size_number theory.2

# size_number theory.2 - Here we are interested in there...

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Here we are interested in there trade-off between the size and number of offspring. We want to know the size of offspring that maximizes individ- ual fitness. DEFINING THE VARIABLES. .. q = fraction of resouces allocated to each offspring. qmin = the fraction of resources required to produce and offspring that has a suvival probability greater than zero. x = exponent that controls the shape of expected fitness a function of q. No = number of offspring, which is equal to 1/q Wo = fitness of individual offspring. In[1]:= q=. qmin=. x=. No=. Wo=. FIRST WE DEFINE THE Number OF OFFSRPING, No (Put the cursor anywhere in the bracket below and press "SHIFT-RETURN." ) In[6]:= No = 1/q Out[6]= 1 ÅÅÅÅ q NEXT WE DEFINE THE FITNESS OF OFFSRPING, Wo (Put the cursor anywhere in the bracket below and press "SHIFT-RETURN." )

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In[7]:= Wo = (q - qmin)^x Out[7]= H q - qmin L x THE FITNESS OF THE PARENT IS THE SIZE (QUALITY) OF OFFSPRING TIMES THE NUMBER OF OFFSPRING. HENCE WE GET, (Put the cursor anywhere in the bracket below and press "SHIFT-RETURN." ) In[8]:= Wp = No*Wo Out[8]= H q - qmin L x ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅ q WE WANT TO KNOW WHERE PARENTAL FITNESS IS AT A LOCAL MAXIMUM. SO WE TAKE THE FIRST PARTIAL DERIVATIVE OF PARENTAL FITNESS WITH RESPECT TO ALLOCATION TO OFFSPRING, AND SET IT EQUAL TO ZERO. FIRST WE TAKE THE DERIVATIVE. .. (Put the cursor anywhere in the bracket below and press "SHIFT-RETURN." ) In[9]:= Firstder = D[Wp, q] Out[9]= - H q - qmin L x ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅ q 2 + H q - qmin L - 1 + x x ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅ q (Put the cursor anywhere in the bracket below and press "SHIFT-RETURN." ) 2 size:number theory.2.nb
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size_number theory.2 - Here we are interested in there...

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