WEBVTT
1
00:00:01.189 --> 00:00:04.309 A:middle L:90%
in this problem. We have two species x and
2
00:00:04.309 --> 00:00:10.419 A:middle L:90%
y and party in part B. Each describes a
3
00:00:10.419 --> 00:00:14.300 A:middle L:90%
different system corresponding to the dynamics between the two species
4
00:00:14.300 --> 00:00:17.839 A:middle L:90%
X and Y, and we were asked to classify
5
00:00:17.839 --> 00:00:22.109 A:middle L:90%
the models in parts A and B according to whether
6
00:00:22.109 --> 00:00:25.739 A:middle L:90%
they describe cooperation between the two species or competition between
7
00:00:25.739 --> 00:00:31.149 A:middle L:90%
the two species. Now what we can understand from
8
00:00:31.149 --> 00:00:34.250 A:middle L:90%
looking at the terms of the model is that the
9
00:00:34.250 --> 00:00:37.479 A:middle L:90%
X Y terms that is, a product terms between
10
00:00:37.689 --> 00:00:45.250 A:middle L:90%
the two species represent encounters between them. So whenever
11
00:00:45.250 --> 00:00:48.250 A:middle L:90%
they interact, we're going to have a term like
12
00:00:48.259 --> 00:00:51.929 A:middle L:90%
extends. Why, poppet? And this comes from
13
00:00:52.130 --> 00:00:55.990 A:middle L:90%
something called the law of Mass Action, which is
14
00:00:56.000 --> 00:01:00.109 A:middle L:90%
implicit in the formulation of the ordinary differential equations for
15
00:01:00.109 --> 00:01:07.780 A:middle L:90%
the model. And we can consider how these encounters
16
00:01:07.780 --> 00:01:15.379 A:middle L:90%
describe competition cooperation no and increase in why he's going
17
00:01:15.379 --> 00:01:23.709 A:middle L:90%
to make DX over DT larger for party increasing.
18
00:01:23.709 --> 00:01:27.129 A:middle L:90%
Why is gonna make the ex already t larger because
19
00:01:27.450 --> 00:01:34.200 A:middle L:90%
of the positive term. 0.1 actually 0.1 x y
20
00:01:34.540 --> 00:01:38.010 A:middle L:90%
. So whenever X and y interact, we have
21
00:01:38.019 --> 00:01:42.739 A:middle L:90%
that the D X over DT growth rate is incremental
22
00:01:42.739 --> 00:01:49.120 A:middle L:90%
by this much. So that means the ex grocery
23
00:01:49.120 --> 00:01:53.459 A:middle L:90%
is gonna be higher because of this interaction. And
24
00:01:53.459 --> 00:01:56.969 A:middle L:90%
an increase in X, conversely, makes d y
25
00:01:56.969 --> 00:02:06.370 A:middle L:90%
over dt larger because of the positive term, 0.0
26
00:02:07.120 --> 00:02:13.719 A:middle L:90%
four x Y. So this means that either species
27
00:02:13.729 --> 00:02:15.680 A:middle L:90%
impacts the growth rate of the other species and a
28
00:02:15.680 --> 00:02:20.169 A:middle L:90%
positive cooperative way. So the system describes a cooperation
29
00:02:20.169 --> 00:02:24.729 A:middle L:90%
model as a whole. Now, for part B
30
00:02:25.620 --> 00:02:30.229 A:middle L:90%
, we see the opposite because an increase in X
31
00:02:31.620 --> 00:02:36.740 A:middle L:90%
part B reduces the growth rate of why so it
32
00:02:36.740 --> 00:02:39.800 A:middle L:90%
causes D Y over dt to go down because of
33
00:02:39.800 --> 00:02:45.530 A:middle L:90%
the negative. Come on, negative interaction term.
34
00:02:46.620 --> 00:02:49.129 A:middle L:90%
So you have a negative coefficient on X and wire
35
00:02:50.060 --> 00:02:52.729 A:middle L:90%
, not negative, because their populations of species.
36
00:02:53.419 --> 00:02:59.340 A:middle L:90%
So this is automatically negative and similarly, an increase
37
00:02:59.340 --> 00:03:02.090 A:middle L:90%
in why reduces the growth rate of X so d
38
00:03:02.090 --> 00:03:06.469 A:middle L:90%
x over DT equals don't because of this term negative
39
00:03:06.469 --> 00:03:14.710 A:middle L:90%
0.0 006 x Y So this system describes a competition
40
00:03:14.710 --> 00:03:19.719 A:middle L:90%
model because the growth rate of one species is in
41
00:03:19.719 --> 00:03:25.699 A:middle L:90%
Worsley, proportional to, um, the increase in
42
00:03:25.699 --> 00:03:30.599 A:middle L:90%
the other species or, more precisely, said we
43
00:03:30.599 --> 00:03:35.090 A:middle L:90%
have that one species increasing causes the growth rate of
44
00:03:35.090 --> 00:03:36.370 A:middle L:90%
the other species to go down