1)A famous model of the spruce budworm population proposed in (Ludwig et al., 1978) (which used a differential equation) involved assuming logistic growth for the budworm, but introducing another term for predation due to birds. The predation term used was g(N) = β2N2 α2+N2 , where N denotes the number of budworms and α and β are two parameters than can be chosen to vary the graph to fit experimental data.

1)Graph g and explain why its shape is reasonable to describe the number of budworms consumed by birds for various sizes of the budworm population. In particular, does it rise and level off as you think it should? How do the values of α and β affect the shape of the graph?

2)Investigate the full model Nt+1 = Nt + r Nt(1 − Nt/K) − β2N2 t α2 + N2 t using MATLAB for a variety of parameter choices (but keep r small to avoid cycles or chaos in the logistic part of the model). Find parameter values that seem to give realistic behavior.

3)What can be said about steady states and their stability?

pur-new-sol

Related Questions