Chapter 12, Population dynamics Video Solutions, A Kinetic View of Statistical Physics

Problem 1

Generalize the two-species competition model to incorporate competitive asymmetry. The evolution of this reaction is now governed by
$$
\begin{aligned}
&\dot{A}=A\left(1-A-\epsilon_{A} B\right) \equiv F_{A}(A, B) \\
&\dot{B}=B\left(1-B-\epsilon_{B} A\right) \equiv F_{B}(A, B)
\end{aligned}
$$
Study the dynamical behavior of these equations as a function of $\epsilon_{A}$ and $\epsilon_{B}$ and enumerate all possible behaviors.

Sana Riaz

Numerade Educator

18:12

Problem 2

Perform a linearized analysis of the Lotka-Volterra equations (12.5) and determine the nature of the local flow near the two fixed points $(0,0)$ and $(1,1)$. Next, investigate the following two generalizations of the Lotka-Volterra model.
(a) The constants on the right-hand side of the Lotka-Volterra model are all distinct:
$$
\begin{aligned}
&\dot{A}=r_{A} A-k_{A B} A B \\
&\dot{B}=-r_{B} B+k_{B A} A B
\end{aligned}
$$
(b) The prey species is subject to self-regulation:
$$
\begin{aligned}
&\dot{A}=A(1-A)-A B \\
&\dot{B}=-B+A B
\end{aligned}
$$
Determine the fixed points for these two models and describe the global flow in the $A-B$ plane.

Mike Gaerlan

Numerade Educator

16:17

Problem 3

Consider the three-species competition model of Example $12.3$ (page 379).
(a) Verify the stability of the coexistence fixed point in the region of parameter space defined by $\alpha+\beta<2$, and the stability of a single-species fixed point in the region $(\alpha>1) \cap(\beta>1)$
(b) Investigate the dynamical behavior for the case where $\alpha$ and $\beta$ lie in the complementary region $[\alpha+\beta<2] \cup[(\alpha>1) \cap(\beta>1)]$. Start by writing the evolution equations for the sum, $S=A+B+C$, and the product, $\Pi=A B C$. Show that by neglecting quadratic terms in the equation for $\dot{S}$, the asymptotic solution is $S \rightarrow 1$. In this approximation, show that $\Pi$ asymptotically decays exponentially with time. Thus a trajectory lies in the triangle defined by $A+B+C=1$, but moves progressively closer to the boundaries of the triangle, since $\Pi \rightarrow 0$. Finally, study the trajectory close to the point $(1,0,0)$ by assuming that termsquadratic in $B$ and $C$ are vanishingly small and $A$ is close to 1 . Use this information to show that the time $\tau$ that a trajectory spends in the neighborhood of the point $(1,0,0)$ is roughly
$$
\tau=\frac{(\alpha+\beta-2)(\beta-\alpha) t}{2(\beta-1)(1-\alpha)}
$$
Thus the time that a trajectory spends near any single-species fixed point scales in proportion to the total elapsed time.

Mike Gaerlan

Numerade Educator

01:21

Problem 4

Use the method of analysis developed to treat the $S I R$ model to determine the fraction of a large population that never learns of a rumor that starts with a single spreader.

Luca Alexander

Numerade Educator

02:25

Problem 5

Starting with the governing equation (12.17) for the birth process, write the evolution equations for the first three moments of the number distribution $M_{j} \equiv\left\langle n^{j}\right\rangle$. Solve these equations and show that their time dependences are $M_{1}=e^{t}, M_{2}=2 e^{2 t}-e^{t}$, and $M_{3}=6 e^{3 t}-6 e^{2 t}+e^{t}$.

Alison Rodriguez

Numerade Educator

01:34

Problem 6

Instead of using the exponential ansatz, find the probability distribution for the birth process using the generating function technique. Try then to get the solution using induction: solve (12.17) for $n=1,2$, and 3 (this can be done thanks to the recurrent nature of the governing equations), then guess $P_{n}$ in the general case, and verify your guess.

Prabhakar Kumar

Numerade Educator

01:34

Problem 7

Prabhakar Kumar

Numerade Educator

04:40

Problem 8

Consider the birth process starting with $n_{0}>1$ particles. Derive the results cited in the text, Eqs (12.23)-(12.24).

Shahab Ullah

Numerade Educator

02:08

Problem 9

Consider the decay process $A \rightarrow \emptyset$.
(a) Use the generating function method to solve the master equation and thereby find $P_{n}(t)$ when initially the system contains $N>1$ particles.
(b) Find $\langle n\rangle$ and $\left\langle n^{2}\right\rangle$ and argue that the fluctuations in the number of particles are small.
(c) Starting with $N$ particles, determine the average time until all particles disappear. Note: It is possible to write $P_{n}(t)$ without any calculation. If you can do so, then skip step (a).

