# Differential Equations: Populations

The population sizes of a prey, X, and a predator, Y (measured in thousands) are given by x and y, respectively. They are governed by the diﬀerential equations

ẋ = −pxy + qx and ẏ = rxy - sy (where p, q, r and s are positive constants (p ≠ r).

In the absence of species Y (i.e. y = 0), how would I ﬁnd a solution for x at time t if x(0) = x0, where x0 > 0. Is this model realistic? Why?

How would I determine all the equilibrium points for this system of diﬀerential equations expressing my answer in terms of p, q, r and s.

How would I classify each equilibrium point using the method of matrices with two real distinct eigenvalues?

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The population sizes of a prey, X, and a predator, Y (measured in thousands) are given by x and y, respectively. They are governed by the diﬀerential equations

ẋ = −pxy + qx and ẏ = rxy - sy (where p, q, r and s are positive constants (p ≠ r).

In the absence of species Y (i.e. y = 0), how would I ﬁnd a solution for x at time t if x(0) = x0, where x0 > 0. Is this ...

#### Solution Summary

The expert determines all the equilibrium points for a system.

Differential equations to describe infection

Use models to describe the population dynamics of disease agents.

Total population is a Constant (T).

A small group of infected individuals are introduced into a large population. Describe spread of infection within population as a function of time. This disease which, after recovery, confers immunity. The population can be divided into three distinct classes.

a) A susceptible class S, who can catch the disease

b) the infected class I - who have the disease and can transmit.

c) the removed class R, who have had the disease or are recovered, immune or isolated until recovered, or indeed, dead.

The gain in the infected class is at a rate proportional to the number of infected and susceptible persons that is rSI, where r > 0 (is a constant) r >0 is called the infection rate. The susceptibles are lost at the same rate.

The rate of removal of infected to the removed class is proportional to the number of infected people, that is aI, where a > 0 is a constant. a >0 is called the removal rate.

The incubation period is short enough to be negligible.

S(0) = So>0, I(0) = Io>), R(0) =0

#1 Formulate the three differential equations describing the rate of change in the number of S, I, and R populations.

A key questions is that given r, a, and S and the initial number of infected individuals (Io) whether the infection will spread or not, and if it does how it develops with time and of course when it will start to decline. The term epidemic means that I(t) >Io for some t > 0

#2 Find the minimum number of suseptibles required to start an epidemic in terms of a and r.

Assume that a=0.5, r=0.005, So = 150 Io = 50, N = 200

#3 Is this epidemic probable or not?

#4 Determine the global maximum number of infected population in this disease process

#5 When does the maximum in the infected class happen?

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