Sub-standard Editions of Sperm Wars – an explanation and apology

A number of people, mainly in the United States, have complained that they obtained substandard copies of Sperm Wars from online booksellers – particularly Amazon.com – and in many cases paid over the odds as well. Some of these people asked for an explanation if not an apology and here I am happy to give both.

Around the time that Basic Books published the second (paperback) edition of my book in the USA, a company calling themselves Diane Books Publishing Company began advertising a look-alike edition on the Internet, including on Amazon, usually for an inflated price. This company owned no rights to publish or sell the book and from all accounts the printing was faint and often skewed, the cover was pale, the paper poor quality and the book badly bound. As one reader complained, it looked no better than a photocopy, which is almost certainly what it was.

I, my agent, and my legitimate publishers complained to Amazon and for a while the version was removed from their site, but it soon returned and at the time of writing (March 2011) is still there ( click here.) Short of taking legal action for breach of copyright there is nothing more I can do except apologise to those people who were tricked into paying out good money for an inferior product. If it helps I can assure everybody that neither I nor my legitimate publishers ever saw a cent of income from the sale of those books (though presumably Amazon and others did.) At least the much larger-scale illegal downloading of Sperm Wars that is currently taking place strikes only at my livelihood, not also at the pockets of people who simply want to read my book.

Types of Criticism of Sperm Wars

Three major groups of readers seem to have taken exception to my book:

1. those uncomfortable with an open, objective discussion of sexual matters;
2. others concerned that anything other than total condemnation of behaviour they judge to be aberrant will have undesirable social consequences; and
3. yet others who believe, hope, or might somewhere have heard that the science behind the book is either non-existent, flimsy, or even discredited.

With the first group, I can only sympathise, maybe apologise, but wonder why they subjected themselves to my book in the first place. With the second group, I simply disagree, maintaining yet again that objective discussion is not the same as condoning, and that understanding is more powerful than both wishful-thinking and blind condemnation. But with the third group, there is an argument to be had. There is good science behind Sperm Wars, no part has yet been discredited, some parts are gaining in support, and the ideas it presents are as current now as they were when I first published the book in 1996.

The Evidence for the Topics Covered in Sperm Wars

In writing Sperm Wars, I opted neither to litter the book with scientific references nor to present the distraction of interpretations that differed from my own; I was not aiming at an audience that would want them, rather at people who just wanted a good straightforward description of a new way of looking at human sex. And despite some of the criticisms I have received, on balance I don’t regret that decision.

To cater for the more scientifically minded, I explained in the Foreword to Sperm Wars that anybody who wanted to see the scientific evidence and learn of any alternative views should consult the academic tome Human Sperm Competition that I had already written with my former student (now a Professor and OBE) Mark Bellis. Our book was crammed not only with tables and graphs but also with pages and pages of references, and there seemed no need to repeat them in a popular version. At the time, the decision seemed reasonable: Human Sperm Competition was still sensibly priced or was available in most libraries; not so reasonable now with it selling for the ludicrous figure of well over £100/$200.

Some readers of Sperm Wars – even some Professional Critics – evidently didn’t notice that they could if they wished find the raw science that they wanted. Others (who did notice) seemed not to believe me and decided rather cynically that I was just providing a clever cover for being totally unable to substantiate anything. But everything – well, almost everything – does have published scientific support that can be appraised. Even before my academic book with Mark Bellis, I and my Manchester team had published a series of peer-reviewed papers and a list of these is provided elsewhere on this website (click here.) But even despite these, a few scientists have decided that aspects of the science behind Sperm Wars are in some way flawed. If you wish to read their arguments (many are also included in the CRITICISM page) and my response, return to the menu and choose a topic.

Oral Sex

watch this video first if you don’t know or can’t remember the story

“The book is controversial … because several of the hypotheses … are not supported by scientific research” – meaning, presumably, that no supporting research has been published.

Generally, this is simply not true, but there is an exception and it is the suggested role of oral sex in sperm warfare. Even now I know of no evidence other than inference from other animals to support this idea. It helps a little that James Kohl and Robert Francoeur quite independently wrote something similar in their 1995 book The Scent of Eros. But even that is no substitute for real data.