Khoobchandra Agrawal

Numerade Educator

03:54

Problem 10

For the birth-death process, compute $\langle n\rangle$ and $\left\langle n^{2}\right\rangle$ that appear in Eq. (12.22).

Jacob Shpiece

Numerade Educator

07:27

Problem 11

Linearization. Let's take a nonlinear ordinary differential equation and turn it into a system of infinitely many linear differential equations. As an example, consider the first-order differential equation $\dot{x}=x^{2}$, subject to the initial condition $x(0)=1$.
(a) Let $x_{1} \equiv x, x_{2} \equiv x^{2}$, and generally $x_{j} \equiv x^{j}$. Show that this transformation recasts the original nonlinear differential equation into a system of first-order linear differential equations
$$
\dot{x}_{j}=j x_{j+1}, \quad j=1,2,3, \ldots
$$The initial conditions are $x_{j}(0)=1$ for all $j$.
(b) Try to solve the equations from (a) using linear techniques. If you succeed, you will get $x_{j}=(1-t)^{-j}$, which is of course much easier to obtain if you merely solve $\dot{x}=x^{2}$
(c) Recall the theorem that a system of linear differential equations with constant coefficients admits a solution on the time interval $0 \leq t<\infty .$ Can you reconcile this theorem with the outcome of our problem, namely the existence of a solution only on a finite time interval $0 \leq t<1 ?$

Mike Gaerlan

Numerade Educator

04:22

Problem 12

Starting with Eq. (12.25) for $P_{n}$ for annihilation, determine the evolution equations for the first three moments of the number distribution $M_{j} \equiv\left\langle n^{j}\right\rangle$. In particular, show that the correct mean-field rate equation is recovered if the time is rescaled by $1 / V$, where $V$ is the volume of the system.

Stanley Enemuo

Numerade Educator

04:22

Problem 13

Stanley Enemuo

Numerade Educator

View

Problem 14

Following the same approach as that leading to Eq. (12.30), compute the average completion time to the state where no particles remain for annihilation $A+A \rightarrow 0$ when the system starts with $N$ particles.

Victor Salazar

Numerade Educator

03:41

Problem 15

Show that the mean-square completion time for coalescence is given by Eq. (12.31). Hint: Take into account that for an exponentially distributed random variable $x$, $\left\langle x^{2}\right\rangle=2\langle x\rangle^{2}$. Hence for an individual transition time $\left\langle t_{n}^{2}\right\rangle=2\left\langle t_{n}\right\rangle^{2}=2 r_{n}^{-2}$.

Khoobchandra Agrawal

Numerade Educator

01:42

Problem 16

Determine the mass of the largest cluster in a finite system in constant-kernel aggregation.

John Nicolle

Numerade Educator

01:10

Problem 17

This is an extended problem to illustrate that the gelation time goes to zero for aggregation in a finite system with homogeneity index $\lambda>1$. Consider the generalized product kernel $K(i, j)=(i j)^{\lambda}$, with $\lambda>1$, for which the master equations are
$$
\frac{d c_{k}}{d t}=\frac{1}{2} \sum_{i+j=k}(i j)^{\lambda} c_{i} c_{j}-k^{\lambda} c_{k} \sum_{i} l^{\lambda} c_{i}
$$
Assume that the total number of particles $N$ is large, so that the master equations (with $c_{k}=N_{k} / N$, where $N_{k}$ is the average number of clusters of mass $k$ ) are a good approximation.
(a) At short times, neglect the loss terms and show that $c_{k} \simeq A_{k} t^{k-1}$, in which the coefficients satisfy the recursion relations
$$
(k-1) A_{k}=\frac{1}{2} \sum_{i+j=k}(i j)^{\lambda} A_{i} A_{j}
$$
for $k \geq 2$ and with $A_{1}=1$This is an extended problem to illustrate that the gelation time goes to zero for aggregation in a finite system with homogeneity index $\lambda>1$. Consider the generalized product kernel $K(i, j)=(i j)^{\lambda}$, with $\lambda>1$, for which the master equations are
$$
\frac{d c_{k}}{d t}=\frac{1}{2} \sum_{i+j=k}(i j)^{\lambda} c_{i} c_{j}-k^{\lambda} c_{k} \sum_{i} l^{\lambda} c_{i}
$$
Assume that the total number of particles $N$ is large, so that the master equations (with $c_{k}=N_{k} / N$, where $N_{k}$ is the average number of clusters of mass $k$ ) are a good approximation.
(a) At short times, neglect the loss terms and show that $c_{k} \simeq A_{k} t^{k-1}$, in which the coefficients satisfy the recursion relations
$$
(k-1) A_{k}=\frac{1}{2} \sum_{i+j=k}(i j)^{\lambda} A_{i} A_{j}
$$
for $k \geq 2$ and with $A_{1}=1$

Dominador Tan

Numerade Educator

01:06

Problem 18

Complete the steps of the derivation of the mean extinction time $(12.60)$ starting from Eq. (12.59).

James Kiss

Numerade Educator