S4m: Interacting populations

Example 1

S4m: Interacting populations (ver. 1)

Two species, greenfish and bluegill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d G }{dt}= -\frac{1}{50} \, B G - \frac{1}{125} \, G^{2} + 8 \, G\]

\[\frac{d B }{dt}= -\frac{1}{100} \, B^{2} - \frac{3}{250} \, B G + 6 \, B\]

Draw an appropriate phase plane. Then, assuming that the current population is 598 greenfish and 98 bluegill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( G , B )\in\left\{ ( 0 , 0 ), ( 250 , 300 ), ( 1000 , 0 ), ( 0 , 600 )\right\}\]

In the long term, only greenfish will survive.

Example 2

S4m: Interacting populations (ver. 2)

Two species, bluegill and greenfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d B }{dt}= -\frac{3}{500} \, B^{2} - \frac{3}{250} \, B G + \frac{18}{5} \, B\]

\[\frac{d G }{dt}= -\frac{1}{100} \, B G - \frac{1}{250} \, G^{2} + 2 \, G\]

Draw an appropriate phase plane. Then, assuming that the current population is 252 bluegill and 102 greenfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( B , G )\in\left\{ ( 0 , 0 ), ( 100 , 250 ), ( 600 , 0 ), ( 0 , 500 )\right\}\]

In the long term, only bluegill will survive.

Example 3

S4m: Interacting populations (ver. 3)

Two species, yellowgill and greenfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d Y }{dt}= -\frac{1}{50} \, G Y - \frac{1}{125} \, Y^{2} + \frac{16}{5} \, Y\]

\[\frac{d G }{dt}= -\frac{1}{100} \, G^{2} - \frac{3}{250} \, G Y + \frac{12}{5} \, G\]

Draw an appropriate phase plane. Then, assuming that the current population is 241 yellowgill and 38 greenfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( Y , G )\in\left\{ ( 0 , 0 ), ( 100 , 120 ), ( 400 , 0 ), ( 0 , 240 )\right\}\]

In the long term, only yellowgill will survive.

Example 4

S4m: Interacting populations (ver. 4)

Two species, bluegill and purplegill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d B }{dt}= -\frac{3}{250} \, B^{2} - \frac{1}{100} \, B P + \frac{12}{5} \, B\]

\[\frac{d P }{dt}= -\frac{1}{125} \, B P - \frac{1}{50} \, P^{2} + \frac{16}{5} \, P\]

Draw an appropriate phase plane. Then, assuming that the current population is 240 bluegill and 39 purplegill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( B , P )\in\left\{ ( 0 , 0 ), ( 100 , 120 ), ( 200 , 0 ), ( 0 , 160 )\right\}\]

In the long term, both species will co-exist.

Example 5

S4m: Interacting populations (ver. 5)

Two species, redfish and greenfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d R }{dt}= -\frac{1}{200} \, G R - \frac{1}{100} \, R^{2} + \frac{1}{2} \, R\]

\[\frac{d G }{dt}= -\frac{9}{1000} \, G^{2} - \frac{3}{500} \, G R + \frac{27}{50} \, G\]

Draw an appropriate phase plane. Then, assuming that the current population is 60 redfish and 9 greenfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( R , G )\in\left\{ ( 0 , 0 ), ( 30 , 40 ), ( 50 , 0 ), ( 0 , 60 )\right\}\]

In the long term, both species will co-exist.

Example 6

S4m: Interacting populations (ver. 6)

Two species, greenfish and redfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d G }{dt}= -\frac{3}{250} \, G^{2} - \frac{1}{100} \, G R + 6 \, G\]

\[\frac{d R }{dt}= -\frac{1}{125} \, G R - \frac{1}{50} \, R^{2} + 8 \, R\]

Draw an appropriate phase plane. Then, assuming that the current population is 601 greenfish and 102 redfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( G , R )\in\left\{ ( 0 , 0 ), ( 250 , 300 ), ( 500 , 0 ), ( 0 , 400 )\right\}\]

In the long term, both species will co-exist.

Example 7

S4m: Interacting populations (ver. 7)

Two species, purplegill and magentafish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d P }{dt}= -\frac{1}{50} \, M P - \frac{1}{125} \, P^{2} + 8 \, P\]

\[\frac{d M }{dt}= -\frac{1}{100} \, M^{2} - \frac{3}{250} \, M P + 6 \, M\]

Draw an appropriate phase plane. Then, assuming that the current population is 601 purplegill and 100 magentafish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( P , M )\in\left\{ ( 0 , 0 ), ( 250 , 300 ), ( 1000 , 0 ), ( 0 , 600 )\right\}\]

In the long term, only purplegill will survive.

Example 8

S4m: Interacting populations (ver. 8)

Two species, purplegill and bluegill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d P }{dt}= -\frac{1}{50} \, B P - \frac{1}{125} \, P^{2} + 8 \, P\]

\[\frac{d B }{dt}= -\frac{1}{100} \, B^{2} - \frac{3}{250} \, B P + 6 \, B\]

Draw an appropriate phase plane. Then, assuming that the current population is 600 purplegill and 101 bluegill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( P , B )\in\left\{ ( 0 , 0 ), ( 250 , 300 ), ( 1000 , 0 ), ( 0 , 600 )\right\}\]

In the long term, only purplegill will survive.

Example 9

S4m: Interacting populations (ver. 9)

Two species, purplegill and redfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d P }{dt}= -\frac{1}{100} \, P^{2} - \frac{1}{250} \, P R + \frac{4}{5} \, P\]

\[\frac{d R }{dt}= -\frac{3}{500} \, P R - \frac{3}{250} \, R^{2} + \frac{36}{25} \, R\]

Draw an appropriate phase plane. Then, assuming that the current population is 100 purplegill and 42 redfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( P , R )\in\left\{ ( 0 , 0 ), ( 40 , 100 ), ( 80 , 0 ), ( 0 , 120 )\right\}\]

In the long term, both species will co-exist.

Example 10

S4m: Interacting populations (ver. 10)

Two species, greenfish and bluegill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d G }{dt}= -\frac{1}{100} \, B G - \frac{3}{250} \, G^{2} + 6 \, G\]

\[\frac{d B }{dt}= -\frac{1}{50} \, B^{2} - \frac{1}{125} \, B G + 8 \, B\]

Draw an appropriate phase plane. Then, assuming that the current population is 599 greenfish and 100 bluegill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( G , B )\in\left\{ ( 0 , 0 ), ( 250 , 300 ), ( 500 , 0 ), ( 0 , 400 )\right\}\]

In the long term, both species will co-exist.

Example 11

S4m: Interacting populations (ver. 11)

Two species, bluegill and redfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d B }{dt}= -\frac{1}{100} \, B^{2} - \frac{1}{250} \, B R + \frac{4}{5} \, B\]

\[\frac{d R }{dt}= -\frac{3}{500} \, B R - \frac{3}{250} \, R^{2} + \frac{36}{25} \, R\]

Draw an appropriate phase plane. Then, assuming that the current population is 100 bluegill and 39 redfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( B , R )\in\left\{ ( 0 , 0 ), ( 40 , 100 ), ( 80 , 0 ), ( 0 , 120 )\right\}\]

In the long term, both species will co-exist.

Example 12

S4m: Interacting populations (ver. 12)

Two species, greenfish and yellowgill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d G }{dt}= -\frac{3}{500} \, G^{2} - \frac{9}{1000} \, G Y + \frac{27}{25} \, G\]

\[\frac{d Y }{dt}= -\frac{1}{100} \, G Y - \frac{1}{200} \, Y^{2} + Y\]

Draw an appropriate phase plane. Then, assuming that the current population is 119 greenfish and 22 yellowgill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( G , Y )\in\left\{ ( 0 , 0 ), ( 60 , 80 ), ( 180 , 0 ), ( 0 , 200 )\right\}\]

In the long term, only greenfish will survive.

Example 13

S4m: Interacting populations (ver. 13)

Two species, bluegill and yellowgill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d B }{dt}= -\frac{1}{100} \, B^{2} - \frac{1}{250} \, B Y + \frac{2}{5} \, B\]

\[\frac{d Y }{dt}= -\frac{3}{500} \, B Y - \frac{3}{250} \, Y^{2} + \frac{18}{25} \, Y\]

Draw an appropriate phase plane. Then, assuming that the current population is 48 bluegill and 20 yellowgill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( B , Y )\in\left\{ ( 0 , 0 ), ( 20 , 50 ), ( 40 , 0 ), ( 0 , 60 )\right\}\]

In the long term, both species will co-exist.

