Visualizing a Continuous Model of Mutualism
by Mike Martin

On this page, plots of direction fields and solution curves are generated for the phase plane associated with a system of two 1st-order autonomous ordinary differential equations. Plots of the solution curves as functions of time are given below the phase plane, too. This page focuses on a particular form of differential equations which are most often used to model the population numbers for two species who benefit mutualistically from each other. The equations are of the form:

M ' ( t ) = a M (t) + b M (t) N (t) - α1 (M (t))2
N ' ( t ) = c N (t) + d M (t) N (t) - α2 (N (t))2

M(0) = M0
N(0) = N0

The populations, M and N, are usually in units relative to a large population, say in thousands or millions.

The user inputs positive values of the parameters a, b, c, d, α1, and α2.  Also, grid bounds, time bounds, and the initial conditions must be specified.

Enter in the value of the parameters and select the plotting bounds, then press the "Plot/Evaluate" button.

a =  

b =  

c =  

d =  

α1 =  

α2 =  

M (0) =  

N (0) =  

0   ≤   t   ≤  

  ≤   M   ≤  

  ≤   N   ≤  


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