Lecture 12(Oct 26)

Lecture Notes

#### Population Growth and Regulation

1. Why care?

• Human population growth is largely governed by the same set of principles as those of other organisms
• The management of renewable resources is an exercise in managing population growth
• Predicting the spread of invasive species
• Understanding threats of extinction and managing species conservation
• Predicting and managing the spread of diseases

2. Factors determining population change

• Birth adds to a population size
• Immigration adds to a population size
• Death subtracts from a population size
• Emigration subtracts from a population size

Typically, these factors are expressed in vital rates, or rate per individual, which measures the per capita growth rate.

3. r - instantaneous rate of increase

• r is measured by (B+I)-(D+E).
• We won’t be using I or E in this lecture.
• r = $\left ( \frac{1}{n} \right )* \left ( \frac{dN}{dt} \right )$
• $\left ( \frac{dN}{dt} \right )$ = rN

$\left ( \frac{dN}{dt} \right )$ is a differential equation showing per capita exponential growth

4. λ (lambda) - per capita geometric rate of increase

• λ is measured by population size at time t divided by population size at time t-1
• λ= $\left ( \frac{N_{t}}{N_{t-1}} \right )$
• λ= $\left ( \frac{N_{t+1}}{N_{t}} \right )$

Nt+1=λNt is a difference equation showing geometric growth

5. Predicting population into the future - mussel example

• There are 2 mussels in year 0 and λ=3. How many mussels will there be in year 5?
• NttNo
• N5=35*2 = 486
• There are 300 mussels in the lake and λ=3. How many mussels will be in the lake next year?
• Nt+1= λN1
• Nt+1= 3*300 = 900

5. Discrete vs Continuous time

• NttNo is a discrete time unit model - there are blocks of time throughout which everything is assumed to be constant. This is one year, in our examples.
• Discrete models should be used when populations have discrete breeding seasons and non-overlapping generations.
• Nt=Noert is a continuous time model - growth is continuous and instantaneous
• Continuous models should be used when populations have overlapping generations and continuous reproduction
• $\left ( \frac{dN}{dt} \right ) = rN \Rightarrow$ Nt=Noert
• For the proof that this equation is right, refer to the Powerpoint

6. r vs λ, a summary

 factor r λ conversion r = logeλ λ = er growing population > 0 > 1 shrinking population < 0 0 < λ < 1 stable population r = 0 λ = 1 equation usage differential difference time model continuous discrete growth pattern exponential geometric

7. What to take from the discussion of r vs λ, so far

• r and λ are, in practice, used interchangeably
• Any relative change in vital rates that shifts r away from zero will cause our population to increase exponentially or spiral to extinction
• The population has no stable equilibrium

8. Population doubling time
- More intuitive to think about rates of population growth
- N(t)/N(0)=e^rt
- 2= e^rt => t= 0.69/r (“rule of 70”, r is per capita)

9. The mathematics of human population growth
- Overall growth rate is a function of births, deaths, current population size and time
- Crude birth rate = births/ 1000 individuals per year
- Crude death rate = deaths/ 1000 individuals per year
- Crude growth rate= CBR – CDR
- Per cent growth rate (%GR)= crude growth rate/ 10
- Doubling time= 70/ %GR

10. Populations reach a limit
- Populations can’t grow exponentially forever, something must restrain population growth
- Negative feedback is the force that constrains population growth, it’s often referred to as density dependence in ecology

11. If populations are bounded…
- Per capita growth rate must decline as density increases
- k: a theoretical limit on the number of organisms that the environment can support
- above k, r must be negative, and population must decline
- below k, r must be positive, and population must increase
- at k, r must be zero

- Adjust r with (K-N)/K, the equation for logistic growth:dN/dt= rN(1-N/K) (density dependent)
- 1- N/K represents the proportion of unutilized resources

13. The logistic equation
- Continuous time model
- Instantaneously response to any difference between N and K
- No time lags, no possibility of population cycles or chaos
- Stable point solution of N=K

14. The S-shaped curve
- Overall rate of population growth changes as N approaches K
- r, k are constants, but accelerating phase changes to decelerating phase at k/2 (at point k/2, growing fastest)
- A simple management model: harvest population to keep it growing at maximal rate

15. Population change and regulation
- Many kinds of factors that influence population change, only a subset of these that act as agents of negative feedback and generate equilibria, which is required for population regulation

16. Factors that affect population size and growth rate
- Biotic: predation, competition, parasitism, mutualism, disease
- Abiotic: temperature, disturbance, precipitation, pollution, salinity, PH

17. Population regulation
- The existence of an equilibrium density towards which a population is attracted
- Regulating factors are those that can generate and maintain such equilibria
- They cause population growth rates to decline toward zero as the population approaches equilibrium
- All regulating factors are biotic (opinion), but not all biotic factors are regulating all of the time, abiotic factors don’t response to population size

Positive density dependence (Allee effect)

• Probability of finding mates increases as population increases
• Ability to defend population against predators increases as population increases.

19. Density-vague dynamics
- There can be considerable noise around the relationship between per capita growth rate and density
- The relationship may be dramatically non-linear
- Argues for the concepts of “floors” and “ceilings” to population variation
- Delayed density dependence

20. Population oscillations
- All populations fluctuate over time
- Our models tell us that populations can exhibit damped oscillations, regular cycling, or become unstable
- Populations may fluctuate for intrinsic reasons, due to environmental variation, or because of time lags
- The magnitude of oscillations increases with an increase in r and an increase in time-lag, which means high rates of growth and long time-lags are destabilizing

21. Recruitment windows
- Episodes of favorable recruitment are seen in age structure of fish populations and forests

 life history attribute r- selected k-selected number of offspring many few size of offspring small large number of reproductive events many few generation time short long parental care poor advanced age at first reproduction young older longevity short lifespan long lifespan dispersal time pre-reproduction post-reproduction recovery & colonization quick recovery, wide colonization slow recovery, poor colonization describes growth rate carrying capacity number fluctuation can fluctuate widely fluctuations are usually small environment unpredictable stable competition poor competitors good competitors
page revision: 8, last edited: 09 Dec 2009 01:47