Example 14

S4m: Interacting populations (ver. 14)

Two species, magentafish and redfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d M }{dt}= -\frac{3}{500} \, M^{2} - \frac{9}{1000} \, M R + \frac{27}{50} \, M\]

\[\frac{d R }{dt}= -\frac{1}{100} \, M R - \frac{1}{200} \, R^{2} + \frac{1}{2} \, R\]

Draw an appropriate phase plane. Then, assuming that the current population is 59 magentafish and 11 redfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( M , R )\in\left\{ ( 0 , 0 ), ( 30 , 40 ), ( 90 , 0 ), ( 0 , 100 )\right\}\]

In the long term, only magentafish will survive.

Example 15

S4m: Interacting populations (ver. 15)

Two species, redfish and yellowgill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d R }{dt}= -\frac{1}{100} \, R^{2} - \frac{1}{200} \, R Y + \frac{5}{2} \, R\]

\[\frac{d Y }{dt}= -\frac{3}{500} \, R Y - \frac{9}{1000} \, Y^{2} + \frac{27}{10} \, Y\]

Draw an appropriate phase plane. Then, assuming that the current population is 301 redfish and 49 yellowgill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( R , Y )\in\left\{ ( 0 , 0 ), ( 150 , 200 ), ( 250 , 0 ), ( 0 , 300 )\right\}\]

In the long term, both species will co-exist.

Example 16

S4m: Interacting populations (ver. 16)

Two species, bluegill and yellowgill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d B }{dt}= -\frac{3}{500} \, B^{2} - \frac{3}{250} \, B Y + \frac{36}{25} \, B\]

\[\frac{d Y }{dt}= -\frac{1}{100} \, B Y - \frac{1}{250} \, Y^{2} + \frac{4}{5} \, Y\]

Draw an appropriate phase plane. Then, assuming that the current population is 98 bluegill and 38 yellowgill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( B , Y )\in\left\{ ( 0 , 0 ), ( 40 , 100 ), ( 240 , 0 ), ( 0 , 200 )\right\}\]

In the long term, only bluegill will survive.

Example 17

S4m: Interacting populations (ver. 17)

Two species, greenfish and magentafish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d G }{dt}= -\frac{3}{500} \, G^{2} - \frac{9}{1000} \, G M + \frac{27}{25} \, G\]

\[\frac{d M }{dt}= -\frac{1}{100} \, G M - \frac{1}{200} \, M^{2} + M\]

Draw an appropriate phase plane. Then, assuming that the current population is 119 greenfish and 22 magentafish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( G , M )\in\left\{ ( 0 , 0 ), ( 60 , 80 ), ( 180 , 0 ), ( 0 , 200 )\right\}\]

In the long term, only greenfish will survive.

Example 18

S4m: Interacting populations (ver. 18)

Two species, purplegill and redfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d P }{dt}= -\frac{3}{250} \, P^{2} - \frac{1}{100} \, P R + 6 \, P\]

\[\frac{d R }{dt}= -\frac{1}{125} \, P R - \frac{1}{50} \, R^{2} + 8 \, R\]

Draw an appropriate phase plane. Then, assuming that the current population is 601 purplegill and 98 redfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( P , R )\in\left\{ ( 0 , 0 ), ( 250 , 300 ), ( 500 , 0 ), ( 0 , 400 )\right\}\]

In the long term, both species will co-exist.

Example 19

S4m: Interacting populations (ver. 19)

Two species, magentafish and purplegill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d M }{dt}= -\frac{1}{100} \, M^{2} - \frac{1}{250} \, M P + \frac{4}{5} \, M\]

\[\frac{d P }{dt}= -\frac{3}{500} \, M P - \frac{3}{250} \, P^{2} + \frac{36}{25} \, P\]

Draw an appropriate phase plane. Then, assuming that the current population is 101 magentafish and 39 purplegill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( M , P )\in\left\{ ( 0 , 0 ), ( 40 , 100 ), ( 80 , 0 ), ( 0 , 120 )\right\}\]

In the long term, both species will co-exist.

Example 20

S4m: Interacting populations (ver. 20)

Two species, purplegill and yellowgill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d P }{dt}= -\frac{3}{250} \, P^{2} - \frac{1}{100} \, P Y + \frac{12}{5} \, P\]

\[\frac{d Y }{dt}= -\frac{1}{125} \, P Y - \frac{1}{50} \, Y^{2} + \frac{16}{5} \, Y\]

Draw an appropriate phase plane. Then, assuming that the current population is 239 purplegill and 42 yellowgill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( P , Y )\in\left\{ ( 0 , 0 ), ( 100 , 120 ), ( 200 , 0 ), ( 0 , 160 )\right\}\]

In the long term, both species will co-exist.

Example 21

S4m: Interacting populations (ver. 21)

Two species, redfish and purplegill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d R }{dt}= -\frac{1}{100} \, P R - \frac{3}{250} \, R^{2} + \frac{6}{5} \, R\]

\[\frac{d P }{dt}= -\frac{1}{50} \, P^{2} - \frac{1}{125} \, P R + \frac{8}{5} \, P\]

Draw an appropriate phase plane. Then, assuming that the current population is 121 redfish and 20 purplegill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( R , P )\in\left\{ ( 0 , 0 ), ( 50 , 60 ), ( 100 , 0 ), ( 0 , 80 )\right\}\]

In the long term, both species will co-exist.

Example 22

S4m: Interacting populations (ver. 22)

Two species, bluegill and magentafish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d B }{dt}= -\frac{1}{125} \, B^{2} - \frac{1}{50} \, B M + \frac{8}{5} \, B\]

\[\frac{d M }{dt}= -\frac{3}{250} \, B M - \frac{1}{100} \, M^{2} + \frac{6}{5} \, M\]

Draw an appropriate phase plane. Then, assuming that the current population is 121 bluegill and 21 magentafish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( B , M )\in\left\{ ( 0 , 0 ), ( 50 , 60 ), ( 200 , 0 ), ( 0 , 120 )\right\}\]

In the long term, only bluegill will survive.

Example 23

S4m: Interacting populations (ver. 23)

Two species, bluegill and greenfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d B }{dt}= -\frac{3}{250} \, B^{2} - \frac{1}{100} \, B G + \frac{6}{5} \, B\]

\[\frac{d G }{dt}= -\frac{1}{125} \, B G - \frac{1}{50} \, G^{2} + \frac{8}{5} \, G\]

Draw an appropriate phase plane. Then, assuming that the current population is 120 bluegill and 18 greenfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( B , G )\in\left\{ ( 0 , 0 ), ( 50 , 60 ), ( 100 , 0 ), ( 0 , 80 )\right\}\]

In the long term, both species will co-exist.

Example 24

S4m: Interacting populations (ver. 24)

Two species, redfish and purplegill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d R }{dt}= -\frac{1}{100} \, P R - \frac{3}{250} \, R^{2} + \frac{12}{5} \, R\]

\[\frac{d P }{dt}= -\frac{1}{50} \, P^{2} - \frac{1}{125} \, P R + \frac{16}{5} \, P\]

Draw an appropriate phase plane. Then, assuming that the current population is 239 redfish and 40 purplegill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( R , P )\in\left\{ ( 0 , 0 ), ( 100 , 120 ), ( 200 , 0 ), ( 0 , 160 )\right\}\]

In the long term, both species will co-exist.

Example 25

S4m: Interacting populations (ver. 25)

Two species, magentafish and yellowgill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d M }{dt}= -\frac{3}{500} \, M^{2} - \frac{9}{1000} \, M Y + \frac{27}{25} \, M\]

\[\frac{d Y }{dt}= -\frac{1}{100} \, M Y - \frac{1}{200} \, Y^{2} + Y\]

Draw an appropriate phase plane. Then, assuming that the current population is 119 magentafish and 19 yellowgill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( M , Y )\in\left\{ ( 0 , 0 ), ( 60 , 80 ), ( 180 , 0 ), ( 0 , 200 )\right\}\]

In the long term, only magentafish will survive.

