Supplement 1. Mathematica code to calculate the overcompensation and consumer–resource stability boundaries of the semi-discrete model.

File List
StabilityAnalysisSemi-DiscreteLogistic.nb - Mathematica code

StabilityAnalysisSemi-DiscreteLogistic.pdf - PDF of the code

Description
Mathematica (version 5) code to calculate the overcompensation and
consumer-resource stability boundaries of the semi-discrete model as a function of mu
and alpha, for different values of rho. The analysis is decsribed in Appendix A.

The code calculates the stability boundaries in sections.

data1 - bottom part of the consumer-resource boundary (heavy solid line in Fig. 2a of the main text);

data2 - top part of the consumer-resource boundary (dotted line in Fig. 2a);

data3 - top of the left arm of the overcompensation boundary (dashed line in
Fig. 2a) until it crosses the consumer-resource boundary;

data4 - bottom of the left arm of the overcompensation boundary (dashed line
in Fig. 2a) until the minimum of the boundary;

data5 - right arm of the overcompensation boundary (dashed line in Fig.
2a).

Some parameter definitions:

J11, J12, J21, J22 - elements of the Jacobian from the linear stability
analysis of the model.

Cond1 - condition for existence of a positive
equilibrium;

Cond2 - transition from stability to overcompensation cycles;

Cond3 - transition from stability to consumer - resource cycles;

Period -
period of population cycles on the consumer - resource stability boundary
(for a discussion of caclulating boundary conditions and the period see Gurney, W. S. C., and R. M. Nisbet. 1998. Ecological Dynamics. Oxford University Press, New York, New York, USA)