \left \{ \frac{\lambda _{1}\lambda _{m}^{D}q_{ 1}^{CD}} {\lambda _{m}^{D} +\lambda _{1}q_{1}^{CD}}, \frac{k_{b}\lambda _{1}\lambda _{f}^{C}p_{1}^{CD}} {\lambda _{f}^{C} +\lambda _{1}rp_{1}^{CD}}\right \}. }$$

(8.44)

Only female parental care is an ESS if

$$\displaystyle{ k_{f}\lambda _{1}q_{1}^{DC}p_{ 1}^{DC} >\max \left \{ \frac{k_{b}\lambda _{1}\lambda _{m}^{C}q_{ 1}^{DC}} {\lambda _{m}^{C} +\lambda _{1}q_{1}^{DC}}, \frac{\lambda _{1}\lambda _{f}^{D}p_{1}^{DC}} {\lambda _{f}^{D} +\lambda _{1}rp_{1}^{DC}}\right \}. }$$

(8.45)

Parental care by both sexes is an ESS if

$$\displaystyle{ k_{b}\lambda _{1}q_{1}^{CC}p_{ 1}^{CC} >\max \left \{\frac{k_{f}\lambda _{1}\lambda _{m}^{D}q_{ 1}^{CC}} {\lambda _{m}^{D} +\lambda _{1}q_{1}^{CC}}, \frac{k_{m}\lambda _{1}\lambda _{f}^{D}p_{1}^{CC}} {\lambda _{f}^{D} +\lambda _{1}rp_{1}^{CC}}\right \}. }$$

(8.46)

Appendix 2: Stability Conditions in Game with Asynchronous Moves

No parental care with unconditional desertion by males, [(D, D), D], is neutrally stable when

$$\displaystyle\begin{array}{rcl} \max \{k_{f}, \frac{k_{b}} {k_{f}}\}& <& \frac{\lambda _{m}^{D}(\lambda _{m}^{C} +\lambda _{1}q_{1}^{DD})} {\lambda _{m}^{C}(\lambda _{m}^{D} +\lambda _{1}q_{1}^{DD})}{}\end{array}$$

(8.47)

$$\displaystyle\begin{array}{rcl} k_{f}& <& \frac{\lambda _{f}^{D}(\lambda _{f}^{C} +\lambda _{1}rp_{1}^{DD})} {\lambda _{f}^{C}(\lambda _{f}^{D} +\lambda _{1}rp_{1}^{DD})}.{}\end{array}$$

(8.48)

This strategy profile is strongly stable when, in addition

$$\displaystyle{ k_{b} < \frac{\lambda _{f}^{D}(\lambda _{f}^{C} +\lambda _{1}rp_{1}^{DD})} {\lambda _{f}^{C}(\lambda _{f}^{D} +\lambda _{1}rp_{1}^{DD})}. }$$

Biparental care with unconditional care from males, [(C, C), C], is neutrally stable when

$$\displaystyle\begin{array}{rcl} \min \{k_{f}, \frac{k_{b}} {k_{f}}\}& >& \frac{\lambda _{m}^{D}(\lambda _{m}^{C} +\lambda _{1}q_{1}^{CC})} {\lambda _{m}^{C}(\lambda _{m}^{D} +\lambda _{1}q_{1}^{CC})}{}\end{array}$$

(8.49)

$$\displaystyle\begin{array}{rcl} \frac{k_{b}} {k_{f}}& >& \frac{\lambda _{f}^{D}(\lambda _{f}^{C} +\lambda _{1}rp_{1}^{CC})} {\lambda _{f}^{C}(\lambda _{f}^{D} +\lambda _{1}rp_{1}^{CC})}.{}\end{array}$$

(8.50)

The condition required for strong stability, i.e.

$$\displaystyle{ k_{b} > \frac{\lambda _{f}^{D}(\lambda _{f}^{C} +\lambda _{1}rp_{1}^{CC})} {\lambda _{f}^{C}(\lambda _{f}^{D} +\lambda _{1}rp_{1}^{CC})}, }$$

is a weaker condition than Condition (8.50). Hence, if [(C, C), C] is neutrally stable, then it is strongly stable.

Finally, it can be shown that [(C, D), C], i.e. biparental care with male care being conditional on the female giving care, is neutrally stable when

$$\displaystyle\begin{array}{rcl} k_{f}& <& \frac{\lambda _{m}^{D}(\lambda _{m}^{C} +\lambda _{1}q_{1}^{CC})} {\lambda _{m}^{C}(\lambda _{m}^{D} +\lambda _{1}q_{1}^{CC})} < \frac{k_{b}} {k_{f}}{}\end{array}$$

(8.51)

$$\displaystyle\begin{array}{rcl} k_{b}& >& \frac{\lambda _{f}^{D}(\lambda _{f}^{C} +\lambda _{1}rp_{1}^{CC})} {\lambda _{f}^{C}(\lambda _{f}^{D} +\lambda _{1}rp_{1}^{CC})}.{}\end{array}$$

(8.52)

This strategy profile is strongly stable when, in addition

$$\displaystyle{ \frac{k_{b}} {k_{f}} > \frac{\lambda _{f}^{D}(\lambda _{f}^{C} +\lambda _{1}p_{1}^{CC}r)} {\lambda _{f}^{C}(\lambda _{f}^{C}(\lambda _{f}^{D} +\lambda _{1}p_{1}^{CC}r))}. }$$