Example 26

S4m: Interacting populations (ver. 26)

Two species, bluegill and yellowgill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d B }{dt}= -\frac{3}{500} \, B^{2} - \frac{9}{1000} \, B Y + \frac{27}{10} \, B\]

\[\frac{d Y }{dt}= -\frac{1}{100} \, B Y - \frac{1}{200} \, Y^{2} + \frac{5}{2} \, Y\]

Draw an appropriate phase plane. Then, assuming that the current population is 302 bluegill and 52 yellowgill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( B , Y )\in\left\{ ( 0 , 0 ), ( 150 , 200 ), ( 450 , 0 ), ( 0 , 500 )\right\}\]

In the long term, only bluegill will survive.

Example 27

S4m: Interacting populations (ver. 27)

Two species, magentafish and redfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d M }{dt}= -\frac{3}{500} \, M^{2} - \frac{3}{250} \, M R + \frac{18}{25} \, M\]

\[\frac{d R }{dt}= -\frac{1}{100} \, M R - \frac{1}{250} \, R^{2} + \frac{2}{5} \, R\]

Draw an appropriate phase plane. Then, assuming that the current population is 48 magentafish and 19 redfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( M , R )\in\left\{ ( 0 , 0 ), ( 20 , 50 ), ( 120 , 0 ), ( 0 , 100 )\right\}\]

In the long term, only magentafish will survive.

Example 28

S4m: Interacting populations (ver. 28)

Two species, yellowgill and redfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d Y }{dt}= -\frac{3}{250} \, R Y - \frac{3}{500} \, Y^{2} + \frac{36}{25} \, Y\]

\[\frac{d R }{dt}= -\frac{1}{250} \, R^{2} - \frac{1}{100} \, R Y + \frac{4}{5} \, R\]

Draw an appropriate phase plane. Then, assuming that the current population is 101 yellowgill and 40 redfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( Y , R )\in\left\{ ( 0 , 0 ), ( 40 , 100 ), ( 240 , 0 ), ( 0 , 200 )\right\}\]

In the long term, only yellowgill will survive.

Example 29

S4m: Interacting populations (ver. 29)

Two species, redfish and bluegill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d R }{dt}= -\frac{9}{1000} \, B R - \frac{3}{500} \, R^{2} + \frac{27}{10} \, R\]

\[\frac{d B }{dt}= -\frac{1}{200} \, B^{2} - \frac{1}{100} \, B R + \frac{5}{2} \, B\]

Draw an appropriate phase plane. Then, assuming that the current population is 299 redfish and 51 bluegill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( R , B )\in\left\{ ( 0 , 0 ), ( 150 , 200 ), ( 450 , 0 ), ( 0 , 500 )\right\}\]

In the long term, only redfish will survive.

Example 30

S4m: Interacting populations (ver. 30)

Two species, bluegill and yellowgill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d B }{dt}= -\frac{3}{250} \, B^{2} - \frac{1}{100} \, B Y + \frac{12}{5} \, B\]

\[\frac{d Y }{dt}= -\frac{1}{125} \, B Y - \frac{1}{50} \, Y^{2} + \frac{16}{5} \, Y\]

Draw an appropriate phase plane. Then, assuming that the current population is 241 bluegill and 38 yellowgill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( B , Y )\in\left\{ ( 0 , 0 ), ( 100 , 120 ), ( 200 , 0 ), ( 0 , 160 )\right\}\]

In the long term, both species will co-exist.

Example 31

S4m: Interacting populations (ver. 31)

Two species, purplegill and bluegill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d P }{dt}= -\frac{1}{100} \, B P - \frac{3}{250} \, P^{2} + \frac{6}{5} \, P\]

\[\frac{d B }{dt}= -\frac{1}{50} \, B^{2} - \frac{1}{125} \, B P + \frac{8}{5} \, B\]

Draw an appropriate phase plane. Then, assuming that the current population is 120 purplegill and 19 bluegill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( P , B )\in\left\{ ( 0 , 0 ), ( 50 , 60 ), ( 100 , 0 ), ( 0 , 80 )\right\}\]

In the long term, both species will co-exist.

Example 32

S4m: Interacting populations (ver. 32)

Two species, yellowgill and redfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d Y }{dt}= -\frac{1}{50} \, R Y - \frac{1}{125} \, Y^{2} + \frac{16}{5} \, Y\]

\[\frac{d R }{dt}= -\frac{1}{100} \, R^{2} - \frac{3}{250} \, R Y + \frac{12}{5} \, R\]

Draw an appropriate phase plane. Then, assuming that the current population is 241 yellowgill and 38 redfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( Y , R )\in\left\{ ( 0 , 0 ), ( 100 , 120 ), ( 400 , 0 ), ( 0 , 240 )\right\}\]

In the long term, only yellowgill will survive.

Example 33

S4m: Interacting populations (ver. 33)

Two species, yellowgill and bluegill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d Y }{dt}= -\frac{1}{50} \, B Y - \frac{1}{125} \, Y^{2} + \frac{8}{5} \, Y\]

\[\frac{d B }{dt}= -\frac{1}{100} \, B^{2} - \frac{3}{250} \, B Y + \frac{6}{5} \, B\]

Draw an appropriate phase plane. Then, assuming that the current population is 121 yellowgill and 22 bluegill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( Y , B )\in\left\{ ( 0 , 0 ), ( 50 , 60 ), ( 200 , 0 ), ( 0 , 120 )\right\}\]

In the long term, only yellowgill will survive.

Example 34

S4m: Interacting populations (ver. 34)

Two species, redfish and greenfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d R }{dt}= -\frac{1}{50} \, G R - \frac{1}{125} \, R^{2} + \frac{16}{5} \, R\]

\[\frac{d G }{dt}= -\frac{1}{100} \, G^{2} - \frac{3}{250} \, G R + \frac{12}{5} \, G\]

Draw an appropriate phase plane. Then, assuming that the current population is 240 redfish and 42 greenfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( R , G )\in\left\{ ( 0 , 0 ), ( 100 , 120 ), ( 400 , 0 ), ( 0 , 240 )\right\}\]

In the long term, only redfish will survive.

Example 35

S4m: Interacting populations (ver. 35)

Two species, purplegill and magentafish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d P }{dt}= -\frac{1}{50} \, M P - \frac{1}{125} \, P^{2} + \frac{8}{5} \, P\]

\[\frac{d M }{dt}= -\frac{1}{100} \, M^{2} - \frac{3}{250} \, M P + \frac{6}{5} \, M\]

Draw an appropriate phase plane. Then, assuming that the current population is 121 purplegill and 20 magentafish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( P , M )\in\left\{ ( 0 , 0 ), ( 50 , 60 ), ( 200 , 0 ), ( 0 , 120 )\right\}\]

In the long term, only purplegill will survive.

Example 36

S4m: Interacting populations (ver. 36)

Two species, purplegill and redfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d P }{dt}= -\frac{1}{100} \, P^{2} - \frac{1}{200} \, P R + \frac{5}{2} \, P\]

\[\frac{d R }{dt}= -\frac{3}{500} \, P R - \frac{9}{1000} \, R^{2} + \frac{27}{10} \, R\]

Draw an appropriate phase plane. Then, assuming that the current population is 300 purplegill and 52 redfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( P , R )\in\left\{ ( 0 , 0 ), ( 150 , 200 ), ( 250 , 0 ), ( 0 , 300 )\right\}\]

In the long term, both species will co-exist.

Example 37

S4m: Interacting populations (ver. 37)

Two species, magentafish and redfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d M }{dt}= -\frac{3}{500} \, M^{2} - \frac{9}{1000} \, M R + \frac{27}{10} \, M\]

\[\frac{d R }{dt}= -\frac{1}{100} \, M R - \frac{1}{200} \, R^{2} + \frac{5}{2} \, R\]

Draw an appropriate phase plane. Then, assuming that the current population is 298 magentafish and 51 redfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( M , R )\in\left\{ ( 0 , 0 ), ( 150 , 200 ), ( 450 , 0 ), ( 0 , 500 )\right\}\]

In the long term, only magentafish will survive.

Example 38

S4m: Interacting populations (ver. 38)

Two species, bluegill and yellowgill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d B }{dt}= -\frac{3}{500} \, B^{2} - \frac{3}{250} \, B Y + \frac{18}{25} \, B\]

\[\frac{d Y }{dt}= -\frac{1}{100} \, B Y - \frac{1}{250} \, Y^{2} + \frac{2}{5} \, Y\]

Draw an appropriate phase plane. Then, assuming that the current population is 48 bluegill and 18 yellowgill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( B , Y )\in\left\{ ( 0 , 0 ), ( 20 , 50 ), ( 120 , 0 ), ( 0 , 100 )\right\}\]

In the long term, only bluegill will survive.

