The document discusses population growth models, including exponential and logistic growth models. It provides examples of using the logistic growth model to model bird population growth over time in a forest area. The model is fitted to data showing the population was 500 in 1990 and 856 in 1997. The carrying capacity of the forest is 7000 birds. The model is then used to predict the population will be 1850 birds in 2007 and the year the population will reach 1000 birds.

1. ENVIRONMENTAL
MODELLING
ENVI SCI 24

2. Population Growth
◦ Models the rate at which the population increases over time
2020 2021 2022
Philippines 109,035,343 111,046,913 ?????
Growth Rate =
Year 1 – Year 2
Year 1
Growth Rate =
111,046,913 - 109,035,343
109,035,343
Growth Rate = 0.018448 or 1.85

3. Population Growth
2020 2021 2022
Philippines 109,035,343 111,046,913 ?????
2022 Population =
[R(Growth Rate)N(Year 2 Population)] + N
2022 Population =
[(0.018)(111,046,913)] + 111,046,913
2022 Population =
113045757.4 or
113045757

4. Factors that affect Population
◦ Development
◦ Immigration
◦ Gender …
◦ Poverty
◦ Natural Resources
◦ Birth rate and Death Rate
◦ And more…

5. Malthusian Theory
◦ Thomas Robert Malthus (1766 – 1834)
◦ 1798, published anonymously the first edition of An Essay on the Principle of Population as It Affects the Future
Improvement of Society, with Remarks on the Speculations of Mr. Godwin, M. Condorcet, and Other Writers
◦ “population growth will always tend to outrun the food supply and the betterment of humankind is impossible without
strict limits on reproduction”

6. Malthusian Theory
◦ “population growth will always tend to outrun the food supply and the betterment of humankind is impossible without
strict limits on reproduction”
Resources
/Production

8. Population Growth Model
◦ Exponential Growth Model ◦ Logistic Growth Model

9. Exponential Growth Model Vs. Logistic Growth
Model
Exponential Growth Model
- Increases at an increasing rate
Has a differential equation of:
Where: Y = Population level
Logistic Growth Model
- Increases exponentially, then increases at a decreasing
rate then no longer increases.
Has a differential equation of:
Where: Y = Population level
L = the limiting factor or the “Carrying
Capacity”
dy
= Ky
dt
dy
=
Population Growth
rate (Population
level / time(t))
dt
dy
= Ky(1- Y/L)
dt
dy
=
Population Growth
rate (Population
level / time(t))
dt

10. Exponential Growth Model & Logistic Growth
Model
◦ Differential equation = “Describes how nature works”
◦ A Mathematical equation that involves variables (x, or y) instead of a number.
◦ Provides “Understanding” Rather than “Computation”

11. Exponential Growth Model
Exponential Growth Model
- Increases at an increasing rate
Has a differential equation of:
Where: Y = Population level
dy
= Ky
dt
dy
=
Population Growth
rate (Population
level / time(t))
dt
y
t

12. Exponential Growth Model
Exponential Growth Model
General Equation/Solution
y
t
2020 (t0) 109,035,343
2021 (t1) 111,046,913
2022 (t2) 113045757
2023
2024
Y = ce^kt
P(t) = Pºe^kt

13. Exponential Growth Model
Exponential Growth Model
2020 (t0) 109,035,343
2021 (t1) 111,046,913
2022 (t2) 113045757
2023 (t3)
2024 (t4)
P(t) = Pºe^kt
P(t) = Population at a given time (t)
Pº= Base Population
e= Statistical constant
(Approximately =
2.71828183)
K = Constant
t = time

14. Exponential Growth Model
Exponential Growth Model
2020 (t0) 109,035,343
2021 (t1) 111,046,913
2022 (t2) 113045757
2023 (t3)
2024 (t4)
P(t) = Population at a given time (t)
Pº= Base Population
e= Statistical constant
(Approximately =
2.71828183)
K = Constant
t = time

15. Exponential Growth Model
Exponential Growth Model
2020 (t0) 109,035,343
2021 (t1) 111,046,913
2022 (t2) 113045757
2023 (t3)
2024 (t4)
P(t) = Population at a given time (t)
Pº= Base Population
e= Statistical constant
(Approximately =
2.71828183)
K = Constant
t = time

16. Exponential Growth Model
Exponential Growth Model
2020 (t0) 109,035,343
2021 (t1) 111,046,913
2022 (t2) 113045757
2023 (t3)
2024 (t4)
P(t) = Population at a given time (t)
Pº= Base Population
e= Statistical constant
(Approximately =
2.71828183)
K = Constant
t = time

17. Logistic Growth Model
Logistic Growth Model
- Increases exponentially, then increases at a decreasing
rate then no longer increases.
Has a differential equation of:
Where: Y = Population level
L = the limiting factor or the “Carrying
Capacity”
dy
= Ky(1- Y/L)
dt
dy
=
Population Growth
rate (Population
level / time(t))
dt