Penis Design

watch this video first if you don’t know or can’t remember the story

Rather as for Oral Sex, when Sperm Wars was written I knew of no evidence beyond simple inference from other animals to support the view that the human penis removes sperm from a woman as well as putting sperm in. But gratifyingly, the hypothesis triggered a brilliant series of experiments by Gordon Gallup and Rebecca Burch in the Departments of Psychology at the Universities of New York at Albany and Oswego that would seem to settle the matter. Using a model vagina and various shapes, sizes and designs of penis, they showed in papers in 2003 and 2004 that the thrusting human penis was perfectly designed for sperm removal.

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Homosexuality

(sorry, no video available)

The explanation for homosexuality in Sperm Wars that many people seem to have found controversial, even offensive, is as follows: evolution maintains the trait in western populations at the level it is found (6% of men; 3% of women; 80% of each of these being bisexual) because at lower levels the behaviour is reproductively advantageous, and at higher levels it is disadvantageous. Compared with exclusive heterosexuals, the suggested advantage is that whereas on average over a lifetime bisexuals produce fewer children (a proportion of which will be exclusively homosexual), they also produce them when younger and so have a shorter generation interval. There are data to support this for lesbians (but not for gay men because it is impossible to know how many children a man has with enough accuracy for this sort of analysis.)

The figures quoted for levels of homosexuality and bisexuality are not mine but taken from surveys by others. The evidence that the trait is genetic is also not mine but can be found in the wider literature. The principle of a balanced polymorphism is well established within evolutionary theory, so too is the evolutionary trade-off between offspring number and generation interval. As a result, any criticisms directed at me should only concern my explanation for the observed levels of homosexuality and bisexuality, not the figures themselves. The supporting data for lesbians is also mine (with Mark Bellis).

When writing Sperm Wars I made no attempt to justify applying to homosexual behaviour the widely accepted trade-off between offspring number and generation interval. But some critics have expressed doubts that it could work. I am grateful therefore to Kevin J. Harrington from San Francisco, California who, after reading Sperm Wars, was moved to provide mathematical proof that it does (the mathematically inclined can read his proof here.)

I have no real insight into how the homosexual sections in Sperm Wars have been received generally by the homosexual/bisexual community, except to point out that one of the best reviews of my book appeared in Gay & Lesbian Humanist magazine. The main objection to my handling of the topic seems to have come from people who find it difficult to accept that homosexual behaviour could possibly be adaptive rather than, in their words, “aberrant.” But objections are not proof, and as far as I am aware nothing has happened scientifically since publication to challenge the explanation for homosexuality that the book contains.

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Rape

(sorry, no video available)

My handling of rape in Sperm Wars has been listed as one of the reasons the book is considered controversial. So controversial, in fact, that my Japanese publishers decided to leave out the relevant chapters (plus also the chapter on prostitution) from their second edition.

The figures I quote for rape are all from surveys and studies by others, and as nobody seems to have accused me of being eclectic I assume that the problem critics have concerns more how I use and explain these figures than the figures themselves. Again, little that is specific has been said, though one much repeated misquotation has me saying that that there is no difference between rape and rough-and-tumble intercourse. What I actually say is this:

“It is short step from rough-and-tumble intercourse, across the boundary of mutual consent, to rape. This is date rape – when intercourse is forced by a man on a woman who has at least found him attractive enough to ‘date’, and often attractive enough to kiss and ‘pet’. It is not the random, predatory rape by a man who is usually a complete stranger to his victim.”

I then go on to give examples of why drawing the line between rough-and-tumble intercourse and date rape is such an acknowledged nightmare for legal systems around the world. My eventual conclusion is that drawing the line is no job for a biologist and should be left to the law, and from that point on I just discuss the biology of rough-and-tumble sex. At no point do I say that there is no line to be drawn, and I make it very clear that predatory rape is a different matter yet again.

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Testis Size and Promiscuity

watch this video first if you don’t know or can’t remember the story

In the Preface to his book Promiscuity, Tim Birkhead describes a lecture I once gave to a small meeting he had organised. According to him, my lecture caused “sniggering incredulity” and “downright indignation” – and later even acrimony – amongst some members of the audience. He says the work presented was “difficult to defend” and “naïve” and was the cause of him losing faith in me as a scientist. That lecture was the first presentation of research that in Sperm Wars became a chapter on the influence of testis size on the chances of a man becoming involved in sperm warfare. Tim, as convener of the meeting, had encouraged the participants to present new work, still in progress, to encourage discussion – which is very much what I did.

