Fokker–Plank System for Movement of Micro-organism Population in Confined Environment

Abstract

We consider self-propelled particles confined between two parallel plates, moving with a constant velocity while their moving direction changes by rotational diffusion. The probability distribution of such micro-organisms in confined environment is singular because particles accumulate at the boundaries. This leads us to distinguish between the probability distribution densities in the bulk and in the boundaries. They satisfy a degenerate Fokker–Planck system and we propose boundary conditions that take into account the switching between free-moving and boundary-contacting particles. Relative entropy property, a priori estimates and the convergence to an unique steady state are established. The steady states of both the PDE and individual based stochastic models are compared numerically.

Appendix: Discrete Relative Entropy Inequality and a Priori Estimate

The semi-discrete relative gap $\omega _{i,j}(t)$ and $\omega _{\pm ,j}(t)$ are defined by

$$\begin{aligned} \left\{ \begin{aligned} \omega _{i,j}(t)=&\frac{p_{i,j}(t)}{q_{i,j}}-1,&\begin{aligned} i\in V_y \setminus \{1\},\quad j&\in V_{\theta _+}, \\ \text {or} \quad i\in V_y \setminus \{2I\},\quad j&\in V_{\theta _-}, \\ \end{aligned}&\\ \omega _{1,j}(t)=&0, \quad \omega _{+,j}(t)=\frac{p_{+,j}(t)}{q_{+,j}}-1,&j\in V_{\theta _+}, \\ \omega _{2I,j}(t)=&0, \quad \omega _{-,j}(t)=\frac{p_{-,j}(t)}{q_{-,j}}-1,&j\in V_{\theta _-}, \end{aligned}\right. \end{aligned}$$

(6.1)

Here we state the discrete relative entropy inequality which is satisfied by discrete density $p_{i,j}(t)$ and $p_{\pm ,j}(t)$.

Theorem 6.1

For any convex function $H\in \mathcal {C}^2(\mathbb {R})$, semi-discrete relative gap $\omega _{i,j}(t)$ and $\omega _{\pm ,j}(t)$ satisfy the relative entropy inequality:

$$\begin{aligned} \frac{\mathrm{d}}{ \, \mathrm{d} t}\left[ \sum _{i\in V_y}\sum _{j\in V_{\theta }}q_{i,j}H(\omega _{i,j})+\sum _{j\in V_{\theta _+}}q_{+,j}H(\omega _{+,j})+\sum _{j\in V_{\theta _-}}q_{-,j}H(\omega _{-,j})\right] \le 0, \end{aligned}$$

(6.2)

where equality holds if and only if $\omega _{i,j}=0$ for any $i \in V_{y}$, $j \in V_{\theta }$ and $\omega _{\pm ,j}=0$ for any $j \in V_{\theta ,\pm }$.

Proof

As in Sect. 2, we divide the proof in several steps.

First step: estimate of relative entropy at BCP. Substitute $p_{+,j}$ by $q_{+,j}\omega _{+,j}+q_{+,j}$ in (4.2d) and employ (4.6d), we deduce that

$$\begin{aligned} q_{+,j}\frac{\mathrm{d}\omega _{+,j}}{ \, \mathrm{d} t}= & {} \frac{D_{\theta }}{\Delta \theta ^2}[(q_{+,j-1}\omega _{+,j-1}+q_{+,j+1}\omega _{+,j+1}-2q_{+,j}\omega _{+,j}) \nonumber \\&+(q_{+,j-1}+q_{+,j+1}-2q_{+,j})] \nonumber \\&+\,V\sin \theta _j q_{2I,j}(\omega _{2I,j}+1) \nonumber \\= & {} \frac{D_{\theta }}{\Delta \theta ^2}[(q_{+,j-1}\omega _{+,j-1}+q_{+,j+1}\omega _{+,j+1}-2q_{+,j}\omega _{+,j})\nonumber \\&-\omega _{2I,j}(q_{+,j-1}+q_{+,j+1}-2q_{+,j})] \nonumber \\= & {} \frac{D_{\theta }}{\Delta \theta ^2}[q_{+,j+1}(\omega _{+,j+1}-\omega _{+,j})-q_{+,j-1}(\omega _{+,j}-\omega _{+,j-1})\nonumber \\&+\,(\omega _{+,j}-\omega _{2I,j})(q_{+,j-1}+q_{+,j+1}-2q_{+,j})]. \end{aligned}$$

