Exploring the Power Law Relationship between Mass and Metabolic Rate from ODE

[schaefera]

I'm trying to find a power law relationship between mass and metabolic rate, given that each of these quantities is defined by a differential equation.

Assuming dM/dt=a*M(t) and dR/dt=b*R(t), where M(t) is mass and R(t) is metabolic rate, I know that I can solve each of these equations to get:
M(t)=c*exp[at] and R(t)=k*exp[bt].

Here is where the first part of my question comes in: Let's say I try to solve both of these for t, and then set them equal to each other. Then I end up with a*ln(M/C)=b*ln(R/K) where C and K are different constants than before, but that's not really important. Is solving for t and setting equal allowed? I'm not sure if I'm looking at a specific time where the two equations are equal in this case, but I can't think of any other way to get rid of the variable.

Otherwise, I would think of dividing the two equations and getting M(t)/R(t)=h*exp[(a-b)t] where h is, again, a new constant that is unimportant. In this case, M=h*R*exp[(a-b)t]... which is different than when I eliminate t.

In either case, I don't see a power law relationship! These are exponential, and not power law, equations unless I'm very mistaken. How can I get to the final product to see the power law in play?

 

[haruspex]

Assuming dM/dt=a*M(t) and dR/dt=b*R(t), where M(t) is mass and R(t) is metabolic rate, I know that I can solve each of these equations to get:
M(t)=c*exp[at] and R(t)=k*exp[bt].

Let's say I try to solve both of these for t, and then set them equal to each other. Then I end up with a*ln(M/C)=b*ln(R/K) where C and K are different constants than before,

Looks to me that c and k are the same as before, but a and b are inverted.

Is solving for t and setting equal allowed?

Seems good to me. Why do you think it's invalid? You want to know the relationship between M and R at each given value of t.

I don't see a power law relationship!

You don't?
a*ln(M/C)=b*ln(R/K)
ln((M/C)^a)=ln((R/K) ^b)
(M/C)^a=(R/K) ^b

 

[schaefera]

Thank you so much! Silly me- I was exponentiating both sides without bringing the constants into the log.

I was unsure about solving for t because it seemed like I'm setting the two sides equal to each other for all t while they might not always be equal.

 

[Chestermiller]

Just divide one differential equation by the other to eliminate the dt.

 

[blue_raver22]

I would like to commend you on your exploration of the relationship between mass and metabolic rate using differential equations. It is important to understand the underlying mathematical principles behind any relationship in order to fully comprehend its behavior and implications.

In response to your first question, solving for t and setting the equations equal to each other is a valid approach. However, this method may not provide a clear understanding of the power law relationship between mass and metabolic rate. Instead, I would suggest looking at the ratio of the two equations, as you have mentioned in your second approach. This will give you a better understanding of how the two quantities are related to each other.

In terms of finding a power law relationship, it is important to note that the power law relationship between mass and metabolic rate is not a direct result of the differential equations themselves, but rather an observation of the behavior of these quantities in biological systems. Therefore, it may be helpful to gather data from various biological systems and analyze it using regression techniques to determine the power law relationship between mass and metabolic rate.

Additionally, it is possible that the power law relationship may not hold for all biological systems, as there are many factors that can influence metabolic rate, such as age, activity level, and environmental conditions. Therefore, it is important to carefully consider the limitations and assumptions of the power law relationship in your analysis.

In summary, while solving the differential equations and finding a power law relationship may not be a straightforward process, it is a valuable exercise in understanding the underlying principles and behavior of mass and metabolic rate in biological systems. I encourage you to continue your exploration and to consider gathering and analyzing data to further support your findings.