2. I’m not sure I understand your last point. Even if a small change pushes the barely tolerable to intolerable you still have the opposite effect on the other side of the curve, where it provides relief. I am assuming here that in most polygenic traits the curve is not so narrow that there aren’t dysfunctional individuals being produced on both extremes.
3. I concede your point that if there is less than perfect competition, this effect doesn’t completely negate any effect on average welfare. It would still make such an effect smaller and less relevant as compared to other considerations, like species composition.
2. tl;dr: The effect would be smaller on the opposite side, assuming something like a normal distribution of temperatures at baseline and a symmetric survival rate (or serious adverse event) function of temperature, centered on the mean temperature, which we would assume by symmetry, and because of the optimization of evolution. So, the animals who benefit would benefit less than the harm to the animals who are made worse off. You’ll get more deaths on the side you’re moving the temperature towards than you’re preventing on the other side.
Assume the survival rate function is unimodal and increasing/nondecreasing to the left of its mean/max and decreasing/nonincreasing to the right of its mean/max, so basically has a shape similar to a normal distribution.
To illustrate with a simplified example, assume the temperature is just constant, and we’re just shifting this constant temperature. The survival rate will be lower anywhere away from the mean, and lower the further away from the mean. We’re assuming at equilibrium, the temperature already maximizes this, and so any shift in temperature, in either direction, will decrease survival.
Survival rate function of temperature, centered at the mean temperature
The same would happen for a uniform distribution of temperatures within some bounded range, initially centered at the centre of the survival function distribution, and shifting the whole temperature distribution. The expected survival rate, i.e. taking the expected value of the survival rate as a function of temperature over the temperature distribution, would be lower if you shift the temperature distribution in any direction. This would basically just be the area under the curve where the temperature distribution is, and this is maximized if the temperature range, holding its width constant, has the same mean/centre as the survival rate function. Intuitively, you could just replace the sharp constant temperature distribution with a very narrow uniform distribution. The result is basically a consequence of Anderson’s theorem (although this particular result only gives greater than or equal to, not strict inequality; there are strict ones like this one for the normal distribution).
If the temperature instead followed a distribution that looked roughly normal (symmetric + unimodal + increasing/nondecreasing to the left and decreasing/nonincreasing to the right of the max), you get the same result, too. You could show this using Anderson’s theorem, approximating the temperature distribution by a mixture of uniform distributions with the same mean and using a convergence theorem (the Monotone Convergence Theorem should do, or the Dominated Convergence Theorem if you want to be less careful with your approximating distributions).
I think the argument still works if the survival rate function is constant at its max for a while so that there’s a comfortable range of temperatures, if you have temperatures falling outside the range where it’s maximized and where the survival rate function is strictly increasing/decreasing. The survival rate function could look like a symmetric trapezoid, and you could calculate integrals/areas for a uniform temperature distribution explicitly.
The area under the curve not in the temperature range would increase if you shift the temperature distribution: if it’s just the two triangles at the ends and assuming the temperature range has width between a and b, as illustrated in the picture, its area would be, for
initial triangle widths w outside the temperature range, w<b−a2,
This is increasing as the absolute value of x increases, so you lose more on the side you lose on than you gain on the side you gain on with a shift. So, the area under the curve in the temperature range and hence the expected survival rate would be decreasing (by the amount A1(x)+A2(x) is increasing) as you shift the temperature range away from its initial mean.
2. I’m not sure I understand your last point. Even if a small change pushes the barely tolerable to intolerable you still have the opposite effect on the other side of the curve, where it provides relief. I am assuming here that in most polygenic traits the curve is not so narrow that there aren’t dysfunctional individuals being produced on both extremes.
3. I concede your point that if there is less than perfect competition, this effect doesn’t completely negate any effect on average welfare. It would still make such an effect smaller and less relevant as compared to other considerations, like species composition.
2. tl;dr: The effect would be smaller on the opposite side, assuming something like a normal distribution of temperatures at baseline and a symmetric survival rate (or serious adverse event) function of temperature, centered on the mean temperature, which we would assume by symmetry, and because of the optimization of evolution. So, the animals who benefit would benefit less than the harm to the animals who are made worse off. You’ll get more deaths on the side you’re moving the temperature towards than you’re preventing on the other side.
Assume the survival rate function is unimodal and increasing/nondecreasing to the left of its mean/max and decreasing/nonincreasing to the right of its mean/max, so basically has a shape similar to a normal distribution.
To illustrate with a simplified example, assume the temperature is just constant, and we’re just shifting this constant temperature. The survival rate will be lower anywhere away from the mean, and lower the further away from the mean. We’re assuming at equilibrium, the temperature already maximizes this, and so any shift in temperature, in either direction, will decrease survival.
The same would happen for a uniform distribution of temperatures within some bounded range, initially centered at the centre of the survival function distribution, and shifting the whole temperature distribution. The expected survival rate, i.e. taking the expected value of the survival rate as a function of temperature over the temperature distribution, would be lower if you shift the temperature distribution in any direction. This would basically just be the area under the curve where the temperature distribution is, and this is maximized if the temperature range, holding its width constant, has the same mean/centre as the survival rate function. Intuitively, you could just replace the sharp constant temperature distribution with a very narrow uniform distribution. The result is basically a consequence of Anderson’s theorem (although this particular result only gives greater than or equal to, not strict inequality; there are strict ones like this one for the normal distribution).
If the temperature instead followed a distribution that looked roughly normal (symmetric + unimodal + increasing/nondecreasing to the left and decreasing/nonincreasing to the right of the max), you get the same result, too. You could show this using Anderson’s theorem, approximating the temperature distribution by a mixture of uniform distributions with the same mean and using a convergence theorem (the Monotone Convergence Theorem should do, or the Dominated Convergence Theorem if you want to be less careful with your approximating distributions).
I think the argument still works if the survival rate function is constant at its max for a while so that there’s a comfortable range of temperatures, if you have temperatures falling outside the range where it’s maximized and where the survival rate function is strictly increasing/decreasing. The survival rate function could look like a symmetric trapezoid, and you could calculate integrals/areas for a uniform temperature distribution explicitly.
The area under the curve not in the temperature range would increase if you shift the temperature distribution: if it’s just the two triangles at the ends and assuming the temperature range has width between a and b, as illustrated in the picture, its area would be, for
A1(x)+A2(x)=12h1(x)(w+x)+12h2(x)(w−x)=c2(w+x)(w+x)+c2(w−x)(w−x)=c(w2+x2)initial triangle widths w outside the temperature range, w<b−a2,
a shift of x ,
so shifted triangle widths of w+x and w−x, and
triangle heights h1(x)=c(w+x),h2(x)=c(w−x), c>0.
This is increasing as the absolute value of x increases, so you lose more on the side you lose on than you gain on the side you gain on with a shift. So, the area under the curve in the temperature range and hence the expected survival rate would be decreasing (by the amount A1(x)+A2(x) is increasing) as you shift the temperature range away from its initial mean.