1
00:00:01,038 --> 00:00:06,795
in this video I will introduce
2
00:00:06,795 --> 00:00:09,353
a modified version
3
00:00:09,353 --> 00:00:11,911
of the logistic equation
4
00:00:11,911 --> 00:00:14,471
called the logistic equation with harvest.
5
00:00:14,471 --> 00:00:17,057
I will use this model to show
an example of a bifurcation
6
00:00:17,057 --> 00:00:19,643
and will construct a bifurcation diagram;
7
00:00:19,643 --> 00:00:23,526
a realy important type of diagram
8
00:00:23,526 --> 00:00:27,409
n the study
of all sort of dynamical systems.
9
00:00:27,409 --> 00:00:30,942
The way I'll do this
10
00:00:30,942 --> 00:00:33,298
might seem a little bit slow or involved.
11
00:00:33,298 --> 00:00:35,654
The bifurcation diagram is a bit abstract
12
00:00:35,654 --> 00:00:38,357
and so I think
building it up step by step,
13
00:00:38,357 --> 00:00:40,159
kind of carefully, is really helpful
14
00:00:40,159 --> 00:00:41,961
for getting an understanding
of what it means.
15
00:00:41,961 --> 00:00:44,697
So I'll be presenting things
16
00:00:44,697 --> 00:00:46,521
perhaps a little slower
than seems necessary
17
00:00:46,521 --> 00:00:48,345
but my experience has been
that doing so really helps
18
00:00:48,345 --> 00:00:50,587
when it comes time
to put all this together
19
00:00:50,587 --> 00:00:52,829
and look at the bifurcation diagram.
20
00:00:52,829 --> 00:00:55,072
Ok, so here is
the logistic equation with harvest.
21
00:00:55,072 --> 00:01:00,029
Without this h term,
22
00:01:00,029 --> 00:01:03,333
this is the regular logistic equation:
23
00:01:03,333 --> 00:01:06,637
r is the growth rate
and k is the carrying capacity.
24
00:01:06,637 --> 00:01:10,986
And I'm going to add another parameter h
25
00:01:10,986 --> 00:01:15,335
which we can interpret as the harvest rate.
26
00:01:15,335 --> 00:01:18,761
So for concreteness I think
that we are modeling some fisheries
27
00:01:18,761 --> 00:01:21,045
in a lake or a bay or something,
28
00:01:21,045 --> 00:01:23,329
and h here is ...
it would just be a number,
29
00:01:23,329 --> 00:01:25,614
like a number of fish.
30
00:01:25,614 --> 00:01:31,591
So a manager might set h at 50:
31
00:01:31,591 --> 00:01:37,568
catch 50 fish a year,
or 100 fish or 3 fish.
32
00:01:37,568 --> 00:01:41,784
Notice that these terms here
are density dependent.
33
00:01:41,784 --> 00:01:46,000
The growth rate depends on
the numbers of fish that are present (P).
34
00:01:46,000 --> 00:01:50,233
The h term (how many we catch)
35
00:01:50,233 --> 00:01:54,466
does not depend
on the number of fish that are present.
36
00:01:54,466 --> 00:01:58,761
That's maybe not
an entirely accurate view of things
37
00:01:58,761 --> 00:02:03,056
but maybe for some fish
that are dumb or slow or easy to catch,
38
00:02:03,056 --> 00:02:07,352
you can catch them equally well
independent of how many there are.
39
00:02:07,352 --> 00:02:10,306
And as you probably guessed
40
00:02:10,306 --> 00:02:13,260
this is a model to not take too seriously
41
00:02:13,260 --> 00:02:15,368
(I don't think that anybody thinks
42
00:02:15,368 --> 00:02:17,476
fisheries are
really, really described by this)
43
00:02:17,476 --> 00:02:19,585
but it's a model that will help us think
through some trade-offs and ideas
44
00:02:19,585 --> 00:02:23,172
and I will talk more about
how we can generalize
45
00:02:23,172 --> 00:02:26,759
from this model towards
the end of this video or the next one.
