Generative eBooks

The Rosenzweig-McArthur model

Generative eBooks — 2017-06-14T14:00:15Z

We come back to preys and predators. We previously studied a basic model for the interaction of sharks and sardines, namely $$ \begincases \dotx= x(1-x) - xy\ \doty=\beta y ( x-\alpha) \endcases $$ where $x$ represents the density of sardines, $y$ the density of sharks, and $\alpha,\beta$ are positive parameters. We observed and proved that there cannot be limit cycles in this model, which means that the populations of sharks and sardines cannot oscillate periodically in a robust way.
A (…)

- Theorems & examples

The idealized pendulum

Generative eBooks — 2017-06-14T13:58:14Z

Consider the motion of the following idealized pendulum: a bob of mass $m$ is attached to one end of a massless rigid rod. The other end of the rod is pivoted so that the mass may swing in a vertical plane. We neglect both the friction of the pivot and air drag.
The swinging of the pendulum is governed by $$ \ddot\theta=-\fracgL \sin \theta $$ where $\theta$ is the angle between the rod and the downward vertical. This equation is derived in all textbooks of classical mechanics.
Phase (…)

- Theorems & examples

Building a limit cycle from scratch

Generative eBooks — 2017-06-14T13:55:26Z

We want to cook up a simple example of a limit cycle, some kind of toy model. Let us look for a system having the circle of radius one centered at $(0,0)$ as an attracting limit cycle. The trick is to think in terms of polar coordinates $(r,\theta)$. The simplest situation would be to have uncoupled equations for the radial and angular motions: $$ \begincases \dotr=f(r)\ \dot\theta=g(\theta). \endcases $$ In the $r$-direction, we want $f(1)=0$, that is $r=1$ to be a fixed point. We also want (…)

- Theorems & examples

Lyapunov's theorem

Generative eBooks — 2017-06-14T13:51:52Z

Suppose that we can find a continuously differentiable, real-valued function $L(\boldsymbolx)$ such that $L(\boldsymbolx)>L(\bar\boldsymbolx)$ for all $\boldsymbolx$ in some neighborhood $U$ of $\bar\boldsymbolx$.
If $\dotL(\boldsymbolx)\leq 0$ for all $\boldsymbolx\in U$, then $\bar\boldsymbolx$ is stable.
If $\dotL(\boldsymbolx)< 0$ for all $\boldsymbolx\in U\backslash\\bar\boldsymbolx\$, then $\bar\boldsymbolx$ is asymptotically stable.
If $\dotL(\boldsymbolx)> 0\thinspace \textfor (…)

- Theorems & examples

Hartman-Grobman Theorem

Generative eBooks — 2017-06-14T13:50:50Z

Suppose $(\bar x,\bar y)$ is a fixed point of a system
$$ \begincases \dotx = f(x,y)\\ \doty= g(x,y). \endcases $$ Assume that the real part of the eigenvalues of the Jacobian matrix
$$ A= \beginpmatrix \frac\partial f\partial x\scriptstyle(\bar x,\bar y) & \frac\partial f\partial y\scriptstyle(\bar x,\bar y)\\ \frac\partial g\partial x\scriptstyle(\bar x,\bar y) & \frac\partial g\partial x\scriptstyle(\bar x,\bar y) \endpmatrix $$ are nonzero. Then there is a small region around (…)

- Theorems & examples

Poincaré-Bendixson theorem

Generative eBooks — 2017-06-14T13:49:26Z

Consider a system
$$ \begincases \dotx=f(x,y)\\ \doty=g(x,y) \endcases $$
where $f$ and $g$ have continuous partial derivatives, and such that solutions exist for all $t$. Let $R$ denote a closed, bounded region of the $xy$-plane which contains no fixed points. Suppose that no solution may leave $R$. Then the system has a periodic solution in the region $R$.

- Theorems & examples

Andronov-Hopf bifurcation theorem

Generative eBooks — 2017-06-14T13:39:32Z

Assume that, near $\mu=\mu^*$, the Jacobian matrix of the vector field evaluated at $(\barx(\mu),\bary(\mu))$ has eigenvalues of the form
$$ a(\mu)\pm i b(\mu) $$
with
$$ a(\mu^*)=0\quad\textand\quad b(\mu^*)\neq 0. $$
Also assume that the real parts of the eigenvalues change signs as $\mu$ is varied through $\mu^*$, i.e.,
$$ \frac\textda\textd\mu(\mu^*)\neq 0. $$
Given these hypotheses, the following possibilities arise: There is a range $\mu^*<\mu<\mu_1$ such (…)

- Theorems & examples

Three examples from Biology

Generative eBooks — 2017-03-06T15:17:12Z

- Part II: Two-dimensional systems: flows in the plane

A model of spread of infectious diseases

Generative eBooks — 2017-03-06T15:15:23Z

We study one of the simplest models used in epidemics. We divide a given population into three disjoint groups. The population of susceptible individuals is denoted by $S$, the infected population by $I$, and the recovered population by $R$. Of course, each of these is a function of time. We make the following assumptions: the total population is constant (no births or deaths), so that $\dotP=0$, where $P(t)=S(t)+I(t)+R(t)$. We denote by $P_0$ this constant; the rate of transmission of the (…)

- Three examples from Biology

The growth of two conflicting populations

Generative eBooks — 2017-03-06T15:09:39Z

We study here the simplest model describing two populations that compete fot the same limited food source or in some way inhibit each other's growth. For example, competition may be for territory which is directly related to food ressources.
The model is given by $$ \begincases \dotx=r_1 \, x\left(1-\fracxK_1-\fracb_12yK_1\right) \ \doty=r_2\, y\left(1-\fracyK_2-\fracb_21xK_2\right) \endcases $$ (…)

- Three examples from Biology