Example 39

S4m: Interacting populations (ver. 39)

Two species, yellowgill and greenfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d Y }{dt}= -\frac{1}{250} \, G Y - \frac{1}{100} \, Y^{2} + 2 \, Y\]

\[\frac{d G }{dt}= -\frac{3}{250} \, G^{2} - \frac{3}{500} \, G Y + \frac{18}{5} \, G\]

Draw an appropriate phase plane. Then, assuming that the current population is 252 yellowgill and 99 greenfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( Y , G )\in\left\{ ( 0 , 0 ), ( 100 , 250 ), ( 200 , 0 ), ( 0 , 300 )\right\}\]

In the long term, both species will co-exist.

Example 40

S4m: Interacting populations (ver. 40)

Two species, bluegill and purplegill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d B }{dt}= -\frac{1}{100} \, B^{2} - \frac{1}{200} \, B P + B\]

\[\frac{d P }{dt}= -\frac{3}{500} \, B P - \frac{9}{1000} \, P^{2} + \frac{27}{25} \, P\]

Draw an appropriate phase plane. Then, assuming that the current population is 119 bluegill and 21 purplegill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( B , P )\in\left\{ ( 0 , 0 ), ( 60 , 80 ), ( 100 , 0 ), ( 0 , 120 )\right\}\]

In the long term, both species will co-exist.

Example 41

S4m: Interacting populations (ver. 41)

Two species, magentafish and redfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d M }{dt}= -\frac{3}{500} \, M^{2} - \frac{9}{1000} \, M R + \frac{27}{50} \, M\]

\[\frac{d R }{dt}= -\frac{1}{100} \, M R - \frac{1}{200} \, R^{2} + \frac{1}{2} \, R\]

Draw an appropriate phase plane. Then, assuming that the current population is 61 magentafish and 12 redfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( M , R )\in\left\{ ( 0 , 0 ), ( 30 , 40 ), ( 90 , 0 ), ( 0 , 100 )\right\}\]

In the long term, only magentafish will survive.

Example 42

S4m: Interacting populations (ver. 42)

Two species, yellowgill and greenfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d Y }{dt}= -\frac{1}{200} \, G Y - \frac{1}{100} \, Y^{2} + Y\]

\[\frac{d G }{dt}= -\frac{9}{1000} \, G^{2} - \frac{3}{500} \, G Y + \frac{27}{25} \, G\]

Draw an appropriate phase plane. Then, assuming that the current population is 118 yellowgill and 20 greenfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( Y , G )\in\left\{ ( 0 , 0 ), ( 60 , 80 ), ( 100 , 0 ), ( 0 , 120 )\right\}\]

In the long term, both species will co-exist.

Example 43

S4m: Interacting populations (ver. 43)

Two species, greenfish and yellowgill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d G }{dt}= -\frac{3}{500} \, G^{2} - \frac{9}{1000} \, G Y + \frac{27}{50} \, G\]

\[\frac{d Y }{dt}= -\frac{1}{100} \, G Y - \frac{1}{200} \, Y^{2} + \frac{1}{2} \, Y\]

Draw an appropriate phase plane. Then, assuming that the current population is 62 greenfish and 12 yellowgill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( G , Y )\in\left\{ ( 0 , 0 ), ( 30 , 40 ), ( 90 , 0 ), ( 0 , 100 )\right\}\]

In the long term, only greenfish will survive.

Example 44

S4m: Interacting populations (ver. 44)

Two species, bluegill and purplegill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d B }{dt}= -\frac{3}{500} \, B^{2} - \frac{9}{1000} \, B P + \frac{27}{50} \, B\]

\[\frac{d P }{dt}= -\frac{1}{100} \, B P - \frac{1}{200} \, P^{2} + \frac{1}{2} \, P\]

Draw an appropriate phase plane. Then, assuming that the current population is 58 bluegill and 8 purplegill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( B , P )\in\left\{ ( 0 , 0 ), ( 30 , 40 ), ( 90 , 0 ), ( 0 , 100 )\right\}\]

In the long term, only bluegill will survive.

Example 45

S4m: Interacting populations (ver. 45)

Two species, bluegill and yellowgill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d B }{dt}= -\frac{1}{100} \, B^{2} - \frac{1}{250} \, B Y + 2 \, B\]

\[\frac{d Y }{dt}= -\frac{3}{500} \, B Y - \frac{3}{250} \, Y^{2} + \frac{18}{5} \, Y\]

Draw an appropriate phase plane. Then, assuming that the current population is 249 bluegill and 101 yellowgill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( B , Y )\in\left\{ ( 0 , 0 ), ( 100 , 250 ), ( 200 , 0 ), ( 0 , 300 )\right\}\]

In the long term, both species will co-exist.

Example 46

S4m: Interacting populations (ver. 46)

Two species, greenfish and magentafish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d G }{dt}= -\frac{1}{100} \, G^{2} - \frac{1}{250} \, G M + \frac{4}{5} \, G\]

\[\frac{d M }{dt}= -\frac{3}{500} \, G M - \frac{3}{250} \, M^{2} + \frac{36}{25} \, M\]

Draw an appropriate phase plane. Then, assuming that the current population is 102 greenfish and 40 magentafish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( G , M )\in\left\{ ( 0 , 0 ), ( 40 , 100 ), ( 80 , 0 ), ( 0 , 120 )\right\}\]

In the long term, both species will co-exist.

Example 47

S4m: Interacting populations (ver. 47)

Two species, bluegill and redfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d B }{dt}= -\frac{1}{100} \, B^{2} - \frac{1}{250} \, B R + 2 \, B\]

\[\frac{d R }{dt}= -\frac{3}{500} \, B R - \frac{3}{250} \, R^{2} + \frac{18}{5} \, R\]

Draw an appropriate phase plane. Then, assuming that the current population is 248 bluegill and 102 redfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( B , R )\in\left\{ ( 0 , 0 ), ( 100 , 250 ), ( 200 , 0 ), ( 0 , 300 )\right\}\]

In the long term, both species will co-exist.

Example 48

S4m: Interacting populations (ver. 48)

Two species, purplegill and magentafish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d P }{dt}= -\frac{1}{50} \, M P - \frac{1}{125} \, P^{2} + \frac{8}{5} \, P\]

\[\frac{d M }{dt}= -\frac{1}{100} \, M^{2} - \frac{3}{250} \, M P + \frac{6}{5} \, M\]

Draw an appropriate phase plane. Then, assuming that the current population is 118 purplegill and 19 magentafish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( P , M )\in\left\{ ( 0 , 0 ), ( 50 , 60 ), ( 200 , 0 ), ( 0 , 120 )\right\}\]

In the long term, only purplegill will survive.

Example 49

S4m: Interacting populations (ver. 49)

Two species, redfish and greenfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d R }{dt}= -\frac{3}{250} \, G R - \frac{3}{500} \, R^{2} + \frac{36}{25} \, R\]

\[\frac{d G }{dt}= -\frac{1}{250} \, G^{2} - \frac{1}{100} \, G R + \frac{4}{5} \, G\]

Draw an appropriate phase plane. Then, assuming that the current population is 101 redfish and 39 greenfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( R , G )\in\left\{ ( 0 , 0 ), ( 40 , 100 ), ( 240 , 0 ), ( 0 , 200 )\right\}\]

In the long term, only redfish will survive.

Example 50

S4m: Interacting populations (ver. 50)

Two species, magentafish and purplegill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d M }{dt}= -\frac{1}{100} \, M^{2} - \frac{1}{200} \, M P + M\]

\[\frac{d P }{dt}= -\frac{3}{500} \, M P - \frac{9}{1000} \, P^{2} + \frac{27}{25} \, P\]

Draw an appropriate phase plane. Then, assuming that the current population is 119 magentafish and 21 purplegill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( M , P )\in\left\{ ( 0 , 0 ), ( 60 , 80 ), ( 100 , 0 ), ( 0 , 120 )\right\}\]

In the long term, both species will co-exist.

