Lecture 8 populations and logistic growth (1)
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The document outlines key concepts related to population dynamics, focusing on density-independent and density-dependent growth, maximum sustained yield, and the logistic growth equations. It also discusses factors affecting population size, the allee effect, and stability properties in population models, including examples and scenarios involving populations like ravens and mayflies. Additionally, it touches on the historical context of seafood populations and the philosophical influence of figures like Ed Ricketts on John Steinbeck.
![Organization & Equations
Logistic
Continuous Discrete
Differential Difference Difference
dN/dt = rN (1-N/K) Nt = K/[1+ [(K – N0)/N0] e -(rt)] Nt+1 = Nt + R0 Nt (1 – Nt/K)](https://image.slidesharecdn.com/lecture8populationsandlogisticgrowth1-140617091439-phpapp02/85/Lecture-8-populations-and-logistic-growth-1-2-320.jpg)
![Organization & Equations
Logistic
Continuous Discrete
Differential Difference Difference
dN/dt = rN (1-N/K) Nt = K/[1+ [(K – N0)/N0] e -(rt)] Nt+1 = Nt + R0 Nt (1 – Nt/K)](https://image.slidesharecdn.com/lecture8populationsandlogisticgrowth1-140617091439-phpapp02/85/Lecture-8-populations-and-logistic-growth-1-6-320.jpg)
![Organization & Equations
Logistic
Continuous Discrete
Differential Difference Difference
dN/dt = rN (1-N/K) Nt = K/[1+ [(K – N0)/N0] e -(rt)] Nt+1 = Nt + R0 Nt (1 – Nt/K)](https://image.slidesharecdn.com/lecture8populationsandlogisticgrowth1-140617091439-phpapp02/85/Lecture-8-populations-and-logistic-growth-1-14-320.jpg)
![What if you want to predict the pop
size at some point in time?
• The integrated form of the differential
logistic equation for continuous
reproduction. The difference equation.
• Nt = K/[1 + [(K – N0)/N0] e-rt]](https://image.slidesharecdn.com/lecture8populationsandlogisticgrowth1-140617091439-phpapp02/85/Lecture-8-populations-and-logistic-growth-1-15-320.jpg)
![How do we
calculate data for
the function line?
Use the difference equation . Example: K = 500; r = 0.5; N0 = 10.
• Nt = K/[1 + [(K – N0)/N0]e-rt]
N1 = K/[1 + [(K – N0)/N0]e-rt] = 500/[1+[(500-10)/10] e-0.5*1 ]= 16
N2 = K/[1 + [(K – N0)/N0]e-rt] = 500/[1+[(500-10)/10] e-0.5*2 ] = 27
• N3 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 42
• N4 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 66
• N5 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 100
• N6 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 145
• N7 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 202
• N8 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 264
• N9 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 324
• N10 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 372
• N11 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 417
• N12 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 446
• N13 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 466
• N14 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 479
• N15 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 487
• N16 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 492
• N17 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 495
• N18 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 497
• N19 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 498
• N20 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 499
• N21 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 499
• N22 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 500
0
100
200
300
400
500
600
0 5 10 15 20 25
Time
Nt](https://image.slidesharecdn.com/lecture8populationsandlogisticgrowth1-140617091439-phpapp02/85/Lecture-8-populations-and-logistic-growth-1-18-320.jpg)
![Organization & Equations
Logistic
Continuous Discrete
Differential Difference Difference
dN/dt = rN (1-N/K) Nt = K/[1+ [(K – N0)/N0] e -(rt)] Nt+1 = Nt + R0 Nt (1 – Nt/K)](https://image.slidesharecdn.com/lecture8populationsandlogisticgrowth1-140617091439-phpapp02/85/Lecture-8-populations-and-logistic-growth-1-25-320.jpg)
![A population with discrete reproduction
• A population of mayflies started with 2 individuals, with a
carry capacity of 100 and an r = 0.2. The population
reproduces once each year. What is the population size at
Nt+1.
