Ball M A, Parker G A
Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 3BX, U.K.
J Theor Biol. 1998 Sep 21;194(2):251-62. doi: 10.1006/jtbi.1998.0756.
We investigate game theory models of ejaculate expenditure by males in a species where there is a risk probability, q, that females will mate twice before laying a given set of eggs. With frequency 1-q females mate just once; then males optimally ejaculate an arbitrary minimum amount of sperm. The paper extends the analysis of Parker et al. (1997) in which males have limited information about the three risk states of the female: 0 (virgins which will mate now but not again), 1 (virgins which will mate now and then once more), and 2 (once-mated females who will mate now but not again). We derive a general structure for finding ESSs under imperfect information about states, and examine two special cases in which males know the overall risk probability q, but have imperfect knowledge of the states (0, 1, 2). In Case 1, males cannot distinguish between states 0 and 1 but have limited information about state 2. As their information increases, so does the difference in sperm allocation, with more going to females assessed as mated (state 2) than to females assessed as virgins (0, 1). This difference decreases with q in a species, but the average ejaculate expenditure increases with q. Even for small amounts of information, the behaviour as q-->0 is different from that predicted for zero information. In Case 2, males have perfect information about state 2, but limited information about states 0, 1. This has a major effect if q is small: males effectively behave as if they had perfect information by giving equal amounts of sperm to females assessed as 1 as those assessed to be 2, while giving a minimum amount to females assessed as 0. Ejaculate expenditures generally increase with the overall species-level risk q. A result common to both cases is that even a small amount of information allows more strategic choice than the zero information case and hence qualitatively different behaviour.
我们研究了一个物种中雄性射精量的博弈论模型，在该物种中，雌性在产下一组特定卵之前有q的概率会进行两次交配。雌性以1 - q的频率只交配一次；此时雄性最优地射出任意最小量的精子。本文扩展了帕克等人（1997年）的分析，其中雄性对雌性的三种风险状态信息有限：0（现在会交配但不再交配的处女）、1（现在会交配且之后还会交配一次的处女）和2（已交配过一次且现在会交配但不再交配的雌性）。我们推导了在关于状态的不完全信息下寻找进化稳定策略（ESSs）的一般结构，并研究了两个特殊情况，即雄性知道总体风险概率q，但对状态（0、1、2）的了解不完美。在情况1中，雄性无法区分状态0和1，但对状态2的信息有限。随着它们信息的增加，精子分配的差异也会增加，分配给被评估为已交配（状态2）的雌性的精子比分配给被评估为处女（0、1）的雌性更多。这种差异在一个物种中会随着q的增加而减小，但平均射精量会随着q的增加而增加。即使对于少量信息，当q趋于0时的行为也与零信息时预测的行为不同。在情况2中，雄性对状态2有完美信息，但对状态0、1的信息有限。如果q很小，这会产生重大影响：雄性实际上的行为就好像它们有完美信息一样，给被评估为1的雌性与被评估为2的雌性等量的精子，而给被评估为0的雌性最小量的精子。射精量通常会随着整个物种层面风险q的增加而增加。两种情况的一个共同结果是，即使是少量信息也比零信息情况允许更多的策略选择，因此行为在性质上有所不同。
