General Monotone Response and Many Competitors

By measuring 5 and Xj/yj in units of S and time in units of D , one obtains the nondimensional system 

These assumptions are really quite mild the / need only be increasing and sufficiently smooth. It is not even required that f, be bounded on IR. From a biological perspective, of course, the model loses relevance for really large values of S. In addition to the Monod functions, other functions that have been suggested include the exponential kinetics AM(1—exp(—S log 2/a)), hyperbolic kinetics m tanh(S log 3/2a), and piece-wise linear kinetics given by mS/2a for S 2a and by w for S 2a. The piecewise linear kinetics fails to satisfy the strict monotonocity of (iii) and fails to satisfy (iv) at one point, but these assumptions could be weakened so as to include this case. 

be the unique solution of fi S) = 1 if one exists otherwise let A, = -1-00. We assume the equations are numbered such that 

This assures that competitor Xi has the least requirements for growth and therefore is favored to win the competition. 

It is easily seen that n is positively invariant for (3.3) indeed, the vector field (3.3) points into the interior of Q on that part of its boundary where = 1. To see this, observe that if x(0) lies on this component of the boundary of Q then 

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