18. Logistic Growth Model
Y =
L
1+be^-kt
General equation/Solution
P(t) = Population at a given time (t)
L= Limiting Factor / Carrying Capactiy
e= Statistical constant (Approximately =
2.71828183)
b = Constant
K = Constant
t = time
P(t) =
L
1+be^-kt

19. Logistic Growth Model
500 birds were released in a forest area in 1990. By 1997,
there were 856 birds on the island. The forest area can
sustain a maximum of 7000 birds.
1. How many birds will there be by 2007?
2. In what year will the population reach 1000?
P(t) =
L
1+be^-kt

20. Logistic Growth Model 500 people birds were released in a forest area in 1990.
By 1997, there were 856 birds on the island. The forest
area can sustain a maximum of 7000 birds.
1. How many birds will there be by 2007?
2. In what year will the population reach 1000?
- Solve for b & k
P(t): L = 7000
P(0) = 500
P(7) = 856

21. Logistic Growth Model
- Solve for b & k
P(t): L = 7000
P(0) = 500
P(7) = 856

22. Logistic Growth Model
- Solve for b & k
P(t): L = 7000
P(0) = 500
P(7) = 856

23. Logistic Growth Model
- Solve for b & k
P(t): L = 7000
P(0) = 500
P(7) = 856

24. Logistic Growth Model
- Solve for b & k
P(t): L = 7000
P(0) = 500
P(7) = 856

25. Logistic Growth Model
- Solve for b & k
P(t): L = 7000
P(0) = 500
P(7) = 856

26. Logistic Growth Model
- Solve for b & k
P(t): L = 7000
P(0) = 500 b = 14
P(7) = 856

27. Logistic Growth Model
- Solve for b & k
P(t): L = 7000
P(0) = 500 b = 14
P(7) = 856

28. Logistic Growth Model
- Solve for b & k
P(t): L = 7000
P(0) = 500 b = 14
P(7) = 856

29. Logistic Growth Model
- Solve for b & k
P(t): L = 7000
P(0) = 500 b = 14
P(7) = 856

30. Logistic Growth Model
- Solve for b & k
P(t): L = 7000
P(0) = 500 b = 14
P(7) = 856

31. Logistic Growth Model
- Solve for b & k
P(t): L = 7000
P(0) = 500 b = 14
P(7) = 856

32. Logistic Growth Model
- Solve for b & k
P(t): L = 7000
P(0) = 500 b = 14
P(7) = 856

33. Logistic Growth Model
P(t): L = 7000
P(0) = 500 b = 14
P(7) = 856 k = 0.095
500 people birds were released in a forest area in 1990.
By 1997, there were 856 birds on the island. The forest
area can sustain a maximum of 7000 birds.
1. How many birds will there be by 2007?
2. In what year will the population reach 1000?

34. Logistic Growth Model
P(t): L = 7000
P(0) = 500 b = 14
P(7) = 856 k = 0.095
P(17) = ?
500 people birds were released in a forest area in 1990.
By 1997, there were 856 birds on the island. The forest
area can sustain a maximum of 7000 birds.
1. How many birds will there be by 2007?
2. In what year will the population reach 1000?

35. Logistic Growth Model
P(t): L = 7000
P(0) = 500 b = 14
P(7) = 856 k = 0.095
P(17) = ?
500 people birds were released in a forest area in 1990.
By 1997, there were 856 birds on the island. The forest
area can sustain a maximum of 7000 birds.
1. How many birds will there be by 2007?
2. In what year will the population reach 1000?

36. Logistic Growth Model
P(t): L = 7000
P(0) = 500 b = 14
P(7) = 856 k = 0.095
P(17) = ?
500 people birds were released in a forest area in 1990.
By 1997, there were 856 birds on the island. The forest
area can sustain a maximum of 7000 birds.
1. How many birds will there be by 2007?
2. In what year will the population reach 1000?

37. Logistic Growth Model
P(t): L = 7000
P(0) = 500 b = 14
P(7) = 856 k = 0.095
P(17) = ?
500 people birds were released in a forest area in 1990.
By 1997, there were 856 birds on the island. The forest
area can sustain a maximum of 7000 birds.
1. How many birds will there be by 2007?
2. In what year will the population reach 1000?

38. Logistic Growth Model
P(t): L = 7000
P(0) = 500 b = 14
P(7) = 856 k = 0.095
P(17) =1850
t = ?
500 people birds were released in a forest area in 1990.
By 1997, there were 856 birds on the island. The forest
area can sustain a maximum of 7000 birds.
1. How many birds will there be by 2007?
2. In what year will the population reach 1000?

39. Logistic Growth Model
P(t): L = 7000
P(0) = 500 b = 14
P(7) = 856 k = 0.095
P(17) =1850
t = ?
500 people birds were released in a forest area in 1990.
By 1997, there were 856 birds on the island. The forest
area can sustain a maximum of 7000 birds.
1. How many birds will there be by 2007?
2. In what year will the population reach 1000?
“PRACTICE COMPUTATION”