Generative eBooks

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

Sharks & Sardines

Generative eBooks — 2017-03-06T15:02:59Z

We come back to the prey-predator model we introduced in the opening part. In dimensionless form it is given by the equations
$$ \begincases \dotx= x(1-x-y)\ \doty=\beta(x-\alpha)y \endcases $$
where $\alpha,\beta$ are positive parameters. Under this form, the carrying capacity of prey is normalized to be equal to one. We are only interested in the positive quadrant since $x,y$ are interpreted as abundances of populations. One can check that the positive quadrant is invariant in (…)

- Three examples from Biology

Interlude: no chaos for two-dimensional systems

Generative eBooks — 2016-06-08T13:00:46Z

A natural question is whether we can classify all possible types of long-term behaviors of solutions of two-dimensional systems. We saw two types of behaviors: as $t\to+\infty$, solutions that tend to a fixed points, or solutions that wrap around a closed trajectory (limit cycle). Are there other types of behaviors? We consider a two-dimensional system $$ \begincases \dotx=f(x,y)\ \doty=g(x,y) \endcases $$ where $f$ and $g$ are continuously differentiable functions.
There is a first (…)

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

Stability of Fixed Points & the Lyapunov method

Generative eBooks — 2016-06-08T13:00:13Z

We investigate the stability and instability properties of fixed points. In other words, what happens if we perturb a system which is sitting at a fixed point? As the reader can guess, we should use the linear approximation of the system, but we expect only to be able to deal with ‘small' perturbations. This is why we shall present another, somewhat more geometric, technique: the method of Lyapunov. Besides, this method gives us a grasp on the size of the basin of attraction of fixed point (…)

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

A bunch of examples with limit cycles

Generative eBooks — 2016-06-08T12:59:37Z

We present various examples where attracting limit cycles show up.
Cycling preys and predators: the Rosenzweig-McArthur model
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 (…)

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

Bifurcations

Generative eBooks — 2016-06-08T12:59:00Z

Here we extend what we have seen on bifurcations in one-dimensional systems. In dimension two, we still find that fixed points can be created or destroyed or destabilized as we vary a parameter. But there is a novelty: periodic solutions are possible, and there are ways to turn them on or off by tuning a parameter. This is the so-called the Poincaré-Andronov-Hopf bifurcation. Let us stress that we just open the door of a broad field about which entire books are written.
Saddle-node (…)

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

Periodic oscillations & limit cycles

Generative eBooks — 2016-06-08T12:57:46Z

We have so far studied the simplest thing a solution of a differential equation can do: be attracted or repelled by a fixed point. The next simplest thing it can do is to behave periodically in time, that is, to trace out a closed curve, called a cycle. We have already met examples of periodic solutions in the opening part of this ebook. In particular, we have seen an example of a stable or attracting limit cycle. It corresponds to self-sustained oscillations which are robust under slight (…)

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

Interlude: for the pleasure of the eyes

Generative eBooks — 2016-06-08T12:56:20Z

All the models we presented so far are motivated by physics or biology. Here you can go through a gallery of phase portraits of two-dimensional systems we find beautiful, without worrying about what they could potentially model.

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