jlwilliams94
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This is a problem that we got in my math 1910 class. I understand parts a, b, and i, but I do not understand part ii because the dk/dlogN changes to dk/dN. I may be missing something implicit, I'm not sure. Here (http://www1.imada.sdu.dk/~hjm/MM503/neuhauserkap5opgaver.pdf) is is a photocopy of the question I found online in case I made a mistake in typing it out. It is #39 on page 6.
Intraspecific Competition: Suppose that a study plot contains N annual plants, each of which produces S seeds that are sown within the same plot. The number of surviving plants in the next year is given by
A(N)=(NS)/(1+(aN)^b) Equation (1)
for some positive constants a and b. This mathematical model incorporates density-dependent mortality. The greater the number of plants in the plot, the lower is the number of surviving offspring per plant, which is given by A(N)/N and is called the net reproduction rate.
a) Show that A(N)/N is a decreasing function of N.
b) The following quantity, called the k-value, can be used to quantify the effects of
intraspecific competition (i.e. competition between individuals of the same species):
k=log[initial density]-log[final density]
Here, “log” denotes the logarithm to base 10. The initial density is the product of the number of plants (N) and the number of seeds each plant produces (S). The final density is given by eqn (1). Use the expression for k and eqn (1) to show that
k=log[NS]-log[(NS)/(1+(aN)^b)]
=log[1+(aN)^b]
We typically plot k versus log N; the slope of the resulting curve is then used to quantify the effects of competition.
i) Show that (dlogN)/dN=1/(Nln(10))
where ln denotes the natural logarithm.
ii) Show that dk/dlogN=(ln10)N[dk/dN]=b/[1+(aN)^-b]
iii) Find lim as N approaches infinity of dk/dlogN
iv) Show that if dk/dlogN<1, then a(N) is increasing; where as if dk/dlogN>1 then A(N) is decreasing. Explain in words what the two inequalities mean with respect to varying the initial density of seeds and observing the number of surviving plants the next year. The first case is called under-compensation and the second case is called over-compensation.
V) The case dk/dlogN=1 is referred to as exact compensation. Suppose that you plot k versus log N and observe that over a certain range of values of N, the slope of the resulting curve is equal to 1. Explain what this means.
Intraspecific Competition: Suppose that a study plot contains N annual plants, each of which produces S seeds that are sown within the same plot. The number of surviving plants in the next year is given by
A(N)=(NS)/(1+(aN)^b) Equation (1)
for some positive constants a and b. This mathematical model incorporates density-dependent mortality. The greater the number of plants in the plot, the lower is the number of surviving offspring per plant, which is given by A(N)/N and is called the net reproduction rate.
a) Show that A(N)/N is a decreasing function of N.
b) The following quantity, called the k-value, can be used to quantify the effects of
intraspecific competition (i.e. competition between individuals of the same species):
k=log[initial density]-log[final density]
Here, “log” denotes the logarithm to base 10. The initial density is the product of the number of plants (N) and the number of seeds each plant produces (S). The final density is given by eqn (1). Use the expression for k and eqn (1) to show that
k=log[NS]-log[(NS)/(1+(aN)^b)]
=log[1+(aN)^b]
We typically plot k versus log N; the slope of the resulting curve is then used to quantify the effects of competition.
i) Show that (dlogN)/dN=1/(Nln(10))
where ln denotes the natural logarithm.
ii) Show that dk/dlogN=(ln10)N[dk/dN]=b/[1+(aN)^-b]
iii) Find lim as N approaches infinity of dk/dlogN
iv) Show that if dk/dlogN<1, then a(N) is increasing; where as if dk/dlogN>1 then A(N) is decreasing. Explain in words what the two inequalities mean with respect to varying the initial density of seeds and observing the number of surviving plants the next year. The first case is called under-compensation and the second case is called over-compensation.
V) The case dk/dlogN=1 is referred to as exact compensation. Suppose that you plot k versus log N and observe that over a certain range of values of N, the slope of the resulting curve is equal to 1. Explain what this means.