How Far Can Chemotactic Cross-diffusion Enforce Exceeding Carrying Capacities?

Abstract

We consider nonnegative solutions of the Neumann initial-boundary value problem for the chemotaxis-growth system

$$\begin{aligned} \left\{ \begin{array}{l} u_t=\varepsilon u_{xx} -(uv_x)_x +ru -\mu u^2, \qquad x\in \Omega , \ t>0, \\ 0=v_{xx}-v+u, \qquad x\in \Omega , \ t>0, \end{array} \right. \quad (\star ) \end{aligned}$$

in $\Omega :=(0,L)\subset \mathbb {R}$ with $L>0, \varepsilon >0, r\ge 0$ and $\mu >0$, along with the corresponding limit problem formally obtained upon taking $\varepsilon \searrow 0$. For the latter hyperbolic–elliptic problem, we establish results on local existence and uniqueness within an appropriate generalized solution concept. In this context we shall moreover derive an extensibility criterion involving the norm of $u(\cdot ,t)$ in $L^\infty (\Omega )$. This will enable us to conclude that in this case $\varepsilon =0$,

• if $\mu \ge 1$, then all solutions emanating from sufficiently regular initial data are global in time, whereas

• if $\mu <1$, then some solutions blow-up in finite time.

The latter will reveal that the original parabolic–elliptic problem ($\star$), though known to possess no such exploding solutions, exhibits the following property of dynamical structure generation: given any $\mu \in (0,1)$, one can find smooth bounded initial data with the property that for each prescribed number $M>0$ the solution of ($\star$) will attain values above $M$ at some time, provided that $\varepsilon$ is sufficiently small. In particular, this means that the associated carrying capacity given by $\frac{r}{\mu }$ can be exceeded during evolution to an arbitrary extent. We finally present some numerical simulations that illustrate this type of solution behavior and that, moreover, inter alia, indicate that achieving large population densities is a transient dynamical phenomenon occurring on intermediate time scales only.