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?

Percentage survivorship is a function of time; there ia negligible decline in survivorship at pH=6, marked decline in survivorship at pH=3.5, resulting in no survivors after just 8 hours.

What is the function for this problem and how do I plot it in Maple?

Hi all,

I have the following expression,

where K and K_P are the controller gains, k is the stiffness, b is the damper, m1 and m2 are masses.

How can I find the condition on variables such that the numerator of this expression is greater that zero?

The conditions should appear as inequalities.

Download Routh-Hurwitz_stability.mw

The rate of resource consumption by organisms,v depends on resource concentration, S. The Blackman model of resource consumption assumes a linear relationship between v and S. Below a threshold concentration, S_k , consumption rate increases linearly with S=0 when v=0; when S=S_k, v reaches its max value, v_max. For S> S_k, v stays at v_max.

The function like this is defined piecewise,

  v= { g(S)   for 0<=S< S_k

         vmax  for S>=Sk

g(S)=( v_max/S_k).S

How do I graph v as a function of S with the graph showing the point where S_k=v_max?

Egg development

It takes 3 days at 20^oC, but almost 20 days at 5^oC for an egg to develop and hatch. When graphed on log -log plot, it says that egg development time (in days) as a function of temperature (in ^oC) is a straight line.

How do I sketch the log-log graph in Maple?

The chemical reaction is given as

A+2B ---> AB₂

The reaction occurs ina closed vessel and the initial concentration of A=[a] and B=[b]

The reaction rate is given by

R(x)= k(a-x)(b-2x)²

where x=[AB₂]

Graph R(x) for the relevant values of x when a=5,b=6 and k=0.3.

Need help creating a proc() for Kaprekar's routine. Ive been able to convert a number (x,n) to ascending/descending order and then defining f to be the sum of them subtracted, but i cant figure out how to put it into a proc so that it continues to loop until kaprekar's number 6174 is reached and stop.

If you have any idea please advise! thanks

Im trying to create a groebner basis to solve a series of equations to make a basis. The values can only be 1 to 4 for a..p which im also unsure if/how to define which may be causing the problem.

 My equations are as follows:

e1 := (((a-1)(a-2))(a-3))(a-4) = 0;
e2 := (((b-1)(b-2))(b-3))(b-4) = 0;
e3 := (((c-1)(c-2))(c-3))(c-4) = 0;
e4 := (((d-1)(d-2))(d-3))(d-4) = 0;
e5 := (((e-1)(e-2))(e-3))(e-4) = 0;
e6 := (((f-1)(f-2))(f-3))(f-4) = 0;
e7 := (((g-1)(g-2))(g-3))(g-4) = 0;
e8 := (((h-1)(h-2))(h-3))(h-4) = 0;
e9 := (((i-1)(i-2))(i-3))(i-4) = 0;
e10 := (((j-1)(j-2))(j-3))(j-4) = 0;
e11 := (((k-1)(k-2))(k-3))(k-4) = 0;
e12 := (((l-1)(l-2))(l-3))(l-4) = 0;
e13 := (((m-1)(m-2))(m-3))(m-4) = 0;
e14 := (((-1+n)(n-2))(n-3))(n-4) = 0;
e15 := (((-1+o)(o-2))(o-3))(o-4) = 0;
e16 := (((p-1)(p-2))(p-3))(p-4) = 0;
e17 := a+b+c+d-10 = 0;
e18 := a*b*c*d-24 = 0;
e19 := e+f+g+h-10 = 0;
e20 := e*f*g*h-24 = 0;
e21 := i+j+k+l-10 = 0;
e22 := i*j*k*l = 24;
e23 := m+n+o+p-10 = 0;
e24 := m*n*o*p-24 = 0;
e25 := a+b+e+f-10 = 0;
e26 := a*b*e*f-24 = 0;
e27 := c+d+g+h-10 = 0;
e28 := c*d*g*h-24 = 0;
e29 := i+j+m+n-10 = 0;
e30 := i*j*m*n-24 = 0;
e31 := k+l+o+p-10 = 0;
e32 := k*l*o*p-24 = 0;
e33 := a+e+i+m-10 = 0;
e34 := a*e*i*m-24 = 0;
e35 := b+f+j+n-10 = 0;
e36 := b*f*j*n-24 = 0;
e37 := c+g+k+o-10 = 0;
e38 := c*g*k*o-24 = 0;
e39 := d+h+l+p-10 = 0;
e40 := d*h*l*p-24 = 0;
e41 := c = 4;
e42 := e = 1;
e43 := j = 2;                  
e44 := p = 3;
 

Hi

This question may have been asked before but I cannot find it!

I am currently have 3 lists:

L:= [1,3,5,2,3]

M:=[4,2,4,6,2]

R:= [5,3,2,1,4]

I would like to be able to come up with a way (I assume a procedure) where I can order L and M based on the numerical order of R. So in this case that would be:

R:=[1,2,3,4,5]

L:=[2,5,3,3,1]

M:=[6,4,2,2,4]

I hope this makes sense? Any help would be greatly appreciated!

Hi All,

I am ploting with the command of:

logplot([1-R(Pp, tp, tf, 'qN(Pp, tp, tf)'), 1/(1+1+tp), .5/(.5+.5+tp), (1-Pp)*Beta(2, 1+tp)/(Pp*Beta(1, 1)+(1-Pp)*Beta(1, 1+tp))], Pp = 0 .. 1, style = pointline, numpoints = 1, gridlines = true, symbolsize = 10, legend = ["Our worst-case prior with Pp", "Flat prior Beta (1,1)", "Jeffrey's prior Beta (0.5,0.5)", "Normalised flat prior Beta (1,1) with Pp"], labels = ["Pp", "Posterior expected pfd"], symbol = [asterisk, diagonalcross, box, circle], title = "Given fixed t_past=100")

and get:

which is ok but the legends are wrong. where are the symbols in the legends? Am I missing something....

how to solve differential equations with dependent functions in maple?