Notation Question (Differential Equations)

In summary, the conversation discusses building the Predator-Prey (Lotka Volterra) model, which starts with the "Population Law of Mass Action" where the rate of change of one population due to interaction with another is proportional to the product of the two populations. The conversation also discusses the use of differential equations and the potential confusion that can arise from using simplified notation. The main points highlighted are that the rate of change due to interaction is proportional to the product of the populations and the sign of the coefficient depends on the predator-prey relationship.
  • #1
dkotschessaa
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I want to show, in a talk, how to build the Predator-Prey (Lotka Volterra) model. It starts off with the "Population Law of Mass Action" i.e. "At time t the rate of change of one population due to interaction with another is proportional to the product of the two populations at that time t"


Is it an abuse of notation to write this (for populations x (predator) and y(the prey)) as:

##\frac{dx}{dt} \propto xy ##
##\frac{dy}{dt} \propto xy ##



I think it's ok, I've just never seen this written for a differential equation.

The next step would be to write:

##\frac{dx}{dt} = axy ##
##\frac{dy}{dt} = bxy ##

For some a and b, then start showing what happens when x starts eating y.

-Dave K
 
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  • #2
dkotschessaa said:
I want to show, in a talk, how to build the Predator-Prey (Lotka Volterra) model. It starts off with the "Population Law of Mass Action" i.e. "At time t the rate of change of one population due to interaction with another is proportional to the product of the two populations at that time t"


Is it an abuse of notation to write this (for populations x (predator) and y(the prey)) as:

##\frac{dx}{dt} \propto xy ##
##\frac{dy}{dt} \propto xy ##



I think it's ok, I've just never seen this written for a differential equation.

That is potentially confusing; the full Lotka-Volterra equations include other terms, so it is not actually true that [itex]dx/dt \propto xy[/itex] etc.

I would start with [tex]
\frac{dx}{dt} = rx + p_{\mathrm{interact}} \\
\frac{dy}{dt} = st + q_{\mathrm{interact}}
[/tex] and then proceed immediately to [tex]
p_{\mathrm{interact}} = axy, \\
q_{\mathrm{interact}} = bxy.
[/tex] I would also make all the coefficients positive and introduce minus signs where necessary.

The important points are that (1) the rate of change due to interaction is proportional to the product of the populations, and (2) the sign of the coefficient depends on who is eating who.
 
  • #3
pasmith said:
That is potentially confusing; the full Lotka-Volterra equations include other terms, so it is not actually true that [itex]dx/dt \propto xy[/itex] etc.

I would start with [tex]
\frac{dx}{dt} = rx + p_{\mathrm{interact}} \\
\frac{dy}{dt} = st + q_{\mathrm{interact}}
[/tex] and then proceed immediately to [tex]
p_{\mathrm{interact}} = axy, \\
q_{\mathrm{interact}} = bxy.
[/tex] I would also make all the coefficients positive and introduce minus signs where necessary.

The important points are that (1) the rate of change due to interaction is proportional to the product of the populations, and (2) the sign of the coefficient depends on who is eating who.


did you mean ##\frac{dy}{dt} = sy + q_{\mathrm{interact}} ## in your second example?

That's a very good way of explaining it though. Thank you.

-Dave K
 

Related to Notation Question (Differential Equations)

What is a differential equation?

A differential equation is an equation that relates one or more functions to their derivatives. It describes how a function changes over time or in relation to another variable.

What is the purpose of using notation in differential equations?

Notation in differential equations is used to represent mathematical concepts and relationships in a concise and standardized way. It allows for a more efficient and accurate communication of ideas and solutions.

What are the different types of notation used in differential equations?

The most common types of notation in differential equations include ordinary differential equations (ODEs), partial differential equations (PDEs), and vector notation. ODEs involve functions of one variable, while PDEs involve functions of multiple variables. Vector notation is used to represent systems of equations involving multiple functions.

How do I solve a differential equation?

Solving a differential equation involves finding the function or functions that satisfy the given equation. This can be done analytically using mathematical techniques such as separation of variables or substitution, or numerically using computer software or algorithms.

What are some real-life applications of differential equations?

Differential equations are used in a variety of scientific fields, including physics, chemistry, biology, and engineering. They are used to model and understand complex systems and phenomena, such as heat transfer, population growth, and fluid dynamics.

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