46
00:02:26,759 --> 00:02:29,975
Ok, so we've got
logistic equation with harvest
47
00:02:29,975 --> 00:02:33,191
and what I am going to do
48
00:02:33,191 --> 00:02:36,474
is analyse the logistic equation
49
00:02:36,474 --> 00:02:39,757
using qualitative techniques
for different values of h.
50
00:02:39,757 --> 00:02:42,411
The first value I am going to do
51
00:02:42,411 --> 00:02:44,181
will be a little bit borring and redundant
52
00:02:44,181 --> 00:02:45,951
because we've already looked at this.
53
00:02:45,951 --> 00:02:51,805
And so this is the case where h is zero
54
00:02:51,805 --> 00:02:57,659
so this means
we aren't harvesting any fish at all.
55
00:02:57,659 --> 00:03:01,695
And this is the same number
as I plugged in before
56
00:03:01,695 --> 00:03:04,386
so the picture is the same.
57
00:03:04,386 --> 00:03:06,180
If the population is in here it's increasing,
58
00:03:06,180 --> 00:03:07,974
if the population is above 100 it's decreasing.
59
00:03:07,974 --> 00:03:14,494
So we have a stable fixed point at 100
60
00:03:14,494 --> 00:03:21,014
and then an unstable fixed point at zero.
61
00:03:21,014 --> 00:03:28,666
I'll draw the phase line for that....
62
00:03:28,666 --> 00:03:33,768
here it is, here's a fixed point .....
63
00:03:33,768 --> 00:03:37,169
there's a fixed point ....
64
00:03:37,169 --> 00:03:39,436
Now I am going to use
a scale that's designed
65
00:03:39,436 --> 00:03:41,703
to sort of line up with what's on this
66
00:03:41,703 --> 00:03:45,558
and I'll make a note to myself that
this was the case when h is 0.
67
00:03:45,558 --> 00:03:48,128
Now I should put a few more arrows on....
68
00:03:48,128 --> 00:03:50,698
Ok, so this is just
what we did in the last video.
69
00:03:50,698 --> 00:03:54,739
Stable fixed point at 100,
unstable fixed point at zero
70
00:03:54,739 --> 00:03:58,780
and this is the case when
we are not harvesting any fish at all.
71
00:03:58,780 --> 00:04:02,771
So now we analized it for h = zero
72
00:04:02,771 --> 00:04:06,762
I am going to do the same thing
but for a different value of h
73
00:04:06,762 --> 00:04:14,133
So here is h = 40
74
00:04:14,133 --> 00:04:21,504
So let me just say a little bit
about this before we do the phase line.
75
00:04:21,504 --> 00:04:27,465
The effect of this h term - just geometrically,
76
00:04:27,465 --> 00:04:31,439
when you subtract
a constant from a function
77
00:04:31,439 --> 00:04:35,413
it moves it down, vertically.
78
00:04:35,413 --> 00:04:39,354
So here this is the graph where h = 0
79
00:04:39,354 --> 00:04:41,981
h = 40, I just subtract 40 from it
80
00:04:41,981 --> 00:04:43,733
so if I took this
and shifted it down by 40
81
00:04:43,733 --> 00:04:45,546
I would get this curve.
82
00:04:47,359 --> 00:04:49,172
So here is the curve
and lets see what the dynamics are now.
83
00:04:49,172 --> 00:04:52,859
What fixed points are there?
What are their stablity?
84
00:04:52,859 --> 00:04:55,596
So now we have positive growth
85
00:04:55,596 --> 00:04:58,333
if we are in between
this number and this number.
86
00:04:58,333 --> 00:05:01,071
So in between here, the fish increase.
87
00:05:01,071 --> 00:05:08,101
If the number of fish is greater than 83 or 84
88
00:05:08,101 --> 00:05:12,787
the number of fish will decrease,
move this way.
89
00:05:12,787 --> 00:05:17,473
And now, if I am
less than, say, 17 or 18 fish,
90
00:05:17,473 --> 00:05:23,489
then I'll move towards zero
and then negative infinity .
91
00:05:23,489 --> 00:05:27,500
So if we start, say, with 15 fish,
92
00:05:27,500 --> 00:05:31,511
that is bad news for the fish
because they will all die out.
