Imagine walking along a path in a large grassy meadow. On the path, the level of

Question

Imagine walking along a path in a large grassy meadow. On the path, the level of disturbance is high and very few (maybe no) plant species grow there. If you look in the distance, far from the path, you see areas of low disturbance. In these undisturbed areas you may see that although plants thrive, there are just one or two dominant species of plants. Finally, if you look in a part of the meadow that lies between the areas of high and low disturbance, it is likely that you will see a large variety of plant species. This common observation, that the number of species is maximized in areas with neither extremely low nor or extremely high disturbances, is called the intermediate disturbance principal. The goal of this project is to explore a mathematical model that explains this principal.

Explanation / Answer

Solution:

We have A=Amax -aE and z=bE
And we have given relation S=Az

a). Now proceeding taking log on both sides
ln(S)=zln(A)

now differentiating both sides with respect to E
d(ln S)/d(E) = d(zlnA)/d(E)

which gives (1/S)(dS/dE)=lnA(dz/dE)+z(d(ln A)/dE)

=b(lnA) + (z(-a))/A (dz/dE=b and d(ln A)/dE=-a/A -by chain rule)

Now taking 1/S on other side
dS/dE=-za(S/A) + bS(lnA)

b). We have A=Amax -aE and z=bE

So if E tends to zero then A wil tend to Amax and z will tend to 0.
then in equation dS/dE= -za(S/A) + bS(lnA) first term will vanish

And finally we will have dS/dE= bS(lnA) which is greater than zero.

So now slope dS/dE >0 at the condition that E tends to zero,

which shows that at the small level of disturbance(E), diversity(S) increases.

c). With the above equations
When E will be very large then the difference (Amax -aE) will be small but positive

And z will tend to 1.
So in equation dS/dE= -za(S/A) + bS(lnA) ,magnitude of first term will be greater than magnitude of second term since small denominator A , So dS/dE <0 ,which proves that slope is negative.

Now we can say that as disturbance(E) becomes very large, Diversity (S) decreases.

d). From b part, we know that if E is very less (nearly 0), then S increases and from c part, we know that if E is very large, then S decreases

So from this knowledge and assuming continuity, we can say that there must be a point in between E=0 and E=very large, where the curve will change its slope from positive to negative.
And that is the relevant value of E where we will find the maxima.

e). Now with all the assumptions, we can say that S=f(E)

So taking A=Amax -aE and z=bE in S=Az  

We have f(E)=S= (Amax -aE)bE

f).   We have S=(Amax -aE)bE

  Differentiating both sides with respect to E

dS/dE=(bE)(Amax -aE)(bE-1) (-a)(b)

putting dS/dE=0(where slope is zero)

we get Amax =aE, which gives E=Amax /a on which the curve will take its maximum value

And which is S=(Amax -(a)(Amax /a))bE=0

So maximum value of S exactly is zero.

Now graph of S, it will be parabola facing downward passing through 0.

  

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