Math 5110/6830
Homework 5.1
This homework is devoted to the Nicholson-Bailey model that we started
considering in class:
Hn+1 = kHn e−aPn
Pn+1 = cHn (1 − e−aPn )
1. Show that the Nicholson-Bailey model has the following steady states
(0,0) and
k ln k
ln k
,
ac(k − 1) a
2. (dV 2.14.17) Suppose that we modify the assumptions of the NicholsonBailey model (that we considered in class) in the following way: if a host has
one encounter with a parasitoid c parasitoid progeny will be produced, but if it
has two or more encounters 2c parasitoid progeny are produced. Write down a
new model incorporating this assumption.
3. One possible modification for the Nicholson-Bailey model is to limit the
growth of the host species in isolation by assuming intra-species competition
and, subsequently, an existence of the carrying capacity (maximum possible
level of the population). Beddington, Free and Lawton realized this in the
following model:
Nn+1 = er(1−Nn /K) Nn e−aPn ,
(1)
Pn+1 = λf Nn [1 − e−aPn ].
a)Determine all fixed points. You may need to find solutions of the system of
equations graphically.
b)(Extra credit) Under what conditions on the model parameters are fixed
points stable? Unstable?
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