Generative eBooks

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