An Optimal Switching Approach for Bird Migration

Migration is a seasonal movement of organisms between seasonal favorable regions. Bird migration, in particular, is remarkable for their long distances, geological barriers involved and predation risks. Each year, billions of birds driven by the optimal living habitats take on journeys that span up to thousands of kilometers, across ocean and continents. Avoiding intraspecific competition and exploiting crucial resources in seasonal favorable regions for breeding were usually considered as two main elements for evolution of bird migration [32]. Various bird species show a wide range of complex migratory behaviors. For example, different species show both high and low route repeatability, consistency and variation in duration of migration [35], and different migratory strategies between spring and fall migration [25]. The route and schedule of migration are affected by many factors, including the presence of geological barriers, predation risk [39], global warming [16], access to climate information at the terminal site [7], and population density dependence [12]. Although bird migrations exhibit complex behaviors, they all serve the same purpose: to optimize survival probability and reproductive success. Stochastic optimal control, an effective tool for modeling population dynamics in biological/ecological systems [19, 40], is usually employed to model bird migration [1, 14, 23, 30].

Optimal control theory is ubiquitous in theoretical and applied ecology. It predicts how an individual agent can navigate a set of risks and reward to maximize a payoff functional, which includes the reproductive gain and the survival rate in a migration process. In general, a type of behavior maximizing the payoff functional will tend to increase in frequency in the population if they have a heritable component through genetics or learning. The payoff functional is often stochastic (e.g., depending on the mean arrival time at the destination following a diffusion process), and is expressed in terms of an expectation. In such a case, stochastic dynamic programming (SDP) can be used to reformulate the stochastic optimization problem into a set of deterministic partial differential equations, to be solved backward in time. Here we follow the convention in ecology and resource management to refer to both the model itself and the solution approach as SDP.

SDP has been widely used in population dynamics [19, 40], particularly in the study of bird migration [1, 14, 23, 30]. Conventional bird migration models based on SDP [30] are usually discretized in space, using a large number of variables to characterize the migration process, which often makes them analytically intractable. For continuous-time models, solving a stochastic optimization problem reduces to analyze the underlying Hamilton-Jacobi-Bellman (HJB in short) equation [11, Chapter 9] by using analytical method (e.g.,[42]) or numerical techniques (e.g., [26]), and the optimal feedback control may be obtained as a byproduct.

The objective of this paper is to propose an optimal switching model of bird migration with stopover decision based on SDP. The optimal switching model [29, Chapter 5] is a decision-making framework in which individuals switch between two or more diffusion processes to maximize objective function. In our model, the migratory birds switch between three states: “detour”, “direct flight”, and “resting at a stopover site”, each is governed by a separate stochastic differential equation (see Section 2). The optimal switching strategy in our model (location and timing to switch between states) is derived by studying a deterministic system of HJB variational inequalities through both analytical and numerical framework, see Section 3 and 4, respectively. Finally, in Section 5, we apply this optimal switching model to answer several specific biological questions by characterizing the optimal strategies of birds under different scenarios.

There are many interesting biological questions arising from animal migration. The first question we discuss in this paper is the effect of the deterioration of stopover site. Climate change can dramatically reshape the migration behaviors of birds [1, 37, 36]. Extensive research has demonstrated the profound influence of temperature increase on bird migration choice [37, 5]. Furthermore, the loss of wetland due to increasing temperature [3] can greatly reduce the number of options of migratory shorebirds which depend heavily on the stopover wetlands along the coast for food supplies. Hence, it is natural to ask that

• •

  Can we quantify and analyze the change in their migratory payoff as the quality of major stopover sites decline, and predict the corresponding change in the migration strategy?

We shall explore this question and present our simulation results in Subection 5.1.

Another important factor influencing the migration process is the weather dynamics [7]. Instead of focusing on the stopover site, we will place more emphasis on the effect of the stochastic weather condition happening at the terminal site. However, in nature, birds can anticipate environmental change based on past experience and perception [7, 6] but only to a limited degree. In particular, they may not be aware of every spontaneous change of climate, especially the condition at the terminal site when it is far away. To this end, we consider the effect of imperfect versus perfect information about the weather condition at the terminal site. Based on the principle of adaptation, we assume that birds have formed a stable strategy according to the average condition of the weather dynamics at the terminal site, which is based on the past experience. But the actual quality of the terminal site may contain unpredictable daily fluctuation due to stochasticity, or be shifted in time due to global climate change. These factors lead to potential mismatch of the perceptual quality of the terminal site versus its actual quality that the bird experiences that particular year.

