Biology Reference
In-Depth Information
(c) What happens to the population described by Eq. (1-23) if:
(i) P(t
0
)
¼
50,
(ii) P(t
0
)
¼
150, and
¼
(iii) P(t
0
)
400?
(d) Describe how the graph in Figure 1-18 will change when the
harvesting rate is greater than 52 cattle/year.
(e) What is the largest number of cattle/year that could be sold while
guaranteeing the existence of a stable equilibrium for the model?
(f) What is the maximal sustainable yield for this model?
VIII. THE VERHULST MODEL FOR DISCRETE
POPULATION GROWTH
For modeling purposes, it is appropriate to hypothesize that many
species reproduce at a uniform rate. Said another way, there is no
preferred time at which reproduction occurs. In these cases, it is often
best to use a continuous model. As discussed in Section II, the use of
discrete models may be appropriate when the population remains
constant throughout intervals of time and then changes with a jump at
the end of an interval. Reproduction in such cases may be synchronized
to environmental stimuli, such as weather, light, or specific chemicals.
The reproduction of bamboo, cicadas, and some species of salmon occurs
in almost perfect synchronization, for example.
The choice between continuous and discrete models is fundamental to
the modeling process and should be based strictly upon the specific
biological problem at hand. It is incorrect to assume that a simple
discretization of a continuous model will lead to an acceptable discrete
model with similar behavior. Instead, such models should be derived
from first principles.
To illustrate this point, we consider a discrete model of population
growth in which the rate of change has the same functional form as that
of the continuous logistic model from the last section:
P
dP
dt
¼
P
K
a 1
:
(1-24)
In the discrete case, instead of representing the population at time t by
P(t), we denote the population size throughout the n-th generation
by P
n.
The change from the n-th to the (n
þ
1)-st generation is then given
by P
n
þ
1
P
n
and the model is:
