§10. Systems of First Order Equations

Systems of first order differential equations arise in many contexts. The qualitative methods developed earlier can be used to understand some of the basic properties of the solutions of a system.

The first example illustrates a situation which gives rise to a system (pair) of differential equations.

Example 10–1. The classical Lotka-Volterra predator-prey model can be described as follows. An ecosystem contains 2 different species A and B. Species A is a predator that feeds on species B and only on species B. Species B depends only on the enviroment for nutrition and nutrition is unlimited. Denote by A(t) and B(t) the population sizes for the 2 species at time t. A possible model is then provided by the system of differential equations

d

dt d

dt

The constants kA and kB represent the growth rates of the individual species populations in the absence of the other species. Thus kA < 0 and kB > 0 here. The constants e and f reflect the interdependency of the 2 species. The product A(t)B(t) measures the chances of interaction between members of the two species. In the present context, the constant e should be positive while f should be negative.

Exercise 10–1. Why is e > 0 reasonable? Why is f < 0 reasonable? Why should kA < 0 and kB > 0 hold?

The system can be expressed in vector form as

d

dt (A(t), B(t)) = (kAA(t) + eA(t)B(t), f A(t)B(t) + kBB(t)) which is reminicent of equations studied earlier.

The Lotka-Volterra system is an example of an autonomous system since the independent variable t appears only through the unknown functions A(t) and B(t).

Qualitative information about the solution of the system can be obtained by methods similar to those used earlier.

Example 10–2. Suppose a Lotka-Volterra model for the predator-prey population is given by

dtd (A(t), B(t)) = (−0.10A(t) + 0.01A(t)B(t), −0.01A(t)B(t) + 0.03B(t))

Copyright 2002 Jerry Alan Veeh. All rights reserved.

§10: Systems of First Order Equations 62

with the initial population A(0) = 5 and B(0) = 30. What is the population dynamics?

In parallel with the analysis in the one dimensional case, the first step in the analysis is to find the equilibrium solutions. This is done by setting the derivative vector equal to the zero vector and solving for A(t) and B(t). In this case, the two equilibrium solutions are A(t) = B(t) = 0 for all t, and A(t) = 3 and B(t) = 10 for all t.

Exercise 10–2. Verify that these are the only two equilibrium solutions.

The second step is to plot the direction field. Since the system is autonomous, the derivative vector depends only on the values of the unknown functions. The direction field is then obtained by plotting a vector of length one proportional to

dtd (A(t), B(t)) at each point in the A-B plane.

Direction Field for Lotka-Volterra Model

As before, once the direction field is drawn, the behavior of a solution starting from any given initial conditions can be sketched by following the arrows.

Solution with A(0)=10 B(0)=30

Example 10–3. The Maple command to draw the direction field for the last example is

DEplot({diff(A(t), t) = −0.1 A(t) + 0.01 A(t) B(t),

diff(B(t), t) = −.01 A(t) B(t) + 0.03 B(t)}, [A(t), B(t)], 0..100, A = 0..30, B = 0..50);