(6.3)

Notice that

$$\begin{aligned} \begin{aligned}&q_{+,j-1}H(\omega _{+,j-1})+q_{+,j+1}H(\omega _{+,j+1})-2q_{+,j}H(\omega _{+,j})\\&\quad =q_{+,j+1}[H(\omega _{+,j+1})-H(\omega _{+,j})]-q_{+,j-1}[H(\omega _{+,j})-H(\omega _{+,j-1})]\\&\qquad +H(\omega _{+,j}) (q_{+,j-1}+q_{+,j+1}-2q_{+,j}), \end{aligned} \end{aligned}$$

(6.4)

by rearranging $H'(\omega _{+,j})$(6.3)$-\frac{D_{\theta }}{\Delta \theta ^2}$(6.4), it follows that

$$\begin{aligned} \begin{aligned}&q_{+,j}\frac{\mathrm{d}H(\omega _{+,j})}{ \, \mathrm{d} t}-\frac{D_{\theta }}{\Delta \theta ^2}[q_{+,j-1}H(\omega _{+,j-1})+q_{+,j+1}H(\omega _{+,j+1})-2q_{+,j}H(\omega _{+,j})] \\&\quad =-\frac{D_{\theta } }{\Delta \theta ^2}q_{+,j+1}[H(\omega _{+,j+1})-H(\omega _{+,j})-H'(\omega _{+,j})(\omega _{+,j+1}-\omega _{+,j})] \\&\qquad -\frac{D_{\theta } }{\Delta \theta ^2}q_{+,j-1}[H'(\omega _{+,j})(\omega _{+,j}-\omega _{+,j-1})-H(\omega _{+,j})+H(\omega _{+,j-1})] \\&\qquad -[H'(\omega _{+,j})(\omega _{+,j}-\omega _{2I,j})-H(\omega _{+,j})+H(\omega _{2I,j})]V\sin \theta _j q_{2I,j}\\&\qquad +H(\omega _{2I,j})V\sin \theta _j q_{2I,j},\\&\quad \le H(\omega _{2I,j})V\sin \theta _j q_{2I,j} \end{aligned}\end{aligned}$$

(6.5)

where equality holds if and only if $\omega _{+,j-1}=\omega _{+,j}=\omega _{+,j+1}=\omega _{2I,j}$.

Sum up (6.5) for $j\in V_{\theta _+}$, we have

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\sum _{j\in V_{\theta _+}}q_{+,j}H(\omega _{+,j})+\frac{D_{\theta }}{\Delta \theta ^2}q_{+,J+1}H(\omega _{+,J+1})+\frac{D_{\theta }}{\Delta \theta ^2}q_{+,2J-1}H(\omega _{+,2J})\nonumber \\&\quad \le \sum _{j\in V_{\theta _+}}H(\omega _{2I,j})V\sin \theta _j q_{2I,j},\end{aligned}$$

(6.6)

where equality holds if and only if $\omega _{+,j}=\omega _{2I,j}=0$ for any $j\in V_{\theta _+}$.

Similarly, we obtain an inequality for $q_{-,j}H(\omega _{-,j})$

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\sum _{j\in V_{\theta _-}}q_{-,j}H(\omega _{-,j})+\frac{D_{\theta }}{\Delta \theta ^2}q_{-,1}H(\omega _{-,1})+\frac{D_{\theta }}{\Delta \theta ^2}q_{-,J}H(\omega _{-,J-1}) \nonumber \\&\qquad \le -\sum _{j\in V_{\theta _-}}H(\omega _{1,j})V\sin \theta _j q_{1,j},\end{aligned}$$

(6.7)

where equality holds if and only if $\omega _{-,j}=\omega _{1,j}=0$ for any $j\in V_{\theta _-}$.