Example 51

S4m: Interacting populations (ver. 51)

Two species, purplegill and yellowgill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d P }{dt}= -\frac{3}{250} \, P^{2} - \frac{1}{100} \, P Y + \frac{6}{5} \, P\]

\[\frac{d Y }{dt}= -\frac{1}{125} \, P Y - \frac{1}{50} \, Y^{2} + \frac{8}{5} \, Y\]

Draw an appropriate phase plane. Then, assuming that the current population is 120 purplegill and 21 yellowgill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( P , Y )\in\left\{ ( 0 , 0 ), ( 50 , 60 ), ( 100 , 0 ), ( 0 , 80 )\right\}\]

In the long term, both species will co-exist.

Example 52

S4m: Interacting populations (ver. 52)

Two species, purplegill and magentafish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d P }{dt}= -\frac{9}{1000} \, M P - \frac{3}{500} \, P^{2} + \frac{27}{25} \, P\]

\[\frac{d M }{dt}= -\frac{1}{200} \, M^{2} - \frac{1}{100} \, M P + M\]

Draw an appropriate phase plane. Then, assuming that the current population is 120 purplegill and 18 magentafish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( P , M )\in\left\{ ( 0 , 0 ), ( 60 , 80 ), ( 180 , 0 ), ( 0 , 200 )\right\}\]

In the long term, only purplegill will survive.

Example 53

S4m: Interacting populations (ver. 53)

Two species, magentafish and bluegill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d M }{dt}= -\frac{1}{200} \, B M - \frac{1}{100} \, M^{2} + \frac{5}{2} \, M\]

\[\frac{d B }{dt}= -\frac{9}{1000} \, B^{2} - \frac{3}{500} \, B M + \frac{27}{10} \, B\]

Draw an appropriate phase plane. Then, assuming that the current population is 301 magentafish and 50 bluegill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( M , B )\in\left\{ ( 0 , 0 ), ( 150 , 200 ), ( 250 , 0 ), ( 0 , 300 )\right\}\]

In the long term, both species will co-exist.

Example 54

S4m: Interacting populations (ver. 54)

Two species, purplegill and yellowgill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d P }{dt}= -\frac{3}{250} \, P^{2} - \frac{1}{100} \, P Y + 6 \, P\]

\[\frac{d Y }{dt}= -\frac{1}{125} \, P Y - \frac{1}{50} \, Y^{2} + 8 \, Y\]

Draw an appropriate phase plane. Then, assuming that the current population is 600 purplegill and 100 yellowgill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( P , Y )\in\left\{ ( 0 , 0 ), ( 250 , 300 ), ( 500 , 0 ), ( 0 , 400 )\right\}\]

In the long term, both species will co-exist.

Example 55

S4m: Interacting populations (ver. 55)

Two species, bluegill and redfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d B }{dt}= -\frac{1}{100} \, B^{2} - \frac{1}{200} \, B R + \frac{5}{2} \, B\]

\[\frac{d R }{dt}= -\frac{3}{500} \, B R - \frac{9}{1000} \, R^{2} + \frac{27}{10} \, R\]

Draw an appropriate phase plane. Then, assuming that the current population is 302 bluegill and 49 redfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( B , R )\in\left\{ ( 0 , 0 ), ( 150 , 200 ), ( 250 , 0 ), ( 0 , 300 )\right\}\]

In the long term, both species will co-exist.

Example 56

S4m: Interacting populations (ver. 56)

Two species, yellowgill and purplegill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d Y }{dt}= -\frac{3}{250} \, P Y - \frac{3}{500} \, Y^{2} + \frac{36}{25} \, Y\]

\[\frac{d P }{dt}= -\frac{1}{250} \, P^{2} - \frac{1}{100} \, P Y + \frac{4}{5} \, P\]

Draw an appropriate phase plane. Then, assuming that the current population is 98 yellowgill and 39 purplegill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( Y , P )\in\left\{ ( 0 , 0 ), ( 40 , 100 ), ( 240 , 0 ), ( 0 , 200 )\right\}\]

In the long term, only yellowgill will survive.

Example 57

S4m: Interacting populations (ver. 57)

Two species, yellowgill and purplegill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d Y }{dt}= -\frac{1}{50} \, P Y - \frac{1}{125} \, Y^{2} + 8 \, Y\]

\[\frac{d P }{dt}= -\frac{1}{100} \, P^{2} - \frac{3}{250} \, P Y + 6 \, P\]

Draw an appropriate phase plane. Then, assuming that the current population is 601 yellowgill and 101 purplegill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( Y , P )\in\left\{ ( 0 , 0 ), ( 250 , 300 ), ( 1000 , 0 ), ( 0 , 600 )\right\}\]

In the long term, only yellowgill will survive.

Example 58

S4m: Interacting populations (ver. 58)

Two species, redfish and bluegill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d R }{dt}= -\frac{1}{250} \, B R - \frac{1}{100} \, R^{2} + \frac{4}{5} \, R\]

\[\frac{d B }{dt}= -\frac{3}{250} \, B^{2} - \frac{3}{500} \, B R + \frac{36}{25} \, B\]

Draw an appropriate phase plane. Then, assuming that the current population is 98 redfish and 38 bluegill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( R , B )\in\left\{ ( 0 , 0 ), ( 40 , 100 ), ( 80 , 0 ), ( 0 , 120 )\right\}\]

In the long term, both species will co-exist.

Example 59

S4m: Interacting populations (ver. 59)

Two species, yellowgill and greenfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d Y }{dt}= -\frac{1}{100} \, G Y - \frac{3}{250} \, Y^{2} + \frac{6}{5} \, Y\]

\[\frac{d G }{dt}= -\frac{1}{50} \, G^{2} - \frac{1}{125} \, G Y + \frac{8}{5} \, G\]

Draw an appropriate phase plane. Then, assuming that the current population is 118 yellowgill and 19 greenfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( Y , G )\in\left\{ ( 0 , 0 ), ( 50 , 60 ), ( 100 , 0 ), ( 0 , 80 )\right\}\]

In the long term, both species will co-exist.

Example 60

S4m: Interacting populations (ver. 60)

Two species, bluegill and redfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d B }{dt}= -\frac{3}{500} \, B^{2} - \frac{9}{1000} \, B R + \frac{27}{50} \, B\]

\[\frac{d R }{dt}= -\frac{1}{100} \, B R - \frac{1}{200} \, R^{2} + \frac{1}{2} \, R\]

Draw an appropriate phase plane. Then, assuming that the current population is 60 bluegill and 9 redfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( B , R )\in\left\{ ( 0 , 0 ), ( 30 , 40 ), ( 90 , 0 ), ( 0 , 100 )\right\}\]

In the long term, only bluegill will survive.

Example 61

S4m: Interacting populations (ver. 61)

Two species, redfish and yellowgill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d R }{dt}= -\frac{1}{125} \, R^{2} - \frac{1}{50} \, R Y + \frac{8}{5} \, R\]

\[\frac{d Y }{dt}= -\frac{3}{250} \, R Y - \frac{1}{100} \, Y^{2} + \frac{6}{5} \, Y\]

Draw an appropriate phase plane. Then, assuming that the current population is 119 redfish and 22 yellowgill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( R , Y )\in\left\{ ( 0 , 0 ), ( 50 , 60 ), ( 200 , 0 ), ( 0 , 120 )\right\}\]

In the long term, only redfish will survive.

Example 62

S4m: Interacting populations (ver. 62)

Two species, redfish and purplegill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d R }{dt}= -\frac{1}{250} \, P R - \frac{1}{100} \, R^{2} + \frac{2}{5} \, R\]

\[\frac{d P }{dt}= -\frac{3}{250} \, P^{2} - \frac{3}{500} \, P R + \frac{18}{25} \, P\]

Draw an appropriate phase plane. Then, assuming that the current population is 49 redfish and 22 purplegill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( R , P )\in\left\{ ( 0 , 0 ), ( 20 , 50 ), ( 40 , 0 ), ( 0 , 60 )\right\}\]

In the long term, both species will co-exist.

Example 63

S4m: Interacting populations (ver. 63)

Two species, redfish and purplegill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d R }{dt}= -\frac{1}{250} \, P R - \frac{1}{100} \, R^{2} + 2 \, R\]

\[\frac{d P }{dt}= -\frac{3}{250} \, P^{2} - \frac{3}{500} \, P R + \frac{18}{5} \, P\]

Draw an appropriate phase plane. Then, assuming that the current population is 248 redfish and 100 purplegill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( R , P )\in\left\{ ( 0 , 0 ), ( 100 , 250 ), ( 200 , 0 ), ( 0 , 300 )\right\}\]

In the long term, both species will co-exist.