As such, we will additionally address these two questions in Subection 5.2:

• •

  Will only knowing the information about the averaged condition at the terminal site decrease the expected payoff significantly if we compare with the same situation but with perfect information?

• •

  How will the migratory payoff be impacted if the optimal arrival date (or peak green-timing) deviates significantly from past experience, due to the global climate change [31]?

We model the bird migration process as a stochastic optimal switching problem. In this model, individuals switch between different states (representing detour flight, direct flight and waiting) where they are governed by a different drift-diffusion process. A value function $V$ representing the expected payoff function when optimal control is exercised. To evaluate $V$, we derive a system of variational inequalities and prove that $V$ is the unique viscosity solution of the system of variational inequalities (see Sections 2.4 and 3).

The essential idea of the model can be presented by a simple qualitative example of a migrating Pacific brant, which travels more than 5000 km from breeding ground at Izembek Lagoon on the Alaska Peninsula to their wintering ground in Southern Baja California or mainland Mexico [30]. Each individual has a $T$-day period to complete the migration to its terminal site. For simplicity, we consider that the bird can not survive unless it reaches the terminal site.

For the model, we model the Pacific as a linear interval $[0,L]$ (with $x=0$ being the starting location, and $x=L$ being the destination). A typical individual can switch between the two states: “detour flight” or “wait” as it moves along the interval $[0,L]$, where “wait” is only possible at one of the $M$ stopover sites. Additionally, as individual departs from one of the $M$ stopover sites, it may choose to embark on a direct flight across the Pacific ocean towards the terminal site (Baja Peninsula), which is modeled by the state “direct flight” which has a greater mean speed and variability comparing to the “detour flight”. See Figure 1 for an illustration. Instead of using a two-dimensional graph, we model the choice of direct flight by the fact that an individual cannot swtich back to “detour flight” or “wait” once it adopts a “direct flight”, and the higher mean speed also reflects that fact that the physical distance of a direct flight (crossing the ocean) is smaller than a detour flight (along the coast).

The whole switching process is shown in Figure 2. Here we label the $i$-th stopover sites by the interval $[L_{i}-\epsilon,L_{i}+\epsilon]$. For x $\in[0,L_{1}-\epsilon)$, where the bird still migrate within the continent and do not reach the ocean, the individual bird continues in the state “detour flight” until it reaches the first stopover site $[L_{1}-\epsilon,L_{1}+\epsilon]$ (representing the first staging area near the ocean). At this point, it can choose either to detour along the coast or to take direct flight over the ocean. If it adopts a direct flight, then it switches to the respective diffusion process which does not terminate until it arrives at the destination. Otherwise, it continues to the next stopover site and the process repeats itself.

We translate the bird migration into mathematical framework for optimal switching problem. Denote $L$ as the distance between starting point and terminal site, and $T<\infty$ as the terminal time for bird migration, then we consider an optimal switching problem on a finite horizon. Fix a probability space $(\varOmega,\mathcal{F},P)$ with a filtration $\mathbb{F}=(\mathcal{F}_{s})_{s\geqslant t}$ satisfying the usual conditions (see, e.g. [11, 29]). The regime space

$$\Pi=\Pi_{m}=\{1,2,....,m\}$$

(with $m=3$ in our case), and describes the state that a bird is staying in. Then denote $(t,x,i)\in[0,T]\times[0,L]\times\Pi$ as the initial conditions, which describes the exact position $x$ of an individual bird at time $t$ and regime $i$.

A switching control is a double sequence $\alpha=(\tau_{n},\iota_{n})_{n\geqslant 1}$, where $(\tau_{n})$ are an increasing sequence of stopping times, $\tau_{n}\rightarrow\infty$ as $n\rightarrow\infty$ a.s., denote the decision on “when to switch”; $\iota_{n}$ are $\mathcal{F}_{\tau_{n}}$-measurable valued in $\Pi$, represent the new value of the regime during time $[\tau_{n},\tau_{n+1})$. The set of all switching controls is denoted as $\mathcal{A}$. Given the initial regime $i$, we can define the controlled switching process among different states as follows:

$$\pi_{s}:=\pi(s,i)=\sum_{n\geq 0}\iota_{n}\chi_{[\tau_{n},\tau_{n+1})}(s),s\geq t,\quad\pi_{t^{-}}=i,$$