Second Step: Estimate of Relative Entropy at FSP Substituting $p_{i,j}$ by $q_{i,j}\omega _{i,j}+q_{i,j}$ in (4.2a) we obtain that for any $i\in V_y$, $j\in V_{\theta _+}$,

$$\begin{aligned} \begin{aligned} q_{i,j}\frac{\mathrm{d}\omega _{i,j}}{ \, \mathrm{d} t}&=-\frac{V\sin \theta _j}{\Delta y} (q_{i,j}\omega _{i,j}-q_{i-1,j}\omega _{i-1,j})\\&\quad +\frac{D_{\theta }}{\Delta \theta ^2}(q_{i,j-1}\omega _{i,j-1}+q_{i,j+1}\omega _{i,j+1}-2q_{i,j}\omega _{i,j}) \\&=-\frac{V\sin \theta _j}{\Delta y} q_{i-1,j}(\omega _{i,j}-\omega _{i-1,j})+\frac{D_{\theta }}{\Delta \theta ^2}q_{i,j+1}(\omega _{i,j+1}-\omega _{i,j})\\&\quad -\frac{D_{\theta }}{\Delta \theta ^2}q_{i,j-1}(\omega _{i,j}-\omega _{i,j-1})\\&\quad +\omega _{i,j}\left[ -\frac{V\sin \theta _j}{\Delta y}(q_{i,j}-q_{i-1,j})+\frac{D_{\theta }}{\Delta \theta ^2}(q_{i,j-1}+q_{i,j+1}-2q_{i,j})\right] \end{aligned} \end{aligned}$$

(6.8)

By (4.6a), the last term is zero. Rearranging $H'(\omega _{i,j})$(6.8), we deduce

$$\begin{aligned}\begin{aligned}&q_{i,j}\frac{\mathrm{d}H(\omega _{i,j})}{ \, \mathrm{d} t}-\frac{D_{\theta }}{\Delta \theta ^2}[q_{i,j-1}H(\omega _{i,j-1})+q_{i,j+1}H(\omega _{i,j+1})-2q_{i,j}H(\omega _{i,j})] \\&\quad =-\frac{V\sin \theta _j}{\Delta y} q_{i-1,j}H'(\omega _{i,j})(\omega _{i,j}-\omega _{i-1,j}) +\frac{2D_{\theta }}{\Delta \theta ^2}q_{i,j}H(\omega _{i,j}) \\&\qquad +\frac{D_{\theta }}{\Delta \theta ^2}q_{i,j+1}[H'(\omega _{i,j})(\omega _{i,j+1}-\omega _{i,j})-H(\omega _{i,j+1})]\\&\qquad -\frac{D_{\theta }}{\Delta \theta ^2}q_{i,j-1}[H'(\omega _{i,j})(\omega _{i,j}-\omega _{i,j-1})+H(\omega _{i,j-1})] \\&\quad =-\frac{V\sin \theta _j}{\Delta y}[q_{i,j}H(\omega _{i,j})-q_{i-1,j}H(\omega _{i-1,j})]\\&\qquad -\frac{V\sin \theta _j}{\Delta y}q_{i-1,j}[H'(\omega _{i,j})(\omega _{i,j}-\omega _{i-1,j})\\&\qquad -\,H(\omega _{i,j})+H(\omega _{i-1,j})] \\&\qquad -\frac{D_{\theta }}{\Delta \theta ^2}q_{i,j+1}[H(\omega _{i,j+1})-H(\omega _{i,j})-H'(\omega _{i,j})(\omega _{i,j+1}-\omega _{i,j})] \\&\qquad -\frac{D_{\theta }}{\Delta \theta ^2}q_{i,j-1}[H'(\omega _{i,j})(\omega _{i,j}-\omega _{i,j-1})-H(\omega _{i,j})+H(\omega _{i,j-1})] \\&\qquad -\frac{V\sin \theta _j}{\Delta y}(q_{i-1,j}-q_{i,j})H(\omega _{i,j})-\frac{D_{\theta }}{\Delta \theta ^2}(q_{i,j-1}+q_{i,j+1}-2q_{i,j})H(\omega _{i,j}). \end{aligned} \end{aligned}$$

