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?
- Nt=λtNo
- 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
- Nt=λtNo 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
12. Adding carrying capacity
- 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 |