(2.1)

where $\chi_{[\tau_{n},\tau_{n+1})}$ is the indicator function, defined by

$$\chi_{B}:=\chi_{B}(y)=\begin{cases}1,&\text{if}~{}y\in B,\\ 0,&\text{otherwise};\end{cases}$$

and we set $\tau_{0}:=t$, $\iota_{0}:=i$ and $t\in[0,T)$ is any fixed. Then we use a Markov process $\{X^{t,x,i}_{s}\}_{s\geq t}$ to represent the whole migration, starting from position $x$ and the state $i$ at an initial time $t$, which satisfies the following stochastic differential equation

$$\begin{split}dX_{s}^{t,x,i}&=\sum_{j=1}^{3}v_{j}\chi_{\{j=\pi_{s}\}}ds+\sum_{j=1}^{3}\sqrt{2\mu_{j}}\chi_{\{j=\pi_{s}\}}dB_{s},\ \ \text{for}~{}0\leq t\leq s<T.\end{split}$$

(2.2)

For the remainder of this article, let the regime state $i=1$ represent the state of detour flight, $i=2$ represent the state of direct flight, and $i=3$ represent the state of waiting, and let $B_{s}$ be a 1-dimensional Brownian motion on the filtered probability space $(\varOmega,\mathcal{F},\mathbb{F},P)$ satisfying the usual conditions. The constants $v_{j}\geq 0$ and $\mu_{j}\geq 0~{}(j=1,2,3)$ represent the velocity and volatility of the regime, respectively, where $v_{i}$ and $\mu_{i}$ are constants satisfying

$$v_{2}>v_{1}>0=v_{3}\quad\text{ and }\quad\mu_{1}>\mu_{2}>0=\mu_{3},$$

which is motivated by the fact that directly flight shortens the route distance to destination ($v_{2}>v_{1}$) while increases stochasticity ($\mu_{2}>\mu_{1}$) owing to weather variations, such as wind direction, wind velocity, etc.

Consequently, for any initial condition $(t,x,i)$ and any given control $\alpha\in\mathcal{A}$, the migration dynamics can alternative be written as:

$$dX^{t,x,i}_{s}=v_{\iota_{n}}ds+\sqrt{2\mu_{\iota_{n}}}dB_{s},\ \ \pi_{s}=\iota_{n},\forall~{}s\in[\tau_{n},\tau_{n+1}),\ \ n\geq 1,$$

(2.3)

where $\pi_{s}$ is defined in (2.1). When $\iota_{n}\neq 3$, from [27, Theorem V.38], we get that there exists a unique strong solution valued in $[0,L]$ to the standard SDE (2.3) for any $(t,x)\in[0,T]\times[0,L]$, denote it by $X^{t,x,i}_{s}$; when $\iota_{n}=3$, we set, for each $(t,x)\in[0,T]\times[0,L]$,

$$X^{t,x,i}_{s}=X^{t,x,i}_{\tau_{n}}\quad\text{ for all }s\in[\tau_{n},\tau_{n+1}].$$

(2.4)

The objective function is a function to be maximized by the individual bird through its migration process subject to the migration dynamics formulated in the previous subsection. Here are some key components for the construction of objective function.

• (H1)

  Running and terminal reward functions. For $j=1,2,3$, we let $f_{j}(t,x)=f(t,x,j):~{}[0,T]\times[0,L]\times\Pi\rightarrow\mathbb{R}$ and $g_{j}(t,x)=g(t,x,j):$ $[0,T]\times[0,L]\times\Pi\rightarrow\mathbb{R}$ be, respectively, the running and terminal reward functions. We assume that they are uniformly Lipschitz continuous in $(t,x)$.

• (H2)

  Mortality/Discount rate. For $j=1,2,3$, we denote the mortality rate associated with birds’ action by $\beta_{j}(t,x)=\beta(t,x,j):~{}[0,T]\times[0,L]\times\Pi\rightarrow\mathbb{R}$ and assume that it is Lipschitz continuous. In reality, the mortality rate for direct flight is considered higher than that for detour flight, reflecting the increased challenges birds face over the sea, such as the inability to rest periodically and greater exposure to adverse weather conditions.