The last two terms offset according to (4.6a), so we deduce that

$$\begin{aligned} \begin{aligned}&q_{i,j}\frac{\mathrm{d}H(\omega _{i,j})}{ \, \mathrm{d} t}+\frac{V\sin \theta _j}{\Delta y}[q_{i,j}H(\omega _{i,j})-q_{i-1,j}H(\omega _{i-1,j})]\\&\quad \le \frac{D_{\theta }}{\Delta \theta ^2}[q_{i,j-1}H(\omega _{i,j-1})+q_{i,j+1}H(\omega _{i,j+1})-2q_{i,j}H(\omega _{i,j})], \end{aligned} \end{aligned}$$

(6.9)

holds for $i \in V_y$, $j \in V_{\theta _+}$, where equality holds if and only if $\omega _{i,j}=\omega _{i,j-1}=\omega _{i,j+1}=\omega _{i-1,j}$.

Similarly, by rearranging (4.2a) and employ (4.6b), we have

$$\begin{aligned} \begin{aligned}&q_{i,j}\frac{\mathrm{d}H(\omega _{i,j})}{ \, \mathrm{d} t}+\frac{V\sin \theta _j}{\Delta y}[q_{i+1,j}H(\omega _{i+1,j})-q_{i,j}H(\omega _{i,j})]\\&\quad \le \frac{D_{\theta }}{\Delta \theta ^2}[q_{i,j-1}H(\omega _{i,j-1})+q_{i,j+1}H(\omega _{i,j+1})-2q_{i,j}H(\omega _{i,j})], \end{aligned}\end{aligned}$$

(6.10)

holds for $i \in V_y$, $j \in V_{\theta _-}$, where equality holds if and only if $\omega _{i,j}=\omega _{i,j-1}=\omega _{i,j+1}=\omega _{i+1,j}$.

For $i \in V_y\setminus \{1,2I\}$, $j\in V_{\theta _0}$, (4.2c) and (4.6a) gives

$$\begin{aligned} q_{i,j}\frac{\mathrm{d}H(\omega _{i,j})}{ \, \mathrm{d} t} \le \frac{D_{\theta }}{\Delta \theta ^2}[q_{i,j-1}H(\omega _{i,j-1})+q_{i,j+1}H(\omega _{i,j+1})-2q_{i,j}H(\omega _{i,j})], \end{aligned}$$

(6.11)

where equality holds if and only if $\omega _{i,j}=\omega _{i,j-1}=\omega _{i,j+1}$.

Third step: special case where $\frac{\partial p}{\partial \theta }$ jumps. On node $(i,j)=(2I,J)$, substitute $p_{i,j}$ by $q_{i,j}\omega _{i,j}+q_{i,j}$ in (4.4c) we obtain

$$\begin{aligned} \begin{aligned} q_{2I,J}\frac{\mathrm{d}\omega _{2I,J}}{\mathrm{d}t}&=\frac{D_{\theta }}{\Delta \theta ^2}\left( q_{2I,J-1}\omega _{2I,J-1}+q_{2I,J+1}\omega _{2I,J+1}-2q_{2I,J}\omega _{2I,J}+\frac{1}{\Delta y}q_{+,J+1}\omega _{+,J+1}\right) \\&=\frac{D_{\theta }}{\Delta \theta ^2}\left[ q_{2I,J-1}(\omega _{2I,J-1}-\omega _{2I,J})+q_{2I,J+1}(\omega _{2I,J+1}-\omega _{2I,J})\right. \\&\left. \quad +\frac{1}{\Delta y}q_{+,J+1}(\omega _{+,J+1}-\omega _{2I,J})\right] \\&\quad +\,\frac{D_{\theta }}{\Delta \theta ^2}\omega _{2I,J}\left( q_{2I,J-1}+q_{2I,J+1}-2q_{2I,J}+\frac{1}{\Delta y}q_{+,J+1}\right) , \end{aligned}\end{aligned}$$

