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LMC.7 - Local Mate Competition(LMC Here we are interested...

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Local Mate Competition (LMC). Here we are interested in the question: how does the number of mates affect the proportion of resources that should be allocated by a hermaphrodite to male and females function in order to maximize individual fitness? DEFINING THE VARIABLES... a i = allocation to male function by the ith hermaphrodite in the population. a = average allocation to male function in the population (this was a* in lecture, but I can't use the star here, because it is used exclusively by Mathematica as the multiplication symbol.) R = Total resource base for reproduction k = the number of mates V = The reproductive value though male fuction W m = Fitness through male function W f = Fitness though female function (assuming Bateman's principle). W i = Fitness of the ith individual (Wf + Wm) FIRST WE DEFINE FITNESS THROUGH FEMALE FUNCTION, W f. (Put the cursor anywhere in the bracket below and press "SHIFT-RETURN." This will store W f in memory) Wf = R*(1 - ai) H 1 - ai L R NEXT DEFINE FITNESS THROUGH MALE FUNCTION, W m. V is the reproductive value of male function, which will be defined shortly. (Put the cursor anywhere in the bracket below and press "SHIFT-RETURN.")
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Wm = V*R*ai aiRV NOW DEFINE THE VARIABLE, V, AS THE RATIO OF EGGS IN THE POPULATION TO SPERM IN THE POPULATION. (Put the cursor anywhere in the bracket below and press "SHIFT-RETURN.") V = (k*(1 - a))/((k - 1)*a + ai) H 1 - a L k ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ai + a H - 1 + k L TOTAL FITNESS OF THE INDIVIDUAL IS THE SUM OF FITNESS THROUGH FEMALE + MALE FUNCTION. (Put the cursor anywhere in the bracket below and press "SHIFT-RETURN.") Wi = Wf + Wm H 1 - ai L R + H 1 - a L aikR ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ai + a H - 1 + k L NOW CALCULATE THE FIRST DERIVATIVE OF INDIVIDUAL FITNESS WITH RESPECT TO ALLOCATION TO MALE FUNCTION. (Put the cursor anywhere in the bracket below and press "SHIFT-RETURN.") 2 LMC.7.nb
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Firstder = D[Wi, ai] - R - H 1 - a L aikR ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅ H ai + a H - 1 + k LL 2 + H 1 - a L kR ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ai + a H - 1 + k L CALCULATE THE SECOND DERIVATIVE OF INDIVIDUAL FITNESS AS A FUNCTION OF ITS ALLOCATION TO MALE FUNCTION.
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