• (H3)

  Switching cost. We denote the cost to switch from regime $k$ to $j$, by $h_{kj}(t,x):~{}[0,T]\times[0,L]\rightarrow\mathbb{R}$ for $k,j\in\{1,2,3\}$. For each $k,j$, we assume $0\leq h_{kj}(t,x)\in C^{1,2}([0,T]\times[0,L])$, where the equality holds iff $k=j$. Moreover, suppose that the function $h_{kj}(t,x)$ satisfy

  $$h_{kj}<h_{kq}+h_{qj},\ \ \text{ for }q\neq k,~{}j,\text{ such that }h_{kq}+h_{qj}<+\infty,$$

  (2.5)

  and

  $$h_{2j}=+\infty,\ \ j=1,~{}3.$$

  (2.6)

  The condition (2.5) means that switching in two steps via an intermediate regime $q$ is more costly than in one step from regime $k$ to $j$; (2.6) implies that the bird cannot switch from direct flight to detour or waiting state.

The arrival time at $x=L$ is a hitting time of the process:

$$\tau=\inf\{s\geq t:X^{t,x,i}_{s}=L\}.$$

The objective function give the expected benefit, given the current state $(t,x,i)\in[0,T]\times[0,L]\times{\Pi}$ and subject to a given control $\alpha=(\tau_{n},\iota_{n})_{n\geq 1}\in\mathcal{A}$:

$$\begin{split}J_{i}(t,x,\alpha)=E^{t,x,i}\bigg{[}\int_{t}^{\tau\wedge T}&e^{-\beta_{\pi_{s}}(s,X_{s}^{t,x,i})(s-t)}f_{\pi_{s}}(s,X_{s}^{t,x,i})ds-\sum_{t\leq\tau_{n}<\tau\wedge T}e^{-\beta_{\pi_{s}}(\tau_{n},X_{s}^{t,x,i})(\tau_{n}-t)}{h_{l_{n-1}l_{n}}}(s,X_{s}^{t,x,i})\\ &+e^{-\beta_{\pi_{s}}(\tau,X_{\tau}^{t,x,i})(\tau-t)}g_{\pi_{s}}(\tau,L)\chi_{\{\tau\leq T\}}\bigg{]},\end{split}$$

(2.7)

where $X_{s}^{t,x,i}$ is the unique strong solution to (2.2) subject to the given control process $\alpha$. Under proper assumptions (see (H1)-(H3)) $J_{i}(t,x,\alpha)$ is well-defined for any initial condition $(t,x,i)$ and any control process $\alpha\in\mathcal{A}$.

Finally, the value function of the optimal control problem is defined as the supremum of the objective function $J_{i}(t,x,\alpha)$ over the set of all admissible control processes $\mathcal{A}$:

$$\begin{split}V_{i}(t,x)=\sup_{\alpha\in\mathcal{A}}J_{i}(t,x,\alpha),\ \ {(t,x,i)\in[0,T]\times[0,L]\times\Pi.}\end{split}$$

(2.8)

By (2.7), it is natural to set the terminal condition

$$V_{i}(T,x)=0\quad\text{ for }x\in[0,L),\\$$

and boundary conditions

$$(V_{i})_{x}(t,0)=0,\quad V_{i}(t,L)=g_{i}(t,L),\quad\text{ for }t\in[0,T],$$

which says that the diffusion process is reflected at the boundary $x=0$ and is absorbing at $x=L$.

The goal of this optimal switching problem is to find a strategy $\alpha^{*}=(\tau^{*}_{n},\iota^{*}_{n})_{n\geq 1}\in\mathcal{A}$ to achieve the maximum of (2.8), $\alpha^{*}$ is called the optimal strategy. A key ingredient of the Bellman’s approach of stochastic optimal control is the dynamic programming principle, which we now recall.

To find the optimal control $\alpha^{*}$, the crucial step is to show, based on the the dynamic programming principle (see Lemma 2.1) and the concept of viscosity solutions (see Definition 3.1), that $(V_{i}(t,x))_{i\in\Pi}$ is the unique viscosity solution to the following system of Hamilton-Jacobi-Bellman (HJB in short) variational inequality with terminal and lateral conditions:

$$\begin{cases}\min[-\partial_{t}V_{i}+\beta_{i}V_{i}-\mathcal{L}_{i}V_{i}-f_{i},V_{i}-\max_{i\neq j}(V_{j}-h_{ij})]=0,&(t,x)\in(0,T)\times(0,L),\\ {(V_{i})_{x}}(t,0)=0,&t\in[0,T],\\ V_{i}(t,L)=g_{i}(t,L),&t\in[0,T],\\ V_{i}(T,x)=0,&x\in[0,L),\end{cases}$$

(2.9)

where $i\in\{1,2,3\}$ and

$$\mathcal{L}_{i}\phi:=v_{i}\phi_{x}+\mu_{i}\phi_{xx}$$

(2.10)

(see details in Proposition 3.2 in Section 3).

