Differential Equations in Models

Models

Models are imitations of reality.

1.  Ignore unimportant factors

2.  Simplify complicated information

Radioactive Decay

Assumptions

We are dealing with a single radioactive substance consisting of atoms each of which, over a fixed period of time, has the same probability of decaying.  This probability is independent of the position of the fixed period of time, depending only on the properties of the particular substance and the length of the fixed period of time.

Variables and Parameters

t = time (indepepndent variable)

r = amount of particular radioactive substance available (dependent variable)

lambda = decay rate (parameter)

Equations

dr/dt = -lambda*r

This is a first order ordinary differential equation. It has one equilibrium solution, r(t) = 0 . There are many solutions, but only one solution satisfying a given initial condition. A qualitative analysis may be performed.  The initial value problem

has a unique solution, obtained by first finding the general solution and then determining the correct particular solution.

Logistic Growth of a Population

Assumptions

Small populations grow at a rate proportional to their size, but for large populations the growth rate is negative.

Variables and Parameters

t = time (independent variable)

P = size of population (dependent variable)

k = growth rate for small populations (parameter)

N = population limit (aka carrying capacity) (parameter)

Equations

dP/dt = kCP

where C is a function of P which is near one when P is near zero, and is negative for N < P .  The simplest such function is C = 1-P/N , which gives us the logistic equation

dP/dt = k*(1-P/N)*P .

Example exercises

Suppose t is measured in years, and P is the number of fish in a lake.  How does the equation change if

1.  100 fish are harvested per year?

2.  1/3 of the population is harvested per year?

3.  the number harvested per year is proportional to the square root of P ?

A Predator-Prey System

Assumptions

Foxes and rabbits occupy the same range.  If there were no foxes, the rabbit population would grow at a rate proportional to its size.  Foxes eat rabbits at a rate proportional to the rate of fox/rabbit interaction.  If there were no rabbits, the population of foxes would decay at a rate proportional to its size.  Foxes are born at a rate proportional to the rate of rabbit eating (which is proportional to the rate of interaction).

Variables and Parameters

t = time (independent variable)

R = size of rabbit population (dependent variable)

F = size of fox population (independent variable)

alpha = growth rate coefficient for rabbits

beta = rabbit eating interactions per interaction

gamma = death rate coefficient of foxes

delta = fox births per interaction

Equations

dR/dt = alpha*R-beta*RF

dF/dt = -gamma*F+delta*RF .

This is a first order system.  Analytic methods do not work to solve this.  Qualitative or numeric methods have to be used.