Question on setting an equation

[Clara Chung]

How to express "a" tends to "b",but "b" increases as "a" tends to "b" in equations?

 

[fresh_42]

How can $a$ tend to $b$, if $b$ runs away from $a$? Who's faster? By which quantities?
I assume you have a special example in mind. Can you tell?

 

[Clara Chung]

How can $a$ tend to $b$, if $b$ runs away from $a$? Who's faster? By which quantities?
I assume you have a special example in mind. Can you tell?

The quantities are arbitrary. I was thinking something in biology like a group of giraffes, their necks tend to grow longer to eat more leaves, but as their necks grow longer, the leaves grow higher too. Sorry for the absurd example.

 

[mfb]

(a-b) goes to zero?
(a-b)/b goes to zero?

 

[fresh_42]

So you are talking about a dynamic system with two functions $x(t)$ and $y(t)$ in a parameterizing variable $t$. In your example - and as usual - $t$ is time, $x(t)$ the size of the giraffes and $y(t)$ the size of trees.
Next there is still the question about the behavior of $x$ and $y$ in time. In addition $x=x(y,t)$ and $y=y(x,t)$, i.e. both functions also depend on the values of the other. The dependencies and behavior are normally expressed by a system of differential equations.

A better example for it might be the sizes of populations in a predator-prey-system which simple cases are described by the Lotka-Volterra equations.

 

[S.G. Janssens]

A better example for it might be the sizes of populations in a predator-prey-system which simple cases are described by the Lotka-Volterra equations.

LV is a very good example to study carefully. It can be analyzed completely, but it is nonlinear. For the simplest linear system of ODE that literally matches the description in the OP, I would imagine something like
$$
\begin{align*}
\frac{da}{dt} &= \mu + \lambda c\\
\frac{db}{dt} &= \mu
\end{align*}
$$
where $c := b - a$ is the difference of $b$ and $a$. Also, $\mu > 0$ and $\lambda > 0$ are parameters and $b(0) > a(0) > 0$ are the initial conditions.

In this case the solution is $b(t) = b(0) + \mu t$ and $c(t) = [b(0) - a(0)] e^{-\lambda t}$. So you can see that $a$ tends to $b$ (because $c$ tends to zero from above) while $b$ increases as well. If you want both processes to occur at an exponential rate, it is easy to modify the above system. (You could try this yourself.) In this case, the relative speed depends on the ratio $\tfrac{\lambda}{\mu}$.

 

[Clara Chung]

LV is a very good example to study carefully. It can be analyzed completely, but it is nonlinear. For the simplest linear system of ODE that literally matches the description in the OP, I would imagine something like
$$
\begin{align*}
\frac{da}{dt} &= \mu + \lambda c\\
\frac{db}{dt} &= \mu
\end{align*}
$$
where $c := b - a$ is the difference of $b$ and $a$. Also, $\mu > 0$ and $\lambda > 0$ are parameters and $b(0) > a(0) > 0$ are the initial conditions.

In this case the solution is $b(t) = b(0) + \mu t$ and $c(t) = [b(0) - a(0)] e^{-\lambda t}$. So you can see that $a$ tends to $b$ (because $c$ tends to zero from above) while $b$ increases as well. If you want both processes to occur at an exponential rate, it is easy to modify the above system. (You could try this yourself.) In this case, the relative speed depends on the ratio $\tfrac{\lambda}{\mu}$.

Thanks for answering, I don't understand db/dt is set to be u. So it is independent of a, how can the effect of b ran away as a approaches b be shown?

 

[S.G. Janssens]

Thanks for answering, I don't understand db/dt is set to be u. So it is independent of a, how can the effect of b ran away as a approaches b be shown?

You can also choose something more complicated for $\tfrac{db}{dt}$ if you like, but what I wanted this system to do is to match literally with

How to express "a" tends to "b",but "b" increases as "a" tends to "b" in equations?

while keeping it as simple as possible. That is, I wanted $b - a$ to become small in time while $b$ itself keeps increasing in time. You can see that this works by solving the system explicitly.

If, instead, you want to model a situation where $a$ approaches $b$ and then, once $b - a$ is small, $b$ runs away from $a$ (so as to increase $b - a$ again), "my" system is not going to work because what are called its "eigenvalues" are real numbers.

So, it depends on a slightly more precise description of what you would like $a$ and $b$ to behave like.

 

[fresh_42]

LV is a very good example to study carefully. It can be analyzed completely, but it is nonlinear. For the simplest linear system of ODE that literally matches the description in the OP, I would imagine something like ...

What I like most of these examples is, that one can draw a lot of them. It's there where I first met vector fields and their behavior, which is far easier to grasp, than a purely topological or algebraic description.

Thanks for answering, I don't understand db/dt is set to be u. So it is independent of a, how can the effect of b ran away as a approaches b be shown?

You can think of and draw $\frac{da}{dt}$ and $\frac{db}{dt}$ as little arrows pointing in the direction where $a$ and $b$ change, i.e. their tangents (varying with $t$ at the spots where $a(t_0)\; , \;b(t_0)$ are at the time $t_0$) and thus resulting in a vector field. (For pictures you may look up "attractor" or "repeller" e.g.)

Here, both $a$ and $b$ depend on $\mu$ and therefore on each other. You cannot change one without influencing the other. Your conditions are satisfied by how @Krylov defined them.

 

[mfb]

@Clara Chung: It would reduce speculations in this thread a lot if you could be clearer in the description of your system.

 

[S.G. Janssens]

What I like most of these examples is, that one can draw a lot of them. It's there where I first met vector fields and their behavior, which is far easier to grasp, than a purely topological or algebraic description.

I very much agree.

I would like to take the liberty to recommend two books on this important perspective that I always recommend: Strogatz' well known Nonlinear Dynamics and Chaos (a nightmare for a rigorous mathematician, but when I was still studying physics I found it fantastic) and Differential Equations, Dynamical Systems, and an Introduction to Chaos by Hirsch, Smale and Devaney (a better book than Strogatz' when you are a little bit (but not much) more advanced in your studies).