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Several mathematical models from theretical biology deal with various forms of cell motility (for example, cancer growth) or evolution of populations.
The equations that make up such models usually contain (among others) a Fickian diffusion term of the form $$- delta Delta u(t,x),$$ where $u$ is the cell (or population) density.
To fix some ideas, consider the following examples.
- A chemotaxis model:
$$partial_t u(t,x) = delta Delta u(t,x) - abla cdot(uKstar u),$$ where $K star u = int_Omega K(x,y)u(t,y)dy$, for a certain kernel $K$.
- A model arising in population dynamics:
$$partial_t u(t,x) = deltaDelta u(t,x) + u(t,x) left(lambda - aint_Omega u(t,y) dy ight).$$
Biologically, what is the meaning of a situation where $delta = 0$ (that is, where there is not Fickian diffusion governed by the Laplacian term)?
Is there any interest in studying such situations?
Can you point out some references on this topic?
Just so we're on the same page…
$u(x,t)$ is a concentration/density function that describes the number of particulate species (bacteria, gas molecules, etc.) at point-in-space $x$ and time $t$. And although it may be an obvious statement to make… with $u$ having the parameters $(x,t)$, this indicates that particulate concentration values are dependent upon both position and time.
$delta$ is the diffusion coefficient of the particulate concentration, defined as a proportionality constant between particulate diffusion and the overall particulate concentration gradient. Particulate diffusion can be regarded as local behavior, whereas the overall particulate concentration gradient describes the behavior of distribution for the entire particulate concentration, and of which may be affected by forces other than the particulate's response to an attractant/repellant gradient, including temperature, pressure, and/or other environmental variables. An in depth understanding of this coefficient can be obtained by reading sections 2.1 & 2.2.
Biologically, what is the meaning of a situation where $delta = 0$?
Well, if you did in fact read the recommended sections, you may already have realized that it's just not possible for $delta = 0$, unless you're considering concentrations of mass that simply can't diffuse throughout one another, but even then $delta$ would be considered as $DNE$.
To reference a portion of 2.1 & 2.2:
Molecular diffusion, strictly speaking, cannot occur under conditions in which both the net flux and the [concentration gradient] are simultaneously zero. If the [concentration gradient] is uniform, then in general fluxes are different for different species, and the net flux is not zero. If the net flux is zero, a small [concentration] gradient must exist in order to counter the tendency for the different species fluxes to be different.
That being said, the diffusion coefficient can be an extremely small value - of which is based on the nature of the particulate(s) undergoing diffusion, and the medium for which they travel throughout - but it's never zero. See here for a series of tables that contain diffusion coefficients for common substances at standard conditions.
And as for the biological meaning to $delta = DNE$… the only conclusion that could be made is that the mass concentrations under consideration can't diffuse with one another, for whatever reason. Whether or not there's meaning beyond this, I believe that would be dependent upon what's specifically being studied.
… that is, where there is not Fickian diffusion governed by the Laplacian term?
This statement however, does have biological meaning to it. When $delta Delta u(t,x) = 0$, this means there is a zero net flux occurring at point-in-space $x$ at time $t$. Which is to say that - the number of particulates of a given species entering a given region of space over a given time interval is the exact same as the number of particulates of the same species leaving that exact same region of space over the exact same time interval. When this is true for all regions of space within the [biological] system, it means that the system is in a steady-state, and/or has reached equilibrium.
Note: Equilibrium and steady-state systems are not the same, however, the differences involved aren't relevant to the scope of this discussion, and so nothing further will be said about this.
When a biological system is in a steady-state, and/or is in equilibrium, that can carry significant implications. Perhaps the most ubiquitous example that also first comes to mind are the steady-states of ion channels, with the effects of cell activation/non-activation being the implications. Another example could be pretty much any chemical reaction that reaches equilibrium.
Is there any interest in studying such situations? Can you point out some references on this topic?
Yes, there are many occurrences where these states are studied, of which you may already be aware of. Regardless, here are a few that seem worthy enough for me to provide in this response:
Steady-state Distribution of Bacteria Chemotactic Toward Oxygen
Equilibrium of Two Populations Subjected to Chemotaxis
The Hodgkin-Huxley Model
The biological meaning of $delta Delta u(t,x) = 0$ usually deals with scenarios where the region(s) in space experiences a flow of mass that has equal incoming & outgoing rates for a given particulate species. Most of the time, this is indicative of a steady-state, and/or equilibrium, and this can be confined to local regions, or globally throughout the system.
And I'd also like to offer a slightly different perspective. Normally, the diffusion coefficient is a temperature & pressure dependent function. However, if you're considering situations with STP/NTP, the diffusion coefficient can truly be treated as a constant. When this is the case, the Laplacian term is analogous to the heat equation. By taking this perspective, the Laplacian term [being zero] is stating that there is no loss or gain in heat (species-specific particulate motion) when considering a region of space over a defined time interval, and the overall model would describe the change(s) in motion (heat) for the entire system. From there, biology could explain as to why the "heat" behaves the way it does, of which would involve the mechanisms & behaviors of an organism with respect to its reaction(s) to an attractant/repellent.
And lastly, for the sake of inclusivity, the other terms in these models generally represent behaviors that are external to local interactions, and each may carry differing effects, such as birth/death rates in population dynamics, or particulate concentration movement that's perpendicular to the axis to which the concentration gradient moves upon, as is the case with the second term in the chemotaxis model. So then, when the Laplacian term is not contributing to any change, i.e., when there is a zero net flux, these other terms will then be the only candidate contributors to any potential changes in the system, whatever those changes may be.