• Nt+1 = Nt + R0 Nt (1 – Nt/K):
• Nt+1 = 2+(.2)(2) [1- (2/100)]
• Nt+1 = 2+.4 [1-.02]
• Nt+1 = 2+(.4)(.98)
– the population is increasing at 98% of the exponential rate
• Nt+1 = 2.392
• Now what?](https://image.slidesharecdn.com/lecture8populationsandlogisticgrowth1-140617091439-phpapp02/85/Lecture-8-populations-and-logistic-growth-1-27-320.jpg)
Lecture 8 populations and logistic growth (1)
- 1. Lecture Outline • Density Independent and Density Dependent growth and the carrying capacity • The differential logistic equation for continuous growth and Maximum Sustained Yield • The difference equation for logistic continuous growth. • The logistic equation for discrete growth • r and extinction
- 2. Organization & Equations Logistic Continuous Discrete Differential Difference Difference dN/dt = rN (1-N/K) Nt = K/[1+ [(K – N0)/N0] e -(rt)] Nt+1 = Nt + R0 Nt (1 – Nt/K)
- 3. Density Independence, page 140 in McConnell • Factors that effect b and d do not depend on the population size. Density Dependence • Factors that effect b and d do depend on population size.
- 4. Recruitment = larvae leave the plankton and attach to solid substrate. • Which is density dependent?
- 5. As N increases, r will decrease to zero. This is density dependence. r N K
- 6. Organization & Equations Logistic Continuous Discrete Differential Difference Difference dN/dt = rN (1-N/K) Nt = K/[1+ [(K – N0)/N0] e -(rt)] Nt+1 = Nt + R0 Nt (1 – Nt/K)
- 7. Logistic growth is density dependent growth. • dN/dt = rN (1-N/K)
- 8. How does (1-N/K) work? • Population 1: K = 100 individuals and N = 5 • Population 2: K = 1500 and N = 1000
- 9. What does the curve look like for continuous populations growing under density dependence?
- 10. • What is the value of K? • What does the upper curve tell us?
- 11. Maximum Sustained Yield… • Fisheries managers tried to harvest enough fish to maintain the maximum growth rate or maximum yield = K/2. • Do you think this worked? Why or Why not?
- 12. Ed Ricketts, John Steinbeck, and Cannery Row “Ricketts was accepting because he just listened and tended to turn whatever people said into something that sounded brilliant.” Ricketts was Steinbeck’s friend and philosophical/intellectual father. 1930s = 500,000 tons of sardines/year 1952 = gone
- 13. Practice • Suppose a population of ravens is growing according to the continuous logistic equation. If K=200 and r = 0.1 individuals/(individual · year-1), what is the maximum growth rate of this population over a year? • dN/dt = rN(1-N/K)
- 14. Organization & Equations Logistic Continuous Discrete Differential Difference Difference dN/dt = rN (1-N/K) Nt = K/[1+ [(K – N0)/N0] e -(rt)] Nt+1 = Nt + R0 Nt (1 – Nt/K)
- 15. What if you want to predict the pop size at some point in time? • The integrated form of the differential logistic equation for continuous reproduction. The difference equation. • Nt = K/[1 + [(K – N0)/N0] e-rt]
- 16. An overview of the procedure. • Create a phase-plane plot with Nt+1 on the Y-axis and Nt on the X-axis • Calculate the function line using the difference equation. • Draw the equilibrium line where Nt+1 = Nt • Use “cob webbing” to determine the stability properties of a population…more in a minute.
- 17. What is a phase plane plot? 0 100 200 300 400 500 600 0 100 200 300 400 500 600 N t Nt+1 Function line Equilibrium line
- 18. How do we calculate data for the function line? Use the difference equation . Example: K = 500; r = 0.5; N0 = 10. • Nt = K/[1 + [(K – N0)/N0]e-rt] N1 = K/[1 + [(K – N0)/N0]e-rt] = 500/[1+[(500-10)/10] e-0.5*1 ]= 16 N2 = K/[1 + [(K – N0)/N0]e-rt] = 500/[1+[(500-10)/10] e-0.5*2 ] = 27 • N3 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 42 • N4 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 66 • N5 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 100 • N6 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 145 • N7 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 202 • N8 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 264 • N9 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 324 • N10 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 372 • N11 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 417 • N12 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 446 • N13 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 466 • N14 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 479 • N15 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 487 • N16 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 492 • N17 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 495 • N18 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 497 • N19 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 498 • N20 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 499 • N21 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 499 • N22 = Nt = K/[1 + [(K – N0)/N0]e-rt] = 500 0 100 200 300 400 500 600 0 5 10 15 20 25 Time Nt
- 19. How do we plot the function line in phase plane? 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Nt Nt+1 Function line Equilibrium line
- 20. What is cobwebbing? 0 100 200 300 400 500 600 0 100 200 300 400 500 600 N t Nt+1
- 21. Stability properties? • What is an equilibrium point?
