METASTABILITY (May 9, 2000)

The equations for chemotaxis that we are using are given by [1]

$$\begin{array}{rcl} u_t &=& d_1 \nabla^2 u - \chi \nabla(u\na... ...\ [2.0ex] v_t &=& d_2 \nabla^2 v - \mu v + \beta u, \end{array}$$ (1)

where u(x,t) is the population and v(x,t) is the concentration of chemotactical substrate. The parameters d₁ and d₂ are the diffusion constants for u and v respectively, $\chi$ is the chemotactic parameter, $\beta$ is the production rate of chemotactic substrate by the population and $\mu$ is the death or decay rate of the chemotactic substrate. In one dimension the model reads

$$\begin{array}{rcl} u_t &=& d_1 u_{xx} - \chi (u v_x)_x \\ [2.0ex] v_t &=& d_2 v_{xx} - \mu v + \beta u. \end{array}$$ (2)

  

Figure: Traveling peak due to non-symmetry of the domain

When starting with an initial peak symmetric w.r.t.  the center of the domain, the stationary solution is reached and it stays put at the center of the domain. However, if we start with an initial peak centered at a different point, it starts to travel towards the boundary. The velocity of the peak increases as it gets closer to the boundary, see figure 1.

  

Figure: Position of the peak as a function of time t for d₁=d₂ ranging from 0.01 to 0.05 (right to left).

A typical plot for the position of the peak as time evolves in given in figure 2. In the figure we plot the trajectory of the peak for increasing diffusion parameter d₁=d₂ corresponding to plots from right to left.

  

Figure: Height of the peak as a function of time t for d₁=d₂ ranging from 0.01 to 0.05 (top to bottom).

The height of the peak also shows interesting behaviour. As it can be seen in figure 3 the height of the peaks first grows and then decays to the stationary shape. After that it travels without changing its shape until it hits the boundary and its height is affected (right-most part of the curves).

  

Figure: Escape time as a function of the diffusion d₁=d₂

In order to check for metastability we want to measure the time it takes for the peak to hit the boundary as a function of the diffusion. This elapsed time t_e, referred from now on as escape time, is depicted in figure 4 as a function of d₁=d₂ in a log-log plot for two different sizes of the domain. From the figure we can observe that the escape time behaves closely as:

$$t_e \propto d_1^\alpha,$$ (3)

for negative $\alpha$. The deviations from the straight fit (see lines in figure 4) can be explained as follows. For small d₁=d₂ the peak is very narrow and almost doesn't feel the boundary and thus the escape time in prone for numerical errors. In fact, for d₁=d₂ small enough the peak doesn't travel at all or travels to some arbitrary spurious position in the domain (not necessarily the center of the domain). On the other hand, the sharp deviation from the strait fit for large d₁=d₂ is due to the fact that the v component is very spread and it is subject to large boundary effects. See for example the left-most snapshot of the traveling peak in figure 1 where the v component is deformed due to the boundary. This interaction with the boundary has the net effect of pulling more rapidly the peak towards the boundary, resulting on a downward deviation of the escape time as a function of d₁=d₂.

0.25 cm Thus the escape time is not exponentially long as d₁ is decreased.

0.25 cm Does this mean that we do not have metastability?