(6.12)

in which the last term is zero by (4.7). Multiply both sides with $H'(\omega _{2I,J})$, it follows that

$$\begin{aligned}&q_{2I,J}\frac{\mathrm{d}H(\omega _{2I,J})}{\mathrm{d}t}-\frac{D_{\theta }}{\Delta \theta ^2}\left[ q_{2I,J+1}H(\omega _{2I,J+1})+q_{2I,J-1}H(\omega _{2I,J-1})-2q_{2I,J}H(\omega _{2I,J})\right] \\&\quad =\frac{D_{\theta }}{\Delta \theta ^2}q_{2I,J-1}[H'(\omega _{2I,J})(\omega _{2I,J-1}-\omega _{2I,J})-H(\omega _{2I,J-1})]\\&\qquad +\frac{D_{\theta }}{\Delta \theta ^2}q_{2I,J+1}[H'(\omega _{2I,J})(\omega _{2I,J+1}-\omega _{2I,J})-H(\omega _{2I,J+1})] \\&\qquad +\frac{D_{\theta }}{\Delta \theta ^2 \Delta y}q_{+,J+1}H'(\omega _{2I,J})(\omega _{+,J+1}-\omega _{2I,J+1})+\frac{2D_{\theta }}{\Delta \theta ^2}q_{2I,J}H(\omega _{2I,J}) \\&\quad =\frac{D_{\theta }}{\Delta \theta ^2}q_{2I,J-1}[H'(\omega _{2I,J})(\omega _{2I,J-1}-\omega _{2I,J})-H(\omega _{2I,J-1})+H(\omega _{2I,J})] \\&\qquad +\frac{D_{\theta }}{\Delta \theta ^2}q_{2I,J+1}[H'(\omega _{2I,J})(\omega _{2I,J+1}-\omega _{2I,J})-H(\omega _{2I,J+1})+H(\omega _{2I,J})] \\&\qquad +\frac{D_{\theta }}{\Delta \theta ^2 \Delta y}q_{+,J+1}[H'(\omega _{2I,J})(\omega _{+,J+1}-\omega _{2I,J})-H(\omega _{+,J+1})+H(\omega _{2I,J})] \\&\qquad +\frac{D_{\theta }}{\Delta \theta ^2 \Delta y}q_{+,J+1}H(\omega _{+,J+1}) \\&\qquad -\frac{D_{\theta }}{\Delta \theta ^2}\left( q_{2I,J-1}+q_{2I,J+1}-2q_{2I,J}+\frac{1}{\Delta y}q_{+,J+1}\right) H(\omega _{2I,J}). \\ \end{aligned}$$

The last term is zero according to (4.7), so we deduce that

$$\begin{aligned} q_{2I,J}\frac{\mathrm{d}H(\omega _{2I,J})}{\mathrm{d}t}&\le \frac{D_{\theta }}{\Delta \theta ^2}\left[ q_{2I,J+1}H(\omega _{2I,J+1})+q_{2I,J-1}H(\omega _{2I,J-1})-2q_{2I,J}H(\omega _{2I,J})\right] \nonumber \\&\quad +\frac{D_{\theta }}{\Delta \theta ^2 \Delta y}q_{+,J+1}H(\omega _{+,J+1})\end{aligned}$$

(6.13)

where equality holds if and only if $\omega _{2I,J-1}=\omega _{2I,J}=\omega _{2I,J+1}=\omega _{+,J+1}$.

Similarly, on node $(i,j)=(2I,2J),(1,0),(1,J)$, we have

$$\begin{aligned} q_{2I,2J}\frac{\mathrm{d}H(\omega _{2I,2J})}{\mathrm{d}t}\le & {} \frac{D_{\theta }}{\Delta \theta ^2}\left[ q_{2I,2J+1}H(\omega _{2I,2J+1})+q_{2I,2J-1}H(\omega _{2I,2J-1})\right. \nonumber \\&\left. -2q_{2I,2J}H(\omega _{2I,2J})\right] +\frac{D_{\theta }}{\Delta \theta ^2 \Delta y}q_{+,2J-1}H(\omega _{+,2J-1})\end{aligned}$$