My hypothesis was that men with larger testes were more likely to become involved in sperm competition, and I presented data that Mark Bellis and I had collected to support this. The lecture did create a stir and attracted various criticisms, mostly constructive, mainly centred on the racial identity of our subjects, our measure of sperm production, and on whether the measure of testis size that we used should be the actual size or size expressed as a percentage of the man’s body weight. It was a simple matter to do as everybody suggested and I had done so with more data, but not published the results, by the time I wrote Sperm Wars. Nothing had changed; our analysis and conclusions had proved robust to all criticisms. The results were eventually published, with even more supporting evidence, in a scientific paper in 1997. Strangely, even three years later in his book, Tim does not acknowledge this and still criticises the work as if it had not changed since he first heard it. The charitable interpretation is that he never read my later paper. But whatever the real reason, his criticism is now largely irrelevant. In September, 2013 a paper in the Proceedings of the National Academy of Sciences of the USA by Mascaro, Hackett and Rilling produced hormonal and neurological evidence that led to essentially the same conclusion: testicular volume is a major component in the trade-off between mating effort and parenting effort in human males.

My testis-size chapter in Sperm Wars seems also to have generated a misunderstanding: some people interpret my explanation as meaning that men with larger testes (and a larger penis) are more successful than men with smaller. That is not what I say, the main point of the chapter being to explain why we find such variation amongst men in size of genitalia. My suggestion is that the situation appears to be yet another balanced polymorphism, evolution fixing the range and frequency of testis (and penis) sizes in humans so that on average men experience the same reproductive success whatever the size of their genitalia. They just do so in different ways as appropriate to their attributes, a conclusion also reached more recently by Mascaro and her colleagues.

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Female Orgasm

watch this video first if you don’t know or can’t remember the story

Sperm Wars contains several chapters concerning the female orgasm and by and large these are based on research carried out by myself and Mark Bellis at Manchester, first published in 1993. Since then, the most detailed critique of that work has been by Elisabeth Lloyd in her 2005 book, The Case of the Female Orgasm: Bias in the Science of Evolution, and various commentators have since uncritically repeated her conclusions.

To Elisabeth’s credit she makes clear early on in her book that she has a bias against an evolutionary approach. Then, having aired her prejudices, she launches with commendable zeal into an attempted demolition of our work. Her attack is almost entirely statistical, complaining about ‘handpicked’ subsets of data, small skewed samples, and our choice of statistical tests. There is nothing constructive here. She doesn’t once say what she would have done differently. Nor does she present any evidence comparable to ours that challenges our findings. Undeterred, she somewhat triumphantly concludes that our work was “worthless” and moves on.

Our paper was published in the prestigious peer-reviewed journal Animal Behaviour. The editors, realising the contentious nature of our article and seeing how dependent it was on our statistics, called in the services of a professional statistician before they would accept our paper for publication. So the published paper had received his all-clear as meeting the necessary standards of statistical care. Of course this does not necessarily mean that he is right and Elisabeth is wrong, but the chances are that it does, not least because he is unbiased and she is not. Her whole book depended on being able to ignore our paper; she simply had to demolish it.

Having admitted to not liking the evolutionary approach, Elisabeth then commits a faux-pas that makes it clear that she doesn’t totally understand it either; at least not the concept of a balanced polymorphism. She secondarily dismisses our work because we demonstrate that women can achieve almost the same range of levels of sperm retention both with and without experiencing orgasms, simply by changing certain sequences of events. For some reason, Elisabeth thinks that this, combined with the variability in the female orgasm, is inconsistent with evolution. Yet I devote a whole chapter of Sperm Wars to explaining why it is not; a chapter that she never mentions. In fact, she does not reference Sperm Wars at all.

After Sperm Wars was published, papers appeared from Dev Singh and his colleagues in Texas and Randy Thornhill and his colleagues at Albuquerque that offer modest evidence in support of the female orgasm story. Dev Singh showed that women desirous of becoming pregnant experience a consistent shift in the frequency and timing of orgasm, and Randy Thornhill has shown something similar when women have sex with men of different characteristics. Elizabeth Lloyd also attacks their papers – as she must – but her attacks depend in large part on first having demolished the paper by Mark Bellis and myself, which she has not. So on balance, far from being discredited, the female orgasm story in Sperm Wars has gained, not lost, ground, at least amongst scientists who favour an evolutionary explanation.

