## Transcribed Text

Fish Population
1
Introduction
this lab, you will use the logistic equation model and study several populations fish
living suburban lake. Using basic numerical and qualitative techniques, you will study how
the trout population evolve over time both with and without harvesting
2 Mathematical Model
Suppose that suburban lake stocked with three types fish: rainbow trout (R). brown
trout (T). and bass (B). the absence any predation harvesting each of these populations
fish may be modeled with the logistic equation
(1)
where g(f) the size of the fish population (in hundreds of fish) at any given time (days), the
parameter >Ois the initial growth population, and >ois the carrying capacity
the population. Keep mind the accuracy any predictions based on the logistic model
depends upon whether the parameters and are constant
Now suppose that the Lake has been opened for fishing (ie "harvesting") and assume for sim-
plicity that humans the only predators the fish. To make sure that the lake over- fished
tomodel how
fishing
It is straightforward to adjust the logistic equation (1) include harvesting. h(f)
the
amount harvesting taking place at time then one way change the model is
Now reasonable assume that amount harvesting depends on the number of nish
in the lake. (After all, there are more fish the lake then easier catch them.) So let's
measure the amount harvesting terms (amount fish) instead terms off Then the
model is
(2)
A reasonable harvesting function could be
pg2
h(g) q+y²
(3)
where the parameters and represent how good the locals are at catching fish.
Finally notice that fwe define the right-hand side of the equation
(4)
then wecan rewrite the differential equation(2)as
s'(t) (g)
Section Questions
1. What are the units of the parameters and L? (Remember what the units of and t are.)
2. Use separation of variables solve(1) analytically (i.e. no harvesting)
3. Solve (1)or the general form the (de not use specific values of
L).
4. Classify the dific erontial equation(2). Isi linear? Isit homogeneous] What sits degree? i t
what does fnon- autonomy mean physically forthis problem)?
5.
Set but DO NOT solve the integnals that one gets when sparation variables done
on model (2) (i.e., with harvesting). Suggest an analytical technique that could be used to
evaluate the integrals.
6. Explore the harvesting function h(g). What happens h(g) as gets very large? What if y
close to 07 Does this make sense physically? (It may help totry graphing h(y).)
3 Numerical Investigation of (1)
Let p= 1.2 and Since there are three species of fish in the lake we would need to
investigate population growth species individually, Suppose fish species have
the same natural growth rate 0.65 but the carrying capacities different each the
species. Rainbow trout have LR =5.4 brown trout have La 8.1. and have LB = 16.3.
Pick one species fish to study i this section Suppose that the lake initialh stocked with
200
your
fish
species
choice.
.
the interval 10 25] with ster sizes -0,0 to
numericall-
the
and
true
solution
(from
2.2
same
graph
. Plot the Toplo the direction field you
may use any program like. provide method. the file dirfield. Instructions for
using
dirfield
are
included
separately.
Section Questions
1. the result Equestion 2.3
for
your chosen
fish
population
classify
their
stability.
2.
Compare
the
"exact"
and
your
fish
population
How
did
step
size
influence
the
accuracy
of
numerical
solution?
3. Using your direction field. describe the behavior of solutions for various initial
conditions
4
Numerical Investigation of (2)
It is difficult (although possible) find the exact solution to equation (2) Furthermore the
solution complicated hard understand Instead using the exact solution to (2) we will
explore
he behavior
qualitatively
As Section let p : 1.2 and g=l. Suppose all the fish species have the same natural
growth rate 0.65 but the capacities are differen each the specles. As before.
rainbow trout have LR 5.4. brown trout have La 8.1. and bass have Ln 16.3
. Explore the behavior the logistic model(2) for cad specles finsh (Le. foreach value o L).
Plot Fasa function and thisplot equilibrium solutions ((2) (Be sure
touse suitable scale when plottings that you don't miss any equilibrium values.)
.
Notice that some values have different number equilibria than others trialand
error find values where fequilibria changes (You'l haveto
try values between the given values LT and LR-) There aretwo values thisrange
when the number dequilibria change. Each these special values calleda hifureation
value,
. Now, plot direction folds Fibe different values 5,4 8.1 and 16.3, Toplot the
like. Weprovide the file
dirfield eparately (Note that you may
(very)lange time intervalto Esolutions,
Section Questions
1.
(a) Describe words what an equilibrium
(b)
Give a
mathematical
definition
Explain
vou
can
this
find the equilibria from the plot /(y).
2. Interpret the function plots and direction fields:
(a) What happening population when f(a) when f(y)=07
(b) Use the information from part stable unstable
(c) Make sure that your function plots and are consistent with each other
Discuss what the equilibria look like your fields Hom can you determine
stability
from
your
direction
fields?
(d) Using your direction fields, what are term behaviors of solutions for
various
initial
How
does
depend
on
the
initial
conditions
(pop-
ulation time 0) and the value L7
3.
Report the two bifurcation values that you
found above.
Describe
how the number equilibria changes each value.
4. Based your results. which species of trout best the lake with? There may
be
various
issues
consider
explain
your
decision.
5.
Discuss
these
questions
briefly
with
your
own
Are
there
any
weaknesses
in
the
model used? How do think could improved? Do you think there are
additional
effects
the
model
should
account
for?

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