1. An analysis of population projections using
the age specific Leslie Population model
TREVOR CHANDLER
CHARLOTTE ROSE HAMILTON
CARESSA MAINLAND
JACOB MORGAN
CHRIS TILLER
Fishing in George’s Bank

2. Reading a Leslie Matrix
f1 f2 f3 f4 f5 f6 f7 f8 f9 f10
s2
s3
s4
s5
s6
s7
s8
s9
s10
Survival rate
s2,1 , s3,2 , … , sn-1, n-2, sn, n-1
Fecundity rate
f1,1 , f1,2 , … , f1, n-1 , f1, n
n1
n2
n3
n4
n5
n6
n7
n8
n9
n10
Population
vector

3. POPULATION PROJECTION
The formula: 𝑥 𝑛 + 1 = 𝐿𝑥 𝑛 for values 𝑛 = 0, 1, 2, … , 𝑛
*assuming 𝑥(0) is given
Can be expressed as
𝑥(𝑛+1)
= 𝐿𝑥(𝑛)
for 𝑛 = 0, 1, 2, … , 𝑛
Population Vector
for year .
f1 f2 f3
s1
s1
n1
n2
n3
n1
n2
n3
Leslie
Matrix
Population
Vector
n
*
n
=

4. GIVEN INFORMATION
Haddock Lifespan:
10 years
Age of sexual maturity: 3
years
Sample Time:
1 year

5. JACOB MORGAN
0
100
0
0
0
0
0
0
0
0
0
*
1900
=
100
0
0
0
0
0
0
0
0
0
PART ONE

6. JACOB MORGAN
10
100
0
0
0
0
0
0
0
0
0
*
1910
=
373.515
208.517
137.363
65.009
18.277
12.277
10.662
2.031
0
0
PART ONE

7. JACOB MORGAN
20
8036.778
3867.200
1935.729
1107.663
597.914
260.765
101.716
32.030
9.176
1.276
* =
100
0
0
0
0
0
0
0
0
0
1920
PART ONE

8. JACOB MORGAN
21
10632.335
5223.905
2629.696
1451.797
775.364
358.748
143.421
40.686
11.211
1.835
* =
100
0
0
0
0
0
0
0
0
0
1921
PART ONE

9. JACOB MORGAN
30
143487.445
69901.418
35623.816
19974.009
10451.780
4707.742
1944.528
579.921
151.167
22.797
* =
100
0
0
0
0
0
0
0
0
0
1930
PART ONE

10. JACOB MORGAN
50
4.679 * e7
2.277 * e7
1.160 * e7
6.512 * e6
3.413 * e6
1.533* e6
631490.921
189130.661
49564.509
7422.769
* =
100
0
0
0
0
0
0
0
0
0
1950
PART ONE

11. 95
2.112 * e13
1.028 * e13
5.233 * e12
2.939 * e12
1.540 * e12
6.920* e11
2.850* e11
8.535* e10
2.237* e10
3.350* e9
* =
100
0
0
0
0
0
0
0
0
0
1995
JACOB MORGAN
PART ONE

12. 100
8.972 * e13
4.367 * e13
2.223 * e13
1.249 * e13
6.544 * e12
2.940* e12
1.211* e12
3.626* e11
9.503* e10
1.423* e10
* =
100
0
0
0
0
0
0
0
0
0
2000
JACOB MORGAN
PART ONE

13. JACOB MORGAN
PART ONE
* 1995 2000
1 2.11 ∗ 1013
8.97 ∗ 1013
2 1.02 ∗ 1013
4.36 ∗ 1013
3 5.23 ∗ 1012
2.2 ∗ 1013
… … …
Approximate ratio 1 : 4
To prove the population model is unstable two
consecutive populations can be scaled to
determine the ratio.

