Population dynamics modelled by a wave function (Mathematical Biology)

Question

The size of an insect population $N(x,t)$ is governed by the partial differential equation:

$\frac {\partial N}{\partial t}=p(N) + \frac {\partial^2 N}{\partial x^2}$

for some function $p(N)$ .

For cases where $p(N) \equiv 0$ is it possible to find a travelling wave solution of the form $N(x-ct)$, for wave speed c, such that $N\rightarrow 0$ as $|x| \rightarrow \infty$

I'm not where to start with this one, but since it's only 2 marks (the previous question was 10) I might be overthinking it . However, I still need to find some maths I need to do. I'm assuming that $p(N) \equiv 0$ implies that:

$\frac {\partial N}{\partial t}= \frac {\partial^2 N}{\partial x^2}$

But I'm not sure what the form $N(x-ct)$ actually means... Do I differentiate $x-ct$ with respect to $t$ and $z$ (which is zero) or what?

Any help is much appreciated. Thanks

Answer 1

Plug $N=\phi(x-c\,t)$ into the equation to get the ODE $$ -c\,\phi'=\phi''. $$ Solve it and find, for each $c\in\mathbb{R}$ the traveling wave solution $$ N=A\,e^{-c(x-ct)}+B. $$ Depending on the sign of $c$ we have $\lim_{x\to+\infty}N=0$ or $\lim_{x\to-\infty}N=0$, but never $\lim_{|x|\to\infty}N=0$.

Answer 2

There are several things wrong with your analysis:

Your equation is not a wave equation, the linear part is a heat equation, or with the non-linear part, a reaction-diffusion equation.

There will be no wave solutions.

---

The solutions of $P(N)=0$ are stable states, for a region of relatively constant values in $x$ direction with a value $N$ close to a root you get the approximate equation $$ \frac{∂N}{∂t}=(N-α) P'(α) $$ with solution $$ N(t)=α+e^{P'(α)t}(N(0)-α) $$ which means that the roots with negative derivative give stable steady states.

The diffusion term strives to reduce the magnitude of the gradient, thus in a mix of steady stable states there will be some balancing. This will occur in the form of growing and shrinking of the regions of stability, seldom in the form of waves.