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Kamikaze Sperm

watch this video first if you don’t know or can’t remember the story

In the words of the author of the original Wikipedia entry for Sperm Wars: “a major focus of the book is a supposed sperm heteromorphism in humans, where less than 1% of a man’s sperm are “egg-getters”, and the rest are infertile “kamikaze sperm” …” The author then goes on to describe killer sperm before finally commenting that the concept of sperm warfare is considered “discredited.”

Heteromorphism (meaning that there are different-looking types of sperm in a single ejaculate) has been a known characteristic of human sperm for decades. There is no “supposed” about it. Why such heteromorphism existed among human sperm had never been explained in an evolutionary sense until Mark Bellis and I published our ‘kamikaze sperm hypothesis’ in 1988.

Also known for decades was that fewer than 1% of sperm in an ejaculate showed any ‘interest’ or ability in fertilising an egg. Again there is no ‘supposed’ about it. My and Mark Bellis’ contribution was simply to dub these sperm ‘egg-getters’. A controversy did arise though from our suggestion as to which sperm these egg-getters might be.

Tens of pages of our book Human Sperm Competition, the scientific foundation for Sperm Wars, was spent explaining that the heads of ‘normal’ sperm vary in size along a continuum, and for convenience could be divided into three: micro, modal, and macro. Our hypothesis (with supporting data) was that these represent different stages or ages in development, and when they pass from being modal to being macro some at least go in search of an egg. Somehow, Roger Short in his review of our book missed all of this and said that our ‘macros’ were about as likely to be egg-getters as they were to contain little men. The reason? He thought, or perhaps wanted to think, that despite everything we had written we were referring to Macrocephalous sperm, a separate large-headed morph which we gave minimal attention and which contains a double (diploid) set of chromosomes; they are of course infertile. But ‘normal’ micro and modal sperm, which have only one (haploid) set of chromosomes, simply cannot grow to become diploid, so Roger cannot possibly have thought that is what we meant. His witty but careless and insulting remark has been picked up and uncritically repeated by others and has caused a great deal of confusion. But as a criticism of our egg-getter hypothesis it is totally vacuous.

The first category of kamikaze sperm to be postulated was blocker sperm (not killer, as some have assumed.) But compared with killers, this category has received little attention since Sperm Wars was published. Tim Birkhead and his colleagues at Sheffield University briefly acknowledged that they may have found evidence of blocking sperm in their 1999 Royal Society paper, but then quickly dismissed the possibility as their predictions had all been based on looking for killer sperm (even though they were using 2-tailed statistics).

For the moment, therefore, there has been no further progress in identifying the characteristics of ‘egg-getter’ and blocker sperm, and the only evidence one way or the other is still that published in Human Sperm Competition and related in Sperm Wars.

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Killer Sperm

watch this video to see a possible killer sperm in action

KILLER SPERM SCIENCE: There is a rumour going around that the concept of killer sperm is dead: “repeatedly squashed” in the words of Tim Birkhead in one of his several attacks; “discredited” according to the anonymous author of the original Wikipedia entry for Sperm Wars. Repeatedly? Search as I might, I can only find one publication that presents new data and that tries (very hard) to kill-off killer sperm. Admittedly as early as 1997 Tim warned that the work was coming, even if he did at the time imply that it had nothing to do with him. Then two years later the work was published – by the Royal Society, no less – with Tim as one of the three authors. Later that year it was summarised in Science and Tim referred to it again in the Preface to his book Promiscuity. One piece of work quoted four times – maybe that’s what Tim meant by “repeatedly” – and it has been uncritically repeated ever since. Even the original Wikipedia author rather underhandedly quoted two of these publications in an attempt, presumably, to double the justification of his/her use of the word “discredited.”

If I have missed other publications of data, somebody please let me know and I will both apologise and re-write this part of my blog accordingly. But until then, as far as I am aware, there is just the single publication, and that is the 1999 Royal Society paper on the work carried out by Tim Birkhead and his two colleagues from Sheffield University.

The first five words of the title of this paper were: “No evidence for killer sperm …”, meaning of course that the paper contained no evidence for killer sperm, not that no evidence existed. We (my team at Manchester) had produced and published plenty and there was nothing in this new paper to challenge that evidence. They hadn’t disproved anything, or discredited anything, they just hadn’t – they claimed – obtained the same results as us. The question is: why not?

