Duarte and Agustí (1998) investigatedthe CO2 balance o f  aquati  c ecosystems. They related thecommunity respiration rates (R) to the gross primary production rates (P) of aquatic ecosystems. (Both quantities were measured in the same units.) They made the following statement:

Our results  confirm the generality of earlier reports that the relation between community respiration rate andgross production is not linear. Community respirationis scaled as the approximate two-thirds power of grossproduction.

(a) Suppose that you obtained data on the gross production and respiration rates of a number of freshwater lakes. How would you display your data graphically to quickly convince an audience that the exponent b in the power equation relating Rand P is indeed approximately 2/3? (Hint: Use an appropriate log transformation in Maple)

(c) The r atio R/P for an ecosystem is important in assessing the global CO2 budget. If respiration exceeds production (i.e., R >P), then the ecosystem acts as a carbon dioxide source, whereas if production exceeds respiration (i.e., P > R), then the ecosystema cts as a carbon dioxide sink. Assume now that the exponent inthe power equation relating R and P is 2/3. Show that the ratio R/P, as a function of P, is continuous for P > 0. Furthermore,show that
           lim R/P =∞
            P→0+

and
            lim R/P= 0
            P→∞


How to use Maple to sketch the graph of the ratio R/P as  afunction of P for P > 0. (Experiment with the graphing calculatorto see how the value of a affects the graph.)

Hyperbolic functions are used in the sciences. The hyperbolic sine , sinh x;the hyperbolic cosine, cosh x; and the hyperbolic tangent, tanh x,defined respectively as

sinh x = e^x − e^−x/ 2, x ∈ R
cosh x = e^x + e^−x/2, x ∈ R
tanh x = e^x − e^−x/e^x + e^−x , x ∈ R

How do I show that these three hyperbolic functions are continuous forall x ∈ R using Maple?

Suppose that an organism reacts to a stimulusonly when the stim ulus exceeds a certain threshold. Assume that
the stimulus is a function of time t and that it is given by s(t) = sin(πt), t ≥ 0.The organism reacts to the stimulus and shows a certain reactio nwhen s(t) ≥ 1/2.

Define a function g(t) such  that g(t) = 0 when the organism shows no reaction at time t and g(t) = 1 when theorganism shows the reaction.

I already got the function g(t) manually.

g(t)= {  1 for 1/6 + 2k<=x<= 5/6 + 2k, k=0,1,2,...

           0  otherwise

  How do I plot s(t) and g(t) in the same coordinate system?
 

N_t+1 = RN_t/ [1+ (R-1/K)N_t] where R> 1, K>0. When N_o >0 , lim _{t approaching infinity}N_t = K for all values of R>0.

Find N_t for t=1,2,3...10 for K= 100 and N_o= 20 when a) R=2, b)R=5 and c)R= 10.

I have managed to get the answers for a), b) and c) manually.

But how do I plot N_t as a function of t for the three choices of R in one coordinate system?

The pH levels of a lake controls the conc. of harmless ammonium ions(NH₄⁺) and toxic ammonia (NH₃) in the lake.

For pH levels <8, conc. of ammonium ions are little affected by pH changes, but decline over many orders of magnitude as pH levels increase beyond pH 8.

Toxic ammonia are negligible at low pH , increase over many orders of magnitude as the pH level increases and reach a high plateau at about pH =10 (after which, NH₃ are little affected by pH changes).

How do I illustrate this graphically?