There is considerable variation in the reproductive strategies that different organisms use. Some organisms produce large...

 

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There is considerable variation in the reproductive strategies that different organisms use. Some organisms produce large numbers of offspring and invest very little in the success of each individual offspring. Others produce few offspring and invest considerable resources in ensuring that each individual survives to adulthood. Biologists use the number of offspring produced by an organism that survive to maturity as a measure of reproductive fitness. In this problem we will explore how reproductive fitness varies with the number of offspring. Suppose that R is a positive constant that denotes the total amount of resources that an organism can provide for its offspring and x denotes the amount of resources an organism invests in each individual off-spring. If an organism produces N offspring, we assume x = . The reproductive fitness F(x) is the expected number of offspring that will survive to maturity and is modeled by Rx k + x where k is a positive constant. The parameters R and k will depend on the particular organism and its habitat. (a) Show that the first-derivative of the fitness function is given by R(k: (k + x) F(x) = F'(x) = (b) Use the function F'(x) to find the intervals of increase and decrease of F and identify the location of a local maximum. (c) The second derivative of the fitness function is -2Rx(3k - x) (k + x) F"(x) = Use the second derivative function to find the intervals of concavity and identify an inflection point. (d) Write up a summary that addresses the following questions: Is the local maximum you found a global maximum for the fitness function? How does the location of the maximum relate to the number of offspring an organism will tend to produce? How does the location of the maximum depend on the parameters k and R? Consider an organism that produces large numbers of offspring. Does this correspond to a "small" or "large" value of k? There is considerable variation in the reproductive strategies that different organisms use. Some organisms produce large numbers of offspring and invest very little in the success of each individual offspring. Others produce few offspring and invest considerable resources in ensuring that each individual survives to adulthood. Biologists use the number of offspring produced by an organism that survive to maturity as a measure of reproductive fitness. In this problem we will explore how reproductive fitness varies with the number of offspring. Suppose that R is a positive constant that denotes the total amount of resources that an organism can provide for its offspring and x denotes the amount of resources an organism invests in each individual off-spring. If an organism produces N offspring, we assume x = . The reproductive fitness F(x) is the expected number of offspring that will survive to maturity and is modeled by Rx k + x where k is a positive constant. The parameters R and k will depend on the particular organism and its habitat. (a) Show that the first-derivative of the fitness function is given by R(k: (k + x) F(x) = F'(x) = (b) Use the function F'(x) to find the intervals of increase and decrease of F and identify the location of a local maximum. (c) The second derivative of the fitness function is -2Rx(3k - x) (k + x) F"(x) = Use the second derivative function to find the intervals of concavity and identify an inflection point. (d) Write up a summary that addresses the following questions: Is the local maximum you found a global maximum for the fitness function? How does the location of the maximum relate to the number of offspring an organism will tend to produce? How does the location of the maximum depend on the parameters k and R? Consider an organism that produces large numbers of offspring. Does this correspond to a "small" or "large" value of k?