This system of PDE facilitates the identification of the associated optimal control $\alpha^{*}$. Precisely, for any $i\neq j$, we define the switching region to be the closed set

$$\mathcal{S}_{ij}=\{(t,x)\in(0,T)\times(0,L):V_{i}(t,x)=(V_{j}-h_{ij})(t,x)\},$$

(2.11)

which means that if an individual in state $i$ is located at location $x$ and time $t$, such that $(t,x)\in\mathcal{S}_{ij}$, then it will switch from regime $i$ to regime $j$. Define

$$\mathcal{S}_{i}=\cup_{j\neq i}\mathcal{S}_{ij}\quad\text{ and }\quad\mathcal{C}_{i}=[0,T]\times[0,L]\setminus\mathcal{S}_{i}.$$

(2.12)

It follows from [28, Lemma 4.2] that

$$\mathcal{S}_{i}=\{(t,x)\in(0,T)\times(0,L):V_{i}(t,x)=\max_{j\neq i}(V_{j}-h_{ij})(t,x)\},$$

then

$$\mathcal{C}_{i}=\{(t,x)\in[0,T]\times[0,L]:V_{i}(t,x)>\max_{j\neq i}(V_{j}-h_{ij})(t,x)\}.$$

By Proposition 3.3, we get that $\mathcal{S}_{i}$ is the region where changing the regime $i$ is optimal. If $(t,x)\in\mathcal{C}_{i}$, then it is optimal to stay in regime $i$ at least for a short time (Proposition 3.3). Hence, $\mathcal{C}_{i}$ is called the continuation region.

Moreover, by (2.5), the switching regions are disjoint, so that

$$\mathcal{S}_{ij}\subset\mathcal{C}_{j},\ \ j\neq i\in\Pi.$$

Roughly speaking, an individual who has just switched from regime $i$ to $j$ does switch immediately from $j$ to another regime. When $i=2$, assumption (H3)implies that there is no switch, this gives $\mathcal{S}_{2}=\emptyset$ and $\mathcal{C}_{2}=[0,T]\times[0,L]$.

Consequently, the optimal strategy for an individual bird/agent can be fully characterized by the switching regions $\mathcal{S}_{12},\mathcal{S}_{13},\mathcal{S}_{31},\mathcal{S}_{32}$, as given in (2.11).

In Sections 2 and 3, we derived the deterministic PDEs of the value functions for the optimal control problem $\eqref{eq: optimal_control}$. For clarity, we recall that:

$$\left\{\begin{aligned} &\min[-\partial_{t}V_{i}+\beta V_{i}-\mathcal{L}_{i}V_{i}-f_{i},V_{i}-\max_{j\neq i}(V_{j}-h_{ij})]=0&(t,x)\in[0,T)\times(0,L),\\ &{(V_{i})_{x}}(t,0)=0&t\in[0,T],\\ &V_{i}(t,L)=G(t)&t\in(0,T],\\ &V_{i}(T,x)=0&x\in[0,L),\end{aligned}\right.$$

(4.14)

where $\mathcal{L}_{i}$ is given in (2.10). Then, we use a finite difference scheme to solve the coupled system (4.14). We give a brief introduction to the process below.

• Step 1: The $(t,x)$ domain is discretized into a lattice with step sizes $\Delta t$ and $\Delta x$. Denote $V_{i,m,n}=V_{i}(m\Delta t,n\Delta x)$, then the parabolic part of the equation can be discretized according to the implicit Euler scheme:

  	$\displaystyle-$	$\displaystyle\frac{V_{i,m+1,n}-V_{i,m,n}}{\Delta t}-v_{i}\frac{V_{i,m,n+1}-V_{i,m,n}}{\Delta x}$		(4.15)
  		$\displaystyle-\mu_{i}\frac{V_{i,m,n+1}-2V_{i,m,n}+V_{i,m,n-1}}{{\Delta x}^{2}}+\beta_{i}V_{i,m,n}-f_{i,m,n}=0.$

• Step 2: Let the solution $(V_{i,m+1,n})_{1\leq i\leq 3,1\leq n\leq N_{x}}$ be given for the $(m+1)$-th time step, we solve (4.15) for the solution at the $m$-th time step and denote the result by $(\hat{V}_{i,m,n})_{1\leq i\leq 3,1\leq n\leq N_{x}}$.