Example 64

S4m: Interacting populations (ver. 64)

Two species, bluegill and yellowgill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d B }{dt}= -\frac{3}{500} \, B^{2} - \frac{9}{1000} \, B Y + \frac{27}{10} \, B\]

\[\frac{d Y }{dt}= -\frac{1}{100} \, B Y - \frac{1}{200} \, Y^{2} + \frac{5}{2} \, Y\]

Draw an appropriate phase plane. Then, assuming that the current population is 298 bluegill and 49 yellowgill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( B , Y )\in\left\{ ( 0 , 0 ), ( 150 , 200 ), ( 450 , 0 ), ( 0 , 500 )\right\}\]

In the long term, only bluegill will survive.

Example 65

S4m: Interacting populations (ver. 65)

Two species, purplegill and greenfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d P }{dt}= -\frac{1}{50} \, G P - \frac{1}{125} \, P^{2} + 8 \, P\]

\[\frac{d G }{dt}= -\frac{1}{100} \, G^{2} - \frac{3}{250} \, G P + 6 \, G\]

Draw an appropriate phase plane. Then, assuming that the current population is 601 purplegill and 98 greenfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( P , G )\in\left\{ ( 0 , 0 ), ( 250 , 300 ), ( 1000 , 0 ), ( 0 , 600 )\right\}\]

In the long term, only purplegill will survive.

Example 66

S4m: Interacting populations (ver. 66)

Two species, magentafish and bluegill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d M }{dt}= -\frac{1}{250} \, B M - \frac{1}{100} \, M^{2} + \frac{4}{5} \, M\]

\[\frac{d B }{dt}= -\frac{3}{250} \, B^{2} - \frac{3}{500} \, B M + \frac{36}{25} \, B\]

Draw an appropriate phase plane. Then, assuming that the current population is 100 magentafish and 41 bluegill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( M , B )\in\left\{ ( 0 , 0 ), ( 40 , 100 ), ( 80 , 0 ), ( 0 , 120 )\right\}\]

In the long term, both species will co-exist.

Example 67

S4m: Interacting populations (ver. 67)

Two species, magentafish and bluegill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d M }{dt}= -\frac{1}{100} \, B M - \frac{3}{250} \, M^{2} + 6 \, M\]

\[\frac{d B }{dt}= -\frac{1}{50} \, B^{2} - \frac{1}{125} \, B M + 8 \, B\]

Draw an appropriate phase plane. Then, assuming that the current population is 601 magentafish and 101 bluegill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( M , B )\in\left\{ ( 0 , 0 ), ( 250 , 300 ), ( 500 , 0 ), ( 0 , 400 )\right\}\]

In the long term, both species will co-exist.

Example 68

S4m: Interacting populations (ver. 68)

Two species, yellowgill and purplegill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d Y }{dt}= -\frac{3}{250} \, P Y - \frac{3}{500} \, Y^{2} + \frac{18}{25} \, Y\]

\[\frac{d P }{dt}= -\frac{1}{250} \, P^{2} - \frac{1}{100} \, P Y + \frac{2}{5} \, P\]

Draw an appropriate phase plane. Then, assuming that the current population is 51 yellowgill and 22 purplegill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( Y , P )\in\left\{ ( 0 , 0 ), ( 20 , 50 ), ( 120 , 0 ), ( 0 , 100 )\right\}\]

In the long term, only yellowgill will survive.

Example 69

S4m: Interacting populations (ver. 69)

Two species, greenfish and redfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d G }{dt}= -\frac{1}{100} \, G^{2} - \frac{1}{200} \, G R + G\]

\[\frac{d R }{dt}= -\frac{3}{500} \, G R - \frac{9}{1000} \, R^{2} + \frac{27}{25} \, R\]

Draw an appropriate phase plane. Then, assuming that the current population is 120 greenfish and 22 redfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( G , R )\in\left\{ ( 0 , 0 ), ( 60 , 80 ), ( 100 , 0 ), ( 0 , 120 )\right\}\]

In the long term, both species will co-exist.

Example 70

S4m: Interacting populations (ver. 70)

Two species, bluegill and magentafish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d B }{dt}= -\frac{3}{500} \, B^{2} - \frac{3}{250} \, B M + \frac{18}{5} \, B\]

\[\frac{d M }{dt}= -\frac{1}{100} \, B M - \frac{1}{250} \, M^{2} + 2 \, M\]

Draw an appropriate phase plane. Then, assuming that the current population is 248 bluegill and 101 magentafish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( B , M )\in\left\{ ( 0 , 0 ), ( 100 , 250 ), ( 600 , 0 ), ( 0 , 500 )\right\}\]

In the long term, only bluegill will survive.

Example 71

S4m: Interacting populations (ver. 71)

Two species, magentafish and bluegill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d M }{dt}= -\frac{1}{250} \, B M - \frac{1}{100} \, M^{2} + 2 \, M\]

\[\frac{d B }{dt}= -\frac{3}{250} \, B^{2} - \frac{3}{500} \, B M + \frac{18}{5} \, B\]

Draw an appropriate phase plane. Then, assuming that the current population is 251 magentafish and 101 bluegill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( M , B )\in\left\{ ( 0 , 0 ), ( 100 , 250 ), ( 200 , 0 ), ( 0 , 300 )\right\}\]

In the long term, both species will co-exist.

Example 72

S4m: Interacting populations (ver. 72)

Two species, yellowgill and redfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d Y }{dt}= -\frac{1}{200} \, R Y - \frac{1}{100} \, Y^{2} + \frac{5}{2} \, Y\]

\[\frac{d R }{dt}= -\frac{9}{1000} \, R^{2} - \frac{3}{500} \, R Y + \frac{27}{10} \, R\]

Draw an appropriate phase plane. Then, assuming that the current population is 299 yellowgill and 48 redfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( Y , R )\in\left\{ ( 0 , 0 ), ( 150 , 200 ), ( 250 , 0 ), ( 0 , 300 )\right\}\]

In the long term, both species will co-exist.

Example 73

S4m: Interacting populations (ver. 73)

Two species, greenfish and purplegill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d G }{dt}= -\frac{3}{500} \, G^{2} - \frac{9}{1000} \, G P + \frac{27}{10} \, G\]

\[\frac{d P }{dt}= -\frac{1}{100} \, G P - \frac{1}{200} \, P^{2} + \frac{5}{2} \, P\]

Draw an appropriate phase plane. Then, assuming that the current population is 300 greenfish and 51 purplegill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( G , P )\in\left\{ ( 0 , 0 ), ( 150 , 200 ), ( 450 , 0 ), ( 0 , 500 )\right\}\]

In the long term, only greenfish will survive.

Example 74

S4m: Interacting populations (ver. 74)

Two species, purplegill and bluegill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d P }{dt}= -\frac{1}{100} \, B P - \frac{3}{250} \, P^{2} + \frac{6}{5} \, P\]

\[\frac{d B }{dt}= -\frac{1}{50} \, B^{2} - \frac{1}{125} \, B P + \frac{8}{5} \, B\]

Draw an appropriate phase plane. Then, assuming that the current population is 122 purplegill and 20 bluegill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( P , B )\in\left\{ ( 0 , 0 ), ( 50 , 60 ), ( 100 , 0 ), ( 0 , 80 )\right\}\]

In the long term, both species will co-exist.

Example 75

S4m: Interacting populations (ver. 75)

Two species, yellowgill and redfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d Y }{dt}= -\frac{1}{50} \, R Y - \frac{1}{125} \, Y^{2} + \frac{16}{5} \, Y\]

\[\frac{d R }{dt}= -\frac{1}{100} \, R^{2} - \frac{3}{250} \, R Y + \frac{12}{5} \, R\]

Draw an appropriate phase plane. Then, assuming that the current population is 240 yellowgill and 42 redfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( Y , R )\in\left\{ ( 0 , 0 ), ( 100 , 120 ), ( 400 , 0 ), ( 0 , 240 )\right\}\]

In the long term, only yellowgill will survive.