93
00:05:31,511 --> 00:05:37,595
So let me draw the phase line
for this situation.
94
00:05:37,595 --> 00:05:41,651
Here is my line and I have
95
00:05:41,651 --> 00:05:45,707
a fixed point there, fixed point there.
96
00:05:45,707 --> 00:05:51,815
In between these two fixed points
the population increases,
97
00:05:51,815 --> 00:05:55,887
above it decreases, and so on,
98
00:05:55,887 --> 00:05:59,959
And this is h = 40.
99
00:05:59,959 --> 00:06:03,086
Note by the way that the steady population
100
00:06:03,086 --> 00:06:06,213
- the carrying capacity used to be 100
101
00:06:06,213 --> 00:06:10,445
and now if I harvest 40 a year
102
00:06:10,445 --> 00:06:13,267
presumably that would move it down to 60
103
00:06:13,267 --> 00:06:15,148
and it doesn't quite work that way
104
00:06:15,148 --> 00:06:16,402
because this has a pretty large growth rate,
105
00:06:18,074 --> 00:06:24,665
so yes you are harvesting
but they are also sort of growing back
106
00:06:24,665 --> 00:06:29,059
part of the way towards the 100
which is what it would want to be,
107
00:06:29,059 --> 00:06:33,453
what the population would go to
if h wasn't here.
108
00:06:33,453 --> 00:06:37,980
So now the new steady state
for this fishery would be 82, 83,
109
00:06:37,980 --> 00:06:40,998
but note also that
there is a critical number of fish
110
00:06:40,998 --> 00:06:43,010
and if you get underneath this population
111
00:06:43,010 --> 00:06:45,022
the population will die off
and it will go down to zero.
112
00:06:45,022 --> 00:06:52,637
Ok, so, not surprising I don't know
anything at all about fish
113
00:06:52,637 --> 00:06:57,713
but this seems like a
not crazy story to tell about fish.
114
00:06:57,713 --> 00:07:02,789
Ok, so that's h = 40
115
00:07:02,789 --> 00:07:06,352
Now I am going to try another h value
so this is h = 65
116
00:07:06,352 --> 00:07:08,728
again notice that
as h gets larger this curve
117
00:07:08,728 --> 00:07:10,312
(it's an upside down parabola)
118
00:07:10,312 --> 00:07:11,896
moves down the axis.
119
00:07:11,896 --> 00:07:17,111
So now this is moved another 25 down
120
00:07:17,111 --> 00:07:22,326
and it is just kind of
barely peeking up above this axis.
121
00:07:22,326 --> 00:07:28,669
So now we are harvesting
a lot of fish a year,
122
00:07:28,669 --> 00:07:32,898
the carrying capacity is 100,
we are harvesting 65 a year.
123
00:07:32,898 --> 00:07:35,717
You might think 'Oh, is that too many?
124
00:07:35,717 --> 00:07:38,536
Is that going to make
the fish die off?' Well, let's see ....
125
00:07:38,536 --> 00:07:43,260
We do have a reasonable growth in here,
126
00:07:43,260 --> 00:07:46,409
if we're between these two values,
127
00:07:46,409 --> 00:07:48,509
the growth rate (that's on the y axis)
is positive
128
00:07:48,509 --> 00:07:50,609
so the population will increase
till around a little less than 70.
129
00:07:50,609 --> 00:07:56,787
If we are above this value
130
00:07:56,787 --> 00:08:00,906
it will decrease and stop at 70
131
00:08:00,906 --> 00:08:03,652
and if we are to the left of here,
132
00:08:03,652 --> 00:08:06,751
it will decrease towards zero:
the fish will die.
133
00:08:09,850 --> 00:08:12,950
So lets draw the phase line for this.
134
00:08:12,950 --> 00:08:17,318
Here is my line and now I have a fixed point here
135
00:08:17,318 --> 00:08:20,230
and there - remember fixed points occur
when the growth rate is zero.
136
00:08:20,230 --> 00:08:23,142
Sso here and here.
137
00:08:23,142 --> 00:08:28,555
In between these two points,
the population grows.