14. PART TWO
CHRIS TILLER
BIRTH RATE: -10%
SURVIVAL RATE: -15%
Environmental Change: Pollution
POLLUTION:
ITS EFFECTS
ON THE LESLIE
MATRIX

15. PART TWO
CHRIS TILLER
4.679 * 107
2.277 * 107
1.160 * 107
6.512 * 106
3.413 * 106
1.533* 106
631490.921
189130.661
49564.509
7422.769
1950
POLLUTION RATE LESLIE MATRIX:
EFFECTIVE 1950
* =
2.092 * 109
9.534 * 108
4.605 * 108
2.519 * 108
1.263 * 108
5.178* 107
1.888* 107
4.302* 106
784162.309
35736.554
40 1990

16. PART TWO
CHRIS TILLER
4.970 * 1012
2.419 * 1012
1.232 * 1012
6.916 * 1011
3.625 * 1011
1.629* 1011
6.707* 1010
2.009* 1010
5.264* 109
7.883* 108
90 YEARS WITHOUT
POLLUTION
2.092 * 109
9.534 * 108
4.605 * 108
2.519 * 108
1.263 * 108
5.178* 107
1.888* 107
4.302* 106
784162.309
35736.554
1990 1990
40 YEARS WITH
POLLUTION

17. CARESSA MAINLAND
SURVIVAL RATE: -20%
Environmental Change: Harvesting
HARVESTING:
ITS EFFECTS
ON THE LESLIE
MATRIX
PART THREE

18. CARESSA MAINLAND
68315.959
40167.392
20744.526
6893.249
2787.123
523.082
78.567
2.9289
0
0
* =
HARVESTING RATE LESLIE MATRIX:
EFFECTIVE 1925
8.972 * e13
4.367 * e13
2.223 * e13
1.249 * e13
6.544 * e12
2.940* e12
1.211* e12
3.626* e11
9.503* e10
1.423* e10
1925 19305
PART THREE

19. CARESSA MAINLAND
1.060 * 1010
600054.980
355705.705
123803.518
37643.600
8204.552
1432.438
62.1433
0
0
* =
HARVESTING RATE LESLIE MATRIX:
EFFECTIVE 1925
8.972 * e13
4.367 * e13
2.223 * e13
1.249 * e13
6.544 * e12
2.940* e12
1.211* e12
3.626* e11
9.503* e10
1.423* e10
1925 195025
PART THREE

20. CARESSA MAINLAND
5.437 * 108
30.076 * 108
1.821* 108
6.341 * 107
1.932 * 107
4.205* 106
732108.071
31866.253
0
0
* =
HARVESTING RATE LESLIE MATRIX:
EFFECTIVE 1925
8.972 * e13
4.367 * e13
2.223 * e13
1.249 * e13
6.544 * e12
2.940* e12
1.211* e12
3.626* e11
9.503* e10
1.423* e10
1925 199570
PART THREE

21. CARESSA MAINLAND
1.087 * 109
6.153 * 108
3.643 *108
1.268 *108
3.864 * 107
8.411 * 106
1.464* 106
63738.539
0
0
* =
8.972 * e13
4.367 * e13
2.223 * e13
1.249 * e13
6.544 * e12
2.940* e12
1.211* e12
3.626* e11
9.503* e10
1.423* e10
1925 2000
75
HARVESTING RATE LESLIE MATRIX:
EFFECTIVE 1925
PART THREE

22. CHARLOTTE ROSE HAMILTON
SURVIVAL RATE: -20%
Environmental Change: Modified Harvesting
HARVESTING:
ITS EFFECTS
ON THE LESLIE
MATRIX
PART FOUR

23. CHARLOTTE ROSE HAMILTON
2.114* 106
924733.193
431488.064
230751.783
112194.460
43623.198
14838.864
2886.215
421.149
0
* =
765011.376
402316.747
223078.560
99962.883
40208.961
12980.357
3742.502
611.267
72.531
0
1940 197030
HARVESTING RATE LESLIE MATRIX:
EFFECTIVE 1940
PART FOUR

24. 8.645* 106
3.782* 106
1.765* 106
943766.571
458775.574
178412.785
60709.927
11804.712
1721.521
0
* =
765011.376
402316.747
223078.560
99962.883
40208.961
12980.357
3742.502
611.267
72.531
0
1940 199050
HARVESTING RATE LESLIE MATRIX:
EFFECTIVE 1940
CHARLOTTE ROSE HAMILTON
PART FOUR

25. TREVOR CHANDLER
PART FIVE
SURVIVAL RATE: -100% (3 – 5)
Environmental Change: Harvest Young
HARVESTING:
ITS EFFECTS
ON THE LESLIE
MATRIX

26. PART FIVE
* =
765011.376
402316.747
223078.560
99962.883
40208.961
12980.357
3742.502
611.267
72.531
0
1940 19411
HARVESTING RATE LESLIE MATRIX:
EFFECTIVE 1940
TREVOR CHANDLER
945320.646
344255.119
193112.038
0
0
0
4543.125
748.500
91.69
0