One reason is that, in scientific terms, they had been fairly naughty. When one group of scientists checks the repeatability of the results of another group of scientists, the accepted protocol is that they try as closely as possible to stick to the same experimental regime. The Sheffield group carried out three series of experiments, and not one of them came even near to being a repeat of ours except in the most basic of senses.

Experiments on killer sperm involve taking ejaculates from two men and mixing them. As a control, sperm from each man are also mixed self with self. The data collected after a succession of time intervals include: how many die; how many form diads (pairs of sperm joined at the head – see video); how the morph frequency changes; and how many are swimming around without an acrosome (the ‘bomb on the head’ that we claimed was the weapon sperm use to kill others). If killer sperm are at work there should be fewer dead sperm, fewer sperm without acrosomes, and fewer sperm joined at the head in the ‘self-self’ mixes than in the ‘self-other’ mixes. To make sure no cheating goes on in obtaining the results the slides are labelled so that the person doing the counting does not know which slide is which.

The Sheffield group did three types of experiment, the first and the third of which bore absolutely no resemblance to anything we had done and even in principle were a very doubtful way of looking for killer sperm. Their second series began as a fairly proper replicate, but even that rapidly departed from our own. Again they stood little if any chance of observing killer sperm in action.

In their first series, they tested sperm in diluted semen on the totally ludicrous grounds that most sperm competition will take place in the vagina while the sperm are in semen. I groaned when I first read this. The authors clearly hadn’t read anything Mark Bellis and I had written. We had shown, both in papers and in our books, that any sperm that hasn’t escaped the semen within say, 45 minutes, was doomed. They were destined to be ejected from the female in the flowback. We had also pointed out that if a second man does have intercourse with a woman while she still has semen inside her, his penis is designed to remove that semen from her vagina before he ejaculates. We have never said that killer sperm are active in semen in the vagina. In fact, I would predict that ‘killer’ behaviour is not even triggered until the sperm are out of the semen and at least in the cervix. The Sheffield group’s first series of experiments was a total waste of time and the lack of killer activity that they recorded was meaningless. And as we had never carried out such experiments at Manchester, it certainly wasn’t a replication of our experiments.

The Sheffield group’s third series was interesting and adventurous, but again was not an attempt to replicate our own experiment. They said, quite rightly, that in order to tell whether sperm are selectively killing sperm from another man, it is necessary for the experimenters to know which man each sperm came from. So for their third series, they dyed the sperm from different males different colours. It was an ambitious project (though they only used sperm from five males) and would only be meaningful if positive results were obtained, because negative results would prove nothing. Presumably, if killer sperm exist, they distinguish same-man sperm from other-man sperm via fairly subtle chemical differences on the sperm’s surface. So bathing them in a completely foreign chemical with totally unknown consequences was very much a leap in the dark. Positive results would show that they hadn’t destroyed the sperms’ ability to separate same-male sperm from foreign male; negative results (such as they obtained) would show only that sperm bathed in dye cannot recognise ‘foreign’ sperm. They tell us nothing about the existence of killer sperm behaviour itself. The experimenters ‘proof’ that their treatment had not affected sperm behaviour was not that the sperm could still detect subtle details of surface chemistry, but simply that the spems’ motility was unimpaired, which is irrelevant. A headless cockerel can run around for ages (so can a headless sperm swim) – but I wouldn’t bet money on either in a fight.

Only the second series of experiments performed by the Sheffield group came close to repeating our experiments. At least in that they used a fluid medium for the sperm that was similar to the one we used in our later experiments, though not our earlier. Early on, we had no choice but to use blood plasma for the sperm to swim around in. Later we used whatever artificial medium was being used in the Manchester IVF clinic, which did change from time to time. The Sheffield group, in their second series of experiments, also used the same medium as their local IVF clinic, so that at least was an attempt to replicate our protocol.

In our experiment, the first signs of killer activity appeared after the sperm had been together for 6 hours, peaked after 12 hours, and were still evident, but the sperm generally becoming less active after 24 hours. Yet the Sheffield group designed their experiment to stop after only 3 hours (their first and third series stopped after 3 hours as well), a time at which even our Manchester results were not significant. The Sheffield group also stopped their experiment after mixing sperm from only 10 men; we used sperm from 132 men.