• Step 3: We update again to account for the variational inequality arising from the switching control by the equation below,

  $$V_{i,m,n}=\max_{j\neq i}(\hat{V}_{i,m,n},\hat{V}_{j,m,n}-h_{ij}).$$

  (4.16)

By taking the procedures above, we can obtain the optimal payoff value $V(t,x)$ for individuals starting at $(t,x)$, which estimates the largest expected value individual can obtain by executing the optimal control.

In this subsection, we show how to represent the optimal strategy through the computation of value functions above. To achieve this goal, we recall the definitions of switching region and continuation region as below:

$$\text{(Switching region)}\qquad\mathcal{S}_{i}=\{(t,x)\in[0,T]\times[0,L]:V_{i}(t,x)=\max_{j\neq i}(V_{j}-h_{ij})(t,x)\}.$$

The complement set of $\mathcal{S}_{i}$ denoted by

$$\text{(Continuation region)}\qquad\mathcal{C}_{i}=\{(t,x)\in[0,T]\times[0,L]:V_{i}(t,x)>\max_{j\neq i}(V_{j}-h_{ij})(t,x)\}.$$

(4.17)

We also denote the closed set representing the switching region from regime $i$ to regime $j$ by $\mathcal{S}_{ij}$:

$$\mathcal{S}_{ij}=\{(t,x)\in[0,T]\times[0,L]:V_{i}(t,x)=(V_{j}-h_{ij})(t,x)\quad\text{for some}\quad j\neq i\}.$$

(4.18)

With these notations, Proposition 3.3 shows that the diffusion process for an individual adopting the optimal switching strategy can be fully characterized by a controlled diffusion process where an individual switch between different diffusion processes according to its physical location $(t,x)$ and its current state $i$, as given by (4.17) and (4.18). To numerically demonstrate this, we follow the following steps:

• •

  First, applying the numerical methods described in sub-section 4.1, we obtain the value functions $V_{i}(t,x)$ for any fixed initial condition $(t,x,i)$.

• •

  Then, we compare $V_{i}(t,x)$ for different regimes $i$ by taking switching cost $h_{ij}$ (cost of switching from regime i to regime j) into consideration to draw the switching regions and continuing regions.

In this section, we discuss how to implement the optimal control $\alpha^{*}$ determined by the switching regions derived in the subsection 4.2 with stochastic simulations. Consider the stochastic processes that model the movement of individuals over time, governed by  (2.3). To analyze this process, we discretize $(t,x)$ domain into a lattice with step sizes $\Delta t\ \text{and }\Delta x$, with absorbing boundaries (e.g. the process will be terminated once individual touch the boundary point $x=L$). Then individual movements will be modeled as a random walk with drift on the discretized domain with the transition probabilities $p_{\ell},p_{r}$ (and the probability $1=p_{\ell}-p_{r}$ to stay put). We first fix spatial step size and then obtain the time step size according to spatial step to maintain numerical stability. After that, we derive for each regime $i$ the transition probabilities $p_{\ell}$, $p_{r}$ for each regime. For simplicity, we present the formulas for $i=1$, as the formulas for the other diffusion regime is similar.

$$u_{t}=\mu_{1}u_{xx}+v_{1}u_{x},$$

let

$$\begin{split}&\Delta x=h,\ \ \Delta t=\frac{\delta}{\mu_{1}+\mu_{2}}h^{2},\\ &\frac{p_{l}+p_{r}}{2}=\frac{\mu_{1}}{\mu_{1}+\mu_{2}},\\ &p_{l}-p_{r}=\frac{v_{1}\Delta t}{\Delta x}=v_{1}\times\Delta x\times\frac{\Delta t}{(\Delta x)^{2}}=\frac{v_{1}\Delta x}{\mu_{1}+\mu_{2}}.\end{split}$$

Then one gets