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

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)

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.

Recruitment = larvae leave the plankton and attach to solid substrate.

• Which is density dependent?

As N increases, r will decrease to zero.

This is density dependence.

r

N K

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)

Logistic growth is density dependent growth.

• dN/dt = rN (1-N/K)

How does (1-N/K) work?

• Population 1: K = 100 individuals and N = 5

• Population 2: K = 1500 and N = 1000

What does the curve look like for continuous populations growing under density dependence?

• What is the value of K?• What does the upper curve tell us?

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?

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/year1952 = gone

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)

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)

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]

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.

What is a phase plane plot?

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N t

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Function line

Equilibrium line

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 ]= 16N2 = 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

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Nt

How do we plot the function line in phase plane?

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Function line

Equilibrium line

What is cobwebbing?

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Stability properties?

• What is an equilibrium point?

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.

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.

What are the stability properties of the Allee effect?

• How many equilibrium points with an Allee effect?

• Which are stable which are unstable?

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)

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?

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?

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)

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.

Kiabab deer herd in northern Arizona

??

Present

Wolves exterminated

What is the relationship between r and the stability of a discrete population?

Pg 130 in McConnell

Dampening, stable limit, complex stable limit, chaos

Time

R0 < 2.02.0 < R0 < 2.5

2.5 < R0 < 2.57 R0 > 2.57

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.

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

Large r and Chaos• Chaos = a non-repeating, drastic fluctuation in

population size with a simple deterministic model…shocking.

Lord Robert May1936 – presentOxford University

Attractor = K

Bifurcation 4 8 Chaos

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)

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 = mammalsOpen cirlce = insectsPlus signs = birdsOpen squares = bony fishA total of 634 populations

The return rates are an approximation of r