I want to find the late time decay of the size of a population of humans which is being eaten by zombies.
Set up
Suppose I have two spatially modulated time dependent populations living in a space $\mathbb{R}^d$, one $h(\mathbf{x},t)$ of humans, and one $z(\mathbf{x},t)$ of zombies.
The human and zombie populations are initially equal in size, and are initialised at a points with seperation $\mathbf{r}$ $$ z(\mathbf{x},t) = \delta(\mathbf{x}+\mathbf{r}) \quad \quad h(\mathbf{x},t) = \delta(\mathbf{x}). $$
Both populations diffusively spread, though the human-zombie symmetry is broken as the humans experience a loss at a rate proportional to zombie density. $$ \partial_t z(\mathbf{x},t) = \tfrac12 \nabla^2 z(\mathbf{x},t) \qquad \qquad \partial_t h(\mathbf{x},t) = \tfrac12 \nabla^2 h(\mathbf{x},t) - \gamma z(\mathbf{x},t) h(\mathbf{x},t) $$ for some $\gamma > 0$.
The distribution of zombies is simple to calculate $$ z(\mathbf{x},t) = \frac{1}{(2\pi t)^{d/2}} \mathrm{exp}\left( - \frac{|\mathbf{x}|^2}{2t} \right) $$ and one finds number of zombies is conserved $Z(t) = \int d \mathbf{x}\, z(\mathbf{x},t) = 1$.
In contrast I have been unable to calculate a form for $h(\mathbf{x},t)$, and the number of humans is monotonically decreasing $$ H(t) = \int d \mathbf{x}\, h(\mathbf{x},t) \qquad \text{satisfies} \qquad \partial_t H(t) < 0. $$
Question
I want to calculate the asymptotic decay of $H(t)$ for $d=1,2$
- For $d=0$ it is straightforward to show that $H(t) = \mathrm{e}^{- \gamma t}$.
- For $d=1$ approximately speaking there is a region $x \in [\sqrt{t} - r, \sqrt{t}]$ which humans can reach by time $t$, whereas zombies cannot. Humans which manage to remain in this region are at very low risk from the zombies. Consequently the decay of the Human population is much slower: Monte Carlo numerics tells me that $H(t) \sim c t^{-2}$ for some constant $c$. However I have been unable to prove this.
- For $d=2$ it seems the region where humans can survive grows faster (specifically the crescent shaped region $\{\mathbf{x}: |\mathbf{x}| < \sqrt{t} , |\mathbf{x} + \mathbf{r}| > \sqrt{t} \}$), and so I expect an even slower power law decay than for $d=1$.