27. TREVOR CHANDLER
PART FIVE
* =
765011.376
402316.747
223078.560
99962.883
40208.961
12980.357
3742.502
611.267
72.531
0
1940 19433
HARVESTING RATE LESLIE MATRIX:
EFFECTIVE 1940
27863.686
133417.2365
204189.260
0
0
0
0
0
136.294
0

28. TREVOR CHANDLER
PART FIVE
* =
765011.376
402316.747
223078.560
99962.883
40208.961
12980.357
3742.502
611.267
72.531
0
1940 196020
HARVESTING RATE LESLIE MATRIX:
EFFECTIVE 1940
342.981
493.109
191.158
0
0
0
0
0
0
0

29. TREVOR CHANDLERTREVOR CHANDLER
SURVIVAL RATE: -100% (5 – 10)
Environmental Change: Harvest elderly
PART FIVE
HARVESTING:
ITS EFFECTS
ON THE LESLIE
MATRIX

30. TREVOR CHANDLER
* =
765011.376
402316.747
223078.560
99962.883
40208.961
12980.357
3742.502
611.267
72.531
0
1940 195010
HARVESTING RATE LESLIE MATRIX:
EFFECTIVE 1940
PART FIVE
5.504* 106
2.946* 106
1.648* 106
738866.137
302113.620
0
0
0
0
0

31. TREVOR CHANDLER
PART FIVE
* =
765011.376
402316.747
223078.560
99962.883
40208.961
12980.357
3742.502
611.267
72.531
0
1940 199050
HARVESTING RATE LESLIE MATRIX:
EFFECTIVE 1940
1.521* 1010
8.108* 1010
4.523* 1010
2.041* 109
8.371* 108
0
0
0
0
0

32. TREVOR CHANDLER
PART FIVE
* =
765011.376
402316.747
223078.560
99962.883
40208.961
12980.357
3742.502
611.267
72.531
0
1940 2040100
HARVESTING RATE LESLIE MATRIX:
EFFECTIVE 1940
3.036* 1014
1.619* 1014
9.031* 1013
4.075* 1013
1.671* 1013
0
0
0
0
0

33. ANALYSIS
- STABILIZATION
- UNCONTROLLED
REPRODUCTION
- HARVESTING
- POLLUTION
TREVOR CHANDLER
CHARLOTTE ROSE HAMILTON
CARESSA MAINLAND
JACOB MORGAN
CHRIS TILLER

34. Sage Mathematical Software 10.12.3.91:8080
T. Shores: Applied Linear Algebra and Matrix Analysis
http://www.montana.edu/screel/Webpages/conservation%20biology/leslie.pdf
http://bandicoot.maths.adelaide.edu.au/Leslie_matrix/leslie.cgi?initial_pop%5B0%5D=100&initial_pop
%5B1%5D=0&initial_pop%5B2%5D=0&initial_pop%5B3%5D=0&initial_pop%5B4%5D=0&initial_pop
%5B5%5D=0&initial_pop%5B6%5D=0&initial_pop%5B7%5D=0&birth_rates%5B0%5D=0&birth_rates
%5B1%5D=0&birth_rates%5B2%5D=1.5&birth_rates%5B3%5D=4.2&birth_rates%5B4%5D=3.8&birth_
rates%5B5%5D=2.5&birth_rates%5B6%5D=1.5&birth_rates%5B7%5D=0&survival_rates%5B0%5D=0.6
5&survival_rates%5B1%5D=0.68&survival_rates%5B2%5D=0.75&survival_rates%5B3%5D=0.7&surviva
l_rates%5B4%5D=0.6&survival_rates%5B5%5D=0.55&survival_rates%5B6%5D=0.4&survival_rates%5B
7%5D=0&Submit+Leslie+Matrix=Submit+Leslie+Matrix
http://www.pelagicos.net/MARS6920_spring2010/lectures/MarConBio_sp2010_Lecture10.PDF
http://www.webpages.uidaho.edu/wlf448/Leslie1.htm
http://amrita.vlab.co.in/?sub=3&brch=65&sim=183&cnt=1
http://www.fao.org/wairdocs/tan/x5939e/x5939e01.htm
SOURCES