(6.14)

$$\begin{aligned} q_{1,0}\frac{\mathrm{d}H(\omega _{1,0})}{\mathrm{d}t}\le & {} \frac{D_{\theta }}{\Delta \theta ^2}\left[ q_{1,1}H(\omega _{1,1})+q_{1,-1}H(\omega _{1,-1})-2q_{1,0}H(\omega _{1,0})\right] \nonumber \\&+\frac{D_{\theta }}{\Delta \theta ^2 \Delta y}q_{-,1}H(\omega _{-,1})\end{aligned}$$

(6.15)

$$\begin{aligned} q_{1,J}\frac{\mathrm{d}H(\omega _{1,J})}{\mathrm{d}t}\le & {} \frac{D_{\theta }}{\Delta \theta ^2}\left[ q_{1,J+1}H(\omega _{1,J+1})+q_{1,J-1}H(\omega _{1,J-1})-2q_{1,J}H(\omega _{1,J})\right] \nonumber \\&+\frac{D_{\theta }}{\Delta \theta ^2 \Delta y}q_{-,J-1}H(\omega _{-,J-1}) \end{aligned}$$

(6.16)

Fourth step: collection and offsets To summarize, sum up (6.9) for $i \in V_y$, $j \in V_{\theta _+}$, (6.10) for $i \in V_y$, $j \in V_{\theta _-}$, (6.11) for $i \in V_y\setminus \{1,2I\}$, $j \in V_{\theta _0}$, together with (6.13)–(6.16), the diffusion term offsets and it follows that

$$\begin{aligned} \begin{aligned}&\frac{\mathrm{d}}{ \, \mathrm{d} t}\sum _{i\in V_y}\sum _{j\in V_{\theta }}q_{i,j}H(\omega _{i,j})+\sum _{j\in V_{\theta _-}}\frac{V\sin \theta _j}{\Delta y}q_{2I,j}H(\omega _{2I,j})-\sum _{j\in V_{\theta _-}}\frac{V\sin \theta _j}{\Delta y}q_{2I,j}H(\omega _{2I,j})\\&\quad \le \frac{D_{\theta }}{\Delta \theta ^2 \Delta y}[q_{+,J+1}H(\omega _{+,J+1})+q_{+,2J-1}H(\omega _{+,2J-1})\\&\qquad +q_{-,1}H(\omega _{-,1})+q_{-,J-1}H(\omega _{+,J-1})]. \end{aligned}\end{aligned}$$

(6.17)

Hence (6.5), (6.7) and (6.17) imply inequality (6.2) in the theorem.

Define

$$\begin{aligned} M(t)=\Delta \theta \Delta y\sum _{i\in V_y}\sum _{j\in V_\theta }q_{i,j}\omega _{i,j}^2(t)+\Delta \theta \sum _{j\in V_{\theta _+}}q_{+,j}\omega _{+,j}^2(t)+\Delta \theta \sum _{j\in V_{\theta _-}}q_{-,j}\omega _{-,j}^2(t).\end{aligned}$$

(6.18)

Choose $H(x)=x^2$ in (6.2) then we have $\mathrm{d}M(t) / \, \mathrm{d} t\le 0$. Suppose the initial data of $\omega _{i,j}$ and $\omega _{\pm ,j}$ satisfies $M(0)<\infty$, then since M(t) decreases monotonically, and it has a lower bound 0, so M(t) converges as t goes to infinity. While the equality of (6.2) holds if and only if $M(t)=0$, so we have $\lim _{t\rightarrow \infty }M(t)=0$, which implies $\lim _{t\rightarrow \infty }\omega _{i,j}(t)=\lim _{t\rightarrow \infty }\omega _{\pm ,j}(t)=0.$ We conclude that $p_{i,j}(t)$ converges to $q_{i,j}$ for any $i \in V_\theta$, $j \in V_y$ as t goes to infinity, and $p_{\pm ,j}(t)$ converges to $q_{\pm ,j}$ for any $j \in V_{j,\pm }$ as t goes to infinity.