138
00:08:28,555 --> 00:08:32,164
To the right, the population decreases
(moves to the left)
139
00:08:32,164 --> 00:08:34,570
and if I am underneath - if I am
140
00:08:34,570 --> 00:08:36,174
below this value a little more than 30 -
141
00:08:36,174 --> 00:08:37,778
the fish die off; bad news for the fish.
142
00:08:37,778 --> 00:08:43,290
Ok, so I'll make a note to myself. h = 65.
143
00:08:43,290 --> 00:08:46,965
So that is the phase line
144
00:08:46,965 --> 00:08:50,640
for this differential equation with h = 65.
145
00:08:50,640 --> 00:08:54,453
Alright, I'll do two more h values.
146
00:08:54,453 --> 00:08:56,995
Next we'll do h = 75
147
00:08:56,995 --> 00:08:59,537
and this is an interesting one
now the curve has been lowered enough.
148
00:08:59,537 --> 00:09:08,844
I am subtracting enough fish evey year
149
00:09:08,844 --> 00:09:18,151
that it just barely touches
this point here.
150
00:09:18,151 --> 00:09:23,759
So what might the phase line
look like for this?
151
00:09:23,759 --> 00:09:27,498
And I'll draw the line in....
152
00:09:27,498 --> 00:09:29,990
Now there is only one fixed point.
153
00:09:29,990 --> 00:09:31,652
It occurs right here in the middle
154
00:09:31,652 --> 00:09:33,314
when have 50 fish; a population of 50.
155
00:09:33,314 --> 00:09:36,442
If I'm above this, I decrease,
156
00:09:36,442 --> 00:09:39,570
and if I'm below this I decrease.
157
00:09:39,570 --> 00:09:42,700
So this fixed point is sort of a odd case one we have not encountered much before there is one in the intermediate homework problem on this
158
00:09:42,700 --> 00:09:53,228
We will call it semi stable perhaps because if you move away here you have a lot of fish you will decrease to this you will get closer to this fixed point
159
00:09:53,228 --> 00:10:01,007
But if you are a little less you have to few fish below 50 you are at 49 and the fish will start to die the population decreases
160
00:10:01,007 --> 00:10:25,127
the population is always decreasing except it happenes to hold steady at this one single point ok that is h = 75. I have a nice phase line for that , and one more. Maybe you can guess what is about to happen
161
00:10:25,127 --> 00:10:33,146
Now h will be 85 so I am harvesting 85 fish a year and this enough of a harvest rate that the parabola is now completely below the x axis
162
00:10:33,146 --> 00:10:43,056
That means that the growth rate is always negative so no mater what the population is it is decreasing towards zero and I guess mathematicaly towards negative infinity
163
00:10:43,056 --> 00:10:57,530
So this realy bad news for the fish So I can draw this phase line I put no dots on this phase line because there are no fixed points no equalibrium points
164
00:10:57,530 --> 00:11:15,110
the yust arrows indicating that no mater what the population is it is always decreasing I will make a note that this h value, the parameter h, is 85
165
00:11:15,110 --> 00:11:22,455
Ok so we looked at this diferental equation and I varied this parameter h and we looked at 5 different values and we saw different behaviors
166
00:11:22,455 --> 00:11:30,359
Most of the time there were two fixed points here there was this special case there was one fixed point and here there were zero fixed point
167
00:11:30,359 --> 00:11:42,836
And this makes sense I think from the point of view fish fisherman fisher folks in at as you harvet more and more fish the population declines.
168
00:11:42,836 --> 00:11:49,820
eventually your harvesting so many fish that the fish population goes to zero
169
00:11:49,820 --> 00:12:04,834
So this way of looking at things is we are analysing the equation one parameter at a time I print out a h value I tell you a h value and hopefully now it not to hard to come up with a phase line for this
170
00:12:04,834 --> 00:12:13,344
What a bifurication diagram will give us is a way of picturing all the posible behaviors of this equation for all posible values for h all at once
171
00:12:13,635 --> 00:12:16,938
it is a nice geometric constuction that you can get a lot of information out
172
00:12:16,938 --> 00:12:20,938
so that will be the topic of the next video