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**

**B**irth adds to a population size**I**mmigration adds to a population size**D**eath subtracts from a population size**E**migration 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= $\left ( \frac{N_{t+1}}{N_{t}} \right )$

N_{t+1}=λN_{t} 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?
- N
_{t}=λ^{t}N_{o} - N
_{5}=3^{5}*2 = 486

- N
- There are 300 mussels in the lake and λ=3. How many mussels will be in the lake next year?
- N
_{t+1}= λN_{1} - N
_{t+1}= 3*300 = 900

- N

**5. Discrete vs Continuous time**

- N
_{t}=λ^{t}N_{o}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.

- N
_{t}=N_{o}e^{rt}is a*continuous*time model - growth is continuous and instantaneous- Continuous models should be used when populations have overlapping generations and continuous reproduction
- $\left ( \frac{dN}{dt} \right ) = rN \Rightarrow$ N
_{t}=N_{o}e^{rt} - For the proof that this equation is right, refer to the Powerpoint

**6. r vs λ, a summary**

factor | r |
λ |

conversion |
r = log_{e}λ |
λ = e^{r} |

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 |