Insofar as neither the Sheffield group nor my own obtained evidence of killer sperm after 3 hours, their experiment was consistent with ours. But as an experiment to test the killer sperm hypothesis as described in Human Sperm Competition and Sperm Wars theirs is totally meaningless. The concept of killer sperm has not been discredited.

KILLER SPERM VIDEO: A second controversy that has arisen over killer sperm concerns the video (at head of section) made in our laboratory by my Manchester team for the Desmond Morris TV programme, The Human Animal.

In the Preface to his book Promiscuity, Tim Birkhead described the film as follows (evidently he first watched the sequence with other behavioural ecologists at a conference in Nottingham.) “At this point we saw a human sperm bumping against what looked like, according to the andrologists I have asked, a sperm that was already dead. Morris continued, ‘This is chemical warfare filmed for the first time … and when the two sperm separate one is left dead.’ Heady stuff, and exactly the sort of ooh-ahh science the media and its susceptible audience love. But wait a minute: even if this really was one sperm killing another, since these sperm were not labelled in any way, how could Baker, or anyone else, know that these sperm were those of different males?”

There are three points here. First, I would say that the active sperm was not “bumping against” the other; the two were somehow joined and had been for 30 minutes before the broadcast sequence began – but you can watch the film and judge for yourself. Second, there is no question – there were several witnesses – that when filming began the two sperm were both alive, but the interaction was half-an-hour long and the BBC, more interested in showing that the survivor was able to swim away and live on, opted to edit the sequence to begin at the point it did. Whether both sperm were still alive by the time the broadcast footage begins is a moot point. Tim’s andrologist friends evidently thought that the sperm on the left was already dead when they first saw it – and I am inclined to agree with them. That is, after all, what killer sperm are about. Nobody can be certain but the sperm had probably been dead for about 10 minutes.

On the third point, there is also total agreement. Nobody can know whether these two sperm were from the same man or different men; we can all agree that this film does not in any scientific sense prove the existence of killer sperm. But equally – and again we should all agree – it doesn’t prove that they do not exist. The possibility still remains that, as Desmond Morris described in his broadcast, this short video of ours really could be the first glimpse of a killer sperm in action.

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Appendix to the section on Homosexuality (only for the mathematically minded)

From Kevin J. Harrington, San Francisco, California (posted on Amazon.com, 9 July 2001 amongst Reader’s comments on the First American Edition of Sperm Wars.) There is no specific allowance in the model for the 20% of people who show homosexual behaviour but who fail to reproduce because they show only homosexual behaviour. But this could be considered part of the reduced number of children produced on average by bisexuals. Otherwise Kevin’s maths look good to me, but if anybody can see a flaw please let me know.

“Suppose one has two populations N and N*, and that the individuals of each population have on average M and M* children, respectively. After n generations, population N will grow to be N(n) = N(0)xM^n, where N(0) is the size of the initial population and N(n) is the size after n generations, with M^n denoting “M raised to the power n.” Similarly, N*(k) = N*(0) x M*^k for the N* population after k generations (having k not equal to n is crucial). Dividing the two equations, one gets N*(k)/N(n) = {N*(0)/N(0)} x {M*^k/M^n} (call this equation I).

If the ratio of the two populations remains stationary over time (e.g., bisexuals vs. heterosexuals), then N*(k)/N(n) = N*(0)/N(0) (call this equation II), PROVIDED that generation k of the N* population is alive at the same time that generation n of the N population is alive. If it takes the N population T years to reproduce and the N* population T* years to reproduce, then if t is the total elapsed number of years, n = t/T and k = t/T* (call these equations III & IV). If we apply equations II, III, and IV to equation I, we get that (M*)^(t/T*) = (M)^(t/T). Taking the logarithm of both sides and using the fact that log(a^b) = b x log(a), we get (t/T*) x log(M*) = (t/T) x log(M), and cancelling the t from both sides and rearranging yields the result T*/T = log(M*)/log(M). That’s the condition that must be satisfied if the ratio of the two populations is to stay the same over time.

So if heterosexuals on average have 4 children and bisexuals on average have 3 children, then log(3)/log(4) = 0.79, so bisexuals must reproduce the next generation on average in a time T* equal to 79% of T if their percentage of the population is to stay the same over time.