- 22. General Concept of Equilibrium • It is a balance between opposing forces…births – deaths. • Non-equilibrium: conditions are constantly changing. A stable balance cannot be achieved. • Equilibrium forces produce a stable balance. • Non-equilibrium forces disrupt a stable balance.
- 23. The Allee Effect: A special case of density dependence. • Some species require a minimum number of individuals or they spiral to extinction. – Some species need a certain group size to find a mate (whales) or to hunt (wolves), care for young (elephants), reproduce (mayflies), and avoid predators (flocking birds). – r is negative below a critical N (Allee density) and then becomes positive as N increases past the critical N.
- 24. What are the stability properties of the Allee effect? • How many equilibrium points with an Allee effect? • Which are stable which are unstable?
- 25. Organization & Equations Logistic Continuous Discrete Differential Difference Difference dN/dt = rN (1-N/K) Nt = K/[1+ [(K – N0)/N0] e -(rt)] Nt+1 = Nt + R0 Nt (1 – Nt/K)
- 26. How to solve for Nt+1 when a population has discrete growth • What happens as r increases in the continuous equation? • What happens as r increases in the discrete equation?
- 27. A population with discrete reproduction • A population of mayflies started with 2 individuals, with a carry capacity of 100 and an r = 0.2. The population reproduces once each year. What is the population size at Nt+1. • Nt+1 = Nt + R0 Nt (1 – Nt/K): • Nt+1 = 2+(.2)(2) [1- (2/100)] • Nt+1 = 2+.4 [1-.02] • Nt+1 = 2+(.4)(.98) – the population is increasing at 98% of the exponential rate • Nt+1 = 2.392 • Now what?
- 28. Your turn. • A population with discrete reproduction has a growth rate of 3.0. They reproduce once per year, K = 1500 and Nt = 20. • What is the population size after two generations? Nt+1 = Nt + R0 Nt (1 – Nt/K)
- 29. Discrete growth has a built-in time lag. • In the continuous growth model we assume that per capita growth changes instantaneously when an individual is born or dies. • Discrete populations do not adjust instantaneously. There is a time lag before N begins to apply a brake to R0 when N overshoots K.
- 30. Kiabab deer herd in northern Arizona ?? Present Wolves exterminated
- 31. What is the relationship between r and the stability of a discrete population? Pg 130 in McConnell
- 32. Dampening, stable limit, complex stable limit, chaos Time R0 < 2.0 2.0 < R0 < 2.5 2.5 < R0 < 2.57 R0 > 2.57
- 33. How do you know the stability characteristics? • The outcome of cobwebbing is determined by the slope of the function line where it crosses the equilibrium line.
- 34. Now we have 4 types of population “stability”. 1. Smooth increase to K…constancy over time. 2. Dampened oscillations…constancy with minor variation around K. 3. Stable cycles… continual variation around K over time. 4. Chaos…this is not stable
- 35. Large r and Chaos • Chaos = a non-repeating, drastic fluctuation in population size with a simple deterministic model…shocking. Lord Robert May 1936 – present Oxford University Attractor = K Bifurcation 4 8 Chaos
- 36. General Meaning • Deterministic, meant that the future behavior was fully determined by the initial conditions, with no random elements involved. • “The phenomenon of deterministic chaos…very simple and purely deterministic laws or equations can give rise to dynamic behavior that not merely looks like random noise, but is so sensitive to initial conditions that long-term prediction is effectively impossible. This ended the Newtonian dream that if the system is simple (very few variables) and orderly (the rules and parameters are exactly known) then the future is predictable.” Robert May (2007) – Theoretical Ecology (2007)
- 37. Sibly et al. 2007. Ecology Letters 10:1-7 • Many return rates are ≈1. Between 0.5 & 2 is considered stable. • <0.5 and disturbance rates may exceed return rates resulting in extinction. >2.0 can result in chaotic behavior and thus, extinction. Closed circles = mammals Open cirlce = insects Plus signs = birds Open squares = bony fish A total of 634 populations The return rates are an approximation of r