Ecology-population growth 2. Clearly define and distinguish r and ractual, as th
ID: 83459 • Letter: E
Question
Ecology-population growth
2. Clearly define and distinguish r and ractual, as these terms relate to population growth
in a single species. You are encouraged to use graphs or equations in your answer. If
you rely only on words, you must be precise! [Study hint: this and several of the
following questions ask you to be precise. What does this mean? Equations are preferable over words because they are necessarily more precise. For example, for a
complete answer, you must say that the population growth rates are per capita or per
individual. Do you see why this detail is necessary?]
Please explain briefly!
2. Clearly define and distinguish r and r actual, as these terms relate to population growth in a single species. You are encouraged to use graphs or equations in your answer. If you rely only on words, you must be precise! Study hint: this and several of the following questions ask you to be precise. What does this mean? Equations are preferable over words because they are necessarily more precise. For example, for a complete answer, you must say that the population growth rates are per capita or per individual. Do you see why this detail is necessary?]Explanation / Answer
r is called the intrinsic or instantaneous rate of increase. It expresses the balance between birth and death processes. Here are some sonditions under which populations may grow exponentially for a short period of time:
r is expressed in numbers per unit time per individual and has the units of 1/time. In a closed population the intrinsic rate of increase is defined as the instantaneous per capita birth rate minus instantaneous death rate. Similarly in open population r is equal to (births + immigration) - (deaths + emigration). In practice, r is somewhat cumbersome to calculate, as its value must be determined by iteration using Euler's implict equation.
erx1xmx
where e is the base of natural logarithms and x subscripts age. Survivorship and fecundity at age x are deignated with 1x & mx, respectively. Provided that the net reproductive rate, R0, is near 1, r can be estimated using the approximate formula,
logeR0/T
where T is the generation time.
r actual is defined as the actual instantaneous rate of increase; it is zero at K, negative above K and positive when the population is below K. Here K is carrying capacity, as the density of organisms at which the net reproductive rate equals unity and the intrinsic rate of increase (r) is zero. The actual instantaneous rate of increase per individual, ra, decreases linearly with increasing population density under assumptions of the Verhulst - Pearl equation. Hypothtical curvilinear relationships between instantaneous rates of increase and population density. concave upward curves have also been postulated. Notice that r in the logistic equation is actually rmax. The equation can be solved for the rate of increase, ractual, which is a variable and a function of r,N and K, by simply factoring out an N:
ractual = ra = dN/NdT = r{(K-N)/K} = rmax - (N-K) rmax.
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