Example 76

S4m: Interacting populations (ver. 76)

Two species, yellowgill and purplegill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d Y }{dt}= -\frac{9}{1000} \, P Y - \frac{3}{500} \, Y^{2} + \frac{27}{10} \, Y\]

\[\frac{d P }{dt}= -\frac{1}{200} \, P^{2} - \frac{1}{100} \, P Y + \frac{5}{2} \, P\]

Draw an appropriate phase plane. Then, assuming that the current population is 302 yellowgill and 52 purplegill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( Y , P )\in\left\{ ( 0 , 0 ), ( 150 , 200 ), ( 450 , 0 ), ( 0 , 500 )\right\}\]

In the long term, only yellowgill will survive.

Example 77

S4m: Interacting populations (ver. 77)

Two species, magentafish and yellowgill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d M }{dt}= -\frac{1}{100} \, M^{2} - \frac{1}{200} \, M Y + \frac{5}{2} \, M\]

\[\frac{d Y }{dt}= -\frac{3}{500} \, M Y - \frac{9}{1000} \, Y^{2} + \frac{27}{10} \, Y\]

Draw an appropriate phase plane. Then, assuming that the current population is 299 magentafish and 48 yellowgill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( M , Y )\in\left\{ ( 0 , 0 ), ( 150 , 200 ), ( 250 , 0 ), ( 0 , 300 )\right\}\]

In the long term, both species will co-exist.

Example 78

S4m: Interacting populations (ver. 78)

Two species, yellowgill and bluegill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d Y }{dt}= -\frac{1}{250} \, B Y - \frac{1}{100} \, Y^{2} + \frac{2}{5} \, Y\]

\[\frac{d B }{dt}= -\frac{3}{250} \, B^{2} - \frac{3}{500} \, B Y + \frac{18}{25} \, B\]

Draw an appropriate phase plane. Then, assuming that the current population is 52 yellowgill and 22 bluegill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( Y , B )\in\left\{ ( 0 , 0 ), ( 20 , 50 ), ( 40 , 0 ), ( 0 , 60 )\right\}\]

In the long term, both species will co-exist.

Example 79

S4m: Interacting populations (ver. 79)

Two species, redfish and purplegill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d R }{dt}= -\frac{1}{250} \, P R - \frac{1}{100} \, R^{2} + \frac{2}{5} \, R\]

\[\frac{d P }{dt}= -\frac{3}{250} \, P^{2} - \frac{3}{500} \, P R + \frac{18}{25} \, P\]

Draw an appropriate phase plane. Then, assuming that the current population is 48 redfish and 20 purplegill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( R , P )\in\left\{ ( 0 , 0 ), ( 20 , 50 ), ( 40 , 0 ), ( 0 , 60 )\right\}\]

In the long term, both species will co-exist.

Example 80

S4m: Interacting populations (ver. 80)

Two species, bluegill and greenfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d B }{dt}= -\frac{3}{500} \, B^{2} - \frac{9}{1000} \, B G + \frac{27}{10} \, B\]

\[\frac{d G }{dt}= -\frac{1}{100} \, B G - \frac{1}{200} \, G^{2} + \frac{5}{2} \, G\]

Draw an appropriate phase plane. Then, assuming that the current population is 302 bluegill and 51 greenfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( B , G )\in\left\{ ( 0 , 0 ), ( 150 , 200 ), ( 450 , 0 ), ( 0 , 500 )\right\}\]

In the long term, only bluegill will survive.

Example 81

S4m: Interacting populations (ver. 81)

Two species, yellowgill and redfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d Y }{dt}= -\frac{1}{250} \, R Y - \frac{1}{100} \, Y^{2} + 2 \, Y\]

\[\frac{d R }{dt}= -\frac{3}{250} \, R^{2} - \frac{3}{500} \, R Y + \frac{18}{5} \, R\]

Draw an appropriate phase plane. Then, assuming that the current population is 249 yellowgill and 100 redfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( Y , R )\in\left\{ ( 0 , 0 ), ( 100 , 250 ), ( 200 , 0 ), ( 0 , 300 )\right\}\]

In the long term, both species will co-exist.

Example 82

S4m: Interacting populations (ver. 82)

Two species, redfish and greenfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d R }{dt}= -\frac{9}{1000} \, G R - \frac{3}{500} \, R^{2} + \frac{27}{10} \, R\]

\[\frac{d G }{dt}= -\frac{1}{200} \, G^{2} - \frac{1}{100} \, G R + \frac{5}{2} \, G\]

Draw an appropriate phase plane. Then, assuming that the current population is 298 redfish and 52 greenfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( R , G )\in\left\{ ( 0 , 0 ), ( 150 , 200 ), ( 450 , 0 ), ( 0 , 500 )\right\}\]

In the long term, only redfish will survive.

Example 83

S4m: Interacting populations (ver. 83)

Two species, greenfish and redfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d G }{dt}= -\frac{1}{100} \, G^{2} - \frac{1}{250} \, G R + \frac{2}{5} \, G\]

\[\frac{d R }{dt}= -\frac{3}{500} \, G R - \frac{3}{250} \, R^{2} + \frac{18}{25} \, R\]

Draw an appropriate phase plane. Then, assuming that the current population is 49 greenfish and 22 redfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( G , R )\in\left\{ ( 0 , 0 ), ( 20 , 50 ), ( 40 , 0 ), ( 0 , 60 )\right\}\]

In the long term, both species will co-exist.

Example 84

S4m: Interacting populations (ver. 84)

Two species, yellowgill and redfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d Y }{dt}= -\frac{3}{250} \, R Y - \frac{3}{500} \, Y^{2} + \frac{18}{5} \, Y\]

\[\frac{d R }{dt}= -\frac{1}{250} \, R^{2} - \frac{1}{100} \, R Y + 2 \, R\]

Draw an appropriate phase plane. Then, assuming that the current population is 249 yellowgill and 100 redfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( Y , R )\in\left\{ ( 0 , 0 ), ( 100 , 250 ), ( 600 , 0 ), ( 0 , 500 )\right\}\]

In the long term, only yellowgill will survive.

Example 85

S4m: Interacting populations (ver. 85)

Two species, yellowgill and bluegill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d Y }{dt}= -\frac{9}{1000} \, B Y - \frac{3}{500} \, Y^{2} + \frac{27}{25} \, Y\]

\[\frac{d B }{dt}= -\frac{1}{200} \, B^{2} - \frac{1}{100} \, B Y + B\]

Draw an appropriate phase plane. Then, assuming that the current population is 121 yellowgill and 18 bluegill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( Y , B )\in\left\{ ( 0 , 0 ), ( 60 , 80 ), ( 180 , 0 ), ( 0 , 200 )\right\}\]

In the long term, only yellowgill will survive.

Example 86

S4m: Interacting populations (ver. 86)

Two species, redfish and purplegill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d R }{dt}= -\frac{1}{50} \, P R - \frac{1}{125} \, R^{2} + \frac{8}{5} \, R\]

\[\frac{d P }{dt}= -\frac{1}{100} \, P^{2} - \frac{3}{250} \, P R + \frac{6}{5} \, P\]

Draw an appropriate phase plane. Then, assuming that the current population is 121 redfish and 21 purplegill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( R , P )\in\left\{ ( 0 , 0 ), ( 50 , 60 ), ( 200 , 0 ), ( 0 , 120 )\right\}\]

In the long term, only redfish will survive.

Example 87

S4m: Interacting populations (ver. 87)

Two species, yellowgill and magentafish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d Y }{dt}= -\frac{1}{250} \, M Y - \frac{1}{100} \, Y^{2} + \frac{4}{5} \, Y\]

\[\frac{d M }{dt}= -\frac{3}{250} \, M^{2} - \frac{3}{500} \, M Y + \frac{36}{25} \, M\]

Draw an appropriate phase plane. Then, assuming that the current population is 100 yellowgill and 39 magentafish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( Y , M )\in\left\{ ( 0 , 0 ), ( 40 , 100 ), ( 80 , 0 ), ( 0 , 120 )\right\}\]

In the long term, both species will co-exist.

Example 88

S4m: Interacting populations (ver. 88)

Two species, redfish and magentafish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d R }{dt}= -\frac{1}{50} \, M R - \frac{1}{125} \, R^{2} + 8 \, R\]

\[\frac{d M }{dt}= -\frac{1}{100} \, M^{2} - \frac{3}{250} \, M R + 6 \, M\]

Draw an appropriate phase plane. Then, assuming that the current population is 601 redfish and 101 magentafish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( R , M )\in\left\{ ( 0 , 0 ), ( 250 , 300 ), ( 1000 , 0 ), ( 0 , 600 )\right\}\]

In the long term, only redfish will survive.

Example 89

S4m: Interacting populations (ver. 89)

Two species, purplegill and bluegill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d P }{dt}= -\frac{1}{100} \, B P - \frac{3}{250} \, P^{2} + 6 \, P\]

\[\frac{d B }{dt}= -\frac{1}{50} \, B^{2} - \frac{1}{125} \, B P + 8 \, B\]

Draw an appropriate phase plane. Then, assuming that the current population is 600 purplegill and 100 bluegill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( P , B )\in\left\{ ( 0 , 0 ), ( 250 , 300 ), ( 500 , 0 ), ( 0 , 400 )\right\}\]

In the long term, both species will co-exist.

Example 90

S4m: Interacting populations (ver. 90)

Two species, bluegill and magentafish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d B }{dt}= -\frac{1}{100} \, B^{2} - \frac{1}{250} \, B M + \frac{4}{5} \, B\]

\[\frac{d M }{dt}= -\frac{3}{500} \, B M - \frac{3}{250} \, M^{2} + \frac{36}{25} \, M\]

Draw an appropriate phase plane. Then, assuming that the current population is 100 bluegill and 40 magentafish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( B , M )\in\left\{ ( 0 , 0 ), ( 40 , 100 ), ( 80 , 0 ), ( 0 , 120 )\right\}\]

In the long term, both species will co-exist.

Example 91

S4m: Interacting populations (ver. 91)

Two species, purplegill and greenfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d P }{dt}= -\frac{1}{50} \, G P - \frac{1}{125} \, P^{2} + 8 \, P\]

\[\frac{d G }{dt}= -\frac{1}{100} \, G^{2} - \frac{3}{250} \, G P + 6 \, G\]

Draw an appropriate phase plane. Then, assuming that the current population is 602 purplegill and 100 greenfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( P , G )\in\left\{ ( 0 , 0 ), ( 250 , 300 ), ( 1000 , 0 ), ( 0 , 600 )\right\}\]

In the long term, only purplegill will survive.

Example 92

S4m: Interacting populations (ver. 92)

Two species, bluegill and redfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d B }{dt}= -\frac{1}{125} \, B^{2} - \frac{1}{50} \, B R + \frac{8}{5} \, B\]

\[\frac{d R }{dt}= -\frac{3}{250} \, B R - \frac{1}{100} \, R^{2} + \frac{6}{5} \, R\]

Draw an appropriate phase plane. Then, assuming that the current population is 118 bluegill and 22 redfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( B , R )\in\left\{ ( 0 , 0 ), ( 50 , 60 ), ( 200 , 0 ), ( 0 , 120 )\right\}\]

In the long term, only bluegill will survive.

Example 93

S4m: Interacting populations (ver. 93)

Two species, yellowgill and greenfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d Y }{dt}= -\frac{3}{250} \, G Y - \frac{3}{500} \, Y^{2} + \frac{18}{25} \, Y\]

\[\frac{d G }{dt}= -\frac{1}{250} \, G^{2} - \frac{1}{100} \, G Y + \frac{2}{5} \, G\]

Draw an appropriate phase plane. Then, assuming that the current population is 48 yellowgill and 20 greenfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( Y , G )\in\left\{ ( 0 , 0 ), ( 20 , 50 ), ( 120 , 0 ), ( 0 , 100 )\right\}\]

In the long term, only yellowgill will survive.

Example 94

S4m: Interacting populations (ver. 94)

Two species, purplegill and magentafish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d P }{dt}= -\frac{3}{250} \, M P - \frac{3}{500} \, P^{2} + \frac{18}{5} \, P\]

\[\frac{d M }{dt}= -\frac{1}{250} \, M^{2} - \frac{1}{100} \, M P + 2 \, M\]

Draw an appropriate phase plane. Then, assuming that the current population is 251 purplegill and 99 magentafish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( P , M )\in\left\{ ( 0 , 0 ), ( 100 , 250 ), ( 600 , 0 ), ( 0 , 500 )\right\}\]

In the long term, only purplegill will survive.

Example 95

S4m: Interacting populations (ver. 95)

Two species, yellowgill and redfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d Y }{dt}= -\frac{1}{200} \, R Y - \frac{1}{100} \, Y^{2} + \frac{1}{2} \, Y\]

\[\frac{d R }{dt}= -\frac{9}{1000} \, R^{2} - \frac{3}{500} \, R Y + \frac{27}{50} \, R\]

Draw an appropriate phase plane. Then, assuming that the current population is 62 yellowgill and 8 redfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( Y , R )\in\left\{ ( 0 , 0 ), ( 30 , 40 ), ( 50 , 0 ), ( 0 , 60 )\right\}\]

In the long term, both species will co-exist.

Example 96

S4m: Interacting populations (ver. 96)

Two species, greenfish and yellowgill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d G }{dt}= -\frac{3}{500} \, G^{2} - \frac{3}{250} \, G Y + \frac{18}{5} \, G\]

\[\frac{d Y }{dt}= -\frac{1}{100} \, G Y - \frac{1}{250} \, Y^{2} + 2 \, Y\]

Draw an appropriate phase plane. Then, assuming that the current population is 251 greenfish and 99 yellowgill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( G , Y )\in\left\{ ( 0 , 0 ), ( 100 , 250 ), ( 600 , 0 ), ( 0 , 500 )\right\}\]

In the long term, only greenfish will survive.

Example 97

S4m: Interacting populations (ver. 97)

Two species, magentafish and yellowgill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d M }{dt}= -\frac{3}{500} \, M^{2} - \frac{9}{1000} \, M Y + \frac{27}{50} \, M\]

\[\frac{d Y }{dt}= -\frac{1}{100} \, M Y - \frac{1}{200} \, Y^{2} + \frac{1}{2} \, Y\]

Draw an appropriate phase plane. Then, assuming that the current population is 58 magentafish and 11 yellowgill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( M , Y )\in\left\{ ( 0 , 0 ), ( 30 , 40 ), ( 90 , 0 ), ( 0 , 100 )\right\}\]

In the long term, only magentafish will survive.

Example 98

S4m: Interacting populations (ver. 98)

Two species, greenfish and purplegill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d G }{dt}= -\frac{3}{500} \, G^{2} - \frac{3}{250} \, G P + \frac{18}{25} \, G\]

\[\frac{d P }{dt}= -\frac{1}{100} \, G P - \frac{1}{250} \, P^{2} + \frac{2}{5} \, P\]

Draw an appropriate phase plane. Then, assuming that the current population is 51 greenfish and 18 purplegill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( G , P )\in\left\{ ( 0 , 0 ), ( 20 , 50 ), ( 120 , 0 ), ( 0 , 100 )\right\}\]

In the long term, only greenfish will survive.

Example 99

S4m: Interacting populations (ver. 99)

Two species, greenfish and yellowgill, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d G }{dt}= -\frac{3}{250} \, G^{2} - \frac{1}{100} \, G Y + \frac{6}{5} \, G\]

\[\frac{d Y }{dt}= -\frac{1}{125} \, G Y - \frac{1}{50} \, Y^{2} + \frac{8}{5} \, Y\]

Draw an appropriate phase plane. Then, assuming that the current population is 122 greenfish and 20 yellowgill, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( G , Y )\in\left\{ ( 0 , 0 ), ( 50 , 60 ), ( 100 , 0 ), ( 0 , 80 )\right\}\]

In the long term, both species will co-exist.

Example 100

S4m: Interacting populations (ver. 100)

Two species, bluegill and greenfish, compete for the same resources. Suppose their populations satisfy the following system of ODEs:

\[\frac{d B }{dt}= -\frac{3}{500} \, B^{2} - \frac{3}{250} \, B G + \frac{36}{25} \, B\]

\[\frac{d G }{dt}= -\frac{1}{100} \, B G - \frac{1}{250} \, G^{2} + \frac{4}{5} \, G\]

Draw an appropriate phase plane. Then, assuming that the current population is 98 bluegill and 42 greenfish, determine the long-term survival of both species.

Answer.

The equilibria of the phase plane are given by

\[( B , G )\in\left\{ ( 0 , 0 ), ( 40 , 100 ), ( 240 , 0 ), ( 0 , 200 )\right\}\]

In the long term, only bluegill will survive.